Fractions and Mixed Numbers as Quotients

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

1. A glass can hold 6 cups of water. How many 1/4 cups will it take to fill the glass?
2. 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?
3. 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?
4. 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?
5. 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?
6. If 7 cups of water are shared among 10 people. How much will each person get?
7. Reduce 45/10 to a mixed number.
8. How will 7 people share 15 pizzas among them equally?
9. A 10-liter pitcher of juice was poured into 7 cups. How much juice was in each cup?
10. 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