How to subtract rational expressions with different denominators - GRE Quantitative Reasoning

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Question

Choose the answer which best simplifies the following expression:

$\frac{2n + 4}{n} - \frac{n^2}{5}$

Answer

To simplify this problem, multiply each term by the denominator of the other over itself:

$\frac{2n + 4}{n}\left(\frac{5}{5}\right) - \frac{n^2}{5}\left(\frac{n}{n}\right)$

$\frac{10n + 20}{5n} - \frac{n^3}{5n}$

Now that both terms share a denominator, you can subtract:

$\frac{-n^3+10n+20}{5n}$

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Question

Choose the answer which best simplifies the following expression:

$\frac{2p^2 + 3p}{2a} - \frac{5p}{3}$

Answer

To simplify this expression, first multiply both terms by the denominator of the other over itself:

$\frac{2p^2 + 3p}{2a}\left(\frac{3}{3}\right) - \frac{5p}{3}\left(\frac{2a}{2a}\right)$

$\frac{6p^2 + 9p}{6a} - \frac{10pa}{6a}$

Now that you have a common denominator, you may subtract:

$\frac{6p^2 +9p - 10pa}{6a}$

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Question

Choose the answer which best simplifies the following expression:

$\frac{2x + a}{4} - \frac{3}{2a}$

Answer

To solve this problem, first multiply each term of the original expression by the denomenator of the other over itself:

$\frac{2x + a}{4}(\frac{2a}{2a}) - \frac{3}{2a}(\frac{4}{4})$

Then you will have two terms with a common denomenator:

$\frac{4ax + 2a^2}{8a} - \frac{12}{8a}$

Combine the terms and simplify for your final answer:

$\frac{4ax +2a^2 - 12}{8a}$

$\frac{2ax + a^2 - 6}{4a}$

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Question

Choose the answer which best simplifies the expression below:

$\frac{2p^2 - 3p}{b} - \frac{b^2}{p}$

Answer

To simplify this problem, multiply each term by the denomenator of the other over itself:

$\frac{2p^2 - 3p}{b}(\frac{p}{p}) - \frac{b^2}{p}(\frac{b}{b})$

Then you will yield terms with a like denomenator, which can be combined:

$\frac{2p^3 - 3p^2}{pb} - \frac{b^3}{pb}$

$\frac{2p^3 - 3p^2 - b^3}{pb}$

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Question

Choose the answer which best simplifies the expression below:

$\frac{xy^2}{m} - \frac{m^2n}{y}$

Answer

To simplify the expression, first multiply each term by the denomenator of the other over itself:

$\frac{xy^2}{m}(\frac{y}{y}) - \frac{m^2n}{y}(\frac{m}{m})$

Then you yield terms with common denomenators, which can be combined:

$\frac{xy^3}{my}-\frac{m^3n}{my}$

$\frac{xy^3 - m^3n}{my}$

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Question

Choose the answer which best simplifies the following expression:

$\frac{x^2 + x + 3}{4a}-\frac{2y^2}{7}$

Answer

To simplify, first multiply each of the terms by the denomenator of the other over itself:

$\frac{x^2 + x + 3}{4a}(\frac{7}{7})-\frac{2y^2}{7}(\frac{4a}{4a})$

Then you will get terms with a common denomenator, which can be combined:

$\frac{7x^2 + 7x + 21}{28a}-\frac{8y^2a}{28a}$

$\frac{7x^2 +7x +21-8y^2a}{28a}$

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Question

Choose the answer which best simplifies the expression below:

$\frac{2n^2 - n}{3a^2}-\frac{-n^2}{b}$

Answer

To simplify, first multiply each of the terms by the denomenator of the other over itself:

$\frac{2n^2 - n}{3a^2}(\frac{b}{b})-\frac{-n^2}{b}(\frac{3a^2}{3a^2})$

You will yield terms with a common denomenator, which can be combined:

$\frac{2n^2b - nb}{3a^2b}-\frac{-3a^2n^2}{3a^2b}$

$\frac{2n^2b-nb+3a^2n^2}{3a^2b}$

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Question

Choose the answer which best simplifies the expression below:

$\frac{2a}{14-y} - \frac{3}{8 +a}$

Answer

To simplify, multiply each of the terms by the denomenator of the other, over itself:

$\frac{2a}{14-y}(\frac{8 +a}{8+a}) - \frac{3}{8 +a}(\frac{14-y}{14-y})$

You will yield terms with a common denomenator, which can be combined:

$\frac{16a +2a^2}{112-8y+14a-ay}-\frac{42-3y}{112-8y+14a-ay}$

$\frac{2a^2 +16a +3y -42}{112 - 8y+14a -ay}$

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