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Chapter 2: Problem 11

Evaluate (a) \(6 \frac{2}{3} \div 4\) (b) \(10 \div 2 \frac{1}{3}\) (c) \(\left(\frac{1}{2}+\frac{1}{3}\right) \div\left(\frac{2}{3}+\frac{1}{5}\right)\) (d) \(\left(6-2 \frac{1}{3}\right) \times\left(4 \frac{1}{2}-1 \frac{3}{4}\right)\) (e) \(\frac{2 \frac{1}{2}+1 \frac{1}{3}}{6 \frac{2}{3}-2 \frac{1}{4}}\)

Short Answer

Expert verified
Question: Evaluate the mixed fraction expressions (a) \(6 \frac{2}{3} \div 4\), (b) \(10 \div 2 \frac{1}{3}\), (c) \(\left(\frac{1}{2}+\frac{1}{3}\right)\div\left(\frac{2}{3}+\frac{1}{5}\right)\), (d) \(\left(6-2 \frac{1}{3}\right) \times\left(4 \frac{1}{2}-1 \frac{3}{4}\right)\), and (e) \(\frac{2 \frac{1}{2}+1 \frac{1}{3}}{6 \frac{2}{3}-2 \frac{1}{4}}\). Answer: (a) \(5\frac{2}{3}\), (b) \(4 \frac{2}{7}\), (c) \(\frac{75}{78}\), (d) \(1\frac{5}{6}\), (e) \(\frac{2}{3}\).

Step by step solution

01

Act 1: Convert mixed fraction to improper fraction

Convert \(6 \frac{2}{3}\) to improper fraction: \(\frac{6*3 + 2}{3} = \frac{20}{3}\). Now, we have: \(\frac{20}{3} \div 4\)
02

Act 2: Rewrite division as multiplication

Rewrite the expression as multiplication by the reciprocal of 4: \(\frac{20}{3} \times \frac{1}{4}\)
03

Act 3: Multiply the fractions

Multiply the numerators and denominators: \(\frac{20*1}{3*4} = \frac{20}{12}\)
04

Act 4: Simplify the fraction

Simplify the fraction by dividing numerator and denominator by their greatest common divisor: \(\frac{20\div4}{12\div4} = \frac{5}{3}\) (a) The answer is \(5\frac{2}{3}\) (mixed fraction form). (b) \(10 \div 2 \frac{1}{3}\)
05

Act 1: Convert mixed fraction to improper fraction

Convert \(2\frac{1}{3}\) to improper fraction: \(\frac{2 * 3 + 1}{3} = \frac{7}{3}\). Now, we have: \(10 \div \frac{7}{3}\)
06

Act 2: Rewrite division as multiplication

Rewrite the expression as multiplication by the reciprocal of \(\frac{7}{3}\): \(10 \times \frac{3}{7}\)
07

Act 3: Multiply the fractions

Multiply \(10 = \frac{10}{1}\) by the fraction: \(\frac{10*3}{1*7} = \frac{30}{7}\) (b) The answer is \(4 \frac{2}{7}\) (mixed fraction form). (c) $\left(\frac{1}{2}+\frac{1}{3}\right) \div\left(\frac{2}{3}+\frac{1}{5}\right)$
08

Act 1: Add fractions

We have two sets of fractions to add: \(\frac{1}{2}+\frac{1}{3}\) and \(\frac{2}{3}+\frac{1}{5}\). Find a common denominator and add the fractions. \(\frac{1}{2}+\frac{1}{3} = \frac{3+2}{6} = \frac{5}{6}\) \(\frac{2}{3}+\frac{1}{5} = \frac{10+3}{15} = \frac{13}{15}\) Now, we have: \(\frac{5}{6} \div \frac{13}{15}\)
09

Act 2: Rewrite division as multiplication

Rewrite the expression as multiplication by the reciprocal of \(\frac{13}{15}\): \(\frac{5}{6} \times \frac{15}{13}\)
10

Act 3: Multiply the fractions

Multiply the numerators and denominators: \(\frac{5*15}{6*13} = \frac{75}{78}\) (c) The answer is \(\frac{75}{78}\) (simplified fraction form). (d) $\left(6-2 \frac{1}{3}\right) \times\left(4 \frac{1}{2}-1 \frac{3}{4}\right)$
11

Act 1: Subtract mixed fractions

We have two sets of mixed fractions to subtract: \(6 - 2\frac{1}{3}\) and \(4\frac{1}{2}- 1\frac{3}{4}\). Convert the fractions to improper fractions and find a common denominator, and then subtract the fractions. \(6 - 2\frac{1}{3} = \frac{18-7}{3} = \frac{11}{3}\) \(4\frac{1}{2}-1\frac{3}{4}=\frac{9-7}{4} = \frac{2}{4}\) Now, we have: \(\frac{11}{3} \times \frac{2}{4}\)
12

Act 2: Multiply the fractions

Multiply the numerators and denominators: \(\frac{11*2}{3*4} = \frac{22}{12}\)
13

Act 3: Simplify the fraction

Simplify the fraction by dividing numerator and denominator by their greatest common divisor: \(\frac{22\div2}{12\div2} = \frac{11}{6}\) (d) The answer is \(1\frac{5}{6}\) (mixed fraction form). (e) \(\frac{2 \frac{1}{2}+1 \frac{1}{3}}{6 \frac{2}{3}-2 \frac{1}{4}}\)
14

Act 1: Add mixed fractions

We have two sets of mixed fractions to add: \(2\frac{1}{2} + 1\frac{1}{3}\) and \(6\frac{2}{3} - 2\frac{1}{4}\). Convert the fractions to improper fractions and find a common denominator, and then add the fractions. \(2\frac{1}{2}+1\frac{1}{3}=\frac{5+4}{6}=\frac{9}{6}\) \(6\frac{2}{3}-2\frac{1}{4}=\frac{44-9}{12}=\frac{35}{12}\) Now, we have: \(\frac{\frac{9}{6}}{\frac{35}{12}}\)
15

Act 2: Rewrite division as multiplication

Rewrite the expression as multiplication by the reciprocal of \(\frac{35}{12}\): \(\frac{9}{6} \times \frac{12}{35}\)
16

Act 3: Multiply the fractions

Multiply the numerators and denominators: \(\frac{9*12}{6*35} = \frac{108}{210}\)
17

Act 4: Simplify the fraction

Simplify the fraction by dividing numerator and denominator by their greatest common divisor: \(\frac{108\div54}{210\div54} = \frac{2}{3}\) (e) The answer is \(\frac{2}{3}\) (simplified fraction form).

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Most popular questions from this chapter

(a) Express \(\frac{3}{5}\) as an equivalent fraction with a denominator of 40 . (b) Express \(\frac{9}{30}\) as an equivalent fraction with a denominator of 10 . (c) Express 6 as an equivalent fraction with a denominator of 4 .

Thermodynamics - Cooling of a liquid. The temperature of a liquid is measured every 20 minutes and the results recorded in the table below. \begin{tabular}{lrllllrr} \hline Time \((\min )\) & 0 & 20 & 40 & 60 & 80 & 100 & 120 \\ Temp \(\left({ }^{\circ} \mathrm{C}\right)\) & 96 & 88 & 81 & 76 & 72 & 70 & 68 \\\ \hline \end{tabular} Calculate the rate of decrease of temperature. in units of \({ }^{\circ} \mathrm{C}\) per minute: (a) in the first 20 minutes (b) in the first 40 minutes (c) in the first 60 minutes (d) in the last 60 minutes (e) in the last 20 minutes In each case express your answer as a fraction in its simplest form.

Express each of the following improper fractions as mixed fractions: (a) \(\frac{20}{3}\) (b) \(\frac{32}{7}\) (c) \(\frac{60}{9}\) (d) \(\frac{102}{50}\) (e) \(\frac{120}{11}\)

Calculate (a) \(\frac{3}{4}+\frac{1}{3}\) (b) \(\frac{1}{2}+\frac{3}{5}\) (c) \(\frac{5}{6}-\frac{1}{3}\) (d) \(\frac{10}{11}-\frac{1}{2}\) (e) \(\frac{4}{7}+\frac{1}{2}-\frac{2}{3}\)

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) \(\frac{16}{21} \times\left(-\frac{3}{4}\right)\)
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