Solving Problems Involving Mixed Operations with Fractions

Worked Example 8.3.1

Simplify the following fractions:

i. \( \frac{3\frac{1}{3} \: \div \: 2 \frac{1}{2}}{4\frac{2}{7}} \)

ii. \(\scriptsize 2 \normalsize \frac{4 }{5}\scriptsize \: \times \: 5 \normalsize \frac{6}{8} \scriptsize \: \div \: \scriptsize 5 \normalsize \frac{1}{9} \)

iii. \(\frac {1}{4} \scriptsize \; of \; \left ( \scriptsize2 \normalsize \frac{3 }{4 } \; \div\; \frac{2}{8} \right) + \scriptsize 1 \normalsize \frac{2 }{3 }\)

i. \( \frac{3\frac{1}{3} \: \div \: 2 \frac{1}{2}}{4\frac{2}{7}} \)

Solution

= \( \large \frac{\frac{10}{3} \: \div \: \frac{5}{2}}{ \frac{30}{7}} \)

= \( \large \frac{\frac{10}{3} \: \times \: \frac{2}{5}}{ \frac{30}{7}} \)

= \( \large \frac{\frac{20}{15}}{ \frac{30}{7}} \)

= \( \large \frac{\frac{4}{3}}{ \frac{30}{7}} \)

= \(\frac{4}{3}\: \div \: \frac{30}{7} \)

= \(\frac{4}{3}\: \times \: \frac{7}{30} \)

= \(\frac{28}{90} \)

= \(\frac{14}{45}\)

ii. \(\scriptsize 2 \normalsize \frac{4 }{5}\scriptsize \: \times \: 5 \normalsize \frac{6}{8} \scriptsize \: \div \: \scriptsize 5 \normalsize \frac{1}{9} \)

Solution

= \(\frac{14 }{5} \; \times \; \frac{46}{8} \; \div \; \frac{46}{9} \)

= \(\frac{14 }{5} \; \times \; \frac{46}{8} \; \times \; \frac{9}{46} \)

= \(\frac{7 }{5} \; \times \; \frac{1}{4} \; \times \; \frac{9}{1} \)

= \(\frac{7 \; \times \; 1 \; \times \; 9 }{5 \; \times \; 4 \; \times \; 1} \)

= \( \frac{63}{20} \scriptsize = 3 \normalsize \frac {3}{20} \)

iii. \(\frac {1}{4} \scriptsize \; of \; \left ( \scriptsize2 \normalsize \frac{3 }{4 } \; \div\; \frac{2}{8} \right) + \scriptsize 1 \normalsize \frac{2 }{3 }\)

Solution

Let’s solve the bracket first  –  Remember BODMAS

\(\frac {1}{4} \scriptsize \: of \: \left ( \normalsize \frac{11 }{4 } \: \div\: \frac{2}{8} \right) + \scriptsize 1 \normalsize \frac{2 }{3 }\)

= \(\frac {1}{4} \scriptsize \: of \: \left ( \normalsize \frac{11 }{4 } \: \times\: \frac{8}{2} \right) + \scriptsize 1 \normalsize \frac{2 }{3 }\)

Next  is to solve  “of” which means multiplication

= \(\frac {1}{4} \scriptsize \: \times \: \normalsize \frac{88 }{8 } + \scriptsize 1 \normalsize \frac{2 }{3 }\)

= \(\frac {1}{4} \scriptsize \: \times \: \scriptsize 11 + \normalsize \frac{5 }{3 } \)

= \(\frac {11}{4} \: + \: \frac{5 }{3} \)

= \(\frac {33 \:+ \:20}{12} = \frac{53 }{12} \)

= \(\scriptsize 4 \frac {5}{12} \)