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Chapter 2: Problem 4

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) \(\frac{4}{7}+\frac{1}{2}-\frac{2}{3}\)

Short Answer

Expert verified
Question: Perform the following additions and subtractions with fractions and provide the answers in their simplest form: (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) \(\frac{4}{7}+\frac{1}{2}-\frac{2}{3}\) Answers: (a) \(\frac{13}{12}\) (b) \(\frac{11}{10}\) (c) \(\frac{1}{2}\) (d) \(\frac{9}{22}\) (e) \(\frac{17}{42}\)

Step by step solution

01

Find the common denominator

The lowest common denominator (LCD) between 4 and 3 is 12. The LCD is the smallest number that both denominators can divide into, and in this case, 12 is the smallest number that both 4 and 3 can divide into.
02

Convert the fractions to their equivalent forms with the LCD

We want fractions with a denominator of 12, so we need to change \(\frac{3}{4}\) and \(\frac{1}{3}\) accordingly: \(\frac{3}{4} \times \frac{3}{3} = \frac{9}{12}\) \(\frac{1}{3} \times \frac{4}{4} = \frac{4}{12}\)
03

Add the fractions

Now that the fractions have the same denominator, we can add them together: \(\frac{9}{12}+\frac{4}{12}=\frac{9+4}{12}=\frac{13}{12}\)
04

Simplify (if possible)

In this case, the result is already in its simplest form, so the final answer is \(\frac{13}{12}\). (b) \(\frac{1}{2}+\frac{3}{5}\)
05

Find the common denominator

The lowest common denominator (LCD) between 2 and 5 is 10.
06

Convert the fractions to their equivalent forms with the LCD

We want fractions with a denominator of 10, so we need to change \(\frac{1}{2}\) and \(\frac{3}{5}\) accordingly: \(\frac{1}{2} \times \frac{5}{5} = \frac{5}{10}\) \(\frac{3}{5} \times \frac{2}{2} = \frac{6}{10}\)
07

Add the fractions

Now that the fractions have the same denominator, we can add them together: \(\frac{5}{10}+\frac{6}{10}=\frac{5+6}{10}=\frac{11}{10}\)
08

Simplify (if possible)

In this case, the result is already in its simplest form, so the final answer is \(\frac{11}{10}\). (c) \(\frac{5}{6}-\frac{1}{3}\)
09

Find the common denominator

The lowest common denominator (LCD) between 6 and 3 is 6.
10

Convert the fractions to their equivalent forms with the LCD

We can leave \(\frac{5}{6}\) as it is since the denominator is already 6. \(\frac{1}{3} \times \frac{2}{2} = \frac{2}{6}\)
11

Subtract the fractions

Now that the fractions have the same denominator, we can subtract them: \(\frac{5}{6}-\frac{2}{6}=\frac{5-2}{6}=\frac{3}{6}\)
12

Simplify (if possible)

We can simplify the result by dividing both the numerator and denominator by 3, resulting in \(\frac{1}{2}\). (d) \(\frac{10}{11}-\frac{1}{2}\)
13

Find the common denominator

The lowest common denominator (LCD) between 11 and 2 is 22.
14

Convert the fractions to their equivalent forms with the LCD

We want fractions with a denominator of 22, so we need to change \(\frac{10}{11}\) and \(\frac{1}{2}\) accordingly: \(\frac{10}{11} \times \frac{2}{2} = \frac{20}{22}\) \(\frac{1}{2} \times \frac{11}{11} = \frac{11}{22}\)
15

Subtract the fractions

Now that the fractions have the same denominator, we can subtract them: \(\frac{20}{22}-\frac{11}{22}=\frac{20-11}{22}=\frac{9}{22}\)
16

Simplify (if possible)

In this case, the result is already in its simplest form, so the final answer is \(\frac{9}{22}\). (e) \(\frac{4}{7}+\frac{1}{2}-\frac{2}{3}\)
17

Finding the common denominator and converting the fractions

The common denominator for all three fractions is 42, and we need to change each fraction accordingly: \(\frac{4}{7} \times \frac{6}{6} = \frac{24}{42}\) \(\frac{1}{2} \times \frac{21}{21} = \frac{21}{42}\) \(\frac{2}{3} \times \frac{14}{14} = \frac{28}{42}\)
18

Adding and subtracting the fractions

Now that the fractions have the same denominator, we can add and subtract them: \(\frac{24}{42}+\frac{21}{42}-\frac{28}{42}=\frac{24+21-28}{42}=\frac{17}{42}\) The final answer for (e) is \(\frac{17}{42}\).

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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) \(-\frac{62}{7}\)

Reliability Engineering - Quality control. (a) Quality control checks find six faulty items from 200 that were tested. Express the number of faulty items as a fraction, in its simplest form, of the total number tested. (b) Improvements to the production process mean that the number of faulty items is halved. Express the number of faulty items now as a fraction of the total number tested.

Express each of the following mixed fractions as an improper fraction: (a) \(3 \frac{2}{3}\) (b) \(5 \frac{2}{5}\) (c) \(7 \frac{1}{2}\) (d) \(-9 \frac{3}{4}\) (e) \(10 \frac{4}{7}\)

(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 .

Express each fraction as an equivalent fraction with a denominator of 32 : (a) \(\frac{1}{16}\) (b) \(\frac{3}{8}\) (c) \(\frac{1}{4}\)
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