Key Concepts

• Divide a mixed number by a mixed number

• Divide a whole number by a mixed number

• Divide a mixed number by a whole number

Introduction:

What is a mixed fraction? 

Combination of a whole number and a proper fraction is known as a mixed fraction. 

How to convert mixed numbers to improper fractions? 

The following steps explain the conversion of mixed number to improper fraction: 

Step 1: Multiply the whole number by the denominator of a fraction. 

Step 2: Add the product to the numerator. 

Step 3: The required result is the improper fraction. 

Mixed number to improper fraction conversion formula: 

Improper fraction =

                             𝐰𝐡𝐨𝐥𝐞 𝐧𝐮𝐦𝐛𝐞𝐫×𝐝𝐞𝐧𝐨𝐦𝐢𝐧𝐚𝐭𝐨𝐫+𝐧𝐮𝐦𝐞𝐫𝐚𝐭𝐨𝐫
                      __________________________________________________
                                                𝐝𝐞𝐧𝐨𝐦𝐢𝐧𝐚𝐭𝐨𝐫

Example: Convert 1^3/4 into an improper fraction. 

Solution:  

From the given mixed number, 

1 is a whole number. 

Multiplying 1 with denominator of the fraction 4, we get 1 × 4 = 4. 

Add 4 to the numerator of the fraction, we get 4 + 3 = 7. 

1^3/4 = (1×4)+3 / 4 =7/4

1.6.1 Divide a mixed number by a mixed number 

How to divide mixed numbers? 

The following steps explain the division of mixed numbers: 

Step 1: Find the estimate for the given mixed numbers. 

Step 2: Convert the mixed numbers to improper fractions. 

Step 3: Flip the divisor of the opposite fraction (reciprocal). 

Step 4: Multiply the two fractions. 

Example 1: Sophia prepared 37 ½ ÷ 6 ¼ liters of juice. She wants to fill the juice in bottles of capacity 6 ¼ liters each. Find the number of bottles required to fill the juice. 

Solution: 

Step 1: 

Estimate 37 ½ ÷6 ¼

Use compatible numbers to estimate the quotient. 

37 ½ ÷ 6 ¼ = 36 ÷ 6 = 6. 

Step 2: 

We have to find the value of 37 ½ ÷6 ¼

Converting the given mixed numbers into improper fractions. 

37 ½ =     (37 x 2) + 1/2 = 74+1 /2 = 75/2

6¼= (6×4)+1/4 = 24+1/4=25/46

Reciprocal of the second fraction (divisor) 25/4 is 4/25

37 ½ ÷ 6 ¼ = 75/2× 4/25 = 75/2 ×4/25 = 6 

Since, 6 is the estimate, the quotient is reasonable. 

∴ Sophia requires 6 bottles to fill the juice. 

Example 2: David has 37 ½ inches of space on his car bumper. He wants to use the bumper space to fit the medium size stickers of 10 ¾ inches. How many stickers can David fit on his car bumper? 

Solution: 

Step 1: 

Estimate 37 ½ ÷ 10 ¾

Use compatible numbers to estimate the quotient. 

37 ½ ÷ 10 ¾ = 36 ÷ 10 = 3.6. 

Step 2: 

We have to find the value of 37 ½ ÷ 10 ¾

Converting the given mixed numbers into improper fractions. 

37 ½ = (37×2)+1/2 = 74+1/2 = 75/2

10 ¾ =(10×4)+3/4 = 40+3/4 = 43/4

Reciprocal of the second fraction (divisor)

43/4 is 4/43

37 ½ ÷ 10 ¾ = 75/2÷43/4=75/2×4/43 = 150/43 = 3.6 

Since, 3.6 is the estimate, the quotient is reasonable. 

∴David can fit 3.6 medium size stickers on his car bumper. 

1.6.2 Divide a whole number by a mixed number 

How to divide a whole number by a mixed number? 

The following steps can explain the whole number division: 

Step 1: First, write the whole number and the mixed number. 

Step 2: Estimate the division using compatible numbers.  

Step 3: Convert the mixed number into an improper fraction. 

Step 4: Change the divisor into a reciprocal fraction. 

Step 5: Multiply the whole number with the reciprocal. 

Step 6: Simplify further to get the answer. 

Example 1: Divide 16 ÷ 1⅗

Solution: 

Step 1: 

Estimate the given numbers using compatible numbers. 

16 ÷ 1⅗ = 16 ÷ 2 = 8. 

Step 2: 

Convert the given mixed number into an improper fraction. 

1⅗ = (1×5)+3/5 = 5+3/5 = 8/5

Write the whole number and mixed number as fractions. 

16 ÷ 1⅗ = 16/1 ÷ 8/5

Multiply the reciprocal of the divisor. 

16 ÷ 1⅗ = 16/1 × 5/8 = 80/8 = 10

Since the estimate 8 is near to the quotient 10. Hence, the answer is reasonable. 

Example 2: Divide 18 ÷ 3⅔

Solution:  

Step 1: 

Estimate the given numbers using compatible numbers. 

18 ÷ 3⅔ = 18 ÷ 3 = 6. 

Step 2: 

Convert the given mixed number into an improper fraction. 

3⅔ = (3×3)+2 / 3 = 9+2 / 3 = 11/3

Write the whole number and mixed number as fractions. 

18 ÷ 3⅔ = 18/1 ÷ 11/3

181÷113

Multiply the reciprocal of the divisor. 

18 ÷ 3⅔ = 18/1 ÷ 11/3 = 18/1 × 3/11 = 54/11 = 4.9

Since the estimate 6 is near to the quotient 4.9. Hence, the answer is reasonable. 

1.6.3 Divide a mixed number by a whole number 

How to divide a mixed number by a whole number? 

The following steps can explain the whole number division: 

Step 1: First, write the mixed number and a whole number. 

Step 2: Estimate the division using compatible numbers.  

Step 3: Convert the mixed number into an improper fraction. 

Step 4: Change the divisor into a reciprocal fraction. 

Step 5: Multiply the mixed number with the reciprocal of a whole number. 

Step 6: Simplify further to get the answer. 

Example 1: Divide 15⅚ ÷ 4 

Solution:  

Step 1: 

Estimate the given numbers using compatible numbers. 

15⅚ ÷ 4 = 16 ÷ 4 = 4. 

Step 2: 

Convert the given mixed number into an improper fraction. 

15⅚ = (15×6)+5 / 6 = 90+5 / 6 = 95/ 6

Write the mixed number and the whole number as factions. 

15⅚ ÷ 4 = 95/6 ÷ 4/1

Multiply the reciprocal of the divisor. 

15⅚ ÷ 4 = 95/6 × 1/4 = 95/24 = 3.9

Since the estimate 4 is near to the quotient 3.9. Hence, the answer is reasonable. 

Example 2: Divide 12⅔ ÷ 6 

Solution:  

Step 1: 

Estimate the given numbers using compatible numbers. 

12⅔ ÷ 6 = 12 ÷ 6 = 2. 

Step 2: 

Convert the given mixed number into an improper fraction. 

12⅔ = (12×3)+2/3 = 36+2/3 = 38/3

Write the mixed number and the whole number as factions. 

12⅔ ÷ 6 = 38/3 ÷ 6/1

Multiply the reciprocal of the divisor. 

12⅔ ÷ 6 = 38/3 × 1/6 = 38/18 = 2.1

Since the estimate 2 is near to the quotient 2.1. Hence, the answer is reasonable. 

What have we learned:

1. Divide 6⁵∕₉ ÷ 1⁷∕₉
2. Mark is constructing a rope ladder with each step measuring 2⅓  feet wide. He has a rope of 21 feet long. How many steps can he construct from the total rope?
3. Divide 2⅝ ÷ 2¼
4. Divide 18 ÷ 3³∕₂
5. Divide 2⅝  ÷ 13
6. Divide 2⅓ ÷ 1⅓
7. Divide 1 ÷ 8⁵∕₉ 
8. Divide 5 ÷ 6⅖ 
9. Divide 1⅖   ÷ 7
10. Divide 2⅓  ÷ 1⅓

What have we learned:

• Understand mixed numbers division.
• Conversion of mixed number to an improper fraction.
• Estimate fractional division by comapatible numbers.
• Divide a mixed number by another mixed number.
• Divide a whole number by a mixed number.
• Divide a mixed number by a whole number.
• Difference between a whole number and a fractional division.

Concept Map:

• Find the estimate for the given mixed numbers.
• Convert the mixed numbers to improper fractions.
• Flip the divisor of the opposite fraction (reciprocal).
• Multiply the two fractions.