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SSAT Upper Level Math : Rational Numbers

Study concepts, example questions & explanations for SSAT Upper Level Math

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All SSAT Upper Level Math Resources

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Example Questions

Example Question #71 : Rational Numbers

Simplify.

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

Possible Answers:

$\frac{x^2+4x+3}{x^2+13x+42}$

$\frac{2x^2+21x+53}{x^2+13x+42}$

$\frac{x^2+3x+2}{x^2+13x+42}$

$\frac{x^2+8x+15}{x^2+13x+42}$

$\frac{2x^2+8x+15}{x^2+13x+42}$

Correct answer:

$\frac{2x^2+21x+53}{x^2+13x+42}$

Explanation:

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 (x+6)(x+7), x^2+7x+6x+42 , x^2 + 13x+42.

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 x+6 and the right fraction numerator is multiplied by x+7 .

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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Example Question #22 : Complex Fractions

Simplify.

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

Possible Answers:

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

None of the above are correct.

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

$\frac{5x^2+20x+14}{x^2+4x+3}$

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

Correct answer:

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

Explanation:

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 (2x^2+2x+6x+6) or 2x^2+8x+6 . 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 2(x+1) and the numerator of right fraction by (x+3) 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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Example Question #23 : Complex Fractions

Simplify.

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

Possible Answers:

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

$\frac{2x+1}{x^4}$

$\frac{3}{x^5}$

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

Correct answer:

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

Explanation:

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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Example Question #71 : Fractions

Simplify.

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

Possible Answers:

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

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

$\frac{20x^2}{x^2+4x+9}$

$\frac{3x^2+4x-5}{2x^2+9x-5}$

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

Correct answer:

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

Explanation:

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 2x^2+3x-2 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 (x+5)(2x-1) or 2x^2-x+10x-5 or 2x^2+9x-5 .

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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Example Question #133 : Number Concepts And Operations

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

Possible Answers:

$\frac{97}{12}$

$\frac{40}{3}$

$\frac{29}{24}$

$\frac{155}{24}$

$\frac{131}{24}$

Correct answer:

$\frac{155}{24}$

Explanation:

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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Example Question #71 : Rational Numbers

Simplify:

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

Possible Answers:

$\frac{3}{10}$

$\frac{13}{12}$

$\frac{52}{75}$

$\frac{1}{5}$

$\frac{3}{7}$

Correct answer:

$\frac{13}{12}$

Explanation:

Add the fractions in the numerator, then divide the sum by the denominator:

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

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

$=\frac{\frac{10}{15} + \frac{3}{15}}{\frac{4}{5}}$

$=\frac{\frac{10+3}{15} }{\frac{4}{5}} = \frac{\frac{13}{15} }{\frac{4}{5}}$

$= \frac{13}{15} \div \frac{4}{5} = \frac{13}{15} \times \frac{5}{4} = \frac{13}{3} \times \frac{1}{4} =\frac{13}{12}$

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Example Question #72 : Rational Numbers

Simplify:

$\frac{\frac{1}{3} + \frac{3}{10}}{\frac{9}{10}}$

Possible Answers:

$\frac{1}{27}$

$\frac{2}{3}$

$\frac{19}{27}$

$\frac{40}{117}$

$\frac{57}{100}$

Correct answer:

$\frac{19}{27}$

Explanation:

Simplify the sum in the numerator, then divide by the denominator:

$\frac{\frac{1}{3} + \frac{3}{10}}{\frac{9}{10}}$

$=\frac{\frac{1\cdot 10}{3 \cdot 10} + \frac{3 \cdot 3}{10 \cdot 3}}{\frac{9}{10}}$

$=\frac{\frac{10}{30} + \frac{9}{30}}{\frac{9}{10}}$

$=\frac{\frac{10+9}{30} }{\frac{9}{10}} = \frac{\frac{19}{30} }{\frac{9}{10}}$

$\frac{19}{30} \div \frac{9}{10} = \frac{19}{30}\times \frac{10}{9} = \frac{19}{3}\times \frac{1}{9} = \frac{19}{27}$

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Example Question #73 : Rational Numbers

Simplify:

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

Possible Answers:

$\frac{2}{5}$

$\frac{5}{18}$

$\frac{3}{5}$

$\frac{4}{15}$

$\frac{1}{4}$

Correct answer:

$\frac{5}{18}$

Explanation:

Rewrite this as a division problem, then solve:

$\frac{1\frac{1}{3}}{4 \frac{4}{5}} = 1\frac{1}{3} \div 4 \frac{4}{5} = \frac{4}{3} \div \frac{24}{5}= \frac{4}{3} \times \frac{5}{24}= \frac{1}{3} \times \frac{5}{6}= \frac{5}{18}$

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Example Question #74 : Rational Numbers

Simplify:

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

Possible Answers:

$6\frac{1}{4}$

$3 \frac{3}{4}$

$4 \frac{1}{2}$

Correct answer:

Explanation:

Rewrite this as a division problem, then solve:

$\frac{5}{1\frac{1}{4}} = 5 \div 1\frac{1}{4}= \frac{5}{1}\div \frac{5}{4} = \frac{5}{1}\times \frac{4}{5}= \frac{1}{1}\times \frac{4}{1} = 4$

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Example Question #75 : Rational Numbers

Simplify:

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

Possible Answers:

$\frac{5}{8}$

$\frac{7}{10}$

$\frac{5}{9}$

$\frac{3}{4}$

$\frac{2}{3}$

Correct answer:

$\frac{2}{3}$

Explanation:

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

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

$= \frac{19}{5} \div \left ( \frac{7}{2}+ \frac{11}{5} \right )$

$=\frac{19}{5} \div \left ( \frac{35}{10}+ \frac{22}{10} \right )$

$=\frac{19}{5} \div \frac{57}{10}$

$=\frac{19}{5} \times \frac{10}{57}$

$=\frac{1 }{1} \times \frac{2}{3} = \frac{2}{3}$

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SSAT Upper Level Math : Rational Numbers

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #71 : Rational Numbers

Example Question #22 : Complex Fractions

Example Question #23 : Complex Fractions

Example Question #71 : Fractions

Example Question #133 : Number Concepts And Operations

Example Question #71 : Rational Numbers

Example Question #72 : Rational Numbers

Example Question #73 : Rational Numbers

Example Question #74 : Rational Numbers

Example Question #75 : Rational Numbers

All SSAT Upper Level Math Resources

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