Multiplying Fractions

We will discuss here about multiplying fractions by a whole number, by a fractional number or by another mixed fractional number.


I. Multiplication of Fractional Number by a Whole Number:

We have learnt             4 × 5                 =          4 times 5

                                                            =          5 + 5 + 5 + 5

                                                            =          20

In the same way 6 × \(\frac{1}{7}\) = 6 times \(\frac{1}{7}\)

                                  = \(\frac{1}{7}\) + \(\frac{1}{7}\) + \(\frac{1}{7}\) + \(\frac{1}{7}\) + \(\frac{1}{7}\) + \(\frac{1}{7}\)

                                  = \(\frac{1 + 1 + 1 + 1 + 1 + 1}{7}\)

                                  = \(\frac{6}{7}\)

        i.e., 6 × \(\frac{1}{7}\) = \(\frac{6 × 1}{7}\) = \(\frac{6}{7}\)

Multiply 4 × \(\frac{3}{5}\)

           4 × \(\frac{3}{5}\) = 4 times 4 × \(\frac{3}{5}\)

                     = \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\)

                     = \(\frac{12}{5}\)

        i.e. 4 × \(\frac{3}{5}\) = \(\frac{4 × 3}{5}\) = \(\frac{12}{5}\)


Product of a whole number and a fractional number =

\(\frac{\textrm{Product of Whole Number × Numerator of the Fractional Number}}{\textrm{Denominator of the Fractional Number}}\)


For examples:

\(\frac{3}{4}\) × 5 = \(\frac{3 × 5}{4}\) = \(\frac{15}{4}\)

\(\frac{6}{7}\) × 2 = \(\frac{6 × 2}{7}\) = \(\frac{12}{7}\)

7 × \(\frac{4}{5}\) = \(\frac{7 × 4}{5}\) = \(\frac{28}{5}\)

4 × \(\frac{3}{11}\) = \(\frac{4 × 3}{11}\) = \(\frac{12}{11}\)


Let us multiply \(\frac{1}{4}\) by 3. We use the rule of repeated addition to find the product.

Multiplying Fractions

We can say that \(\frac{1}{4}\) of 3 = \(\frac{3}{4}\)

To multiply a fraction by a whole number, we multiply the numerator of the fraction by the whole number and reduce the fraction to the lowest terms, if so required.


For example:

(i) Multiply 1\(\frac{2}{9}\) by 25

Solution:

1\(\frac{2}{9}\) × 25

= \(\frac{1 × 9 + 2}{9}\)

= \(\frac{11}{9}\) × 25

= \(\frac{11 × 25}{9}\)

= \(\frac{275}{9}\)

= 30\(\frac{5}{9}\)


(ii) Multiply 2/3 by 7

Solution:

2/3 × 7

= (2 × 7)/3

= 14/3

= 4 2/3


We simply multiply the numerator of the fractional number by the whole number. The denominator remains the same.


(iii) Multiply 20 × \(\frac{4}{5}\)

20 × \(\frac{4}{5}\) = \(\frac{20 × 4}{5}\)

           = \(\frac{2 × 2 × \not 5 × 2 × 2}{\not 5}\)

           = 16

Prime Factors of 20, 4 and 5

20 = 2 × 2 × 5

 4 = 2 × 2

 5 = 5 × 1


(iv) Multiply 2\(\frac{3}{5}\) by 6

Solution:

2\(\frac{3}{5}\) × 6

= (2 × 5 + 3)/5 × 6

= (10 + 3)/5 × 6

= 13/5 × 6

= (13 × 6)/5

= 78/5

= 15\(\frac{3}{5}\)



We change the mixed numbers into improper fractions and then simply multiply the numerator of the fractional number by the whole number. The denominator remains the same.


II. Multiplication of Fractional Number by Another Fractional Number:

For example:

(i) Multiply 2/5 by 4/5

Solution:

2/5 × 4/5

= (2 × 4)/(5 × 5)

= 8/25


Step I: We multiply the numerators.

Step II: We multiply the denominators.

Step III: We write the fraction in the simplest form.



(ii) Multiply 8/9 by 7/10

Solution:

8/9 × 7/10

= (8 × 7)/(9 × 10)

= 56/90

We simply multiply the numerators of the fractional numbers and then multiply the denominators of the fractional numbers. Write the fraction in the simplest form.


(iii) Multiply \(\frac{4}{7}\) × \(\frac{2}{5}\)

Multiply the numerators to get the numerator of the product and

Multiply the denominators to get the denominator of the product.

Reduce the product to the lowest terms.

Therefore, \(\frac{4}{7}\) × \(\frac{2}{5}\) = \(\frac{4 × 2}{7 × 5}\) = \(\frac{8}{35}\)


III: Product of More than Two Fractions:

For examples:

(i) Multiply \(\frac{9}{10}\) × \(\frac{2}{5}\) × \(\frac{3}{7}\)

H.C.F. of 54 and 350

Method I: \(\frac{9}{10}\) × \(\frac{2}{5}\) × \(\frac{3}{7}\) = \(\frac{9 × 2 × 3}{10 × 5 × 7}\) = \(\frac{54}{350}\)

H.C.F. of 54 and 350 is 2

\(\frac{54 ÷ 2}{350 ÷ 2}\) = \(\frac{27}{175}\)

Therefore, \(\frac{9}{10}\) × \(\frac{2}{5}\) × \(\frac{3}{7}\) = \(\frac{27}{175}\)


Method II: \(\frac{9}{10}\) × \(\frac{2}{5}\) × \(\frac{3}{7}\) = ?

9 = 3 × 3 

2 = 2 × 1

3 = 3 × 1   

10 = 2 × 5

 5 = 5 × 1

 7 = 7 × 1

Prime Factors of 9, 2, 3, 10, 5 and 7

Write the numbers as the products of prime factors.

Cancel the numbers common in numerator and denominator.

Therefore, \(\frac{9}{10}\) × \(\frac{2}{5}\) × \(\frac{3}{7}\) = \(\frac{3 × 3 × \not 2 × 1 × 3 × 1}{\not 2 × 5 × 5 × 1 × 7 × 1}\)

                                    = \(\frac{27}{175}\)


(v) Multiply \(\frac{4}{7}\), \(\frac{3}{11}\) and \(\frac{5}{8}\).

Solution:

To multiply two or more fractions, we multiply the numerators of given fractions to find the new numerator of the product and multiply the denominators to get the denominator of the product.

Hence, \(\frac{4}{7}\) × \(\frac{3}{11}\) × \(\frac{5}{8}\) = \(\frac{4 × 3 × 5}{7 × 11 × 8}\)

                             = \(\frac{60}{616}\)


(vi) Multiply \(\frac{10}{21}\) × \(\frac{5}{24}\) × \(\frac{3}{50}\)


\(\frac{10}{21}\) × \(\frac{5}{24}\) × \(\frac{3}{50}\)


\(\frac{10 × 5 × 3}{21 × 24 × 50}\)


= \(\frac{\not 2 × \not 5 × \not 5 × \not 3}{\not 3 × 7 × 3 × \not 2 × 2 × 2 × 2 × \not 5  × \not 5}\)


\(\frac{1}{168}\)

Prime Factors of 10, 5, 3, 21, 24 and 50

10 = 2 × 5

5 = 5 × 1

3 = 3 × 1

21 = 3 × 7

24 = 3 × 2 × 2 × 2

50 = 2 × 5 × 5


III. Multiplication of a Mixed Number by Another Mixed Number:

For Example:

(i) Multiply 2 1/3 by 1 ¾

Solution:

2 1/3 × 1 ¾

= 7/3 × 7/4

= 49/12

= 4 1/12


We change the mixed numbers into improper fractions and then we multiply as usual.



(ii) Multiply 1  7/9 by 3 5/11

Solution:

1  7/9 × 3 5/11

= 16/9 × 38/11

= (16 × 38)/(9 × 11)

= 608/99

= 6 14/99



We change the mixed numbers into improper fractions and then we multiply as usual.


(iii) Multiply 11\(\frac{7}{8}\) by 3\(\frac{1}{24}\)

Solution:

Let us first convert mixed numbers into improper fractions.

11\(\frac{7}{8}\) = \(\frac{11 × 8 + 7}{8}\) = \(\frac{95}{8}\)

3\(\frac{1}{24}\) = \(\frac{3 × 24 + 1}{24}\) = \(\frac{73}{24}\)

Now, \(\frac{95}{8}\) × \(\frac{73}{24}\) = \(\frac{95 × 73}{8 × 24}\)

                     = \(\frac{6935}{192}\)

                     = 36\(\frac{23}{192}\)


(iv) Multiply 3\(\frac{1}{2}\) × 2\(\frac{1}{5}\)

3\(\frac{1}{2}\) × 2\(\frac{1}{5}\) = \(\frac{7}{2}\) × \(\frac{11}{5}\)

              = \(\frac{7 × 11}{2 × 5}\)

              = \(\frac{77}{10}\)

              = 7\(\frac{7}{10}\)

Mixed to Improper


Questions and Answers on Multiplying Fractions:

I. Find the product:

(i) \(\frac{5}{19}\) × 1

(ii) \(\frac{6}{7}\) × 5

(iii) \(\frac{9}{14}\) × 6

(iv) \(\frac{4}{13}\) × 0

(v) \(\frac{1}{7}\) × \(\frac{5}{6}\)

(vi) 1\(\frac{1}{10}\) × 8

(vii) \(\frac{1}{7}\) × \(\frac{8}{1}\)

(viii) \(\frac{1}{3}\) × \(\frac{7}{5}\) × \(\frac{2}{9}\)

(ix) \(\frac{4}{15}\) × \(\frac{10}{21}\)

(x) \(\frac{1}{2}\) of 100

(xi) \(\frac{1}{3}\) of 60

(xii) \(\frac{4}{5}\) of \(\frac{8}{11}\)


Answers:

I. (i) \(\frac{5}{19}\)

(ii) 4\(\frac{2}{7}\)

(iii) 3\(\frac{6}{7}\)

(iv) 0

(v) \(\frac{5}{42}\)

(vi) 8\(\frac{4}{5}\)

(vii) 1\(\frac{1}{7}\)

(viii) \(\frac{14}{135}\)

(ix) \(\frac{8}{63}\)

(x) 50

(xi) 20

(xii) \(\frac{32}{55}\)


II. Multiply and write the product in lowest terms.

(i) \(\frac{1}{2}\) × 40

(ii) \(\frac{1}{3}\) × 150

(iii) \(\frac{2}{7}\) × 21

(iv) \(\frac{7}{38}\) × 0

(v) \(\frac{31}{65}\) × 1

(vi) 8 × \(\frac{17}{24}\)

(vii) \(\frac{3}{7}\) × \(\frac{7}{15}\)

(viii) \(\frac{9}{32}\) × \(\frac{8}{36}\)

(ix) \(\frac{11}{15}\) × \(\frac{45}{88}\)

(x) \(\frac{2}{10}\) ×\(\frac{3}{22}\) ×\(\frac{40}{30}\)

(xi) \(\frac{1}{6}\) ×\(\frac{2}{5}\) ×\(\frac{3}{4}\)

(xii) 3\(\frac{1}{7}\) ×\(\frac{21}{44}\)


Answers:

II. (i) 20

(ii) 50

(iii) 6

(iv) 0

(v) \(\frac{31}{65}\)

(vi) \(\frac{17}{3}\)

(vii) \(\frac{1}{5}\)

(viii) \(\frac{1}{16}\)

(ix) \(\frac{3}{8}\)

(x) \(\frac{2}{55}\)

(xi) \(\frac{1}{20}\)

(xii) 1\(\frac{1}{2}\)


III. Find the Product and Reduce it the Lowest Terms:

(i) 4\(\frac{1}{3}\) × 2\(\frac{1}{3}\)

(ii) 6 × 5\(\frac{1}{2}\)

(iii) 1\(\frac{1}{7}\) × 2\(\frac{1}{4}\)

(iv) \(\frac{3}{4}\) × \(\frac{1}{3}\) × \(\frac{2}{6}\)

(v) 1\(\frac{1}{2}\) × 5\(\frac{2}{3}\) × 4\(\frac{1}{5}\)

(vi) \(\frac{7}{9}\) × \(\frac{10}{15}\) × \(\frac{3}{21}\)

(vii) \(\frac{16}{48}\) × \(\frac{12}{24}\) × \(\frac{15}{30}\)

(viii) \(\frac{19}{38}\) × \(\frac{2}{4}\) × \(\frac{8}{20}\)

(ix) \(\frac{6}{42}\) × 1\(\frac{1}{5}\) × \(\frac{15}{50}\)


Answer:

III. (i) 10\(\frac{1}{9}\)

(ii) 33

(iii) 2\(\frac{4}{7}\)

(iv) \(\frac{1}{12}\)

(v) 35\(\frac{7}{10}\)

(vi) \(\frac{2}{27}\)

(vii) \(\frac{1}{12}\)

(viii) \(\frac{1}{10}\)

(ix) \(\frac{9}{175}\)


IV. Simplify. (use Prime Factorisation)

(i) \(\frac{7}{9}\) × \(\frac{18}{21}\) × \(\frac{6}{10}\)

(ii) \(\frac{24}{36}\) × \(\frac{81}{27}\) × \(\frac{5}{10}\)

(iii) \(\frac{10}{12}\) × \(\frac{12}{14}\) × \(\frac{14}{20}\)

(iv) \(\frac{15}{16}\) × \(\frac{32}{30}\) × \(\frac{1}{4}\)

(v) \(\frac{1}{2}\) × \(\frac{4}{8}\) × \(\frac{16}{6}\)

(vi) \(\frac{13}{22}\) × \(\frac{11}{26}\) × \(\frac{4}{6}\)


Answer:

IV. (i) \(\frac{2}{5}\)

(ii) 1

(iii) \(\frac{1}{2}\)

(iv) \(\frac{1}{4}\) 

(v) \(\frac{2}{3}\)

(vi) \(\frac{1}{6}\)


V. Multiply:

(i) 4 × \(\frac{6}{11}\)

(ii) \(\frac{8}{13}\) × 3

(iii) \(\frac{2}{5}\) × 10

(iv) \(\frac{5}{7}\) × 5

(v) 8 × \(\frac{5}{6}\)

(vi) \(\frac{7}{12}\) × 2

(vii) 15 × \(\frac{1}{4}\)

(viii) 19 × \(\frac{1}{3}\)


Answer:

V. (i) 2\(\frac{2}{11}\)

(ii) 1\(\frac{11}{13}\)

(iii) 4

(iv) 3\(\frac{4}{7}\)

(v) 6\(\frac{2}{3}\)

(vi) 1\(\frac{1}{6}\)

(vii) 3\(\frac{3}{4}\)

(viii) 6\(\frac{1}{3}\)


VI. Find the given quantity.

(i) \(\frac{1}{7}\) of 28 kg apples

(ii) \(\frac{2}{15}\) of $300

(iii) \(\frac{5}{9}\) of 54 km

(iv) \(\frac{2}{5}\) of 70 chairs


Answers:

VI. (i) 4 kg apples

(ii) $40

(iii) 30 km

(iv) 28 chairs

Multiplying Fractions Examples

VII: Word problems on Multiplying Fractions:

1. 2\(\frac{1}{5}\) m of cloth is required to make a shirt. Ron wants to make 25 shirts, what length of cloth does he need?

Answer: 55 m of cloth


2. \(\frac{3}{4}\) cups of milk is required to make a cake of 1 kg. How many cups of milk is required to make a cake of 4\(\frac{1}{2}\) kg?

Answer: 3\(\frac{3}{8}\) cups


3. Shelly bought 16\(\frac{3}{4}\) liters of juice. If the cost of 1 liter juice is $8, find the total cost of juice?

Answer:  $134


4. The weight of each bag is 4\(\frac{1}{4}\) Kg. What would be the weight of 36 such bags?

Answer: 153 kg


5. Sam works for 6\(\frac{2}{8}\) hours each day. For how much time will she work in a month if she works for 24 days in a month?

Answer: 150 hours

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