## Key Concepts

- Whole numbers
- Unit fractions
- Dividing whole numbers by unit fractions
- Dividing unit fractions by whole numbers

## Unit fractions

### What is a unit fraction?

Any fraction with numerator 1 is defined as a unit fraction. In these types of fractions, we consider only one part of the whole, which is equally divided into a finite number of parts. Unit denotes one. Therefore, they are known as unit fractions.

**Example:**

If Abbey eats 1 slice of a large pizza containing 8 slices. Then Abbey ate 1/8 of pizza.

## Whole numbers

### What are whole numbers?

Whole numbers are referred to as the set of natural numbers along with ‘0’. The set of whole numbers is denoted by W.

Whole numbers, W = {0, 1, 2, 3…………}

### Dividing whole numbers by unit fractions

To divide a whole number by a fraction, follow the steps listed below:

**Step 1:** Find the reciprocal of the given fraction.

**Step 2:** Multiply the given whole number by the reciprocal of the fraction.

**Step 3:** The resultant product will be the required answer.

**Example:** Divide 4 by 1/3.

**Solution:** Observe that 4 here is a whole number, where 1/3 is a unit fraction.

**Step 1:** Find the reciprocal of the given fraction.

Reciprocal of 1/3 is 3/1.

**Step 2:** Multiply the given whole number 4 by the reciprocal of the fraction, i.e., 3/1.

4×3/1

= 4×3 = 12.

### Dividing unit fractions by non-zero whole numbers

To divide a unit fraction by a whole number, follow the steps listed below:

**Step 1: **Find the reciprocal of the given whole number.

**Step 2: **Multiply the given unit fraction by the reciprocal of the whole number.

**Step 3:** The resultant product will be the required answer.

**Example: **Divide 1/8 by 9.

**Solution: **We are asked to find the value of 1/8 ÷ 9. Observe that 9 is a whole number, whereas 1/8 is a unit fraction.

**Step 1: **Find the reciprocal of the given whole number.

Reciprocal of 9 is 1/9 .

**Step 2: **Multiply the given unit fraction 1/8

by the reciprocal of the whole number, i.e.,1/9 .

1 / 8 ×1 / 9 = 1×1/ 8×9 = 1/72.

### Dividing whole numbers by unit fractions

**Example 1: **There are four shipments of gas cans to auto supply stores. If each store received 1/10 shipment, how many stores are there in all?

**Method – I**

**Solution: **We observe that 4 shipments must be divided equally and shared among supply stores, where each store gets 1/10 of a shipment, i.e., 4÷ 1/10.

**Step 1: **Divide each of the 4 shipments into 1/10 equal parts. Each part of 1 shipment is 1/10.

**Step 2: **Since, there are 10 tenths in each whole, there are 4×10 = 40 tenths in 4 wholes.

So, we can conclude that 4 ÷1/10 = 40. This explains that 40 stores can be supplied by 4 shipments.

**Method – II**

**Solution: **We will use a number line to find how many 1/10 ’s are there in 4.

**Step 1: **We can observe there are ten 1/10’s in between each whole number.

**Step 2: **There are ten 1/10’s in 1 whole, twenty 1/10’s in 2 wholes, thirty 1/10’s in 3 whole and forty 1/10’s in 4 wholes.

So, we conclude that 4 ÷ 1/10

= 40.

Hence, 40 supply stores receive gas cans from 4 shipments.

**Example 2:** Kiara is putting sugar into a container. The container can hold 5 cups of sugar. How many scoops of sugar she needs, if each scoop can fill 1/3 of a cup?

**Method – I**

**Solution: **We observe that 5 cups of sugar equals 1 container. We must divide 5 by 1/3cups to find how many 1/3cups are required to fill the container, i.e., 5÷ 1/3.

**Step 1: **Divide each of the 5 cups into 3 equal parts. Each part of 1 cup is 1/3.

**Step 2: **Since, there are five thirds in each whole, there are 5×3 = 15 thirds in 5 wholes.

So, we can conclude that 5÷1/3= 15. This explains that fifteen scoops are required to fill one container of sugar.

**Method – II**

**Solution: **We will use a number line to find how many 1/3’s are there in 5.

**Step 1: **We can observe there are three 1/3’s in between each whole number.

**Step 2: **There are three 1/3’s in 1 whole, six 1/3’s in 2 wholes, nine 1/3’s in 3 whole and fifteen 1/3’s in 5 wholes.

So, we conclude that 5÷1/ 3

= 15.

Hence, fifteen scoops are required to fill the container.

### Dividing unit fractions by non-zero whole numbers

**Example 1:** Paul collected 1/2 a pound of strawberries. He must divide them equally among six wooden baskets. How many pounds of strawberries did Paul put in each wooden basket?

**Method – I (Area Model)**

**Solution: **We understand that 1/2 pound must be divided equally among the six wooden baskets, i.e., 1/2 ÷6.

**Step 1: **Divide one pound into two equal parts. Each part of 1 pound is 1/2.

**Step 2: **Divide1/2 of a pound into six equal parts.

**Step 3: **Each part of 1/2 of a pound is equal to 1/12 of a pound.

Hence, each of the six baskets will have 1/12 of a pound.

**Method – II**

**Solution: **We are asked to divide 1/2 of a pound among six wooden baskets equally. Observe that 6 here is a whole number, where 1/2 is a unit fraction.

**Step 1: **Find the reciprocal of the given whole number.

Reciprocal of 6 is 1/6

**Step 2: **Multiply the given unit fraction 1/2 by the reciprocal of the whole number, i.e.,1/6 .

1/2 ×1/6 = 1×1/2×6 = 1/12.

Hence, each of the six wooden baskets will have 1/12 of a pound.

**Example 2: **On the last day of the exam, the teacher had 1/5 of a bundle of blank papers left. She gave the papers to 10 of her students equally. How much of the bundle did every student take home?

**Method – I (Area Model)**

**Solution:** We understand that 1/5 of a bundle must be divided equally among the 10 students, i.e., 1/5÷10.

**Step 1: **Divide one bundle into five equal parts. Each part of 1 bundle is 1/5.

**Step 2: **Divide 1/5 of a bundle into ten equal parts.

**Step 3: **Each part of 1/5 of a bundle is equal to 1/10.

Hence, each of the 10 students will get 1/50 of a bundle.

**Method – II**

**Solution: **We are asked to divide 1/5 of a bundle among ten friends equally. Observe that 10 here is a whole number, where 1/5 is a fraction.

**Step 1: **Find the reciprocal of the given whole number.

Reciprocal of 10 is 1/10.

**Step 2: **Multiply the given unit fraction 1/5 by the reciprocal of the whole number, i.e.,1/10 .

15 ×1/10 = 1×1 / 5×10 =1/50.

Hence, each of the ten friends will get 1/50 of a bundle.

## Exercise

- Kate uses 2 packets of milk powder per day to feed her little bay. How many days will 1/3 of a packet will last?
- A florist used 1/2 a basket of flowers to decorate 3 windows. How many baskets of flowers are used to decorate each window?
- An oil factory uses 1/4 of a tin of peanuts to prepare 2 drums of peanut oil. How many tins of peanut are used to prepare each drum?
- Jake prepares 1/3 pound of cake. He makes it into three equal pieces; what is the weight of each piece in pounds?
- Wilson has a tray of cherries. His daughter ate 1/2 of the tray in three equal parts. How much of the tray did she eat each time?
- A dog’s food bowl holds 4 cups of dog food. If each scoop can fill 1/4 of a cup. How many scoops are required to fill the bowl completely?
- A vat of cement can hold 2 tons. If a bucket with 1/8 of ton capacity is used to empty the vat, find the number of buckets it takes to empty the vat.
- Maddy has 4 storage box partitions. Each box has a partition that takes up 1/3 of the box. How many total partitions are there in the box?
- Ruby is painting 4 walls of her room. If all the rooms take 1/2 a gallon of paint. How much paint was used for each room?
- Cake batter is poured equally into 8 containers. If the total cake batter poured is 1/3 of gallon. How much batter is poured into each container

### Concept Map

### What have we learned

- Whole numbers
- Unit fractions
- Dividing whole numbers by unit fractions
- Dividing unit fractions by non-zero whole numbers

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