### 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:

- Divide 6⁵∕₉ ÷ 1⁷∕₉
- 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?
- Divide 2⅝ ÷ 2¼
- Divide 18 ÷ 3³∕₂
- Divide 2⅝ ÷ 13
- Divide 2⅓ ÷ 1⅓
- Divide 1 ÷ 8⁵∕₉
- Divide 5 ÷ 6⅖
- Divide 1⅖ ÷ 7
- 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.