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?

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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}$=?$

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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}$=?$

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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}$=?$

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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}$=?$

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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}$=?$

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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}$=?$

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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}$=?$

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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}$=?$

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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}$=?$

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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 $\frac{1}{7}$

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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 )$

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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 $\frac{2}{3}$ .

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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}$)$ .

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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$\frac{10}{20}$ .

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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?

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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?

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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?

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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?

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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?

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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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