Relationships to Divide Whole Numbers by Fractions

Key Concepts

■ Divide whole number by fractions

■ Divide fractions by whole numbers

Introduction:

We already know that a fraction is a part of a whole.  

Let us consider an example of a watermelon cut into 5 equal parts and out of which 3 parts are left. 

The fraction can be represented as 3/5

. 

The above figure can be divided into 3 equal parts and each part can be represented into one part out of the 5 parts as shown. 

i.e.,1/5× 3 =3/5

1.4.1 Divide whole number by fractions 

How to divide whole numbers by fractions? 

The following steps explains division of a whole number by a fraction: 

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

Step 2: Write the reciprocal of the fraction. 

Step 3: Multiply the whole number with reciprocal and the required product is the answer. 

Example 1: A cardboard is 3 feet long. If the board is cut into pieces and each piece is 3/4 feet long to make shelves. How many shelves can be made from the board? 

Solution: 

Repeated subtraction method: 

If we write the given whole number 3 as a fraction with denominator 4. 

Then the total board measure can be written as

12/4feet. 

Given that each shelf is 3/4 feet. 

Use repeated subtraction to divide until remainder becomes zero. 

12/4 − 3/4 = 9/4

9/4 − 3/4 = 6/4

6/4 − 3/4 = 3/4

3/4 −3/4 = 0

∴4 shelves can be made from the board. 

Number line method: 

Using number line to divide 3 by 3/4

Reciprocal of 3/4 is 4/3

3 ÷ 3/4 = 3 × 4/3 = 4 

∴4 shelves can be made from the board. 

1.4.2 Divide fractions by whole numbers 

How to divide fractions by whole numbers? 

The following steps explain division of a fraction by a whole number: 

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

Step 2: Convert the whole number into fraction. 

Step 3: Write the reciprocal of the fraction. 

Step 4: Multiply the whole number with reciprocal of the fraction and the required product is the answer. 

Example 1: Divide 4/6÷2

Solution:  

Division of fractions can be explained using the following area model. 

From the above area model, 4/6 can be represented using a rectangle. 4 parts out of 6, shaded in blue, represents the fraction 4/6. 

The given fraction 4/6 when divided by a whole number 2, gives the result 2/ 6 as shown above in the rectangle shaded by yellow. 

i.e., 4 / 6÷2=4/6×1/2=2/6

(since, reciprocal of 2 is 1 / 2) 

Example 2: Divide 1/4÷3

Solution:  

Division of fractions can be explained using the following area model. 

From the above figure, 

The reciprocal of 3 is 1/3

1 / 4÷3 =

1/4 ×1/3 = 1/2

1.4.3 Divide fractions by whole numbers 

Reciprocal of a number: 

Two numbers whose product is one are known as reciprocal of each other. 

How to find the reciprocal of a fraction? 

The reciprocal of a fraction is the interchange of numerator and denominator of the other fraction. 

Example 1: Divide 4 ÷ 2/3

Solution: 

Use patterns of division and multiplication to divide the given whole number by a fraction. 

4÷2/3=4×3/2

            =4 / 1×3 / 2

            =12 / 2 or  6

Example 2: Divide 14 ÷ 4/7

Solution:  

14 ÷ 4/7 =14 / 1÷4 / 7

=14 / 1×7 / 4

=98 / 4

Exercise:

1. Henrieta prepared 60 donuts for a party.The donuts are divided equally among 14 guests, how many donuts will each guest have? Also find the leftover donuts.

2. A container has 12 boxes of Oranges.Total boxes costs $184. How much each box cost?

3. A farmer is shipping 2,384 bananas. There are 70 crates in total, each crate has equal number of bananas. Find the number of bananas in each crate.

4. Divide 86 + 4.

5. Divide 232 + 40.

6. Use the division algorithm to divide 809.40 4. 8.

7. Divide 140 + S to find the decimal quotient

8. Divide 128.8 ÷ 1.4.

9. Divide 14.7 ÷ 2.1.

10. Divide 1.296 ÷ 0.108.

What have we learned:

■ Understand fraction division using area models.

■ Understand fraction division using number line.

■ Divide whole number by fractions.

■ Divide fractions by whole numbers.

■ Use relationships to divide whole numbers by fractions.