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Math Trailhead

Reed College Department of Mathematics & Statistics

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Worksheet Fractions

1.

Add the fractions, and reduce your answer.
\begin{equation*} \frac{12}{6}+\frac{14}{5} \end{equation*}
The reduced answer is
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Answer.
\(4{\textstyle\frac{4}{5}}\)
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2.

Subtract as indicated: (Note: Your answer must be a fraction.)
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a) \(-\frac{3}{5} - (-\frac{7}{15}) =\)
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b) \(\frac{1}{6} - \frac{7}{8} =\)
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Answer 1.
\(\frac{-2}{15}\)
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Answer 2.
\(\frac{-17}{24}\)
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3.

Evaluate each expression if \(a=-{\frac{1}{6}},\ b={\frac{7}{9}},\ c= -2 \frac{1}{3},\ d=4\frac{1}{2}\text{.}\)
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Your answer should be a reduced fraction or a mixed number.
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a) \(4a=\)
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b) \(a+cd=\)
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c) \(9d+\frac{7}{8}=\)
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d) \(d(b+6)=\)
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Answer 1.
\(-{\frac{2}{3}}\)
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Answer 2.
\(-10{\textstyle\frac{2}{3}}\)
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Answer 3.
\(41{\textstyle\frac{3}{8}}\)
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Answer 4.
\(30{\textstyle\frac{1}{2}}\)
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Solution.
Solution
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a) \(4a=4\times -{\frac{1}{6}}=-{\frac{2}{3}}\)
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b) \(a+cd= -{\frac{1}{6}} - 2{\textstyle\frac{1}{3}}\times 4{\textstyle\frac{1}{2}}= -10{\textstyle\frac{2}{3}}\)
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c) \(9d+\frac{7}{8}=\(9\times 4{\textstyle\frac{1}{2}}+\frac{7}{8} = 41{\textstyle\frac{3}{8}}\)
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d) \(d(b+6)=4{\textstyle\frac{1}{2}}({\frac{7}{9}}+6) = 30{\textstyle\frac{1}{2}}\)
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4.

Put in simplest form.
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For example if \(\frac{3}{7}\) were the answer you would put 3/7 in the answer box.
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1) \(\frac{16}{18}\)=
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2) \(\frac{28}{40}\)=
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2) \(\frac{25}{5}\)=
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Answer 1.
\({\frac{8}{9}}\)
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Answer 2.
\({\frac{7}{10}}\)
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Answer 3.
\(5\)
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Solution.
Solution
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1) To put \(\frac{16}{18}\) into simplest form we must divide both numerator and denominator by 2
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(which is their greatest common factor) to get answer \(\frac{8}{9}\) which we put in the answer box as \(8/9\text{.}\)
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2) To put \(\frac{28}{40}\) into simplest form we must divide both numerator and denominator
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by 4 to get answer \(\frac{7}{10}\) which we put in the answer box as \(7/10\text{.}\)
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5.

Write the following rational number in simplest form:
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\(\displaystyle\frac{11968000}{2156000} =\) /
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Hint:
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\(11968000 =\) \(2^5 \cdot 17 \cdot 4^2 \cdot 5^3 \cdot 11\)
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\(2156000 =\) \(5^3 \cdot 11 \cdot 2^5 \cdot 7^2\)
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Hint.
Consider the provided factorizations for 11968000 and 2156000. The greatest common factor will be the product of factors that are in both.
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Answer 1.
\(272\)
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Answer 2.
\(49\)
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6.

Express the rational number \(\displaystyle \frac{-104}{-136}\) as a reduced fraction.
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Answer:
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Answer.
\({\frac{13}{17}}\)
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7.

Add the following and express your answer as a single fraction. No decimals or mixed fractions.
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\(\displaystyle\frac{21}{6} - \frac{11}{6} =\)
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Remember, to enter a fraction of the form \(\frac{a}{b}\text{,}\) type a / b.
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Hint.
First you will need to get a common denominator by finding the least common multiple.
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Answer.
\(\frac{21\cdot 6-11\cdot 6}{6\cdot 6}\)
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8.

Add the following and express your answer as a single fraction. No decimals or mixed fractions.
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\(\displaystyle\frac{21}{6} - \frac{14}{5} =\)
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Remember, to enter a fraction of the form \(\frac{a}{b}\text{,}\) type a / b.
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Hint.
First you will need to get a common denominator by finding the least common multiple.
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Answer.
\(\frac{21\cdot 5-14\cdot 6}{6\cdot 5}\)
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9.

i) \(\frac{1}{4}-\frac{1}{5}\) =.
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ii) \(\frac{1}{5}-\frac{1}{6}\) =.
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iii) \(\frac{1}{7}-\frac{1}{8}\) =.
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The three problems above all satisfy a rule. It is
  • \(\displaystyle \frac{1}{a}-\frac{1}{a+1}=\frac{2a+1}{a (a+1)}\)
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  • \(\displaystyle \frac{1}{a}-\frac{1}{a+1}=\frac{1}{2a+1}\)
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  • \(\displaystyle \frac{1}{a}-\frac{1}{a+1}=\frac{a}{2a+1}\)
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  • \(\displaystyle \frac{1}{a}-\frac{1}{a+1}= \frac{1}{a(a+1)}\)
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Answer 1.
\({\frac{1}{20}}\)
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Answer 2.
\({\frac{1}{30}}\)
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Answer 3.
\({\frac{1}{56}}\)
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Solution.
Solution
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\(\frac{1}{a}-\frac{1}{a+1}=\frac{a+1}{a(a+1)}-\frac{a}{a(a+1)}= \frac{1}{a(a+1)}\)
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10.

Write each sum in simplest form (as a mixed number).
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\(15 \frac {5}{8} + 16 \frac{1}{6}\)=.
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\(\frac {3}{5} + 8 \frac{4}{7}\)=.
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\(17 \frac {3}{8} + 8 \frac{3}{4}\)=.
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\(27 \frac {3}{5} + 16 \frac{4}{5}\)=.
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\(9 \frac {2}{3} + 21 \frac{2}{9}\)=.
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\(16 \frac {5}{9} + 38 \frac{4}{9}\)=.
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\(24 \frac {8}{9} + 27 \frac{14}{15}\)=.
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\(84 \frac {1}{3} + 66 \frac{3}{4}\)=.
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Answer 1.
\(31{\textstyle\frac{19}{24}}\)
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Answer 2.
\(9{\textstyle\frac{6}{35}}\)
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Answer 3.
\(26{\textstyle\frac{1}{8}}\)
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Answer 4.
\(44{\textstyle\frac{2}{5}}\)
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Answer 5.
\(30{\textstyle\frac{8}{9}}\)
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Answer 6.
\(55\)
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Answer 7.
\(52{\textstyle\frac{37}{45}}\)
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Answer 8.
\(151{\textstyle\frac{1}{12}}\)
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Solution.
Solutions
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The first few [the steps before reaching the answer are shown]:
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\(15 \frac {5}{8} + 16 \frac{1}{6}=31+\frac{19}{24}\text{.}\)
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\(\frac {3}{5} + 8 \frac{4}{7}=8+\frac{41}{35}\text{.}\)
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\(15 \frac {3}{8} + 8 \frac{3}{4}=25+\frac{9}{8}\text{.}\)
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11.

\(3\frac{5}{24}+1\frac{11}{24}+ 3\frac{13}{24}=\)
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Answer.
\(8{\textstyle\frac{5}{24}}\)
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Solution.
Solution
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Since the denominators of the fractions are all the same ( 24) we can start by changing each mixed number
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to an improper fraction with denominator 24 and then add them
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\(3\frac{5}{24}+1\frac{11}{24}+ 3\frac{13}{24}= \frac{77}{24}+ \frac{35}{24}+ \frac{85}{24}=\frac{197}{24}=8 \frac{5}{24}\)
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12.

Add these together: \(\displaystyle{ 4 + \frac{1}{9}}\)
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When needed, use an improper fraction in your answer. Don’t use a mixed number.
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Answer.
\({\frac{37}{9}}\)
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Solution.
When doing arithmetic with fractions, it is helpful to rewrite any integers as fractions:
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\(\displaystyle{ 4 = \frac{4}{1} }\)
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Next, to add two fractions, we need to find a common denominator. In this case, it is simply the second denominator \(9\text{.}\) We will rewrite the first fraction:
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\(\displaystyle{ \begin{aligned}\frac{4}{1} \amp = \frac{4 \cdot 9}{1 \cdot 9}\\ \amp = \frac{36}{9} \end{aligned}}\)
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Finally, we add the numerators and keep the denominator unchanged. The whole process is:
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\(\displaystyle{\begin{aligned} 4 + \frac{1}{9} \amp = \frac{36}{9} + \frac{1}{9} \\ \amp = \frac{36 + 1}{9} \\ \amp = \frac{37}{9} \end{aligned} }\)
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The answer to this question is \({{\frac{37}{9}}}\text{.}\)
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13.

Reduce the fraction \(\displaystyle{ \frac{3}{30} }\text{.}\)
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Answer.
\({\frac{1}{10}}\)
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Solution.
There are at least two methods to reduce the fraction \(\displaystyle{ \frac{3}{30} }\text{.}\)
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Method 1:We find a number that divides into both the numerator \(3\text{,}\) and the denominator \(30\text{.}\)
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Check the first few prime numbers one by one: \(2, 3, 5, 7, \ldots\)
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In this case, \(3\) goes into both the numerator and denominator of \(\displaystyle{ \frac{3}{30} }\text{.}\) We divide \(3\) into both numbers, and we have:
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\(\displaystyle{ \begin{aligned} \frac{3}{30} \amp = \frac{3 \div 3}{30\div 3}\\ \amp = \frac{1}{10}\end{aligned} }\)
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Next, check again whether any prime number divides into both the numerator and denominator. We need to keep trying until no prime numbers divide into both numbers.
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In this case, \(\displaystyle{\frac{1}{10}}\) is the final answer.
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Method 2:
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A second method to reduce fraction is to prime factor both the numerator and the denominator, and then cancel out factors in pairs: one from the numerator and one from the denominator.
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\(\displaystyle{\begin{aligned}[t] \frac{3}{30} \amp = \frac{1 \cdot 3}{2 \cdot 3 \cdot 5} \\ \amp =\frac{1}{2 \cdot 5}\\ \amp = \frac{1}{10} \end{aligned} }\)
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Notice that when the numerator’s only prime factor, \(3\text{,}\) is canceled, we have to leave a \(1\) in the numerator.
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14.

Evaluate the following.
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  1. \(\displaystyle{ \frac{-20}{-10}= }\)
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  2. \(\displaystyle{ \frac{24}{-4}= }\)
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  3. \(\displaystyle{ \frac{-42}{6}= }\)
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Answer 1.
\(2\)
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Answer 2.
\(-6\)
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Answer 3.
\(-7\)
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Solution.
The rules for dividing positive numbers are the same as those for multiplication:
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\(\displaystyle{ \text{positive} \div \text{positive} = \text{positive} }\text{,}\)
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\(\displaystyle{ \text{positive} \div \text{negative} = \text{negative} }\text{,}\)
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\(\displaystyle{ \text{negative} \div \text{positive} = \text{negative} }\text{,}\)
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\(\displaystyle{ \text{negative} \div \text{negative} = \text{positive} }\text{.}\)
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The solutions are:
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  1. \(\displaystyle \displaystyle{ \frac{-20}{-10}={2}, }\)
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  2. \(\displaystyle \displaystyle{ \frac{24}{-4}={-6}, }\)
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  3. \(\displaystyle \displaystyle{ \frac{-42}{6}={-7}. }\)
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15.

Combine the fractions, and reduce your answer.
\begin{equation*} \left(3\div\frac{8}{5}\right)-\frac{8}{5} \end{equation*}
The reduced answer is /
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Answer 1.
\(11\)
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Answer 2.
\(40\)
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17.

Here is an expression with negative exponents.
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\(\displaystyle\left(\frac{7}{4}\right)^{-2}=\)
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Evaluate the expression; in other words, write the answer without using exponents.
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Answer.
\({\frac{16}{49}}\)
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Solution.
We evaluate the expression by remembering that \(x^{-n}\) is the same thing as \(\displaystyle{\frac{1}{x^n}}\) for any non-zero, real value of \(x\)
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\(\begin{aligned} \displaystyle\left(\frac{7}{4}\right)^{-2}\amp =\displaystyle\frac{7^{-2}}{4^{-2}}\\ \amp = \displaystyle\frac{\displaystyle\frac{1}{7^{2}}}{\displaystyle\frac{1}{4^{2}}} \\ \amp = \displaystyle\frac{1}{7^{2}}\cdot \frac{4^{2}}{1} \\ \amp = \frac{4^{2}}{7^{2}}\\ \amp = \displaystyle\frac{16}{49} \end{aligned}\)
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Remember that when dividing by a fraction we multiply by its reciprocal.
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18.

Evaluate this expression:
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\(\displaystyle{ {{\frac{6}{7}}}+8\cdot{{\frac{4}{21}}}= }\)
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Answer.
\({\frac{50}{21}}\)
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Solution.
According to the order of operations, the multiplication has higher priority than the addition.
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You might need to go back to previous units to review how to add fractions.
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\(\begin{aligned}[t] {{\frac{6}{7}}}+8\cdot{{\frac{4}{21}}} \amp = {{\frac{6}{7}}} + {{\frac{32}{21}}} \\ \amp = {{\frac{50}{21}}} \end{aligned}\)
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19.

Evaluate the following expressions:
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  1. \(\displaystyle{ {{\frac{4}{9}}}-\left({{\frac{2}{3}}}\right)^{3}= }\)
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  2. \(\displaystyle{ {{\frac{4}{9}}}-\left(-{{\frac{2}{3}}}\right)^{3}= }\)
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Answer 1.
\({\frac{4}{27}}\)
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Answer 2.
\({\frac{20}{27}}\)
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Solution.
You might need to go back to previous units to review how to do arithmetic with fractions.
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  1. \(\displaystyle \begin{aligned}[t] {{\frac{4}{9}}}-\left({{\frac{2}{3}}}\right)^{3} \amp = {{\frac{4}{9}}}-{{\frac{2}{3}}}\cdot{{\frac{2}{3}}}\cdot{{\frac{2}{3}}} \\ \amp = {{\frac{4}{9}}}-{{\frac{8}{27}}} \\ \amp = {{\frac{4}{27}}} \end{aligned}\)
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  2. \(\displaystyle \begin{aligned}[t] {{\frac{4}{9}}}-\left(-{{\frac{2}{3}}}\right)^{3} \amp = {{\frac{4}{9}}}-\left(-{{\frac{2}{3}}}\right)\cdot\left(-{{\frac{2}{3}}}\right)\cdot\left(-{{\frac{2}{3}}}\right) \\ \amp = {{\frac{4}{9}}}-\left({-{\frac{8}{27}}}\right) \\ \amp = {{\frac{20}{27}}} \end{aligned}\)
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20.

Evaluate the following expressions:
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  1. \(\displaystyle{ \left( -{{\frac{3}{8}}} \right) ^{2}= }\)
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  2. \(\displaystyle{ - \left( {{\frac{3}{8}}} \right) ^{2}= }\)
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  3. \(\displaystyle{ - \left( -{{\frac{3}{8}}} \right) ^{2}= }\)
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Answer 1.
\({\frac{9}{64}}\)
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Answer 2.
\(-{\frac{9}{64}}\)
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Answer 3.
\(-{\frac{9}{64}}\)
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Solution.
Notice the difference in the position of the negative symbol.
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  1. \(\displaystyle{\begin{aligned}[t] \left( -{{\frac{3}{8}}} \right) ^{2} \amp = \left( -{{\frac{3}{8}}} \right) \cdot \left( -{{\frac{3}{8}}} \right) \\ \amp = {{\frac{9}{64}}} \end{aligned}}\)
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    In Part a, parentheses have higher priority than exponents.
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  2. \(\displaystyle{\begin{aligned}[t] -\left( {{\frac{3}{8}}} \right) ^{2} \amp = -\left( {{\frac{3}{8}}} \right) \cdot \left( {{\frac{3}{8}}} \right) \\ \amp = {-{\frac{9}{64}}} \end{aligned}}\)
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    In Part b, exponents have higher priority than the negative symbol (which is, in essence, multiplication).
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  3. \(\displaystyle \displaystyle{\begin{aligned}[t] -\left( -{{\frac{3}{8}}} \right) ^{2} \amp = -\left( -{{\frac{3}{8}}} \right) \cdot \left( -{{\frac{3}{8}}} \right) \\ \amp = \left( {{\frac{3}{8}}} \right) \cdot \left( -{{\frac{3}{8}}} \right) \\ \amp = {-{\frac{9}{64}}} \end{aligned}}\)
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The following pattern shows why the negative symbol means “negative one times”:
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\(\displaystyle{\begin{aligned}[t] -2 \amp = -1 \cdot 2 \\ -3 \amp = -1 \cdot 3 \\ -4 \amp = -1 \cdot 4 \\ \amp \vdots \end{aligned} }\)
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21.

Simplify. Your answer should be a reduced fraction, with no common factors in numerator and denominator.
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\(\displaystyle \frac{\frac{3}{5} + \frac{2}{3}}{\frac{3}{4} - \frac{1}{5}} =\)
Answer 1.
\(76\)
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Answer 2.
\(33\)
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Solution.
Solution: We obtain
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\begin{equation*} \frac{\frac{3}{5} + \frac{2}{3}}{\frac{3}{4} - \frac{1}{5}} = \frac{\frac{3 \cdot 3+5 \cdot 2}{15}}{\frac{5 \cdot 3-4 \cdot 1}{20}} =\frac{\frac{19}{15}}{\frac{11}{20}} =\frac{19}{15} \times \frac{20}{11} = \frac{76}{33} \end{equation*}
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22.

Courtney walks 3 laps around a \(\frac{1}{4}\)-mile track.
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[1 mile is 5280 feet].
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How far does she walk?
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Solution.
Solution
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A quarter mile is \(\frac{1}{4}\times 5280=1320\) ft.
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So three laps is \(3\times 1320=3960 ft\text{.}\)
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