## Key Concepts

- Mixed Numbers
- Improper Fractions
- Reciprocal
- Writing an improper fraction as a mixed number
- Fractions and mixed numbers as quotients
- Using multiplications to divide

## Fractions and mixed numbers as quotients

### Mixed fractions or mixed numbers

**What is a mixed fraction or a mixed number?**

A mixed fraction can be defined as a combination of a whole number and a fractional part.

For example, 5 2/9. Here, 5 is whole number and 2/9 is the fractional part. **Mixed fractions are often referred to as mixed numbers.**

### Expressing an improper fraction as a mixed fraction

#### Improper fraction

An improper fraction is a fraction that has a numerator greater than or equal to the denominator.

Example 1: Express the fraction 𝟕𝟓/𝟒 as a mixed fraction.

**Solution:**

7/5 is an improper fraction. Where, the numerator 75 is greater than the denominator 4.

**Step 1:** Divide the numerator with the denominator.

**Step 2: **Find the remainder.

**Step 3:** Arrange the numbers in the following manner, quotient remainder/divisor

*.*

Mixed fraction = quotient remainder/divisor

= 18 3/4

### How can you show a quotient using a fraction or a mixed number

**Example 1:** If 7 yards of a ribbon is to be shared among the two girls equally, how many yards of ribbon would each girl have?

**Solution: **We observe that 7 yards of a ribbon must be divided equally among 2 girls, i.e., 7÷2 or

7/2

**Step 1: **Divide each of the 7 yards into two equal parts. Each part of 1 yard is 1/2 or 1 ÷ 2.

**Step 2: **Each girl gets 3 yards plus 1/2 of a yard, or 3 + 1/2 = 3 1/2 of the total ribbon.

Hence, each girl gets 3 1/2 yards.

**Example 2: **Estimate how much will each of the 4 cousins get if 5 pies are shared equally among them.

**Solution:** We observe that 5 pies must be divided equally among 4 cousins, i.e., 5 ÷ 4 or 5/4

**Step 1: **Divide each pie into four equal parts. Each part of one pie is 1/4 or 1 ÷ 4.

**Step 2: **Each cousin gets 1 complete pie plus 1/4 of a pie, or 1+1/4 = 1 1/4 of the total pies.

Hence, each cousin gets 1 1/4 pie.

### Use multiplication to divide

**How is dividing by a fraction related to multiplication:**

To divide a whole by fraction, one should be aware of the concept of reciprocal.

**What is a reciprocal?**

We are familiar that all the fractions consist of a numerator and a denominator. To find the reciprocal of any fraction, just interchange the positions of the numerator and the denominator. The resultant fraction can be termed as a reciprocal of the given fraction.

**Reciprocal of the fraction:**

**Reciprocal of whole number or natural numbers:**

To find the reciprocal of the whole number, take the given whole number as the numerator and 1 as the denominator. Now, interchange the positions of the numerator and the denominator.

**Division of a whole number by a fraction:**

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. The resultant product will be the required answer.

**For example: **Divide 3 by 2/5

**Solution: **Observe that 3 here is a whole number, where 2/5 is a fraction.

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

Reciprocal of 2/5 is 5/2

**Step 2: **Multiply the given whole number 3 by the reciprocal of the fraction, i.e., 5/2.

3 × 5/2 = 15/2

**Example 1: **If 1/5 of a syrup cup is required to feed each cat in a pet shop. How many cats can be fed using 4 bottles of a syrup?

**Solution: **Here, we observe that 4 is a whole number and 1/5 is a fraction. So, to find how many 1/5 are there in 4 bottles. We divide 4 by 1/5.

#### Method-I

**Step 1: **Divide 4 into 1/5 parts by taking any shape.

**Step 1: **Divide 4

**Step 2: **Since, there are 5 fifths in each whole, there are 5 × 4 = 20 fifths in 4 wholes.

So, we can conclude that 4÷15 = 20, which explains that 20 cats can be fed using 4 bottles of syrup.

#### Method – II

**Solution: **Here, we observe that 4 is a whole number and 1/5 is a fraction. So, to find how many 1/5 are there in 4 bottles. We divide 4 by 1/5.

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

Reciprocal of 1/5 is 5/1

**Step 2: **Multiply the whole number 4 by the reciprocal of the fraction 1/5 i.e., 5/1.

4×51 = 20.

**Example 2: **A wall of length 8 feet must be constructed. John wants to find how many bricks of length

1/4 feet are required to be laid in the first row.

**Solution: **Here, we observe that 8 is a whole number and 1/4 is a fraction. So, to find how many 1/4 feet bricks are required. We divide 8 by 1/4.

#### Method-I

**Step 1: **Divide each foot into 1/4 parts by taking a straight line.

**Step 2: **Since, there are 4 fourths in each whole, there are 8 × 4 = 32 fourths in 8 wholes.

So, we can conclude that 8÷14 = 32, which explains that 32 bricks can be laid in the first layer.

#### Method – II

**Solution: **Here, we observe that 8 is a whole number and 1/4 is a fraction. So, to find how many

1/4 feet of bricks can be laid. We divide 8 by 1/4.

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

Reciprocal of 1/4 is 4/1

**Step 2: **Multiply the whole number 8 by the reciprocal of the fraction 1/4 ,i.e., 4/1.

8 × 4/1 = 32

## Exercise

- A glass can hold 6 cups of water. How many 1/4 cups will it take to fill the glass?
- A dog’s food bowl hold 20 ounces of dog food. The spoon used to serve the food holds

1/3 of an ounce. How many spoons will it take to fill the bowl up to the rim? - A jar holds 30 ounces of ice cream, if each scoop can hold 1/5 of an ounce for one serving.

How many servings can be made? - A container can hold 2 tons of rice bags. If a machine at the mart can empty 1/6 of the ton

in one trip. How many trips are made to empty the container? - The painter used 1/6 of a gallon of black paint to cover one panel of concrete fence. If there are 8 gallons of gray paint, how many panels can be painted in all?
- If 7 cups of water are shared among 10 people. How much will each person get?
- Reduce 45/10 to a mixed number.
- How will 7 people share 15 pizzas among them equally?
- A 10-liter pitcher of juice was poured into 7 cups. How much juice was in each cup?
- If the weight of a bag filled with beans is 60 pounds. I all the beans are transferred into 13 smaller bags, what is the weight of each smaller bag?

### What have we learned

- Fractions and mixed numbers as quotients
- Using multiplications to divide

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