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Go through the Math in Focus Grade 4 Workbook Answer Key Chapter 1 Practice 3 Comparing Numbers to 10,000 to finish your assignments.

Math in Focus Grade 4 Chapter 1 Practice 3 Answer Key Comparing Numbers to 10,000

Write > or < in each Example 15,408 12,508 > means greater than.
< means less than.

Question 1.
63,809 36,908
Answer: >

Explanation:
– The three comparison symbols are less than (<), greater than (>) and equal to (=).
– The less than and greater than symbols are used based on two numbers given to compare.
– A bigger number greater than a smaller number is represented as:
– Bigger Number > Smaller Number
– A smaller number less than a greater number is represented as:
– Smaller Number < Greater Number
Rules to compare numbers:
There are certain rules, based on which it becomes easier to compare numbers. These rules are:
* Numbers with more digits
* Numbers starting with a larger digit
Rule 1: Numbers with more digits
When we compare numbers, then check if both the numbers are having the same number of digits or not. If a number has more digits, then it is greater than the other number.
Examples:
– 33 > 3
– 400 > 39
– 5555 > 555
– 10000 > 9999
Rule 2:Numbers starting with a larger digit
This rule is applicable when two numbers are having the same number of digits. In such cases, we need to check the digit at the leftmost place, whichever is greater. Therefore, the number with a greater digit at the leftmost place of the number is greater than the other number.
Examples:
– 323>232 [323 is greater than 232]
– 343<434 [343 is less than 434]
Thus, in the above examples, we can see that, when we compare the two numbers, though the number of digits is the same, one number is greater than/less than the other number.
The above-given numbers: 63,809 and 36,908
63,809 and 36,908 both the numbers have an equal number of digits, therefore, we will compare the left-most digit of both the numbers. Thus we will compare the left-most digit of both numbers.
6 > 3
Therefore,
63,809 > 36,908
Hence, 63,809 is greater than 36,908.

Question 2.
86,415 86,591
Answer: <

Explanation:
Check the rules in question 1.
The above-given numbers are 86,451 and 86,591
86,451 and 86,591 both the numbers have an equal number of digits, therefore, we will compare the left-most digit of both the numbers. So here rule 2 is applicable.
But the left-most digit is also the same for both numbers, i.e.,8. So, we need to check the second left-most digit.
4 < 5
Therefore,
86,415 < 86,591
Hence, 86,415 is smaller than 86,591.

Question 3.
45,638 8,594
Answer: >

Explanation:
The above-given numbers are 45,638 and 8,594
45,638 and 8,594
Since 45,638 has 5 digits and 8,594 has four digits, therefore, according to rule 1, the number with more digits is greater than the number with fewer digits.
Hence, 45,638 is greater than 8,594.

Question 4.
60,960 69,999
Answer: <

Explanation:
60,960 and 69,999  both the numbers have an equal number of digits, therefore, we will compare the left-most digit of both the numbers.
But the left-most digit is also the same for both numbers, i.e.,6. So, we need to check the second left-most digit.
0 < 9
60,960 < 69,999
Hence, 60,960 is smaller than 69,999.

Compare the eight numbers in Exercises 1 to 4.

Question 5.
Which number is the greatest? ________________
Answer:86,591
The above-given numbers are 63,809, 36,908, 86,415, 86,591, 45,638, 8,594, 60,960, 69,999.
The greatest number is 86,591
Now compare all the numbers according to the rules.

Question 6.
Which number is the least? ________________
Answer:8,594
The above-given numbers are 63,809, 36,908, 86,415, 86,591, 45,638, 8,594, 60,960, 69,999.
The least number is 8,594
Now compare all the numbers according to the rules.

Order these numbers.

Example

Order these numbers.

Question 7.
Begin with the least:

Answer: 79,631   96,137   97,136
Explanation:
Now compare all the numbers
79,631  96,137 and 97,136 all the numbers have an equal number of digits, therefore, we will compare the left-most digit of both the numbers.
Therefore,
7 < 9
Now easily we can say 79,631 is the least.

Question 8.
Begin with the greatest:

Answer: 81,074  80,000  9,469
Explanation:
Apply rule 1:
When we compare numbers, then check if both the numbers are having the same number of digits or not. If a number has more digits, then it is greater than the other number.
So that we can easily say 9,469 is the least number.
Now compare 80,000 and 81,074 both the numbers have an equal number of digits, therefore, we will compare the left-most digit of both the numbers. Here applying rule 2.
But the left-most digit is also the same for both numbers, i.e.,8. So, we need to check the second left-most digit.
0 < 1
Therefore,
80,000 < 81,074
Hence, 80,000 is smaller than 81,074
The numbers from greatest to least:
81,074  80,000  9,469

Write the missing numbers.

Example
1,000 more than 82,586 is ____
17,312 is 40,000 less than 57,312.

Question 9.
10,000 more than 56,821 is ________________
Answer: 66,821
Explanation:
We are looking for a new number which is 10,000 more than 56,821.
We will get the new number by adding 10,000 to 56,821.
We write it down as:
56,821+10,000=66,821
And finally the solution for: What number is 10,000 more than 56,821?
is 66,821.

Question 10.
____ is 50,000 less than 79,895.
Answer:29,895
Explanation:
We are looking for a new number which is 50000 less than 79895.
We will get the new number by subtracting 50000 from 79895.
We write it down as:
79,895-50,000=29,895
And finally the solution for: What number is 50000 less than 79895?
is 29,895.

Question 11.
2,000 less than 18,563 is ________________
Answer:16,563
Explanation:
We are looking for a new number which is 2,000 less than 18,563.
We will get the new number by subtracting 2,000 from 18,563.
We write it down as:
18,563-2,000=16,563
And finally the solution for: What number is 2000 less than 18563?
is 16,563

Question 12.
_____ is 3,000 more than 48,200.
Answer:51,200
Explanation:
We are looking for a new number which is 3,000 more than 48,200.
We will get the new number by adding 3,000 to 48,200.
We write it down as:
48,200+3,000=51,200
And finally the solution for: What number is 3000 more than 48200?
is 51200.

Use the number line to count on or back.

Count on in steps of 4,000 from 20,000. Then write the number that you land on. The first one has been done for you.

Question 13.

Answer:

Question 14.

Answer: 36,000

Explanation:
– A number line is a pictorial representation of numbers on a straight line. It’s a reference for comparing and ordering numbers. It can be used to represent any real number that includes every whole number and natural number. Just to recollect, the whole number is a set of numbers that include all counting numbers (1, 2, 3,4,5,6 …….) and zero (0), whereas the natural number is the set of all counting numbers i.e. 1, 2, 3, 4, 5, 6……..
– Starting point at 2000 and ending point is 50,000
– We count by 1’s here.
– In question 13, step 1 is done and from there 3 steps are forwarded.
– from 24,000 we need forward 3 steps by using counting in steps of 4000
– After 3 steps then we get 36,000
– 24,000+4,000=28,000
– 28,000+4,000=32,000
– 32,000+4,000=36,000

Count back in steps of 3,000 from 50,000. Then write the number that you land on.

Question 15.
6 steps
Answer:32,000

Explanation:
– A number line can be used to record the stages.
– Count back, you need to subtract from the larger number to the smaller number in stages.
– When you use a number line to subtract, you’ll move to the left, instead of the right, on the line.
– 50,000-3000=47,000
– 47,000-3000=44,000
– 44,000-3,000=41,000
– 41,000-3000=38,000
– 38,000-3,000=35,000
– 35,000-3,000=32,000

Question 16.
8 steps
Answer:26,000

Explanation:
– A number line can be used to record the stages.
– Count back, you need to subtract from the larger number to the smaller number in stages.
– When you use a number line to subtract, you’ll move to the left, instead of the right, on the line.
– 50,000-3000=47,000
– 47,000-3000=44,000
– 44,000-3,000=41,000
– 41,000-3000=38,000
– 38,000-3,000=35,000
– 35,000-3,000=32,000
– 32,000-3000=29,000
– 29,000-3000=26,000

Continue or complete the number patterns. Then write the rule for each pattern.

Example

Question 17.

Rule: ____
Answer:
A Number Pattern refers to a sequence of numbers that follow a certain order in mathematics. Patterns typically describe the inverse relationship between numbers. The sequences of numbers can also be called patterns.
Types of number patterns:
1. Arithmetic Sequences
2. Geometric Sequences
The above-given numbers are in the arithmetic pattern.
Arithmetic sequence:
Arithmetic sequences are sequences of numbers with constant differences between consecutive terms in mathematics. Common differences between consecutive terms are called common differences.
Here are some examples of arithmetic numbers:
1, 4, 7, 10, 13, 16, …… is an arithmetic sequence because the difference between consecutive terms is 3.
1, 4, 8, 11, 15, 18, …… is not an arithmetic sequence because the difference between consecutive terms is not a constant.
The above-given numbers are 96,500   86,500   76,500  –  –
Now, we need to find the missing term in the sequence.
Here, we can use the subtraction process to figure out the missing terms in the patterns.
In the pattern, the rule used is “subtracting 10,000 to the previous term to get the next term”.
In the given above, take the second term (86,500). If we subtract “10,000” to the second term (86,500), we get the third term of 76,500.
Similarly, we can find the unknown terms in the sequence.
First missing term: The previous term is 76,500. Therefore, 76,500-10,000 = 66,500
Second missing term: The previous term is 66,500. So, 66,500-10,000 = 56,500
Hence, the complete arithmetic pattern is 96,500   86,500   76,500   66,500    56,500

Question 18.
39,860 ____ 41,860 ____ 43,860
Rule: ________________
Answer:
The above-given numbers are 39,860  –  41,860    –    43,860
Now, we need to find the missing term in the sequence.
Here, we can use the addition process to figure out the missing terms in the patterns.
In the pattern, the rule used is “adding 1000 to the previous term to get the next term”.
In the given above, take the first term (39,860). If we add “1000” to the first term (86,500), we get the second term.
Similarly, we can find the unknown terms in the sequence.
First missing term: The previous term is 39,860. Therefore, 39,860+1,000 = 40,860
The previous term is 40,860. So, 40,860+1,000 = 41,860 (already given)
Second missing term: The previous term is 41,860. Therefore, 41,860+1000=42,860
Hence, the complete arithmetic pattern is 39,860   40,860   41,860    42,860

Question 19.

Rule: ___________
Answer:
The above-given numbers are 25,000    20,000    15,000   –   –
Now, we need to find the missing term in the sequence.
Here, we can use the subtraction process to figure out the missing terms in the patterns.
In the pattern, the rule used is “subtracting 5000 to the previous term to get the next term”.
In the given above, take the second term (20,000). If we subtract “5000” to the second term (20,000), we get the third term of 15,000
Similarly, we can find the unknown terms in the sequence.
First missing term: The previous term is 15,000. Therefore, 15,000-5,000 = 10,000
Second missing term: The previous term is 10,000. Therefore, 10,000-5000=5,000
Hence, the complete arithmetic pattern is 25,000    20,000    15,000   10,000   5,000

Question 20.

Rule: ________________
Answer: subtract 500 from backwards.
Explanation:
The above-given numbers are –   10,349   10,849  –   –   12,349
Now, we need to find the missing term in the sequence.
Here, we can use the addition process to figure out the missing terms in the patterns.
In the pattern, the rule used is “adding 500 to the previous term to get the next term”.
Here we don’t know the first term but given the second and third term. By using these 2 terms we can write the number pattern. First of all, check the difference between the 2nd and 3rd numbers. The difference is 500
If we subtract 500 from 10,349 we get 9,849. Therefore, the first term is 9,849. From there we can add 500 to every next number (or) another solution is by using the given terms we got the difference 500. We can calculate from the backwards means the last number is 12,349. We start from 12,349-500=11,849
Similarly, we can find the unknown terms in the sequence.
First missing term: The previous term is 12,349. Therefore, 12,349-500 = 11,849
Second missing term: The previous term is 11,849 Therefore, 11,849-500=11,349
Hence, the complete arithmetic pattern is 9,849  10,349   10,849   11,849  11,349

Question 21.

Rule: ________________
Answer:
The above-given numbers are 93,308   94,313   95,318   –   –
Now, we need to find the missing term in the sequence.
Here, we can use the addition process to figure out the missing terms in the patterns.
In the pattern, the rule used is “adding 1005 to the previous term to get the next term”.
In the given above, take the second term (94,313). If we add “1005” to the second term (94,313), we get the third term of 95,318
Similarly, we can find the unknown terms in the sequence.
First missing term: The previous term is 95,318. Therefore, 95,318+1005 = 96,323
Second missing term: The previous term is 96,323. Therefore, 96,323+1005=97,328
Hence, the complete arithmetic pattern is 93,308   94,313   95,318   96,323   97,328

Question 22.

Rule: ________________
Answer:
The above-given numbers are 85,765   87,775   89,985   91,995   –   –
Now, we need to find the missing term in the sequence.
Here, we can use the addition process to figure out the missing terms in the patterns.
In the pattern, the rule used is “adding 2010 to the previous term to get the next term”.
In the given above, take the third term (89,985). If we add “2010” to the third term (89,989), we get the fourth term of 91,995
Similarly, we can find the unknown terms in the sequence.
First missing term: The previous term is 91,995. Therefore, 91,995+2010 = 94,005
Second missing term: The previous term is 96,323. Therefore, 94,005+2010=96,015
Hence, the complete arithmetic pattern is 85,765   87,775   89,985   91,995   94,005    96,015

Go through the Math in Focus Grade 4 Workbook Answer Key Chapter 2 Practice 2 Factors to finish your assignments.

Math in Focus Grade 4 Chapter 2 Practice 2 Answer Key Factors

Find the missing factors.

Example 12 1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
The factors of 12 are
1, 2, 3, 4, 6, and 12.

Question 1.
70
1 × ____ = 70
2 × ____ = 70
3 × ____ = 70
4 × ____ = 70
The factors of 70 are 1, 2, 5, 7, ___, ____
___ and ___
Answer:
Factor definition:
When a number is said to be a factor of any other second number, then the first number must divide the second number completely without leaving any remainder. In simple words, if a number (dividend) is exactly divisible by any number (divisor), then the divisor is a factor of that dividend. Every number has a common factor that is one and the number itself.
1 × 70= 70
2 × 35= 70
3 × 23.3= 70
4 × 17.5= 70
Point to remember: Fractions cannot be considered as factors for any number.
The numbers that divide 70 exactly without leaving a remainder are the factors of 70. As 70 is an even composite number, it has many factors other than 1 and 70. Hence, the factors of 70 are 1, 2, 5, 7, 10, 14, 35, and 70. Similarly, the negative factors of 70 are -1, -2, -5, -7, -10, -14, -35, and -70.
1 × 70= 70
2 × 35= 70
5 × 14=70
7 × 10=70
Factors of 70: 1, 2, 5, 7, 10, 14, 35, and 70.

Find the factors of each number.

Question 2.
40
The factors of 40 are
_________
Answer: 1, 2, 4, 5, 8, 10, 20, and 40.
Explanation:
The Factors of 40 are all the integers (positive and negative whole numbers) that you can evenly divide into 40. 40 divided by a factor of 40 will equal another Factor of 40.
How to find the factors of 40:
Since the Factors of 40 are all the numbers that you can evenly divide into 40, we simply need to divide 40 by all numbers up to 40 to see which ones result in an even quotient. When we did that, we found that these calculations resulted in an even quotient:
40 ÷ 1 = 40
40 ÷ 2 = 20
40 ÷ 4 = 10
40 ÷ 5 = 8
40 ÷ 8 = 5
40 ÷ 10 = 4
40 ÷ 20 = 2
40 ÷ 40 = 1
The Positive Factors of 40 are therefore all the numbers we used to divide (divisors) above to get an even number. Here is the list of all Positive Factors of 40 in numerical order:
1, 2, 4, 5, 8, 10, 20, and 40.

Question 3.
63
The factors of 63 are
___________
Answer:
Explanation:
The Factors of 63 are all the integers (positive and negative whole numbers) that you can evenly divide into 63. 63 divided by a factor of 63 will equal another Factor of 63.
How to find the factors of 63:
Since the Factors of 63 are all the numbers that you can evenly divide into 63, we simply need to divide 63 by all numbers up to 63 to see which ones result in an even quotient. When we did that, we found that these calculations resulted in an even quotient:
63 ÷ 1 = 63
63 ÷ 3 = 21
63 ÷ 7 = 9
63 ÷ 9 = 7
63 ÷ 21 = 3
63 ÷ 63 = 1
The Positive Factors of 63 are therefore all the numbers we used to divide (divisors) above to get an even number. Here is the list of all Positive Factors of 63 in numerical order:
1, 3, 7, 9, 21, and 63.

Divide. Then answer each question.

Question 4.
65 ÷ 5 = ____
Answer:13
There are four important terms used in division. These are dividend, divisor, quotient and remainder.
– Dividend: The number to be divided by another number is called the dividend.
– Divisor: The number by which we divide another number (dividend) into equal parts is called the divisor.
– Quotient: The result of division is called a quotient.
– Reminder: The leftover number after division is called the remainder.
– The pictorial representation of the above terminology is given below.

Division formula:
Dividend=Divisor×Quotient+Remainder.
After division, we can put all the values in the formula to verify or check whether our division is correct or not.
the above-given question is 65 ÷ 5 = 13
Here, dividend =65, Divisor =5, quotient =13 and remainder =0
Let us put all the above values in the formula,
Dividend=Divisor×Quotient+Remainder
⇒65 = 5 × 13 + 0
⇒ 65 ÷ 5 = 13
Hence, our division is correct.

Question 5.
46 ÷ 4 = ___
Is 5 a factor of 65?
Answer:
Explanation:
There are four important terms used in division. These are dividend, divisor, quotient and remainder.
– Dividend: The number to be divided by another number is called the dividend.
– Divisor: The number by which we divide another number (dividend) into equal parts is called the divisor.
– Quotient: The result of division is called a quotient.
– Reminder: The leftover number after division is called the remainder.
– The pictorial representation of the above terminology is given below.

Division formula:
Dividend=Divisor×Quotient+Remainder.
After division, we can put all the values in the formula to verify or check whether our division is correct or not.
the above-given question is 46 ÷ 4 = 11.5
Here, dividend =46, Divisor =4, quotient =11.5 and remainder =0
Let us put all the above values in the formula,
Dividend=Divisor×Quotient+Remainder
⇒46 = 4 × 11.5 + 0
⇒ 65 ÷ 5 = 11.5
Hence, our division is correct.
Another question is there, is 5 is a factor of 65:
Yes, 5 is a factor of 65.
– The 4 factors of 65 are:
1, 5, 13, 65
– The factor pairs of 65 are:
1 × 65 = 65
5 × 13 = 65

Question 5.
46 ÷ 4 = ___
Is 4 a factor of 46? ____
Answer: 11.5
Explanation:
There are four important terms used in division. These are dividend, divisor, quotient and remainder.
– Dividend: The number to be divided by another number is called the dividend.
– Divisor: The number by which we divide another number (dividend) into equal parts is called the divisor.
– Quotient: The result of division is called a quotient.
– Reminder: The leftover number after division is called the remainder.
– The pictorial representation of the above terminology is given below.

Another question is there is 4 a factor of 46:
No 4 is not a factor of 46
Therefore, the 4 factors of 46 are:
1, 2, 23, 46.
– The factor pairs of 46 are:
1 × 46 = 46
2 × 23 = 46

Find the common factors of each pair of numbers.

Question 6.

Answer:

Explanation:
– To find common factors of two numbers, first, list out all the factors of two numbers separately and then compare them. Now write the factors which are common to both the numbers. These factors are called common factors of given two numbers.
– As we know, the factors are the numbers that divide the original number completely. But how to check if two or more numbers have common factors between them.
Follow the below steps to find the common factors.
* Write the factors of the given numbers.
* Find the common factor present in them.
Let us check the factors of the two numbers, i.e., 10 and 15.
Factors of 10 = 1, 2, 5, 10
Factors of 15 = 1, 3, 5, 15
Clearly, we can see, the common factors between 10 and 15 are 1, 5.

Question 7.

Answer:

Explanation:
– To find common factors of two numbers, first, list out all the factors of two numbers separately and then compare them. Now write the factors which are common to both the numbers. These factors are called common factors of given two numbers.
– As we know, the factors are the numbers that divide the original number completely. But how to check if two or more numbers have common factors between them.
Follow the below steps to find the common factors.
* Write the factors of the given numbers.
* Find the common factor present in them.
Let us check the factors of the two numbers, i.e., 24 and 36.
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Clearly, we can see, the common factors between 24 and 36 are 1, 2, 3, 4, 6, 12.

Divide. Then answer each question.

Question 8.
18 ÷ 4 = ___ 16 ÷ 4 = ____
Is 4 a common factor of 18 and 16? ____
Answer:
18 ÷ 4 = 4.5
16 ÷ 4 = 4
Explanation:
There are four important terms used in division. These are dividend, divisor, quotient and remainder.
– Dividend: The number to be divided by another number is called the dividend.
– Divisor: The number by which we divide another number (dividend) into equal parts is called the divisor.
– Quotient: The result of division is called a quotient.
– Reminder: The leftover number after division is called the remainder.
– The pictorial representation of the above terminology is given below.

Another question is there is 4 a common factor of 18 and 16:
No 4 is not a factor of 46
Therefore, the factors of 18 and 16 are:
factors of 18: 1, 2, 3, 6, 9, 18
factors of 16: 1, 2, 4, 8, 16.
The common factors are 1, 2

Question 9.
42 ÷ 3 = ____ 84 ÷ 3 = ____
Is 3 a common factor of 42 and 84? ____
Answer:
42 ÷ 3 = 14
84 ÷ 3 = 28
Yes, 3 is a common factor for both the numbers.
Explanation:
There are four important terms used in division. These are dividend, divisor, quotient and remainder.
– Dividend: The number to be divided by another number is called the dividend.
– Divisor: The number by which we divide another number (dividend) into equal parts is called the divisor.
– Quotient: The result of division is called a quotient.
– Reminder: The leftover number after division is called the remainder.
– The pictorial representation of the above terminology is given below.

Another question is there is 3 a common factor of 42 and 84:
Therefore, the factors of 42 and 84 are:
factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
The common factors are 1, 2, 3, 6, 7, 14, 21, 42
Therefore, 3 is a common factor for 42 and 84.

Look at the numbers 80, 27, 40, 62, 36, and 55. Then fill in the blanks.

Question 10.
Which of the numbers have 2 as a factor? _________
Answer:80, 40, 62, 36
The above-given numbers are 80, 27, 40, 62, 36, and 55.
we have to find out the numbers having 2 as a factor.
80, 40, 62, 36 are the factors of 2.
80/40 = 2
40/20 = 2
62/32 = 2
36/18 = 2
Factors are the numbers, that can divide a number exactly.  Hence, after division, there is no remainder left. Factors are the numbers you multiply together to get another number. Thus, a factor is the divisor of another number.

Question 11.
Which of the numbers have 5 as a factor? ______
Answer: 80, 40, 55
Factors can be calculated by using multiplication and division.
– Since multiplication of two numbers results in a product such that the two numbers become the factors of the product. Thus, to find the factors we need to follow the below steps:
* If we need to find the factors of a number say N, then write the multiplication of two numbers in different ways, such that the resulting value is equal to N
* All the individual numbers, that results in the product equal to N are the factors
5 × 12 = 80
5 × 8 = 40
5 × 11 = 55
Factors are the numbers, that can divide a number exactly.  Hence, after division, there is no remainder left. Factors are the numbers you multiply together to get another number. Thus, a factor is the divisor of another number.

Question 12.
Which of the numbers have both 2 and 5 as factors? ______
Answer:
The numbers 80, 40 have 2 and 5
40 × 2 =80
5 × 8 = 40
5 × 16 = 80
Factors are the numbers, that can divide a number exactly.  Hence, after division, there is no remainder left. Factors are the numbers you multiply together to get another number. Thus, a factor is the divisor of another number.

Each set of numbers is all the factors of a number. Find each number.

Question 13.

Answer:

Explanation:
– The factors of a number are defined as the numbers which when multiplied will give the original number, by multiplying the two factors we get the result as the original number. The factors can be either positive or negative integers.
– Factors of 8 are all the integers that can evenly divide the given number 8. Now let us study how to calculate all factors of 8.
– According to the definition of factors of 8, we know that factors of 8 are all the positive or negative integers that divide the number 8 completely. So let us simply divide the number 8 by every number which completely divides 8 in ascending order till 8.
8 ÷ 1 = 8
8 ÷ 2 = 4
8 ÷ 3 = not divides completely
8 ÷ 4 = 2
8 ÷ 5 = not divides completely
8 ÷ 6 = not divides completely
8 ÷ 7 = not divides completely
8 ÷ 8 = 1
So all factors of 8: 1, 2, 4, and 8.

Question 14.

Answer:

Explanation:
– When we divide every number with 12 up to that number itself, the numbers that evenly divide with 12 are the factors of 12.
– This is a very simple and straightforward method. The entire process takes only 5 steps.
– To start off here is step number 1:
Step 1: Consider the number 12
Step 2: Divide it with all the numbers starting from 1 to 12
Step 3: Capture the results
: 12/1 = 12
: 12/2 = 6
: 12/3 = 4
: 12/4 = 3
:  12/5 = 2.4
: 12/6 = 2
: 12/7 = 1.7
: 12/8 = 1.5
: 12/9 = 1.3
:  12/10 = 1.2
: 12/11 = 1.09
: 12/12 = 1
Step 4: Filter out the positive integer quotient for the above, rejecting the decimals.
Step 5: The factors of 12 are 1, 2, 3, 4, 6, 12.

Question 15.

Answer:

Explanation:
The factors of 6 are the numbers that divide 6 exactly without leaving the remainder. In other words, the factors of 6 are the numbers that are multiplied in pairs resulting in an original number 6. As 6 is an even composite number, it has many factors other than 1 and 6. Thus, the factors of 6 are 1, 2, 3 and 6. Similarly, the negative pair factors of 6 are -1, -2, -3 and -6.
Factors of 6: 1, 2, 3 and 6.
Factors of 6 by division method:
– In the division method, the factors of 6 can be found by dividing 6 by different integers. If the integers divide 6 exactly and leave the remainder 0, then those integers are the factors of 6. Now, let us discuss how to find the factors of 6 using the division method.
* 6/1 = 6 (Factor is 1 and Remainder is 0)
* 6/2 = 3 (Factor is 2 and Remainder is 0)
* 6/3 = 2 (Factor is 3 and Remainder is 0)
* 6/6 = 1 (Factor is 6 and Remainder is 0)
– If we divide the number 6 by any numbers other than 1, 2, 3 and 6, then it leaves the remainder. Hence, the factors of 6 are 1, 2, 3 and 6.

Question 16.

Answer:

Explanation:
The factors of 16 are the numbers that divide the number 16 completely without leaving any remainder. As the number 16 is a composite number, it has more than one factor. The factors of 16 are 1, 2, 4, 8 and 16. Similarly, the negative factors of 16 are -1, -2, -4, -8 and -16.
– Factors of 16: 1, 2, 4, 8 and 16.
How to calculate factors of 16:
Go through the following steps to calculate the factors of 16.
* First, write the number 16.
* Find the two numbers, which gives the result as 16 under the multiplication, say 2 and 8, such as 2 × 8 = 16.
* We know that 2 is a prime number that has only two factors, i.e., 1 and the number itself (1 and 2). So, it cannot be further factorized.
* Look at the number 8, which is a composite number but not a prime number. So it can be further factorized.
* 8 can be factored as 2 x 2 x 2 x 1.
* Therefore, the factorization of 16 is written as, 16 = 2 × 2 × 2 × 2 x 1.

Find the greatest common factor of each pair of numbers.

Example
12 and 28

Method 1
The factors of 12 are 1, 2, 3, 4, 6, and 12.
The factors of 28 are 1, 2, 4, 7, 14, and 28.
The common factors of 12 and 28 are 1, 2, and 4.
The greatest common factor of 12 and 28 is 4.

Method 2

2 × 2 = 4
The greatest common factor of 12 and 28 is 4.

3 and 7 have no common factor other than 1.

Question 17.
16 and 30
_________
Answer: GCF of 16 and 30 is 2.
Explanation:
GCF of 16 and 30 is the largest possible number that divides 16 and 30 exactly without any remainder. The factors of 16 and 30 are 1, 2, 4, 8, 16 and 1, 2, 3, 5, 6, 10, 15, 30 respectively. There are 3 commonly used methods to find the GCF of 16 and 30 – long division, prime factorization, and Euclidean algorithm.
– The GCF of two non-zero integers, x(16) and y(30) is the greatest positive integer m(2) that divides both x(16) and y(30) without any remainder.
Methods to find GCF of 16 and 30:
– The methods to find the GCF of 16 and 30 are explained below.
* Long Division Method
* Using Euclid’s Algorithm
* Listing Common Factors
# GCF of 16 and 30 by long division method:

GCF of 16 and 30 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.
Step 1: Divide 30 (larger number) by 16 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (16) by the remainder (14).
Step 3: Repeat this process until the remainder = 0.
The corresponding divisor (2) is the GCF of 16 and 30.
# GCF of 16 and 30 by listing common factors:
Factors of 16: 1, 2, 4, 8, 16
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
There are 2 common factors of 16 and 30, which are 1 and 2. Therefore, the greatest common factor of 16 and 30 is 2.

Find the greatest common factor of the numbers.

Question 18.
21 and 54
___________
Answer: 3
The Greatest Common Factor (GCF) for 21 and 54, notation GCF(21,54), is 3.
Explanation:
To find the GCF of 21 and 54, we first need to find the factors of 21 and 54 and then choose the greatest common factor that is dividable by both 21 and 54.
– The factors of 21 are 1,3,7,21;
– The factors of 54 are 1,2,3,6,9,18,27,54.
So, as we can see, the Greatest Common Factor or Divisor is 3, because it is the greatest number that divides evenly into all of them.

Find all the factors. Then list the prime numbers.

Example
13
The factor of 13 are 1 and 13. 13 is a prime number.

A prime number has only 2 factors, 1 and itself.

Question 19.
12 ____
Answer:
The factors of 12 are the numbers that divide 12 exactly without leaving any remainder. As 12 is an even composite number, it has many factors other than 1 and 12. The factors of 12 can be positive or negative. Hence, the factors of 12 are 1, 2, 3, 4, 6 and 12. Similarly, the negative factors of 12 are -1, -2, -3, -4, -6 and -12.
Factors of 12: 1, 2, 3, 4, 6 and 12.
Now we need to write the prime numbers in the factors of 12.
prime numbers: A prime number is a positive integer having exactly two factors. If p is a prime, then its only factors are necessarily 1 and p itself. Any number that does not follow this is termed a composite number, which can be factored into other positive integers. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers other than 1 and the number itself.
Therefore, the prime numbers are 2 and 3.

Question 20.
7 ______
Answer:
The factors of 20 are the numbers that divide 20 exactly without leaving a remainder. In other words, the numbers are multiplied in pairs resulting in the number 20 being the factor of 20. As 20 is an even composite number, it has many factors other than 1 and 20. Thus, the factors of 20 are 1, 2, 4, 5, 10 and 20. Similarly, the negative factors of 20 are -1, -2, -4, -5, -10 and -20.
Factors of 20: 1, 2, 4, 5, 10 and 20.
Now we need to write the prime numbers in the factors of 20.
prime numbers: A prime number is a positive integer having exactly two factors. If p is a prime, then its only factors are necessarily 1 and p itself. Any number that does not follow this is termed a composite number, which can be factored into other positive integers. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers other than 1 and the number itself.
Therefore, the prime numbers are 2 and 5.

Question 21.
19 ____
Answer:
The numbers that divide 19 completely without leaving a remainder are the factors of 19. In other words, the numbers which are multiplied in pairs, resulting in an original number, are the factors of 19. As number 19 is a prime number, it has only two factors: one and the number itself. Hence, the factors of 19 are 1 and 19, and its negative factors are -1 and -19.
Factors of 19: 1 and 19.
Now we need to write the prime numbers in the factors of 19.
prime numbers: A prime number is a positive integer having exactly two factors. If p is a prime, then its only factors are necessarily 1 and p itself. Any number that does not follow this is termed a composite number, which can be factored into other positive integers. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers other than 1 and the number itself.
Therefore, the prime number is 19.

Question 22.
24 ______
Answer:
By the definition of factors, we know, a factor can divide a given number into an equal number of parts. Therefore, factors of 24 are such whole numbers that can divide 24 into an equal number of parts.  These factors cannot be a fraction.
Factors of 24: 1, 2, 3, 4, 6, 8, 12 and 24.
Now we need to write the prime numbers in the factors of 24.
prime numbers: A prime number is a positive integer having exactly two factors. If p is a prime, then its only factors are necessarily 1 and p itself. Any number that does not follow this is termed a composite number, which can be factored into other positive integers. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers other than 1 and the number itself.
Therefore, the prime numbers are 2, 3.

Question 23.
11 ____
Answer:
In Mathematics, the numbers that divide 11 completely without leaving any remainder are the factors of 11. In other words, the factors of 11 are the numbers that are multiplied in pairs and result in the original number 11. As 11 is a prime number, it has only two factors, such as one and the number itself. Hence, the factors of 11 are 1 and 11, and the negative factors of 11 are -1 and -11.
Factors of 11: 1 and 11.
Now we need to write the prime numbers in the factors of 11.
prime numbers: A prime number is a positive integer having exactly two factors. If p is a prime, then its only factors are necessarily 1 and p itself. Any number that does not follow this is termed a composite number, which can be factored into other positive integers. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers other than 1 and the number itself.
Therefore, the prime number is 11.

Question 24.
63 ______
Answer:
The factors of 63 are the numbers that divide the number 63 exactly without leaving any remainder. In other words, the pair factors of 63 are the numbers that are multiplied in pairs resulting in the original number 63. Since the number 63 is a composite number, 63 has more than two factors. Thus, the factors of 63 are 1, 3, 7, 9, 21 and 63.
Now we need to write the prime numbers in the factors of 63.
prime numbers: A prime number is a positive integer having exactly two factors. If p is a prime, then its only factors are necessarily 1 and p itself. Any number that does not follow this is termed a composite number, which can be factored into other positive integers. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers other than 1 and the number itself.
Therefore, the prime numbers are 3, 7.

Question 25.
Look at the given numbers in Exercises 19-24.
The prime numbers are _____
Explain your reasoning. _________
Answer:
The above-given numbers are 12, 7, 19, 24, 11, 63.
To that we need to write the prime numbers:
Definition: A prime number is a positive integer having exactly two factors. If p is a prime, then its only factors are necessarily 1 and p itself. Any number that does not follow this is termed a composite number, which can be factored into other positive integers. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers other than 1 and the number itself.
Properties of prime numbers:
Some of the properties of prime numbers are listed below:
– Every number greater than 1 can be divided by at least one prime number.
– Every even positive integer greater than 2 can be expressed as the sum of two primes.
– Except for 2, all other prime numbers are odd. In other words, we can say that 2 is the only even prime number.
– Two prime numbers are always coprime to each other.
– Each composite number can be factored into prime factors and individually all of these are unique in nature.
Now from the above definition and properties, we can write the prime numbers:
The prime numbers are:7, 11, 19

Find all the factors. Then list the composite numbers.

Example
18
The factors of 18 are 1, 2, 3, 6,
9 and 18
18 is a composite number.

18 has factors other than 1 and itself, so it is a composite number.

Question 26.
20 _______________
Answer:
The factors of 20 are the numbers that divide 20 exactly without leaving a remainder. In other words, the numbers are multiplied in pairs resulting in the number 20 being the factor of 20. As 20 is an even composite number, it has many factors other than 1 and 20. Thus, the factors of 20 are 1, 2, 4, 5, 10 and 20. Similarly, the negative factors of 20 are -1, -2, -4, -5, -10 and -20.
Now we need to write the composite numbers:
Definition: In Mathematics, composite numbers are numbers that have more than two factors.
therefore, the composite numbers are 4, 10, and 20.
Hence, these numbers are having more than two factors.

Question 27.
15 ______
Answer:
Factors of 15 divide the original number, wholly. A number or an integer that divides 15 evenly without leaving a remainder, then the number is a factor of 15. As the number 15 is an odd composite number, it has more than two factors. Thus, the factors of 15 are 1, 3, 5 and 15.
Now we need to write the composite numbers:
Definition: In Mathematics, composite numbers are numbers that have more than two factors.
therefore, 15 is the composite number.
Hence, this number is having more than two factors.

Question 28.
5 _______________
Answer:
Factors of 5 are the real numbers that can divide the original number, uniformly. If ‘x’ is the factor of 5, then ‘x’ divides 5 into equal parts and there is no remainder left.
The factors of 5 are 1 and 5.
Now we need to write the composite numbers:
Definition: In Mathematics, composite numbers are numbers that have more than two factors.
therefore, there is no composite number.
Question 29.
17 _____
Answer:
The numbers that divide 17 completely and leave the remainder 0, then the numbers are the factors of 17. In other words, if two numbers are multiplied together and result in 17, then the numbers are the factors of 17. As 17 is a prime number, it has only two factors, such as 1 and the number itself. Hence, the factors of 17 are 1 and 17.
Now we need to write the composite numbers:
Definition: In Mathematics, composite numbers are numbers that have more than two factors.
therefore, there is no composite number.

Question 30.
33 _________________
Answer:
Go through the steps given below to learn how to find the factors of 33.
Step 1: First, write the number 33 in your notebook.
Step 2: Find the two numbers, which on multiplication gives 33, suppose 3 and 11, such as 3 × 11 = 33.
Step 3: We know that 3 is a prime number that has only two factors, i.e. 1 and the number itself. So, we cannot factorize it further.
3 = 1 × 3
Step 4:Also, 11 is a prime number and cannot be factored further.
11 = 1 × 11
Step 5: Therefore, the factorization of 33 gives us 1 × 3 × 11.
The unique numbers which are obtained from the above expression are 1, 3, 11, and 33.
Now we need to write the composite numbers:
Definition: In Mathematics, composite numbers are numbers that have more than two factors.
therefore, 33 is the composite number.
Hence, this number is having more than two factors.

Question 31.
27 _____
Answer:
The steps to find the factors are:
1. First, write the number 27 in your notebook.
2. Find the two numbers, which on multiplication gives 27, say 3 and 9, such as 3 × 9 = 27.
3. We know 3 is a prime number that has only two factors, i.e. 1 and the number itself. So, we cannot further factorize it. (i.e.) 3 = 1 × 3
4. But 9 is not a prime number and can be further factorized. (i.e.) 9 = 3 × 3 × 1
5. Therefore, the factorization of 27 gives us, 1 × 3 × 3 × 3.
6. Write down all the unique numbers which are obtained here.
7. Factors of 27: 1, 3, 9 and 27.
Now we need to write the composite numbers:
Definition: In Mathematics, composite numbers are numbers that have more than two factors.
therefore, 9 and 27 are the composite number.
Hence, these numbers are having more than two factors.

Question 32.
Look at the given numbers in Exercises 26-31.
The composite numbers are ____
Explain your reasoning. _____
Answer:
The above-given numbers are 20, 15, 5, 17, 33, 27
To that we need to write the composite numbers:
Def 1: In Mathematics, composite numbers are numbers that have more than two factors.
Def 2: The numbers which can be generated by multiplying the two smallest positive integers and contain at least one divisor other than the number ‘1’ and itself are known as composite numbers. These numbers always have more than two factors.
properties of composite numbers:
The properties of composite numbers are easy to remember.
– Composite numbers have more than two factors
– Composite numbers are evenly divisible by their factors
– Each composite number is a factor in itself
– The smallest composite number is 4
– Each composite number will include at least two prime numbers as its factors (Eg. 10 = 2 x 5, where 2 and 5 are prime numbers)
– Composite numbers are divisible by other composite numbers also
From the above definition and properties we can write the composite numbers:
Therefore, the composite numbers are 20, 15, 27, 33

Use the method given below to find prime numbers.

Question 33.
Find the prime numbers between 1 and 50.

Step 1
1 is neither prime nor composite. So, 1 has been circled. As 2 is the first prime number, it has been underlined. Next, cross out all the numbers that can be divided by 2.

Step 2
3 is the next prime number. Underline it.
Then, cross out all the numbers that can be divided by 3.
Keep underlining the prime numbers and crossing out the numbers that can be divided by the prime numbers until you reach 50.
The prime numbers are ____
Answer:
A prime number is a positive integer having exactly two factors. If p is a prime, then its only factors are necessarily 1 and p itself. Any number that does not follow this is termed a composite number, which can be factored into other positive integers. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers other than 1 and the number itself.
The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

Question 34.
Find two prime numbers between 60 and 90. ____
Answer:
First of all the prime numbers between 60 and 90 are 61, 67, 71, 73, 79, 83, 89.

Question 35.
Find two composite numbers between 60 and 90. ____
Answer:
The composite numbers between 0 and 90 are: 60, 62, 63, 64, 65, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90.
The two composite numbers: 70 and 80
Two composite numbers between 60 and 90 are 70 and 80. All numbers which end in a zero are evenly divisible by 10.

Question 36.
Are there more prime numbers from 1 to 25 or from 26 to 50?
Answer:
YES. There are 9 prime numbers from 1 to 25. There are only 6 prime numbers from 26 to 50.
The prime numbers between 1 to 25 are: 2, 3, 5, 7, 11, 13, 17, 19, 23.
The prime numbers between 26 to 50: 29, 31, 37, 41, 43, 47, 53, 59.

Go through the Math in Focus Grade 4 Workbook Answer Key Chapter 2 Practice 3 Multiples to finish your assignments.

Math in Focus Grade 4 Chapter 2 Practice 3 Answer Key Multiples

Fill in the table with the multiples of each given number.

Example

Question 1.

Answer: 7, 14, 21, 28, 35

Explanation:
– A multiple is a numerical value that is generated when a natural number is multiplied by another natural number or counting number.
– Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, … and so on. How do we get to know a number is multiple of another number? If a number is multiple of another number, then it is evenly divisible by the original number. For example, 14 divided by 7 is equal to 2, thus it is a multiple of 7.
– Any number that can be denoted in the form 7n where n is a natural number is a multiple of 7. So if two values p and q are there, we say that q is a multiple of p if q = np for some natural number n.
7 x 1=7
7 x 2=14
7 x 3=15
7 x 4=28
7 x 5=35
If a number is multiple of 7, then it is evenly divisible by 7.
14 is a multiple of 7 because 14/7 = 2
21 is a multiple of 7 because 21/7= 3
28 is a multiple of 7 because 28/7=4
35 is a multiple of 7 because 35/7=5.
Therefore, the first five multiples of 7 are 7, 14, 21, 28, 35.

Question 2.

Answer:8, 16, 24, 32, 40

Explanation:
The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80… and so on. It is a sequence where the difference between each next number and the preceding number, i.e. two consecutive results, is 8. Multiples are the numbers that give products of any number multiplied by other natural numbers.
The multiples of 8 are the numbers that are generated when 8 is multiplied by any natural number. That means any number that can be expressed in the form of 8n where n is an integer is a multiple of 8. As we know, if two values, p and q, are there, we say that q is a multiple of p if q = np for some integer n.
Thus, as per the definition given above, the multiple of 8 is obtained by multiplying some integer with 8.
Take the first five multiples of 8: 8, 16, 24, 32, 40.
8 × 1 = 8; 8 multiplied by 1 to get 8
8 × 2 = 16; 8 multiplied by 2 to get 16
8 × 3 = 24; 8 multiplied by 3 to get 24
8 × 4 = 32; 8 multiplied by 4 to get 32
8 × 5 = 40; 8 multiplied by 5 to get 40

Question 3.

Answer: 9, 18, 27, 36, 45.

Explanation:
Any number that can be denoted in the form 9n where n is a natural number, is a multiple of 9. So if two values p and q are there, we say that q is a multiple of p if q = np. In other words, the multiples of 9 are the numbers that leave no remainder (i.e. Remainder = 0), when it is divided by 9.
Thus, as per the definition given above, the multiple of 9 is obtained by multiplying some integer with 9.
Take the first five multiples of 9: 9, 18, 27, 36, 45.
9 × 1 = 9; 9 multiplied by 1 to get 9
9 × 2 = 18; 9 multiplied by 2 to get 18
9 × 3 = 27; 9 multiplied by 3 to get 27
9 × 4 = 36; 9 multiplied by 4 to get 36
9 × 5 = 45; 9 multiplied by 5 to get 45
Therefore, the first five multiples of 9 are 9, 18, 27, 36, 45.

Fill in the blanks.

Question 4.
The first multiple of 9 is ____
Answer: 9
Explanation:
Any number that can be denoted in the form 9n where n is a natural number, is a multiple of 9. So if two values p and q are there, we say that q is a multiple of p if q = np. In other words, the multiples of 9 are the numbers that leave no remainder (i.e. Remainder = 0), when it is divided by 9.
Thus, as per the definition given above, the multiple of 9 is obtained by multiplying some integer with 9.
Take the first five multiples of 9: 9, 18, 27, 36, 45.
9 × 1 = 9; 9 multiplied by 1 to get 9
9 × 2 = 18; 9 multiplied by 2 to get 18
9 × 3 = 27; 9 multiplied by 3 to get 27
9 × 4 = 36; 9 multiplied by 4 to get 36
9 × 5 = 45; 9 multiplied by 5 to get 45
Therefore, the first multiple of 9 are 9.

Question 5.
The second multiple of 8 is ____
Answer: 16
The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80… and so on. It is a sequence where the difference between each next number and the preceding number, i.e. two consecutive results, is 8. Multiples are the numbers that give products of any number multiplied by other natural numbers.
The multiples of 8 are the numbers that are generated when 8 is multiplied by any natural number. That means any number that can be expressed in the form of 8n where n is an integer is a multiple of 8. As we know, if two values, p and q, are there, we say that q is a multiple of p if q = np for some integer n.
Thus, as per the definition given above, the multiple of 8 is obtained by multiplying some integer with 8.
Take the first five multiples of 8: 8, 16, 24, 32, 40.
8 × 1 = 8; 8 multiplied by 1 to get 8
8 × 2 = 16; 8 multiplied by 2 to get 16
8 × 3 = 24; 8 multiplied by 3 to get 24
8 × 4 = 32; 8 multiplied by 4 to get 32
8 × 5 = 40; 8 multiplied by 5 to get 40
Therefore, the second multiple of 8 is 16.

Question 6.
The first twelve multiples of 7 are ____
Answer: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84.

Explanation:
– A multiple is a numerical value that is generated when a natural number is multiplied by another natural number or counting number.
– Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, … and so on. How do we get to know a number is multiple of another number? If a number is multiple of another number, then it is evenly divisible by the original number. For example, 14 divided by 7 is equal to 2, thus it is a multiple of 7.
– Any number that can be denoted in the form 7n where n is a natural number is a multiple of 7. So if two values p and q are there, we say that q is a multiple of p if q = np for some natural number n.
7 x 1=7
7 x 2=14
7 x 3=15
7 x 4=28
7 x 5=35
If a number is multiple of 7, then it is evenly divisible by 7.
14 is a multiple of 7 because 14/7 = 2
21 is a multiple of 7 because 21/7= 3
28 is a multiple of 7 because 28/7=4
35 is a multiple of 7 because 35/7=5.
Therefore, the first twelve multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84.

Question 7.
The seventh multiple of 7 is ____
Answer: 49
Explanation:
– A multiple is a numerical value that is generated when a natural number is multiplied by another natural number or counting number.
– Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, … and so on. How do we get to know a number is multiple of another number? If a number is multiple of another number, then it is evenly divisible by the original number. For example, 14 divided by 7 is equal to 2, thus it is a multiple of 7.
– Any number that can be denoted in the form 7n where n is a natural number is a multiple of 7. So if two values p and q are there, we say that q is a multiple of p if q = np for some natural number n.
7 x 1=7
7 x 2=14
7 x 3=15
7 x 4=28
7 x 5=35
7 x 6=42
7 x 7=49
If a number is multiple of 7, then it is evenly divisible by 7.
14 is a multiple of 7 because 14/7 = 2
21 is a multiple of 7 because 21/7= 3
28 is a multiple of 7 because 28/7=4
35 is a multiple of 7 because 35/7=5.
42  is a multiple of 7 because 42/7=6
49 is a multiple of 7 because 49/7=9
Therefore, the first seven multiples of 7 are 7, 14, 21, 28, 35, 42, 49
Hence, the seventh multiple is 49.

Question 8.
The twelfth multiple of 7 is ____.
Answer: 84.
Explanation:
– A multiple is a numerical value that is generated when a natural number is multiplied by another natural number or counting number.
– Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, … and so on. How do we get to know a number is multiple of another number? If a number is multiple of another number, then it is evenly divisible by the original number. For example, 14 divided by 7 is equal to 2, thus it is a multiple of 7.
– Any number that can be denoted in the form 7n where n is a natural number is a multiple of 7. So if two values p and q are there, we say that q is a multiple of p if q = np for some natural number n.
7 x 1=7
7 x 2=14
7 x 3=15
7 x 4=28
7 x 5=35
7 x 6=42
7 x 7=49
7 x 8=56
7 x 9=63
7 x 10=70
7 x 11=77
7 x 12=84
If a number is multiple of 7, then it is evenly divisible by 7.
14 is a multiple of 7 because 14/7 = 2
21 is a multiple of 7 because 21/7= 3
28 is a multiple of 7 because 28/7=4
35 is a multiple of 7 because 35/7=5.
42 is a multiple of 7 because 42/7=6.
49 is a multiple of 7 because 49/7=7.
56 is a multiple of 7 because 56/7=8.
63 is a multiple of 7 because 63/7=9.
70 is a multiple of 7 because 70/7=10.
77 is a multiple of 7 because 77/7=11.
84 is a multiple of 7 because 84/7=12.
Therefore, the first twelve multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84.
Hence, the twelfth multiple is 84.

Check (✓) the correct box and fill in the blank when necessary.

Question 9.
Is 32 a multiple of 6?
Yes, it is the ___ multiple of 6.
No, it is not a multiple of 6.
Answer:
32 is not a multiple of 6
Explanation:
If we divide 32 and 6 then we get 5.333
no, it is not a multiple of 6.

When we divide any multiple with 6 then it should leave the remainder 0 then that number is the multiple of 6.

Question 10.
Is 63 a multiple of 9?
Yes, it is the ___ multiple of 9.
No, it is not a multiple of 9.
Answer: yes, 63 is the multiple of 9.
Explanation:

63 is a multiple of 9 because 63/9=7
Any number that can be denoted in the form 9n where n is a natural number, is a multiple of 9. So if two values p and q are there, we say that q is a multiple of p if q = np. In other words, the multiples of 9 are the numbers that leave no remainder (i.e. Remainder = 0), when it is divided by 9.
Thus, as per the definition given above, the multiple of 9 is obtained by multiplying some integer with 9.
Take the first ten multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
9 × 1 = 9; 9 multiplied by 1 to get 9
9 × 2 = 18; 9 multiplied by 2 to get 18
9 × 3 = 27; 9 multiplied by 3 to get 27
9 × 4 = 36; 9 multiplied by 4 to get 36
9 × 5 = 45; 9 multiplied by 5 to get 45
9 × 6 = 54; 9 multiplied by 6 to get 54
9 × 7 = 63; 9 multiplied by 7 to get 63
9 × 8 = 72; 9 multiplied by 8 to get 72
9 × 9 = 81; 9 multiplied by 9 to get 81
9 × 10 = 90; 9 multiplied by 10 to get 90
Therefore, 63 is a multiple of 9.

Use the numbers in the boxes to make your lists.

Question 11.
Multiples of 3 ____
Answer: 15, 24, 30, 63, 84
Explanation:
Any number that can be denoted in the form 3n where n is an integer is the multiples of 3. So if two values p and q are there, we say that q is a multiple of p if q = np for some integer n. In other words, the multiples of 3 are the numbers that leave no remainder (i.e. Remainder = 0), when it is divided by 3.
The above-given numbers are 30, 84, 15, 63, 56, 24.
To know which number is multiple of 3, we need to divide the given numbers by 3.
30 is the multiple of 3 because 30/3=10
84 is the multiple of 3 because 84/3=28
15 is the multiple of 3 because 15/3=5
63 is the multiple o 3 because 63/3=21
24 is the multiple of 3 because 24/3=8
56 is not the multiple of 3 because 56/3=18.6667
It is in decimals and the remainder does not leave with zero so 56 is not a multiple of 3.
Therefore, the multiples of 3 are 15, 24, 30, 63, 84.

Question 12.
Multiples of 8 _____
Answer: 24, 56
Explanation:
The multiples of 8 are the numbers that are generated when 8 is multiplied by any natural number. That means any number that can be expressed in the form of 8n where n is an integer is a multiple of 8. As we know, if two values, p and q, are there, we say that q is a multiple of p if q = np for some integer n.
The above-given numbers are 30, 84, 15, 63, 56, 24.
To know which number is multiple of 3, we need to divide the given numbers by 8.
30 is not the multiple of 8 because 30/8=3.75
84 is not the multiple of 8 because 84/8=10.5
15 is not the multiple of 8 because 15/8=1.875
63 is not the multiple of 8 because 63/8=7.875
It is in decimals and the remainder does not leave with zero. so these are all not the multiples of 8.
24 is the multiple of 8 because 24/8=3
56 is the multiple of 8 because 56/8=7
Therefore, the multiples of 8 are 24, 56.

Each shaded area shows some of the multiples of a number. Write the number in the box to the left of each shaded area.

Question 13.

Answer:

Explanation:
In simple words, the multiples of 2 are the numbers that leave no remainder (i.e. Remainder = 0), when it is divided by 2. It means the numbers that are exactly divided by 2 are the multiples of 2.
First 10 Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
In math sentences, we can write as:
2/2=1; remainder 0.
4/2=2; remainder 0.
6/2=3; remainder 0.
8/2=4; remainder 0.
10/2=5; remainder 0.
Since the numbers are exactly divided by 2, the numbers 2, 4, 6, 8, 10 are the multiples of 2.

Question 14.

Answer:

Explanation:
Any number that can be denoted in the form 3n where n is an integer is the multiples of 3. So if two values p and q are there, we say that q is a multiple of p if q = np for some integer n. In other words, the multiples of 3 are the numbers that leave no remainder (i.e. Remainder = 0), when it is divided by 3.
First 10 Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
9/3=3; remainder 0.
15/3=5; remainder 0.
18/3=6; remainder 0.
27/3=9; remainder 0.
81/3=27; remainder 0.
Since the above-given numbers are exactly divided by 3, the numbers 9, 15, 18, 27, 81 are the multiples of 3.

Question 15.

Answer:

Explanation:
Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, … and so on. How do we get to know a number is multiple of another number? If a number is multiple of another number, then it is evenly divisible by the original number. For example, 14 divided by 7 is equal to 2, thus it is a multiple of 7.
14/7=2; remainder 0.
21/7=3; remainder 0.
28/7=4; remainder 0.
63/7=9; remainder 0.
49/7=7; remainder 0.
Since the above-given numbers are exactly divided by 7, the numbers 14, 21, 28, 63, 49 are the multiples of 7.

Find the common multiples and the least common multiple.

1 × 2 = 2
2 × 2 = 4
3 × 2 = 6
4 × 2 = 8
5 × 2 = 10
6 × 2 = 12
7 × 2 = 14
8 × 2 = 16
9 × 2 = 18

1 × 3 = 3
2 × 3 = 6
3 × 3 = 9
4 × 3 = 12
5 × 3 = 15
6 × 3 = 18

A common multiple is shared by two or more numbers.

A common multiple that is less than all the others is called the least common multiple.

The first three common multiples of 2 and 3 are 6, 12, and 18
The least common multiple of 2 and 3 is 6.

Question 16.
The first 14 multiples of 5 are 5, 10, 15, 20, 25, 30, 35, ____
The first 10 multiples of? are 7, 14, 21, 28, 35, 42, _____
The first two common multiples of 5 and 7 are _____________
The least common multiple of 5 and 7 is __________
Answer: 35, 70
Explanation:
– The common multiples of 5 and 7 include 35, 70, 105, 140, 175, 210, 245, 280, and so on. The least common multiple is 35. A multiple is a result of multiplying a number by an integer.
– Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, and so on.
– Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, and so on.
– The quantities of 35, 70, 105, 140 are some of the common multiples of 7 and 5.
– A common multiple is defined as a quantity into which each of two or more quantities may be divided with zero remainders.
But in the question asked only the first two common multiples of 5 and 7:
The first two common multiples are 35, 70
The least common multiple of 5 and 7 is 35.

Question 17.
The first 1 5 multiples of 4 are ____________________________
The first 1 2 multiples of 5 are ____________________________
The first three common multiples of 4 and 5 are ____
The least common multiple of 4 and 5 is ____.
Answer:
Multiples of 4:
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80.
we can say that the multiples of 4 are the results in the multiplication table of 4, as both are the same.
The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100.
All numbers which can be divided or are a product of 5 are multiples of 5.
From the above numbers we need to write common multiples and least common multiple:
The first three common multiples of 4 and 5 are: 20, 40, 60
The least common multiple of 4 and 5 is 20.

Write the first ten multiples of each number. Then find the least common multiple.

Question 18.
8 and 5
8 ________
5 ________
The least common multiple of 8 and 5 is _____
Answer:
Multiples of 8:
The multiples of 8 are the numbers that are generated when 8 is multiplied by any natural number. That means any number that can be expressed in the form of 8n where n is an integer is a multiple of 8. As we know, if two values, p and q, are there, we say that q is a multiple of p if q = np for some integer n.
– Thus, as per the definition given above, the multiple of 8 is obtained by multiplying some integer with 8.
– The first ten multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
Multiples of 5:
The multiples of 5 are the numbers that are generated when 5 is multiplied by any natural number. That means any number that can be expressed in the form of 5n where n is an integer is a multiple of 5. As we know, if two values, p and q, are there, we say that q is a multiple of p if q = np for some integer n.
– As per the definition given above, the multiple of 5 is a number obtained by multiplying some integer with 5.
– The first ten multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
For example, 10, 20, 25 and 30 are all multiples of 5 for the following reasons.
5 × 2 = 10; 5 multiplied by 2 to get 10
5 × 4 = 20; 5 multiplied by 4 to get 20
5 × 5 = 25; 5 multiplied by 5 to get 25
5 × 6 = 30; 5 multiplied by 6 to get 30
From the above multiples we need to write the least common multiple:
The least common multiple of 8 and 5 are: 40
Since 40 is the first number they have in common, 40 is the least common multiple of 8 and 5.

Question 19.
6 and 9
6 _________
9 ________
The least common multiple of 6 and 9 is _____
Answer:
Multiples of 6:
The multiples for 6 are the numbers that are generated when 6 is multiplied by any natural number, such as 1, 2, 3, 4, 5, 6, 7, etc. That means any number that can be expressed in the form of 6n where n is an integer is a multiple of 6. As we know, if two values, p and q, are there, we say that q is a multiple of p if q = np for some integer n.
– As per the definition given above, the multiple of 6 is obtained by multiplying some integer with 6.
– The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.
Multiples of 9:
Any number that can be denoted in the form 9n where n is a natural number, is a multiple of 9. So if two values p and q are there, we say that q is a multiple of p if q = np. In other words, the multiples of 9 are the numbers that leave no remainder (i.e. Remainder = 0), when it is divided by 9.
First 10 Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90.
From the above multiples we need to write the least common multiple:
The least common multiple of 6 and 9 are: 18
Since 18 is the first number they have in common, 18 is the least common multiple of 6 and 9.

Question 20.
12 and 15
12 _____
15 ________
The least common multiple of 12 and 15 is ____
Answer:
Multiples of 12:
The multiples of 12 are the numbers that are generated when 12 is multiplied by any natural number. That means any number that can be expressed in the form of 12n where n is an integer is a multiple of 12. As we know, if two values, p and q, are there, we say that q is a multiple of p if q = np for some integer n.
– Thus, as per the definition given above, the multiple of 12 is obtained by multiplying some integer with 12.
– The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 120
Multiples of 15:
Any number that can be denoted in form 15n where n is an integer is a multiple of 15. So if two values p and q, we say that q is a multiple of p if q = np for some integer n.
– The multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150.
From the above multiples we need to write the least common multiple:
The least common multiple of 12 and 15 are: 60
Since 60 is the first number they have in common, 60 is the least common multiple of 12 and 15.

Fill in the blanks. More than one answer is possible.

Question 21.
12 is the least common multiple of 3 and ____
Answer: 4
Explanation:
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 26, 40.
– When we compare the two lists to see what they have in common, we get the answer to “What are the common multiples of 3 and 4?”
– 12, 24, 36, 48, etc.
– Since 12 is the first number they have in common, 12 is the least common multiple of 3 and 4.

Question 22.
32 is the least common multiple of 8 and ___
Answer: 32
Explanation:
– Common multiples of 8 and 32 are numbers that both 8 and 32 can be divided into evenly with no remainder.
– To find the common multiples of 8 and 32, we compare the list of multiples of 8 with the list of multiples of 32 to see what they have in common.
– To create a list of multiples of 8, we multiply 8 by 1, 8 by 2, and so on like this:
8 x 1 = 8
8 x 2 = 16
8 x 3 = 24
8 x 4 = 32
…
– Similarly, to create a list of multiples of 32, we multiply 32 by 1, 32 by 2, and so on like this:
32 x 1 = 32
32 x 2 = 64
32 x 3 = 96
32 x 4 = 128
…
– When we compare the two lists to see what they have in common, we get the answer to “What are the common multiples of 8 and 32?”
32, 64, 96, 128, etc.
– Since 32 is the first number they have in common, 32 is the least common multiple of 8 and 32.

Question 23.
24 is the least common multiple of 6 and ____
Answer: 8
Explanation:
– Common multiples of 6 and 8 are numbers that both 6 and 8 can be divided into evenly with no remainder.
– To find the common multiples of 6 and 8, we compare the list of multiples of 6 with the list of multiples of 8 to see what they have in common.
– To create a list of multiples of 6, we multiply 6 by 1, 6 by 2, and so on like this:
6 x 1 = 6
6 x 2 = 12
6 x 3 = 18
6 x 4 = 24
…
– Similarly, to create a list of multiples of 8, we multiply 8 by 1, 8 by 2, and so on like this:
8 x 1 = 8
8 x 2 = 16
8 x 3 = 24
8 x 4 = 32
…
– When we compare the two lists to see what they have in common, we get the answer to “What are the common multiples of 6 and 8?”
24, 48, 72, 96 etc.
– Since 24 is the first number they have in common, 24 is the least common multiple of 6 and 8.

Question 24.
15 is the least common multiple of 3 and ____
Answer:5
Explanation:
– Common multiples of 3 and 5 are numbers that both 3 and 5 can be divided into evenly with no remainder.
– To find the common multiples of 3 and 5, we compare the list of multiples of 3 with the list of multiples of 5 to see what they have in common.
– To create a list of multiples of 3, we multiply 3 by 1, 3 by 2, and so on like this:
3 x 1 = 3
3 x 2 = 6
3 x 3 = 9
3 x 4 = 12
…
– Similarly, to create a list of multiples of 5, we multiply 5 by 1, 5 by 2, and so on like this:
5 x 1 = 5
5 x 2 = 10
5 x 3 = 15
5 x 4 = 20
…
– When we compare the two lists to see what they have in common, we get the answer to “What are the common multiples of 3 and 5?”
15, 30, 45, 60, etc.
– Since 15 is the first number they have in common, 15 is the least common multiple of 3 and 5.

Question 25.
60 is the least common multiple of 15 and ___.
Answer: 20
Explanation:
– Common multiples of 15 and 20 are numbers that both 15 and 20 can be divided into evenly with no remainder.
– To find the common multiples of 15 and 20, we compare the list of multiples of 15 with the list of multiples of 20 to see what they have in common.
– To create a list of multiples of 15, we multiply 15 by 1, 15 by 2, and so on like this:
15 x 1 = 15
15 x 2 = 30
15 x 3 = 45
15 x 4 = 60
…
– Similarly, to create a list of multiples of 20, we multiply 20 by 1, 20 by 2, and so on like this:
20 x 1 = 20
20 x 2 = 40
20 x 3 = 60
20 x 4 = 80
…
-When we compare the two lists to see what they have in common, we get the answer to “What are the common multiples of 15 and 20?”
60, 120, 180, 240, etc.
– Since 60 is the first number they have in common, 60 is the least common multiple of 15 and 20.

Go through the Math in Focus Grade 4 Workbook Answer Key Chapter 4 Practice 2 Using a Table to finish your assignments.

Math in Focus Grade 4 Chapter 4 Practice 2 Answer Key Using a Table

Use the data in the table to complete the sentences below.

The table shows pictures at rows, columns, and intersections.

Example

Question 1.
___________ is at the intersection of Row E and Column 1.
Answer:
is at the intersection of Row E and Column 1.

Question 2.
is in Row __________ and Column ____.
Answer:
is in Row B and Column 6.

Question 3.
is in Row __________ and Column __________
Answer:
is in Row A and Column 2.

Question 4.
is in Row __________ and Column __________
Answer:
is in Row D and Column 4.

The table shows part of Bill’s class schedule from Monday through Wednesday.

Use the table to answer the questions.

Question 5.
What class does Bill have between 10:00 A.M. and 11:00 A.M. on Mondays? __________
Answer:
Math.

Explanation:
Bill has Math class between 10:00 A.M. and 11:00 A.M. on Mondays

Question 6.
What class does Bill have between 9:00 A.M. and 10:00 A.M. on Wednesdays? ____
Answer:
History.

Explanation:
Bill has history class between 9:00 A.M. and 10:00 A.M. on Wednesdays

Question 7.
His lunch break on Wednesday is between ____________________
Answer:
12:00 P.M to 1:P.M.

Explanation:
Bill’s lunch break on Wednesday is between 12:00 P.M to 1:P.M.

Question 8.
His Math class on _________________ is between 11:00 A.M. and 1 2:00 P.M.
Answer:
Wednesday.

Explanation:
His Math class on Wednesday is between 11:00 A.M. and 1 2:00 P.M.

Question 9.
His Geography class on ________________ is between 10:00 A.M. and 11:00 A.M.
Answer:
Tuesday.

Explanation:
His Geography class on Tuesday is between 10:00 A.M. and 11:00 A.M.

Maria and Vinny collected stamps from three different countries: Singapore, Malaysia, and Thailand. The number of stamps collected is shown in the table below.

Complete the table, and answer the questions.

Question 10.
How many Thailand stamps did Vinny collect? ____
Answer:
29 Stamps.

Explanation:
Vinny collected 52-23 which is 29 Thailand stamps.

Question 11.
How many Thailand stamps did Maria and Vinny collect altogether? ____
Answer:
52 stamps.

Explanation:
The total number of Thailand stamps Maria and Vinny collect altogether is 23+29 which is 52 stamps.

Question 12.
How many more Malaysia stamps than Singapore stamps did Maria and Vinny collect altogether? ____
Answer:
14 more stamps.

Explanation:
The number of Malaysia stamps did Maria and Vinny collect altogether is 60 stamps and the number of Singapore stamps did Maria and Vinny collect altogether is 46 stamps. So the number of many Malaysia stamps than Singapore stamps is 60-46 which is 14 stamps.

Question 13.
Who collected more stamps: Maria or Vinny? ____
Answer:
Maria.

Explanation:
The total number of stamps did Maria collected is 15+42+43 which is 80 stamps and Vinny collected 31+18+29 which is 78 stamps. So Maria collected more stamps.

Question 14.
How many stamps did they collect altogether? ____
Answer:
158 stamps.

Explanation:
The number of stamps did they collect altogether is 78+80 which is 158 stamps.

The table shows the number of quarters and nickels that five students saved.

Complete the table, and answer the questions.

Question 15.
Who saved the greatest amount? ____
Answer:
Chin.

Explanation:
The greatest amount saved by Chin.

Question 16.
Who saved the least amount? ____
Answer:
$2.85.

Explanation:
The least amount saved is $2.85.

Question 17.
How many more coins did Chin collect than Ernest? ____
Answer:
13 more coins did Chin collect than Ernest.

Explanation:
The total number of coins collected by Chin is 43 and the total number of coins collected by Ernest is 30. So there are 43-30 which is 13 more coins did Chin collect than Ernest.

Question 18.
How much more must Bernard save in order to have the same amount as Dawn? ____
Answer:
Bernard needs to save $2.85.

Explanation:
The total amount did Bernard have is $2.85 and the total amount did Dawn have is $5.7. So Bernard needs to save $5.7-$2.85 which is $2.85.

Question 19.
Which two students saved a total of less than $7.50? How much less ? ____
Answer:
Bernard and Ernest.

Explanation:
The two students who saved a total of less than $7.50 is Bernard and Ernest which is $2.85+$4.5 which is $7.35.

Question 20.
Which two students collected the same number of coins?
Answer:
Dawn and Ernest.

Explanation:
Dawn and Ernest are the two students who collected the same number of coins.

Question 21.
Of the two students in Exercise 20, who saved more money? How much more?
Answer:
Chin $5.75.

Explanation:
More money was saved by Chin which is $5.75.

Practice the problems of Math in Focus Grade 4 Workbook Answer Key Chapter 6 Practice 2 Subtracting Fractions to score better marks in the exam.

Math in Focus Grade 4 Chapter 6 Practice 2 Answer Key Subtracting Fractions

Find the equivalent fraction. Complete the model. Then subtract.

Example

Question 1.

Answer:

Explanation:
\(\frac{2}{3}\) – \(\frac{2}{9}\)
= (6 – 2) ÷ 9
= 4 ÷ 9 or \(\frac{4}{9}\)

Subtract. Write each answer in simplest form.
Question 2.

Answer:

Explanation:
\(\frac{8}{10}\) – \(\frac{1}{5}\)
= (8 – 2) ÷ 10
= 6 ÷ 10
= 3 ÷ 5 or \(\frac{3}{5}\)

Question 3.

Answer:

Explanation:
\(\frac{7}{12}\) – \(\frac{1}{4}\)
= (7 – 3) ÷ 12
= 4 ÷ 12
= 1 ÷ 3 or \(\frac{1}{3}\)

Question 4.
The difference between \(\frac{7}{8}\) and \(\frac{1}{4}\) is
Answer:
The difference between \(\frac{7}{8}\) and \(\frac{1}{4}\) is

Explanation:
Difference:
\(\frac{7}{8}\) – \(\frac{1}{4}\)
= (7 – 2) ÷ 4
= 5 ÷ 4 or \(\frac{5}{4}\)

Question 5.
The difference between \(\frac{7}{12}\) and \(\frac{1}{3}\) is
Answer:
The difference between \(\frac{7}{12}\) and \(\frac{1}{3}\) is

Explanation:
\(\frac{7}{12}\) – \(\frac{1}{3}\)
= (7 – 4) ÷ 12
= 3 ÷ 12
= 1 ÷ 4 or \(\frac{1}{4}\)

Practice the problems of Math in Focus Grade 4 Workbook Answer Key Chapter 6 Practice 5 Renaming Improper Fractions and Mixed Numbers to score better marks in the exam.

Math in Focus Grade 4 Chapter 6 Practice 5 Answer Key Renaming Improper Fractions and Mixed Numbers

Express each improper fraction as a mixed number.

Example

Question 1.

Answer:

Explanation:
\(\frac{12}{7}\) = \(\frac{7}{7}\) + \(\frac{5}{7}\)
= 1 + \(\frac{5}{7}\)
= 1\(\frac{5}{7}\)

Question 2.

Answer:

Explanation:
\(\frac{9}{4}\) = \(\frac{8}{4}\) + \(\frac{1}{4}\)
= 2 + \(\frac{1}{4}\)
= 2\(\frac{1}{4}\)

Question 3.

Answer:

Explanation:
\(\frac{13}{6}\) = \(\frac{12}{6}\) + \(\frac{1}{6}\)
= 2 + \(\frac{1}{6}\)
= 2\(\frac{1}{6}\)

Express each improper fraction as a whole number or a mixed number in simplest form. Show your work.

Question 4.
\(\frac{17}{4}\) = ____
Answer:
\(\frac{17}{4}\) = 4\(\frac{1}{4}\)

Explanation:
\(\frac{17}{4}\) = 4\(\frac{1}{4}\)  {[(4 × 4) + 1] ÷ 4}

Question 5.
\(\frac{29}{6}\) = ____
Answer:
\(\frac{29}{6}\) = 4\(\frac{5}{6}\)

Explanation:
\(\frac{29}{6}\) = 4\(\frac{5}{6}\) {[(4 × 6) + 5] ÷ 6}

Express each improper fraction as a whole number or a mixed number in simplest form. Show your work.

Question 6.

Answer:

Explanation:
\(\frac{9}{6}\)  =  \(\frac{6}{6}\)   + \(\frac{7}{6}\)
= 1 + \(\frac{3}{6}\)
= 1 \(\frac{3}{6}\)

Question 7.
\(\frac{12}{4}\) = ____
Answer:
\(\frac{12}{4}\) = 3.

Explanation:
\(\frac{12}{4}\) = \(\frac{3}{4}\) + \(\frac{9}{4}\)
= (3 + 9) ÷ 4
= 12 ÷ 4
= 3.

Question 8.
\(\frac{21}{3}\) = ____
Answer:
\(\frac{21}{3}\) = 7.

Explanation:
\(\frac{21}{3}\) = \(\frac{17}{3}\) + \(\frac{4}{3}\)
= (17 + 4) ÷ 3
= 21 ÷ 3
= 7.

Question 9.

Answer:

Explanation:
\(\frac{14}{4}\)  = 2 + \(\frac{2}{4}\)
= 2 \(\frac{2}{4}\)

Question 10.

Answer:

Explanation:
\(\frac{15}{6}\)  = 2 + \(\frac{3}{6}\)
= 2 \(\frac{3}{6}\)

Express each mixed number as an improper fraction.

Example

Question 11.

Answer:

Explanation:
3\(\frac{5}{9}\) = 3 + \(\frac{5}{9}\)
=  \(\frac{27}{9}\) + \(\frac{5}{9}\)
=  \(\frac{32}{9}\)

Question 12.

Answer:

Explanation:
2\(\frac{5}{8}\) = 2 + \(\frac{5}{8}\)
=  \(\frac{16}{8}\) + \(\frac{5}{8}\)
=  \(\frac{21}{8}\)

Question 13.

Answer:

Explanation:
4\(\frac{2}{7}\) = 4 + \(\frac{2}{7}\)
=  \(\frac{28}{7}\) + \(\frac{2}{7}\)
=  \(\frac{30}{7}\)

Express each mixed number as an improper fraction.

Example

Question 14.
2\(\frac{3}{8}\) = ____
Answer:
2\(\frac{3}{8}\) = 19 ÷ 8 or \(\frac{19}{8}\)

Explanation:
2\(\frac{3}{8}\) = (16 + 3) ÷ 8     (2 × 8 = 16)
= 19 ÷ 8 or \(\frac{19}{8}\)

Question 15.
3\(\frac{3}{4}\) = ____
Answer:
3\(\frac{3}{4}\) = 15 ÷ 4 or \(\frac{15}{4}\)

Explanation:
3\(\frac{3}{4}\) =  (12 + 3) ÷ 4
= 15 ÷ 4 or \(\frac{15}{4}\)

Question 16
6\(\frac{2}{5}\) = ____
Answer:
6\(\frac{2}{5}\) = 32 ÷ 5 or \(\frac{32}{5}\)

Explanation:
6\(\frac{2}{5}\) = (30 + 2) ÷ 5    (6 × 5 = 30)
= 32 ÷ 5 or \(\frac{32}{5}\)

Question 17.
2\(\frac{4}{7}\) = ____
Answer:
2\(\frac{4}{7}\) = 18 ÷ 7 or \(\frac{18}{7}\)

Explanation:
2\(\frac{4}{7}\) = (14 + 4) ÷ 7   (2 × 7 = 14)
= 18 ÷ 7 or \(\frac{18}{7}\)

Express each mixed number as an improper fraction and each improper fraction as a mixed or whole number. Then solve the riddle.

Question 18.

Answer:

Explanation:
\(\frac{9}{7}\) = 1 + \(\frac{2}{7}\)
= 1 \(\frac{2}{7}\)

Question 19.

Answer:

Explanation:
\(\frac{15}{6}\) = 2 + \(\frac{3}{6}\)
= 2\(\frac{3}{6}\)

Question 20.

Answer:

Explanation:
3\(\frac{5}{8}\) = \(\frac{24}{8}\) + \(\frac{5}{8}\)
= (24 + 5) ÷ 8
= \(\frac{29}{8}\) or 29 ÷ 8.

Question 21.

Answer:

Explanation:
5\(\frac{3}{5}\) = \(\frac{25}{5}\) + \(\frac{3}{5}\)
= (25 + 3) ÷ 5
= \(\frac{28}{5}\) or 28 ÷ 5.

Which two animals can look behind without turning their heads? Write the letters which match the answers to find out.

Answer:

Explanation:
2 = a.
\(\frac{16}{7}\) = i.
2\(\frac{1}{2}\) = o.

This handy Math in Focus Grade 5 Workbook Answer Key Chapter 12 Angles provides detailed solutions for the textbook questions.

Math in Focus Grade 5 Chapter 12 Answer Key Angles

Math Journal

Check the box for each correct statement. Then explain your answer.

Question 1.
\(\overleftrightarrow{X Y}\) is a line.

Answer:

Explanation:
The XY is a line and the angle on a straight line is 180°, known as straight angle.
Angle POQ is 90° as per the given information in the diagram.
180° – 90° = 90°
Angle ∠XOP and ∠YOQ are equal angle = 45°

Question 2.
\(\over left right arrow{A B}\) and \(\over left right arrow{C D}\) meet at O.

Answer:

Explanation:
\(\over left right arrow{A B}\) is a line and \(\over left right arrow{C D}\) is a line are crossed at O, the opposite angles are same.
So, m∠e = m∠h and
m∠f + m∠g = m∠j   are true statements.

Put On Your Thinking Cap!

Challenging Practice

Find the unknown angle measures. Explain.

Question 1.
\(\over left right arrow{G J}\) is a line. ∠LHK is a right angle. Find the measure of ∠LHJ.

Answer: 65°
Explanation:
Given information
GJ is a straight line, ∠LHK = 90°
∠JHK = 180° – ∠JHK
= 180° – 155° = 25°
∠JHK = ∠LHK – ∠JHK
= 90° – 25° = 65°

Question 2.
\(\over left right arrow{M N}\) and \(\over left right arrow{X Y}\) meet at O and m ∠a = m ∠b. Find the measure of ∠c.

Answer: 135°
Explanation:
\(\over left right arrow{X Y}\) is a line and \(\over left right arrow{M N}\) is a line are crossed at O, the opposite angles are same.
given information m∠a = m∠b and ∠XOP = 90
as XOY is a straight line, the angle is 180°
90° + m∠a + m∠b = 180°
∠c = 180° -∠XOM= 180 – 45° = 135°

Question 3.
\(\over left right arrow{A C}\) is a line. ∠ABE and ∠DBF are right angles. Find the measure of ∠FBC.

Answer: 26°
Explanation:
∠ABE and ∠DBF = 90°
∠EBF = ∠DBF – ∠DBE
=  90° – 26° = 64°
∠ABD = ∠ABE – ∠DBE
= 90° – 26° = 64°
∠FBC = 180 – (∠ABE + ∠EBF)
= 180° – (90° + 64°)
=180° – 154° = 26°

Question 4.
\(\over left right arrow{A B}\) and \(\over left right arrow{W X}\) meet at O. ∠YOX are right angles. Find the measures of ∠AOX and ∠COY.

Answer:
∠AOX  = 124°
∠COY  =  56°
Explanation:
90° – 56° = 34°
90° – 34° = 56°
∠COY = ∠COB – ∠BOY
= 90° – 34° = 56°
∠AOX = ∠WOX – ∠AOW
=180°- 56° = 124°

Put on Your Thinking cap!

Problem Solving

Solve.

Question 1.
\(\over left right arrow{J K}\) and \(\over left right arrow{L M}\) are lines.
Check the box for each correct statement.

Answer:
\(\over left right arrow{J K}\) and \(\over left right arrow{L M}\) are lines.

Explanation:
\(\over left right arrow{J K}\) is a line and \(\over left right arrow{L M}\) is a line are crossed at O, the opposite angles are same.
so, m∠r + m∠s = m∠p + m∠q is the wrong statement.

Question 2.
\(\over left right arrow{A B}\), \(\over left right arrow{C D}\), and \(\over left right arrow{E F}\) meet at O. Find the sum of the measures of ∠AOC, ∠FOD, and ∠BOE.
m∠AOC + m∠FOD + m∠BOE = _____

Answer: 180°
Explanation:
m∠AOC = 45°

m∠FOD = 45°

m∠BOE = 90°

m∠AOC + m∠FOD + m∠BOE = 180°

Question 3.
ABCD is a square. \(\over right arrow{B E}\) is a ray. Find the measure of ∠x.

Answer: 135°
Explanation:
As BE is a straight line and the angle is 180° at point D
and a square is ABCD is with 90° angle formed at point D
the ange ∠EDC is 135° angle ∠CDB is 45° and ∠ADB is also 45°
now the
∠x =  ∠EDB – ∠ADB
=  180° – 45°  =  135°

Question 4.
How many degrees does the hour hand of a clock turn between 3 P.M. and 7:30 P.M.?
Answer: 135°
Explanation:

Hour hand at 3PM is at 3 and 7:30 as shown in the clock diagram
Total angle is 360°, keep it in mind
and 360° divide in 12 parts
each part is of 30°
from hours hand 3 to 7 :30 its 135°

Question 5.
\(\over left right arrow{A B}\) is a line. The measures of ∠a and ∠b are whole numbers.

If the measure of ∠b is twice that of ∠a, find the measures of ∠a and ∠b.
Answer:
∠a = 60°
∠b = 120°

Explanation:
Here the hint is angle ∠b is twice that of ∠a, and The AB is a line and the angle on a straight line is 180, known as straight angle
∠b + ∠a = 180°
2∠a + ∠a = 180°
3∠a = 180°
∠a = 180°/3 = 60°
∠a = 60°
∠a = 60°
∠b = 2x∠a = 120°

This handy Math in Focus Grade 3 Workbook Answer Key Chapter 17 Practice 2 Right Angles provides detailed solutions for the textbook questions.

Math in Focus Grade 3 Chapter 17 Practice 2 Answer Key Right Angles

Look at these angles. Use a piece of folded paper
to help you answer the questions.

Question 1.
Which angle is less than a right angle?
Angle ________________
Answer:

Angle  C, D and E are angles less then a right angle.
Explanation:
Angle less than a right angles are Acute angles.

Question 2.
Which angle is greater than a right angle?
Angle ________________
Answer:

Angle A is greater then a right angle.
Explanation:
Angle greater than a right angles are Obtuse angles.

Question 3.
Which angles are the same size as right angles?
Angles _________________
Answer:

Angle A and F are the same as right angles.
Explanation:
The angle bounded by two lines perpendicular to each other.
A right triangle has one angle exactly equal to 90 degrees.

Mark all the right angles in each figure.

Question 4.

Answer:

Explanation:
The above figure has,
One right angle is there, which is marked blue color.
The angle bounded by two lines perpendicular to each other.
A right triangle has one angle exactly equal to 90 degrees.

Question 5.

Answer:

Explanation:
The above figure has,
One right angle is there, which is marked blue color.
The angle bounded by two lines perpendicular to each other.
A right triangle has one angle exactly equal to 90 degrees.

Question 6.

Answer:

Explanation:
The above figure has,
One right angle is there, which is marked blue color.
The angle bounded by two lines perpendicular to each other.
A right triangle has one angle exactly equal to 90 degrees.

Question 7.

Answer:

Explanation:
The above figure has,
One right angle triangles , the angle is marked blue color.
The angle bounded by two lines perpendicular to each other.
A right triangle has one angle exactly equal to 90 degrees.

Question 8.

Answer:

Explanation:
The above figure has,
Two right angle are there, which are marked blue color.
The angle bounded by two lines perpendicular to each other.
A right triangle has one angle exactly equal to 90 degrees.

Question 9.

Answer:

Explanation:
The given figure has,
Two right angle are there, which are marked in blue color.
The angle bounded by two lines perpendicular to each other.
A right triangle has one angle exactly equal to 90 degrees.

This handy Math in Focus Grade 5 Workbook Answer Key Chapter 15 Surface Area and Volume provides detailed solutions for the textbook questions.

Math in Focus Grade 5 Chapter 15 Answer Key Surface Area and Volume

Math Journal

This rectangular container is \(\frac{2}{5}\)-filled with water.
How much more water is needed to increase the height of the water level to 3 centimeters?

Show two methods of solving this problem. Which method do you prefer? Why?
Answer:
Given rectangular tank has volume 8 cm X 10 cm X 5 cm = 400 cm³,
Method 1:
Knowing the volume of tank and water filled in it
as the rectangular container is \(\frac{2}{5}\)-filled with water.
\(\frac{2}{5}\) X 400 cm³ = 160 cm³, filled with water.
as 1 cubic centimeter is equal to 1 milliliters,

Put on Your Thinking Cap!

Challenging Practice

Question 1.
A rectangular tank is half-filled with water.
Another 650 cubic centimeters of water are needed to make it \(\frac{3}{5}\) full.
How much water will be in the tank when it is \(\frac{3}{5}\) full?
Answer:
3,900 cubic centimeters of water will be in the tank when it is \(\frac{3}{5}\) full,

Explanation:
Given a rectangular tank is half-filled with water.
Another 650 cubic centimeters of water are needed to make it \(\frac{3}{5}\) full.
Let x cubic centimeters of water so \(\frac{1}{2}\)x + \(\frac{3}{5}\)x = 650 cm³ + x,
\(\frac{5x + 6x}{10}\) = 650 cm³ + x,
\(\frac{11x}{10}\) = 650 cm³ + x,
11 x = 10 X (650 cm³ + x),
11 x =  6500 cm³ + 10 x,
11x – 10 x = 6,500 cm³,
x = 6,500 cm³,
So, water will be in the tank when it is \(\frac{3}{5}\) full is
\(\frac{3}{5}\) x 6,500 cm³ = \(\frac{3 X 6,500}{5}\) cm³,
\(\frac{19,500}{5}\) cm³ = 3,900 cm³,
therefore 3,900 cubic centimeters of water will be in the tank when it is \(\frac{3}{5}\) full.

Question 2.
A cube has a surface area of 21 6 square centimeters.
A second cube has edges that are 3 times as long.
How much greater is the surface area of the second cube than the first cube?
Answer:
1,728 cm² greater is the surface area of the second cube than the first cube,

Explanation:
Given a cube has a surface area of 216 square centimeters as
surface area of cube is 6 X a²= 216 cm²,
a² = 36 cm² = 6 cm X 6 cm,
edge a    = 6 cm,
A second cube has edges that are 3 times long as
6 cm X 3 = 18 cm, Surface area of second cube is 6 x (18)² =
6 X 18 cm X 18 cm = 1,944 cm², Now compairing much greater is the
surface area of the second cube than the first cube is 1,944 cm² – 216 cm² = 1,728 cm².

Put on Your Thinking Cap!

Problem Solving

A prism has a square base whose edges each measure 5 centimeters.
The ratio of its height to its width is 4 : 1.
Find the volume of the rectangular prism in cubic centimeters.

Answer:
Given a prism has a square base whose edge each measure 5 centimeters.
height = 4 cm
width = 1 cm
Volume of the rectangular prism = 2l × 2w × 2h
V = 2 × 5 × 2 × 1 × 2 × 4
V = 80 cubic centimeters

This handy Math in Focus Grade 5 Workbook Answer Key Chapter 15 Practice 5 Volume of a Rectangular Prism and Liquid provides detailed solutions for the textbook questions.

Math in Focus Grade 5 Chapter 15 Practice 5 Answer Key Volume of a Rectangular Prism and Liquid

Write the length, width, and height of each rectangular prism or cube.

Example

Answer:
Volume of given rectangular prism is 240 cm³,

Explanation:
Given Length = 5 cm, Width = 8 cm and Height = 6 cm,
Volume of the rectangular prism is lwh = 5 cm X 8 cm X 6 cm = 240 cm³.

Question 1.

Length = _____12_______ ft
Width = _______12_____ ft
Height = _______20_____ ft
Answer:
Volume of the recangular prism is cube is 2,880 ft³,

Explanation:
Given rectangular prism has length 12 ft, width 12 ft and height 20 ft,
so volume of the rectangular prism is 12 ft X 12 ft X 20 ft = 2,880 ft³.

Question 2.

Length = ____12________ in.
Width = ______25______ in.
Height = _______16_____ in.
Answer:
Volume of the recangular prism is 4,800 in³,

Explanation:
Given rectangular prism has length 12 in, width 25 in and height 16 in,
so volume of the rectangular prism is 12 in X 25 in X 16 in = 4,800 in³.

Question 3.

Length = _____18_______ m
Width = ______18______ m
Height = ______18______ m
Answer:
Volume of the cube is 5832 m³,

Explanation:
Given cube has length 18 m, width 18 m and height 18 m,
so volume of the cube is 18 m X 18 m X 18 m = 5832  m³.

Find the volume of each rectangular prism.

Question 4.

The length of the rectangular prism is _______12_____ centimeters.
The width of the rectangular prism is _______5_____ centimeters.
The height of the rectangular prism is _______9_____ centimeters.
Volume of the rectangular prism = length × width × height
= _____12_______ × _____5_______ ×______9______
= ______540______ cm³
Answer:
Volume of the recangular prism is cube is 540 cm³,

Explanation:
Given rectangular prism has length 12 cm, width 5 cm and height 9 cm,
so volume of the rectangular prism is 12 cm X 5 cm X 9 cm = 540 cm³.

Question 5.

Volume = _____24_______in³
Answer:
Volume of the recangular prism is 24 in³,

Explanation:
Given rectangular prism has length 3 in, width 4 in and height 2 in,
so volume of the rectangular prism is 3 in X 4 in X 2 in = 24 in³.

Question 6.

Volume = _____120_______ yd³,
Answer:
Volume of the recangular prism is 120 yd³,

Explanation:
Given rectangular prism has length 5 yd, width 8 yd and height 3 yd,
so volume of the rectangular prism is 5 yd X 8 yd X 3 yd = 120 yd³.

Find the volume of each rectangular prism or cube.

Question 7.

Volume = _____120_______ft³,
Answer:
Volume of the recangular prism is cube is 120 ft³,

Explanation:
Given rectangular prism has length 4 ft, width 6 ft and height 5 ft,
so volume of the rectangular prism is 4 ft X 6 ft X 5 ft = 2,880 ft³.

Question 8.

Volume = ______4,096______m³,
Answer:
Volume of the cube is 4,096 m³,

Explanation:
Given cube has length 18 m, width 18 m and height 18 m,
so volume of the cube is 16 m X 16 m X 16 m = 4,096 m³.

Question 9.

Volume = _____384_______cm³,

Answer:
Volume of the recangular prism is cube is 384 cm³,

Explanation:
Given rectangular prism has length 4 cm, width 24 cm and height 4 cm,
so volume of the rectangular prism is 4 cm X 24 cm X 4 cm = 384 cm³.

Question 10.

Volume = _____189_______ in³,
Answer:
Volume of the recangular prism is 189 in³,

Explanation:
Given rectangular prism has length 7 in, width 9 in and height 3 in,
so volume of the rectangular prism is 7 in X 9 in X 3 in = 189 in³.

Question 11.

Volume = ______448______ in³,
Answer:
Volume of the recangular prism is 448 in³,

Explanation:
Given rectangular prism has length 8 in, width 8 in and height 7 in,
so volume of the rectangular prism is 8 in X 8 in X 7 in = 448 in³.

Question 12.

Volume = _____120_______ m³,
Answer:
Volume of the recangular prism is 120 m³,

Explanation:
Given rectangular prism has length 8 m, width 3 m and height 5 m,
so volume of the rectangular prism is 8 m X 3 m X 5 m = 120 m³.

Find the volume of each rectangular prism.

Question 13.
Length = 5 cm
Width = 12 cm
Height = 9 cm
Volume = ______540_______ cm³,
Answer:
Volume of the recangular prism is 540 cm³,

Explanation:
Given rectangular prism has length 5 cm, width 12 cm and height 9 cm,
so volume of the rectangular prism is 5 cm X 12 cm X 9 cm = 540 cm³.

Question 14.
Length = 10 in.
Width = 25 in.
Height = 14 in.
Volume = ______3,500_______ in³,
Answer:
Volume of the recangular prism is 3,500 in³,

Explanation:
Given rectangular prism has length 10 in, width 25 in and height 14 in,
so volume of the rectangular prism is 10 in X 25 in X 14 in = 3,500 in ³.

Question 15.
Length = 7 m
Width = 12 m
Height = 8 m
Volume = _____672________ m³,
Answer:
Volume of the recangular prism is 672 m³,

Explanation:
Given rectangular prism has length 7 m, width 12 m and height 8 m,
so volume of the rectangular prism is 7 m X 12 m X 8 m = 672 m³.

Question 16.
Length = 24 ft
Width = 10 ft
Height = 15 ft
Volume = ______3,600_______ ft³,
Answer:
Volume of the recangular prism is 3,600 ft³,

Explanation:
Given rectangular prism has length 24 ft, width 10 ft and height 15 ft,
so volume of the rectangular prism is 24 ft X 10 ft X 15 ft = 3,600 ft³.

Solve. Show your work.

Question 17.
Find the volume of a cube with edges measuring 9 centimeters.
Answer:
Volume of the cube is 729 cm³,

Explanation:
Given cube has edges measuring 9 centimeters each
means length 9 cm, width 9 cm and height 9 cm,
so volume of the cube is 9 cm X 9 cm X 9 cm = 729 cm³.

Question 18.
A rectangular prism has a width of 8 feet and a height of 5 feet.
Its length is twice its width.
Find the volume of the rectangular prism.
Answer:
Volume of the recangular prism is 640 ft³,

Explanation:
Given rectangular prism has width 8 feet and height 5 feet,
its length is twice its width so length is 2 X 8 feet = 16 feet,
so volume of the rectangular prism is 16 ft X 8 ft X 5 ft =640 ft³.

Question 19.
The base of a rectangular prism is a square whose sides each measure 9 inches.
The height of the rectangular prism is 11 inches. Find its volume.
Answer:
Volume of the recangular prism is 891 in³,

Explanation:
Given the base of a rectangular prism is a square whose sides are each measure 9 inches.
The height of the rectangular prism is 11 inches.
So the rectangular prism has length 9 inches, width 9 inches and height 11 inches,
therefore, volume of the rectangular prism is 9 inches X 9 inches X 11 inches = 891 in³.