Operations with Fractions - SSAT Upper Level Quantitative

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Question

Janice had $\frac{3}{4}$ of a cup of milk. If she used $\frac{1}{12}$ cup to make some scrambled eggs, how much of a cup of milk does she have left?

Answer

Subtract the two fractions. To subtract them, they must have a common denominator. Multiply the numerator and denominator of $\frac{3}{4}$ by to create a common denominator of , then subtract the numerators and reduce the resulting fraction.

$\frac{3}{4}-\frac{1}{12}=\frac{9}{12}-\frac{1}{12}=\frac{8}{12}=\frac{2}{3}$

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Question

What is the result of this operation?

$\frac{3}{4}-\frac{1}{4}=?$

Answer

Since the denominators are exactly the same, we can just subtract the tops.

So

$\frac{3}{4}-\frac{1}{4}=\frac{2}{4}$

By reducing we get

$\frac{1}{2}$

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Question

What is the result of this operation?

$\frac{2}{7}-\frac{1}{7}=?$

Answer

Since the denominators are the same, we can just subtract the numerators.

So

$\frac{2}{7}-\frac{1}{7}=\frac{1}{7}$

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Question

What is the result of this operation?

$\frac{4}{9}-\frac{1}{9}=?$

Answer

Since the denominators are the same, we can just subtract the numerators.

$\frac{4}{9}-\frac{1}{9}=\frac{3}{9}$

By reducing our result, we get

$\frac{1}{3}$

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Question

What is the result of this operation?

$\frac{2}{9}-\frac{2}{9}=?$

Answer

Since the denominators are the same, we can just subtract the numerators.

$\frac{2}{9}-\frac{2}{9}=\frac{0}{9}$

The result can just be reduced to

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Question

What is the result of this operation?

$\frac{1}{6}-\frac{1}{3}=?$

Answer

Since the denominators are different, we need to transform them into the same denominator.

So

$\frac{1}{6}-\frac{1}{3}=?$

becomes

$\frac{1}{6}-\frac{12}{32}=?$

$\frac{1}{6}-\frac{2}{6}=?$

Now subtract the numerators

$\frac{1}{6}-\frac{2}{6}=\frac{-1}{6}$

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Question

What is the result of the operation

$\frac{4}{5}-\frac{3}{5}=?$

Answer

Since the denominators are the exact same, we can just subtract the numerators.

$\frac{4}{5}-\frac{3}{5}=\frac{1}{5}$

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Question

What is the result of this operation

$\frac{4}{13}-\frac{2}{13}=?$

Answer

Since the denominators are the exact same, we can just subtract the numerators.

$\frac{4}{13}-\frac{2}{13}=\frac{2}{13}$

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Question

What is the result of this operation

$\frac{6}{6}-\frac{3}{6}=?$

Answer

Since the denominators are the same, we can just subtract the numerators.

$\frac{6}{6}-\frac{3}{6}=\frac{3}{6}$

which we can reduce to

$\frac{1}{2}$

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Question

What is the result of this operation

$\frac{2}{7}-\frac{3}{7}=?$

Answer

Since the denominators are the same, we can just subtract the numerators.

$\frac{2}{7}-\frac{3}{7}=\frac{-1}{7}$

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Question

Multiply these fractions:

$\frac{2}{3}\cdot \frac{1}{7}$

Answer

To multiply the fractions, simply multiply the numerators together and the denominators together.

$\frac{2}{3}\cdot \frac{1}{7}=\frac{2\cdot 1}{3\cdot 7}=\frac{2}{21}$

Since this fraction is in its simplest form, that is the final answer.

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Question

Simplify the expression.

$\left ( {\frac{1}{2} + \frac{1}{4}}\right ) * \left ({\frac{1}{2} - \frac{1}{4}}\right )$

Answer

$\left ( {\frac{1}{2} + \frac{1}{4}}\right ) * \left ({\frac{1}{2} - \frac{1}{4}}\right )$

Start with the terms in parentheses, using the order of operations. Find the least common denominator.

$\left ( {\frac{1* 2}{2* 2} + \frac{1}{4}}\right ) \left ({\frac{1 2}{22} - \frac{1}{4}}\right )$

$\left ( {\frac{2}{4} + \frac{1}{4}}\right ) \left ({\frac{ 2}{4} - \frac{1}{4}}\right )$

Add the fractions in parentheses.

$\left ( {\frac{2+1}{4} } \right ) * \left ({\frac{2-1}{4} } \right )$

Multiply the fractions and simplify.

${\frac{3}{4} } {\frac{1}{4} } = \frac{3 1}{4 * 4} = \frac{3}{16}$

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Question

Simplify the expression $\frac{9}{10}\times \frac{2}{3}$ .

Answer

Beginning with $\frac{9}{10}\times \frac{2}{3}$ , we multiply across both numerators and denominators for both terms:

$=\frac{9\times 2}{10\times 3}$

$=\frac{18}{30}$

Reducing the fraction to its simplest form by dividing by the Greatest Common Factor:

$\frac{18}{30}$

$=\frac{18\div 6}{30\div 6}$

$=\frac{3}{5}$

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Question

Simplify the expression $\frac{4}{3}\times(-\frac{3}{4})$ .

Answer

Multiplying across the numerators and denominators of both terms and simplifying:

$\frac{4}{3}\times(-\frac{3}{4})=-\frac{12}{12}=-1$

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Question

Simplify the expression $\frac{12}{5}\times\frac{10}{20}$ .

Answer

$\frac{12}{5}\times\frac{10}{20}$

First, multiply straight across. Multiply the numerators together and then multiply the denominators together.

$=\frac{12\times 10}{5\times 20}$

$=\frac{120}{100}$

From here, simplify the expression by factoring out the Greatest Common Factor.

In this case the Greatest Common Factor is 20.

$=\frac{120\div 20}{100\div 20}$

$=\frac{6}{5}$

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Question

Steven runs a pizza shop. On Monday, he used $\frac{4}{5}$ of a bag of flour. On Tuesday, he used $\frac{1}{6}$ the amount he used on Monday. How much of a bag of flour did he use on Tuesday?

Answer

To find how many bags of flour he used on Tuesday, multiply $\frac{4}{5}$ by $\frac{1}{6}$ .

To multiply fractions, multiply the numerators together, then multiply the denominators together.

$\frac{4}{5}\times \frac{1}{6}=\frac{4\times1}{5\times6}=\frac{4}{30}=\frac{2}{15}$

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Question

In Mary's class, $\frac{1}{12}$ of the class owns a dog, and of those in the class that own a dog, $\frac{2}{7}$ also owns a cat. What fraction of the class owns both a dog and a cat?

Answer

Multiply $\frac{1}{12}$ by $\frac{2}{7}$ to find out how much of the class have both a cat and a dog.

$\frac{1}{12}\times \frac{2}{7}=\frac{1\times2}{12\times7}=\frac{2}{84}=\frac{1}{42}$

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Question

In the morning, Tina ate $\frac{1}{4}$ of a cake. For her afternoon snack she ate $\frac{5}{6}$ as much as she did in the morning. What fraction of the cake did Tina eat for her afternoon snack?

Answer

Multiply $\frac{1}{4}$ by $\frac{5}{6}$ to find how much cake Tina ate in the afternoon.

$\frac{1}{4}\times\frac{5}{6}=\frac{5\times1}{4\times6}=\frac{5}{24}$

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Question

A certain recipe for a batch of chocolate cookies calls for $\frac{9}{10}$ of a cup of sugar to be used. If Mike only wants to make $\frac{3}{4}$ of a batch, how much of a cup of sugar should he use?

Answer

Multiply $\frac{9}{10}$ by $\frac{3}{4}$ to find out how much sugar he needs to use.

$\frac{9}{10}\times \frac{3}{4}=\frac{9\times3}{10\times4}=\frac{27}{40}$

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Question

At an arcade, $\frac{1}{6}$ of the games are fighting games. Of those games, $\frac{3}{7}$ are Japanese games. What fraction of the games at the arcade are Japanese fighting games?

Answer

To find what fraction of the games are Japanese fighting ones, multiply $\frac{1}{6}$ by $\frac{3}{7}$ .

$\frac{1}{6}\times \frac{3}{7}=\frac{1\times3}{6\times7}=\frac{3}{42}=\frac{1}{14}$

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