Rational Numbers - SSAT Upper Level Quantitative (Math)

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Question

Add:

$\frac{\frac{1}{5}+\frac{5}{6}}{\frac{1}{4}+\frac{2}{5}}+\frac{5}{6}=?$

Answer

First, add the fractions found in the numerator and denominator of the complex fraction. To add two fractions, they need to have a common denominator. You can create a common denominator by multiplying the numerator and denominator of one fraction by as a fraction made up of the denominator of the other fraction. For example, if you need to create a common denominator for $\frac{1}{3}$ and $\frac{1}{2}$ , you would multiply $\frac{1}{3}$ by $\frac{2}{2}$ and $\frac{1}{2}$ by $\frac{3}{3}$ . Since this just multiplies each fraction by , it doesn't change the fractions' values, just the way in which they are represented.

$\frac{1}{5}+\frac{5}{6}=(\frac{1}{5}\cdot \frac{6}{6})+(\frac{5}{6}\cdot\frac{5}{5})=\frac{6}{30}+\frac{25}{30}=\frac{31}{30}$

$\frac{1}{4}+\frac{2}{5}=(\frac{1}{4}\cdot\frac{5}{5})+(\frac{2}{5}\cdot\frac{4}{4})=\frac{5}{20}+\frac{8}{20}=\frac{13}{20}$

Now, divide those two fractions.

$\frac{31}{30}\div\frac{13}{20}=\frac{31}{30}\times\frac{20}{13}=\frac{62}{39}$

Finally, add $\frac{5}{6}$ :

$\frac{62}{39}+\frac{5}{6}=(\frac{62}{39}\cdot \frac{6}{6})+(\frac{5}{6}\cdot\frac{39}{39})=\frac{372}{234}+\frac{195}{234}=\frac{567}{234}\div 3=\frac{189}{78}\div 3=\frac{63}{26}$

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Question

Solve.

$\frac{6}{7x}+\frac{3}{2x^{2}}$

Answer

Find the least common denominator between the two fractions. In our case both and go into $14x^{2}$ thus, this is our common denominator.

The top is multiplied by 2x for the left fraction and to get:

$14x^{2} \rightarrow \frac{6}{7x}\cdot \frac{2x}{2x}=\frac{12x}{14x^2}$ .

For the right fraction, the top is multiplied by 7.

$\frac{3}{2x^2}\cdot \frac{7}{7}=\frac{21}{14x^2}$

To solve we need to add these two fractions together.

Final answer should read,

$\frac{12x}{14x^2}+\frac{21}{14x^2}=\frac{12x+21}{14x^2}$ .

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Question

Solve.

$\frac{6}{x^{2}+6x+9}+\frac{x-2}{x+3}$

Answer

By inspection, the denominator of the left fraction is a perfect square $\left ( x+3\right )^2$ . Since the right fraction already has a , just multiply top and bottom by so the denominators of both fraction are the same.

$\frac{6}{x^2+6x+9}+\frac{(x-2)(x+3)}{(x+3)(x+3)}$

$\frac{6}{x^2+6x+9}+\frac{x^2+3x-2x-6}{x^2+6x+9}$

The top should read with a bottom of $\left ( x+3\right )^2$ .

After simplification, the answer is revealed.

$\frac{6+x^2+x-6}{x^2+6x+9}=\frac{x^2+x}{x^2+6x+9}$

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Question

Simplify.

$\frac{1-x}{x-1}+\frac{x-1}{1-x}$

Answer

If you can recognize that $\frac{1-x}{x-1}$ is -1 then the answer is quick.

Otherwise just find the least common denominator which is .

The numerator of the new fraction will be . It may be tempting to think $\frac{2x^{2}-4x+2}{-x^2+2x-1}$ should have been the right answer, however, the question does say simplify and yes this can be simplified. If you factor the out of the numerator, it's . The bottom does look very similar to the top only the signs are flipped. This means if you factor out a on the bottom, the quadratic equation should match and cancel out leaving you with the answer of .

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Question

Solve.

$\frac{\frac{1}{2}+\frac{1}{7}}{\frac{1}{5}+\frac{1}{8}}+\frac{1}{3}$

Answer

First focus on the big fraction specificall,y the numerator. We need to find a common denominator in the numerator in order to add the two fractions and simplify the expression.

The top should have a common denominator of .

$\frac{1}{2}\cdot \frac{7}{7}+\frac{1}{7}\cdot \frac{2}{2}=\frac{7}{14}+\frac{2}{14}$

Therefore the top should become $\frac{9}{14}$ .

Now lets look at the denominator of the big fraction. We again need to find a common denominator for the two components. In this case it will be .

$\frac{1}{5}\cdot \frac{8}{8}+\frac{1}{8}\cdot \frac{5}{5}=\frac{8}{40}+\frac{5}{40}$

Therefore the bottom should become $\frac{13}{40}$ .

Since we need to have a single fraction added to a $\frac{1}{3}$ , we need to multiply top and bottom by $\frac{40}{13}$ so we can add the fractions easier.

$\frac{\frac{9}{14}}{\frac{13}{40}}=\frac{9}{14}\cdot \frac{40}{13}=\frac{360}{182}=\frac{180}{91}$

So far, the new equation to solve is $\frac{180}{91}$ $+\frac{1}{3}$ .

Same thing, find the least common denominator and solve to arrive at the final answer.

$\frac{180}{91}\cdot \frac{3}{3}+\frac{1}{3}\cdot \frac{91}{91}=\frac{540}{273}+\frac{91}{273}=\frac{631}{273}$

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Question

Solve.

$1+\frac{1}{1+\frac{1}{x}}$

Answer

Although intimidating, just focus on the inner fraction and work your way up.

Common denominator is so that should lead to

$1+\frac{1}{1\cdot \left(\frac{x}{x} \right )+\frac{1}{x}} \rightarrow$ $1+\frac{1}{\frac{x+1}{x}}$ .

By multiplying everything by $\frac{x}{x+1}$ this should lead to $1+\frac{x}{x+1}$ .

Common denominator is now and this should arrive at,

$1\cdot \frac{x+1}{x+1}+\frac{x}{x+1}=\frac{x+1+x}{x+1}=\frac{2x+1}{x+1}$

which is the final answer when everything is simplified.

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Question

Simplify.

$\frac{1}{1+\frac{1}{1+x}}+\frac{2}{2+\frac{2}{2+x}}$

Answer

Work on each fraction and start from the bottom and work your way up.

For the left fraction, it should be

$\frac{1}{\frac{x+2}{x+1}}$ since is the common denominator.

Then multiply by the inverse to get $\frac{x+1}{x+2}$ .

For the right, it's the same concept as the final fraction to get is $\frac{x+2}{x+3}$ .

Finally find the common denominator of these new complex fractions and solve to get the final answer.

$\frac{x+1}{x+2}\cdot \frac{x+3}{x+3}+\frac{x+2}{x+3}\cdot \frac{x+2}{x+2}$

$\frac{x^2+4x+3}{x^2+5x+6}+\frac{x^2+4x+4}{x^2+5x+6}$

$\frac{2x^2+8x+7}{x^2+5x+6}$

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Question

Simplify.

$\frac{1}{x}+\frac{2}{2x}+\frac{3}{3x}+\frac{4}{4x}+\frac{5}{5x}$

Answer

By careful inspection, the fractions all reduce to $\frac{1}{x}$ .

$\frac{2}{2x}=\frac{2\cdot 1}{2\cdot x}=\frac{1}{x}$

$\frac{3}{3x}=\frac{3\cdot 1}{3\cdot x}=\frac{1}{x}$

$\frac{4}{4x}=\frac{4\cdot 1}{4\cdot x}=\frac{1}{x}$

$\frac{5}{5x}=\frac{5\cdot 1}{5\cdot x}=\frac{1}{x}$

All the fractions have values in the numerator and denominator that cancel out.

There are five of these so multiply $\frac{1}{x}$ by to get the final answer.

Answer should be $\frac{5}{x}$ .

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Question

Simplify.

$\frac{x^2+5x+6}{x^2+9x+14}+\frac{x^2+6x+5}{x^2+7x+6}$

Answer

Don't try to find the least common denominator as it will take a lot of time and a definite mistake in arithmetic. Instead try to see if these quadratics can simplify.

Upon inspection, we get

$\frac{(x+2)(x+3)}{(x+2)(x+7)}+\frac{(x+1)(x+5)}{(x+1)(x+6)}$ .

Remember, to break down the quadratic equation by finding the binomials, two numbers that are factors of c must add up to make b.

With the cancellations, we should get

$\frac{x+3}{x+7}+\frac{x+5}{x+6}$ .

The common denominator will be

Remember when foiling, you multiply the numbers/variables that first appear in each binomial, followed by multiplying the outer most numbers/variables, then multiplying the inner most numbers/variables and finally multiplying the last numbers/variables. The left fraction numerator is multiplied by and the right fraction numerator is multiplied by .

Overall, the new fraction should read

$\frac{x^2+6x+3x+18+x^2+7x+5x+35}{x^2+13x+42}$ .

After simplification, answer should be $\frac{2x^2+21x+53}{x^2+13x+42}$ .

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Question

Simplify.

$\frac{2x^2+2x-4}{x^2+2x-3}+\frac{x^2-3x-10}{2x^2-8x-10}$

Answer

Try to simplify the fraction rather than finding the least common denominator. By breaking the quadratic equation into its factors, it becomes:

$\frac{2(x+2)(x-1)}{(x-1)(x+3)}+\frac{(x-5)(x+2)}{2(x-5)(x+1)}$ .

Remember, to break down the quadratic equation by finding the binomials, two numbers that are factors of c must add up to make b. Also, if the values of a, b, and c in the quadratic equation can be reduced, then factor out that divisor to make the quadratic easier to factor.

Upon cancelling, we should get

$\frac{2(x+2)}{(x+3)}+\frac{(x+2)}{2(x+1)}$ .

The common denominator is $(x+3)\left ( 2x+2\right )$ or or . Remember when foiling, you multiply the numbers/variables that first appear in each binomial, followed by multiplying the outer most numbers/variables, then multiplying the inner most numbers/variables and finally multiplying the last numbers/variables. Then multiply the numerator of left fraction by and the numerator of right fraction by and the new fraction should look like,

$\frac{4(x^2+x+2x+2)+x^2+2x+3x+6}{2x^2+8x+6}$ .

Upon distributing the four and simplifying the fraction, answer is shown.

$\frac{5x^2+17x+14}{2x^2+8x+6}$

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Question

Simplify.

$\frac{2}{x^2}+\frac{1}{x^3}$

Answer

Least common denominator is $x^{3}$ . Just multiply the left fraction numerator by and solve.

$\frac{2}{x^2}\cdot \frac{x}{x}+\frac{1}{x^3}=\frac{2x}{x^3}+\frac{1}{x^3}=\frac{2x+1}{x^3}$

Therefore the answer is $\frac{2x+1}{x^3}$ .

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Question

Simplify.

$\frac{x^2-5x}{x^2-25}+\frac{x^2+x-2}{2x^2+3x-2}$

Answer

Try to simplify the fractions rather than trying to find the least common denominator. It should look like this:

$\frac{x(x-5)}{(x-5)(x+5)}+\frac{(x-1)(x+2)}{(2x-1)(x+2)}$ .

Remember, to break down the quadratic equation by finding the binomials, two numbers that are factors of c must add up to make b. The may not be easy to factor but when put into the quadratic formula

$\left(-b\pm \frac{\sqrt{b^2-4ac}}{2a}\right)$ , we get roots of and $\frac{1}{2}$ . Then set equations equal to so that when solving for , the answer is the root.

Now after cancellations, we get $\frac{x}{x+5}+\frac{x-1}{2x-1}$ .

The common denominator is or or .

Remember when foiling, you multiply the numbers/variables that first appear in each binomial, followed by multiplying the outer most numbers/variables, then multiplying the inner most numbers/variables and finally multiplying the last numbers/variables. For the left fraction numerator you multiply by what it's lacking and same with the right fraction numerator.

The overall fraction should look like

$\frac{2x^2-x+x^2-x+5x-5}{2x^2+9x-5}$

Once simplified, the answer will be shown.

$\frac{3x^2+3x-5}{2x^2+9x-5}$

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Question

Add: $\frac{\frac{2}{3}}{\frac{1}{8}} + \frac{3}{\frac{8}{3}}$

Answer

Rewrite the fractions using a division sign.

$\frac{\frac{2}{3}}{\frac{1}{8}} + \frac{3}{\frac{8}{3}}= \left(\frac{2}{3}\div \frac{1}{8}\right)+\left(3\div \frac{8}{3}\right)$

To change the division sign to a multiplication sign, take the reciprocal of the second terms.

$\left(\frac{2}{3}\div \frac{1}{8}\right)+\left(3\div \frac{8}{3}\right)=\left(\frac{2}{3}\times 8\right)+\left(3\times \frac{3}{8}\right)$

Simplify.

$\left(\frac{2}{3}\times 8\right)+\left(3\times \frac{3}{8}\right)= \frac{16}{3}+\frac{9}{8}$

Find the LCD, and rewrite the fractions to add the numerators.

$\frac{16}{3}+\frac{9}{8} = \frac{128}{24}+ \frac{27}{24}= \frac{155}{24}$

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Question

Find the arithmetic mean of the following five numbers.

$\frac{2}{5}, \frac{1}{5},\frac{1}{2},\frac{3}{10}, \frac{2}{5}$

Answer

Add the numbers, then divide by or, equivalently, multiply by $\frac{1}{5}$ .

$\frac{\frac{2}{5}+ \frac{1}{5}+\frac{1}{2}+\frac{3}{10}+ \frac{2}{5}}{5}$

$\frac{1}{5} \left(\frac{2}{5}+ \frac{1}{5}+\frac{1}{2}+\frac{3}{10}+ \frac{2}{5}\right )$

Find the common denominator of the terms we are adding.

$\frac{1}{5} \left(\frac{4}{10}+ \frac{2}{10}+\frac{5}{10}+\frac{3}{10}+ \frac{4}{10}\right )$

$\frac{1}{5} \left(\frac{4+2+5+3+4}{10} \right)$

Multiply and simplify.

$\frac{1}{5} \left(\frac{18}{10} \right)= \frac{1}{5} \left(\frac{9}{5} \right) = \frac{9}{25}$

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Question

Solve,

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

Answer

Since the denominators for the fractions are the same, keep the denominator and add the numerators.

$\frac{2}{13}+\frac{4}{13}=\frac{2+4}{13}=\frac{6}{13}$

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Question

Scott gave $\frac{1}{4}$ of the chocolate chip cookies he made to Cindy, and he gave $\frac{1}{6}$ of the cookies to Stephanie. What fraction of his chocolate chip cookies did he give away?

Answer

This question wants you to add $\frac{1}{4}$ and $\frac{1}{6}$ . First, convert both fractions so that they share the same denominator.

$\frac{1}{4}=\frac{3}{12}$

$\frac{1}{6}=\frac{2}{12}$

$\frac{1}{4}+\frac{1}{6}=\frac{3}{12}+\frac{2}{12}=\frac{5}{12}$

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Question

A poll was conducted in a class to see what fraction of the class plays sports. $\frac{3}{7}$ of the class plays basketball, and $\frac{1}{4}$ of the class plays soccer. The rest of the class do not play any sports. What fraction of the class plays a sport?

Answer

To find what fraction of the class plays a sport, add together $\frac{3}{7}$ and $\frac{1}{4}$ .

First, convert both fractions so that the denominators are the same.

$\frac{3}{7}=\frac{12}{28}$

$\frac{1}{4}=\frac{7}{28}$

Now, you can add them together.

$\frac{3}{7}+\frac{1}{4}=\frac{12}{28}+\frac{7}{28}=\frac{19}{28}$

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Question

Peter ate $\frac{2}{3}$ of a pie for breakfast, then ate $\frac{1}{7}$ of the pie as a morning snack. How much of the pie did Peter eat?

Answer

To find how much of the pie Peter ate, you will need to add together $\frac{2}{3}$ and $\frac{1}{7}$ .

Start by converting both fractions so that the denominators are the same.

$\frac{2}{3}=\frac{14}{21}$

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

Now, you can add the fractions.

$\frac{2}{3}+\frac{1}{7}=\frac{14}{21}+\frac{3}{21}=\frac{17}{21}$

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Question

On a given week, Jeremy spends $\frac{1}{5}$ of his time working on homework and $\frac{1}{4}$ of his time doing chores. What fraction of his time is spent doing homework and doing chores?

Answer

To find how much time Jeremy spends doing his homework and his chores, add together $\frac{1}{5}$ and $\frac{1}{4}$ .

First, convert both fractions so that they have the same denominator.

$\frac{1}{5}=\frac{4}{20}$

$\frac{1}{4}=\frac{5}{20}$

Now, you can add the fractions together.

$\frac{1}{5}+\frac{1}{4}=\frac{4}{20}+\frac{5}{20}=\frac{9}{20}$

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Question

Timothy spends $\frac{2}{9}$ of his weekly allowance on comic books and $\frac{1}{5}$ of his weekly allowance on candy. What fraction of his weekly allowance does he spend on comic books and candy?

Answer

To find out how much of his weekly allowance Timothy spends on candy and comic books, add $\frac{2}{9}$ and $\frac{1}{5}$ together.

To do so, you need to first convert both fractions so that they have the same denominator.

$\frac{2}{9}=\frac{10}{45}$

$\frac{1}{5}=\frac{9}{45}$

Now you can add together the fractions.

$\frac{2}{9}+\frac{1}{5}=\frac{10}{45}+\frac{9}{45}=\frac{19}{45}$

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