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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{6}{2}+\frac{3}{11} \end{equation*}
The reduced answer is
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Answer.
\(3{\textstyle\frac{3}{11}}\)
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2.

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

Evaluate each expression if \(a=-{\frac{1}{5}},\ b={\frac{5}{7}},\ c= -4 \frac{1}{4},\ d=5\frac{1}{2}\text{.}\)
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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.
\(-{\frac{4}{5}}\)
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Answer 2.
\(-23{\textstyle\frac{23}{40}}\)
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Answer 3.
\(50{\textstyle\frac{3}{8}}\)
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Answer 4.
\(36{\textstyle\frac{13}{14}}\)
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Solution.
Solution
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a) \(4a=4\times -{\frac{1}{5}}=-{\frac{4}{5}}\)
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b) \(a+cd= -{\frac{1}{5}} - 4{\textstyle\frac{1}{4}}\times 5{\textstyle\frac{1}{2}}= -23{\textstyle\frac{23}{40}}\)
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c) \(9d+\frac{7}{8}=\(9\times 5{\textstyle\frac{1}{2}}+\frac{7}{8} = 50{\textstyle\frac{3}{8}}\)
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d) \(d(b+6)=5{\textstyle\frac{1}{2}}({\frac{5}{7}}+6) = 36{\textstyle\frac{13}{14}}\)
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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) \(\frac{14}{16}\)=
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2) \(\frac{32}{36}\)=
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2) \(\frac{56}{8}\)=
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Answer 1.
\({\frac{7}{8}}\)
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Answer 2.
\({\frac{8}{9}}\)
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Answer 3.
\(7\)
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Solution.
Solution
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1) To put \(\frac{14}{16}\) 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 \(\frac{7}{8}\) which we put in the answer box as \(7/8\text{.}\)
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2) To put \(\frac{32}{36}\) into simplest form we must divide both numerator and denominator
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by 4 to get answer \(\frac{8}{9}\) which we put in the answer box as \(8/9\text{.}\)
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5.

Write the following rational number in simplest form:
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\(\displaystyle\frac{1604597904}{2023445151} =\) /
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Hint:
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\(1604597904 =\) \(3^3 \cdot 17 \cdot 4^2 \cdot 7^5 \cdot 13\)
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\(2023445151 =\) \(7^5 \cdot 13 \cdot 3^3 \cdot 7^3\)
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Hint.
Consider the provided factorizations for 1604597904 and 2023445151. 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.
\(343\)
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6.

Express the rational number \(\displaystyle \frac{-112}{-238}\) as a reduced fraction.
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Answer:
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Answer.
\({\frac{8}{17}}\)
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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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\(\displaystyle\frac{25}{7} - \frac{15}{7} =\)
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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.
\(\frac{25\cdot 7-15\cdot 7}{7\cdot 7}\)
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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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\(\displaystyle\frac{19}{8} - \frac{18}{7} =\)
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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.
\(\frac{19\cdot 7-18\cdot 8}{8\cdot 7}\)
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9.

i) \(\frac{1}{2}-\frac{1}{3}\) =.
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ii) \(\frac{1}{3}-\frac{1}{4}\) =.
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iii) \(\frac{1}{6}-\frac{1}{7}\) =.
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The three problems above all satisfy a rule. It is
  • \(\displaystyle \frac{1}{a}-\frac{1}{a+1}=\frac{a}{2a+1}\)
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  • \(\displaystyle \frac{1}{a}-\frac{1}{a+1}=\frac{1}{2a+1}\)
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  • \(\displaystyle \frac{1}{a}-\frac{1}{a+1}= \frac{1}{a(a+1)}\)
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  • \(\displaystyle \frac{1}{a}-\frac{1}{a+1}=\frac{2a+1}{a (a+1)}\)
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Answer 1.
\({\frac{1}{6}}\)
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Answer 2.
\({\frac{1}{12}}\)
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Answer 3.
\({\frac{1}{42}}\)
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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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\(19 \frac {3}{8} + 18 \frac{5}{6}\)=.
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\(\frac {4}{5} + 7 \frac{6}{7}\)=.
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\(13 \frac {3}{8} + 6 \frac{3}{4}\)=.
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\(26 \frac {4}{5} + 16 \frac{4}{5}\)=.
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\(8 \frac {1}{3} + 26 \frac{8}{9}\)=.
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\(26 \frac {8}{9} + 28 \frac{4}{9}\)=.
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\(34 \frac {5}{9} + 37 \frac{14}{15}\)=.
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\(84 \frac {1}{3} + 56 \frac{3}{4}\)=.
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Answer 1.
\(38{\textstyle\frac{5}{24}}\)
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Answer 2.
\(8{\textstyle\frac{23}{35}}\)
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Answer 3.
\(20{\textstyle\frac{1}{8}}\)
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Answer 4.
\(43{\textstyle\frac{3}{5}}\)
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Answer 5.
\(35{\textstyle\frac{2}{9}}\)
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Answer 6.
\(55{\textstyle\frac{1}{3}}\)
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Answer 7.
\(72{\textstyle\frac{22}{45}}\)
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Answer 8.
\(141{\textstyle\frac{1}{12}}\)
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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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\(19 \frac {3}{8} + 18 \frac{5}{6}=37+\frac{29}{24}\text{.}\)
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\(\frac {4}{5} + 7 \frac{6}{7}=7+\frac{58}{35}\text{.}\)
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\(19 \frac {3}{8} + 6 \frac{3}{4}=19+\frac{9}{8}\text{.}\)
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11.

\(2\frac{9}{24}+3\frac{16}{24}+ 1\frac{13}{24}=\)
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Answer.
\(7{\textstyle\frac{7}{12}}\)
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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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\(2\frac{9}{24}+3\frac{16}{24}+ 1\frac{13}{24}= \frac{57}{24}+ \frac{88}{24}+ \frac{37}{24}=\frac{182}{24}=7 \frac{14}{24}\)
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12.

Add these together: \(\displaystyle{ -3 + \frac{3}{7}}\)
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When needed, use an improper fraction in your answer. Don’t use a mixed number.
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Answer.
\(-{\frac{18}{7}}\)
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Solution.
When doing arithmetic with fractions, it is helpful to rewrite any integers as fractions:
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\(\displaystyle{ -3 = \frac{-3}{1} }\)
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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 \(7\text{.}\) We will rewrite the first fraction:
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\(\displaystyle{ \begin{aligned}\frac{-3}{1} \amp = \frac{-3 \cdot 7}{1 \cdot 7}\\ \amp = \frac{-21}{7} \end{aligned}}\)
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Finally, we add the numerators and keep the denominator unchanged. The whole process is:
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\(\displaystyle{\begin{aligned} -3 + \frac{3}{7} \amp = \frac{-21}{7} + \frac{3}{7} \\ \amp = \frac{-21 + 3}{7} \\ \amp = -\frac{18}{7} \end{aligned} }\)
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The answer to this question is \({-{\frac{18}{7}}}\text{.}\)
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13.

Reduce the fraction \(\displaystyle{ \frac{7}{56} }\text{.}\)
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Answer.
\({\frac{1}{8}}\)
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Solution.
There are at least two methods to reduce the fraction \(\displaystyle{ \frac{7}{56} }\text{.}\)
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Method 1:We find a number that divides into both the numerator \(7\text{,}\) and the denominator \(56\text{.}\)
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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, \(7\) goes into both the numerator and denominator of \(\displaystyle{ \frac{7}{56} }\text{.}\) We divide \(7\) into both numbers, and we have:
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\(\displaystyle{ \begin{aligned} \frac{7}{56} \amp = \frac{7 \div 7}{56\div 7}\\ \amp = \frac{1}{8}\end{aligned} }\)
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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, \(\displaystyle{\frac{1}{8}}\) 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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\(\displaystyle{\begin{aligned}[t] \frac{7}{56} \amp = \frac{1 \cdot 7}{2 \cdot 2 \cdot 2 \cdot 7} \\ \amp =\frac{1}{2 \cdot 2 \cdot 2}\\ \amp = \frac{1}{8} \end{aligned} }\)
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Notice that when the numerator’s only prime factor, \(7\text{,}\) is canceled, we have to leave a \(1\) in the numerator.
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14.

Evaluate the following.
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  1. \(\displaystyle{ \frac{-24}{-4}= }\)
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  2. \(\displaystyle{ \frac{54}{-6}= }\)
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  3. \(\displaystyle{ \frac{-56}{8}= }\)
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Answer 1.
\(6\)
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Answer 2.
\(-9\)
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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. \(\displaystyle \displaystyle{ \frac{-24}{-4}={6}, }\)
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  2. \(\displaystyle \displaystyle{ \frac{54}{-6}={-9}, }\)
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  3. \(\displaystyle \displaystyle{ \frac{-56}{8}={-7}. }\)
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15.

Combine the fractions, and reduce your answer.
\begin{equation*} \left(9\div\frac{2}{9}\right)-\frac{2}{9} \end{equation*}
The reduced answer is /
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Answer 1.
\(725\)
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Answer 2.
\(18\)
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17.

Here is an expression with negative exponents.
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\(\displaystyle\left(\frac{5}{2}\right)^{-2}=\)
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Evaluate the expression; in other words, write the answer without using exponents.
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Answer.
\({\frac{4}{25}}\)
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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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\(\begin{aligned} \displaystyle\left(\frac{5}{2}\right)^{-2}\amp =\displaystyle\frac{5^{-2}}{2^{-2}}\\ \amp = \displaystyle\frac{\displaystyle\frac{1}{5^{2}}}{\displaystyle\frac{1}{2^{2}}} \\ \amp = \displaystyle\frac{1}{5^{2}}\cdot \frac{2^{2}}{1} \\ \amp = \frac{2^{2}}{5^{2}}\\ \amp = \displaystyle\frac{4}{25} \end{aligned}\)
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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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\(\displaystyle{ {{\frac{2}{3}}}+2\cdot{{\frac{2}{3}}}= }\)
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Answer.
\(2\)
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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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\(\begin{aligned}[t] {{\frac{2}{3}}}+2\cdot{{\frac{2}{3}}} \amp = {{\frac{2}{3}}} + {{\frac{4}{3}}} \\ \amp = {2} \end{aligned}\)
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19.

Evaluate the following expressions:
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  1. \(\displaystyle{ {2}-\left({{\frac{1}{2}}}\right)^{3}= }\)
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  2. \(\displaystyle{ {2}-\left(-{{\frac{1}{2}}}\right)^{3}= }\)
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Answer 1.
\({\frac{15}{8}}\)
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Answer 2.
\({\frac{17}{8}}\)
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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. \(\displaystyle \begin{aligned}[t] {2}-\left({{\frac{1}{2}}}\right)^{3} \amp = {2}-{{\frac{1}{2}}}\cdot{{\frac{1}{2}}}\cdot{{\frac{1}{2}}} \\ \amp = {2}-{{\frac{1}{8}}} \\ \amp = {{\frac{15}{8}}} \end{aligned}\)
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  2. \(\displaystyle \begin{aligned}[t] {2}-\left(-{{\frac{1}{2}}}\right)^{3} \amp = {2}-\left(-{{\frac{1}{2}}}\right)\cdot\left(-{{\frac{1}{2}}}\right)\cdot\left(-{{\frac{1}{2}}}\right) \\ \amp = {2}-\left({-{\frac{1}{8}}}\right) \\ \amp = {{\frac{17}{8}}} \end{aligned}\)
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20.

Evaluate the following expressions:
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  1. \(\displaystyle{ \left( -{{\frac{5}{6}}} \right) ^{2}= }\)
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  2. \(\displaystyle{ - \left( {{\frac{5}{6}}} \right) ^{2}= }\)
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  3. \(\displaystyle{ - \left( -{{\frac{5}{6}}} \right) ^{2}= }\)
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Answer 1.
\({\frac{25}{36}}\)
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Answer 2.
\(-{\frac{25}{36}}\)
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Answer 3.
\(-{\frac{25}{36}}\)
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Solution.
Notice the difference in the position of the negative symbol.
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  1. \(\displaystyle{\begin{aligned}[t] \left( -{{\frac{5}{6}}} \right) ^{2} \amp = \left( -{{\frac{5}{6}}} \right) \cdot \left( -{{\frac{5}{6}}} \right) \\ \amp = {{\frac{25}{36}}} \end{aligned}}\)
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    In Part a, parentheses have higher priority than exponents.
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  2. \(\displaystyle{\begin{aligned}[t] -\left( {{\frac{5}{6}}} \right) ^{2} \amp = -\left( {{\frac{5}{6}}} \right) \cdot \left( {{\frac{5}{6}}} \right) \\ \amp = {-{\frac{25}{36}}} \end{aligned}}\)
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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. \(\displaystyle \displaystyle{\begin{aligned}[t] -\left( -{{\frac{5}{6}}} \right) ^{2} \amp = -\left( -{{\frac{5}{6}}} \right) \cdot \left( -{{\frac{5}{6}}} \right) \\ \amp = \left( {{\frac{5}{6}}} \right) \cdot \left( -{{\frac{5}{6}}} \right) \\ \amp = {-{\frac{25}{36}}} \end{aligned}}\)
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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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\(\displaystyle \frac{\frac{2}{5} + \frac{2}{3}}{\frac{1}{4} - \frac{1}{5}} =\)
Answer 1.
\(64\)
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Answer 2.
\(3\)
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Solution.
Solution: We obtain
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\begin{equation*} \frac{\frac{2}{5} + \frac{2}{3}}{\frac{1}{4} - \frac{1}{5}} = \frac{\frac{3 \cdot 2+5 \cdot 2}{15}}{\frac{5 \cdot 1-4 \cdot 1}{20}} =\frac{\frac{16}{15}}{\frac{1}{20}} =\frac{16}{15} \times \frac{20}{1} = \frac{64}{3} \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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