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Chapter 2: Problem 6

Calculate (a) \(\frac{6}{7} \times \frac{14}{27}\) (b) \(\frac{7}{10} \times \frac{4}{5} \times \frac{30}{49}\) (c) \(\frac{8}{9} \times \frac{18}{25}\) (d) \(\left(-\frac{4}{5}\right) \times\left(-\frac{3}{4}\right)\) (e) \(\frac{16}{21} \times\left(-\frac{3}{4}\right)\)

Short Answer

Expert verified
Question: Multiply the given fractions and simplify the results. a) \(\frac{6}{7} \times \frac{14}{27}\) b) \(\frac{7}{10} \times \frac{4}{5} \times \frac{30}{49}\) c) \(\frac{8}{9} \times \frac{18}{25}\) d) \(\left(-\frac{4}{5}\right) \times\left(-\frac{3}{4}\right)\) e) \(\frac{16}{21} \times\left(-\frac{3}{4}\right)\) Answer: a) \(\frac{7}{63}\) b) \(\frac{10}{98}\) c) \(\frac{8}{225}\) d) \(\frac{3}{5}\) e) \(-\frac{4}{7}\)

Step by step solution

01

Simplify the Fractions

Before multiplying the fractions, we can simplify the fractions by cancelling out common factors from the numerator and the denominator. In this case, we can simplify the fractions as follows: $$ \frac{6}{7} \times \frac{14}{27} = \frac{6 \div 6}{7} \times \frac{14 \div 2}{27 \div 3} = \frac{1}{7} \times \frac{7}{9} $$
02

Multiply the Simplified Fractions

Now, multiply the numerators together and the denominators together as follows: $$ \frac{1}{7} \times \frac{7}{9} = \frac{1 \times 7}{7 \times 9} = \frac{7}{63} $$ #b) Multiply the fractions \(\frac{7}{10} \times \frac{4}{5} \times \frac{30}{49}\) #
03

Simplify the Fractions

We can simplify the fractions first as following: $$ \frac{7}{10} \times \frac{4}{5} \times \frac{30}{49} = \frac{7 \div 7}{10 \div 5} \times \frac{4 \div 2}{5 \div 5} \times \frac{30 \div 6}{49} $$ $$ = \frac{1}{2} \times \frac{2}{1} \times \frac{5}{49} $$
04

Multiply the Simplified Fractions

Now, multiply the fractions: $$ \frac{1}{2} \times \frac{2}{1} \times \frac{5}{49} = \frac{1 \times 2 \times 5}{2 \times 1 \times 49} = \frac{10}{98} $$ #c) Multiply the fractions \(\frac{8}{9} \times \frac{18}{25}\) #
05

Simplify the Fractions

In this case, we can simplify the fractions as follows: $$ \frac{8}{9} \times \frac{18}{25} = \frac{8 \div 2}{9} \times \frac{18 \div 9}{25} = \frac{4}{9} \times \frac{2}{25} $$
06

Multiply the Simplified Fractions

Now, we can multiply the fractions: $$ \frac{4}{9} \times \frac{2}{25} = \frac{4 \times 2}{9 \times 25} = \frac{8}{225} $$ #d) Multiply the fractions \(\left(-\frac{4}{5}\right) \times\left(-\frac{3}{4}\right)\) #
07

Apply the Rule of Signs

The rule of signs states that \(\displaystyle "(-a) \times (-b) = a \times b".\) Apply the rule of signs to the fractions: $$ \left(-\frac{4}{5}\right) \times\left(-\frac{3}{4}\right) = \frac{4}{5} \times \frac{3}{4} $$
08

Multiply the Fractions

Now, multiply the fractions: $$ \frac{4}{5} \times \frac{3}{4} = \frac{4 \times 3}{5 \times 4} = \frac{12}{20} $$
09

Simplify the Result

Simplify the resulting fraction: $$ \frac{12}{20} = \frac{12 \div 4}{20 \div 4} = \frac{3}{5} $$ #e) Multiply the fractions \(\frac{16}{21} \times\left(-\frac{3}{4}\right)\) #
10

Multiply the Fractions

Multiply the fractions directly: $$ \frac{16}{21} \times \left(-\frac{3}{4}\right) = -\frac{16 \times 3}{21 \times 4} $$
11

Simplify the Result

We can simplify the fraction as follows: $$ -\frac{16 \times 3}{21 \times 4} = -\frac{48}{84} $$ $$ -\frac{48 \div 12}{84 \div 12} = -\frac{4}{7} $$ So, the final answer is \(-\frac{4}{7}\).

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Most popular questions from this chapter

Express the following improper fractions as mixed fractions: (a) \(\frac{5}{4}\) (b) \(\frac{6}{5}\) (c) \(\frac{12}{5}\) (d) \(\frac{47}{3}\) (e) \(-\frac{62}{7}\)

Express the following mixed fractions as improper fractions: (a) \(1 \frac{1}{2}\) (b) \(2 \frac{1}{3}\) (c) \(3 \frac{1}{4}\) (d) \(3 \frac{2}{5}\) (e) \(-10 \frac{2}{5}\)

Calculate the following, expressing your answer as an improper fraction: (a) \(1 \frac{2}{5}+2 \frac{3}{4}\) (b) \(4 \frac{1}{3}+6 \frac{3}{7}\) (c) \(2 \frac{4}{5}-1 \frac{2}{3}\) (d) \(6 \frac{1}{2}-3 \frac{3}{4}+2 \frac{1}{5}\) (e) \(3 \frac{1}{3}-4 \frac{4}{9}\)

Express each of the following fractions in their simplest form: (a) \(\frac{12}{60}\) (c) \(\frac{27}{81}\) (d) \(\frac{6}{92}\) (e) \(\frac{377}{390}\)

Calculate the following, expressing your answer as a mixed fraction: (a) \(3 \frac{2}{3} \div 1 \frac{1}{2}\) (b) \(6 \frac{1}{4} \div 2 \frac{2}{3}\) (c) \(10 \frac{3}{4} \div 2 \frac{1}{5}\) (d) \(-12 \frac{2}{5} \div 3 \frac{1}{4}\) (e) \(10 \frac{5}{6} \div\left(-4 \frac{1}{2}\right)\)
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