GRE Math : Simplifying Fractions

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Example Question #1 : Simplifying Fractions

For how many integers, $\displaystyle x$, between 26 and 36 is it true that

$\frac{3}{x}_{,}\frac{4}{x}_{,}$ $\displaystyle \frac{3}{x}_{,}\frac{4}{x}_{,}$ and $\frac{5}{x}$ $\displaystyle \frac{5}{x}$ are all in lowest terms?

Possible Answers:

$\displaystyle 1$

$\displaystyle 4$

$\displaystyle 2$

$\displaystyle 3$

$\displaystyle 5$

Correct answer:

$\displaystyle 2$

Explanation:

If $\displaystyle x$ is even, then $\frac{4}{x}$ $\displaystyle \frac{4}{x}$ is not in lowest terms, since both $\displaystyle x$ and 4 are divisible by 2. Therefore, the only possibilities are 27,29,31,33, and 35. But $\frac{3}{27}= \frac{1}{9}$ $\displaystyle \frac{3}{27}= \frac{1}{9}$, $\frac{3}{33}= \frac{1}{11}$ $\displaystyle \frac{3}{33}= \frac{1}{11}$, and $\frac{5}{35}= \frac{1}{7}$ $\displaystyle \frac{5}{35}= \frac{1}{7}$, so only two integers satisfy the given condition: 29 and 31.

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Example Question #1 : Simplifying Fractions

Mrs. Lawrence's class has $\displaystyle 35$ students, $\displaystyle 14$ of which are girls. If you were to choose a student at random, what's the probability that the student chosen would be a boy?

Possible Answers:

$\frac{2}{5}$ $\displaystyle \frac{2}{5}$

$\frac{3}{5}$ $\displaystyle \frac{3}{5}$

$\frac{2}{7}$ $\displaystyle \frac{2}{7}$

Correct answer:

$\frac{3}{5}$ $\displaystyle \frac{3}{5}$

Explanation:

In order to find out the probability of choosing a boy, you must first find out how many boys there are. Since there are $\displaystyle 14$ girls out of $\displaystyle 35$ students, $\displaystyle 21$ students should be boys.

Therefore, the probability of choosing a boy is,

$\frac{21}{35}$ $\displaystyle \frac{21}{35}$.

However, this is not one of the answer choices; therefore, you must reduce the fraction.

In order to reduce a fraction, you have to find their GCM, or greatest common multiple. This is the biggest number that will go into both the numerator (21) $\displaystyle (21)$ and denominator (35) $\displaystyle (35)$. The largest number is $\displaystyle 7$. Divide both the top and bottom by $\displaystyle 7$, and you will get the answer:

$\frac{3}{5}$ $\displaystyle \frac{3}{5}$

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Example Question #1 : Simplifying Fractions

Simplify the fraction:

$\frac{0.0003(0.4)(0.0002)}{0.00000006(0.01)}$ $\displaystyle \frac{0.0003(0.4)(0.0002)}{0.00000006(0.01)}$

Possible Answers:

$\displaystyle 4$

0.04 $\displaystyle 0.04$

$\displaystyle 40$

0.4 $\displaystyle 0.4$

Correct answer:

$\displaystyle 40$

Explanation:

To simplify the fraction

$\frac{0.0003(0.4)(0.0002)}{0.00000006(0.01)}$ $\displaystyle \frac{0.0003(0.4)(0.0002)}{0.00000006(0.01)}$

It may be helpful to write it in terms of scientific notation:

$\frac{3\cdot 10^{-4}(4\cdot 10^{-1})(2\cdot 10^{-4})}{6\cdot 10^{-8}(1\cdot 10^{-2})}$ $\displaystyle \frac{3\cdot 10^{-4}(4\cdot 10^{-1})(2\cdot 10^{-4})}{6\cdot 10^{-8}(1\cdot 10^{-2})}$

$\frac{24\cdot 10^{-4-1-4}}{6\cdot 10^{-8-2}}$ $\displaystyle \frac{24\cdot 10^{-4-1-4}}{6\cdot 10^{-8-2}}$

$\frac{24\cdot 10^{-9}}{6\cdot 10^{-10}}$ $\displaystyle \frac{24\cdot 10^{-9}}{6\cdot 10^{-10}}$

$4\cdot10^{-9+10}$ $\displaystyle 4\cdot10^{-9+10}$

$4\cdot10$ $\displaystyle 4\cdot10$

$\displaystyle 40$

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Example Question #1063 : Gre Quantitative Reasoning

Simply the fraction:

$\frac{4a^3+12a^2b+12ab^2+4b^3}{3a^2+6ab+3b^2}$ $\displaystyle \frac{4a^3+12a^2b+12ab^2+4b^3}{3a^2+6ab+3b^2}$

Possible Answers:

$\frac{4}{3}(a+b)$ $\displaystyle \frac{4}{3}(a+b)$

$\frac{4}{3(a+b)}$ $\displaystyle \frac{4}{3(a+b)}$

$\frac{3}{4(a+b)}$ $\displaystyle \frac{3}{4(a+b)}$

$\frac{3}{4}(a+b)$ $\displaystyle \frac{3}{4}(a+b)$

Correct answer:

$\frac{4}{3}(a+b)$ $\displaystyle \frac{4}{3}(a+b)$

Explanation:

To simplify the fraction

$\frac{4a^3+12a^2b+12ab^2+4b^3}{3a^2+6ab+3b^2}$ $\displaystyle \frac{4a^3+12a^2b+12ab^2+4b^3}{3a^2+6ab+3b^2}$

Begin by factoring out common terms in the numerator and denominator:

$\frac{4(a^3+3a^2b+3ab^2+b^3)}{3(a^2+2ab+b^2)}$ $\displaystyle \frac{4(a^3+3a^2b+3ab^2+b^3)}{3(a^2+2ab+b^2)}$

Now notice that the coefficients for the polynomials in the numerator and denominator match the form of Pascal's Triangle:

(a+b)^n $\displaystyle (a+b)^n$

$\displaystyle n=1: 1,1$

$\displaystyle n=2:1,2,1$

$\displaystyle n=3:1,3,3,1$

$\displaystyle n=4:1,4,6,4,1$

The fraction can be rewritten as:

$\frac{4(a+b)^3}{3(a+b)^2}$ $\displaystyle \frac{4(a+b)^3}{3(a+b)^2}$

And finally reduced to

$\frac{4}{3}(a+b)$ $\displaystyle \frac{4}{3}(a+b)$

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GRE Math : Simplifying Fractions

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #1 : Simplifying Fractions

Example Question #1 : Simplifying Fractions

Example Question #1 : Simplifying Fractions

Example Question #1063 : Gre Quantitative Reasoning

All GRE Math Resources

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