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PSAT Math : Complex Fractions

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Example Question #1 : Complex Fractions

Simplify:

$\frac{1- \frac{2}{x}}{1+ \frac{2}{x^{2}}}$ $\displaystyle \frac{1- \frac{2}{x}}{1+ \frac{2}{x^{2}}}$

Possible Answers:

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

$\frac{ x^{2}-2 }{ x^{2}+2}$ $\displaystyle \frac{ x^{2}-2 }{ x^{2}+2}$

$\frac{ x^{2}+2x}{ x^{2}+2}$ $\displaystyle \frac{ x^{2}+2x}{ x^{2}+2}$

$\frac{x-2}{x+2}$ $\displaystyle \frac{x-2}{x+2}$

$\frac{ x^{2}-2x}{ x^{2}+2}$ $\displaystyle \frac{ x^{2}-2x}{ x^{2}+2}$

Correct answer:

$\frac{ x^{2}-2x}{ x^{2}+2}$ $\displaystyle \frac{ x^{2}-2x}{ x^{2}+2}$

Explanation:

Simplify the numerator and the denominator, then divide, as follows:

$\frac{1- \frac{2}{x}}{1+ \frac{2}{x^{2}}}$ $\displaystyle \frac{1- \frac{2}{x}}{1+ \frac{2}{x^{2}}}$

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

$=\frac{\frac{ x}{x} - \frac{2}{x}}{\frac{x^{2}}{x^{2}} + \frac{2}{x^{2}}}$ $\displaystyle =\frac{\frac{ x}{x} - \frac{2}{x}}{\frac{x^{2}}{x^{2}} + \frac{2}{x^{2}}}$

$=\frac{\frac{ x-2}{x} }{\frac{x^{2}+2}{x^{2}} }$ $\displaystyle =\frac{\frac{ x-2}{x} }{\frac{x^{2}+2}{x^{2}} }$

$= \frac{ x-2}{x} \div \frac{x^{2}+2}{x^{2}}$ $\displaystyle = \frac{ x-2}{x} \div \frac{x^{2}+2}{x^{2}}$

$= \frac{ x-2}{x} \cdot \frac{x^{2}}{x^{2}+2}$ $\displaystyle = \frac{ x-2}{x} \cdot \frac{x^{2}}{x^{2}+2}$

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

$= \frac{ x^{2}-2x}{ x^{2}+2}$ $\displaystyle = \frac{ x^{2}-2x}{ x^{2}+2}$

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Example Question #82 : Fractions

Evaluate:

$\frac{3+\frac{1}{4}}{3-\frac{1}{4}}$ $\displaystyle \frac{3+\frac{1}{4}}{3-\frac{1}{4}}$

Possible Answers:

$\frac{132}{16}$ $\displaystyle \frac{132}{16}$

$\frac{11}{13}$ $\displaystyle \frac{11}{13}$

$\displaystyle 2$

$\frac{13}{11}$ $\displaystyle \frac{13}{11}$

$\frac{16}{132}$ $\displaystyle \frac{16}{132}$

Correct answer:

$\frac{13}{11}$ $\displaystyle \frac{13}{11}$

Explanation:

Simplify the numerator and the denominator, then divide, as follows:

$\frac{3+\frac{1}{4}}{3-\frac{1}{4}}$ $\displaystyle \frac{3+\frac{1}{4}}{3-\frac{1}{4}}$

$=\frac{ \frac{12}{4}+\frac{1}{4}}{\frac{12}{4}-\frac{1}{4}}$ $\displaystyle =\frac{ \frac{12}{4}+\frac{1}{4}}{\frac{12}{4}-\frac{1}{4}}$

$=\frac{ \frac{13}{4}}{\frac{11}{4} }$ $\displaystyle =\frac{ \frac{13}{4}}{\frac{11}{4} }$

$= \frac{13}{4}\div \frac{11}{4}$ $\displaystyle = \frac{13}{4}\div \frac{11}{4}$

$= \frac{13}{4}\cdot \frac{4}{11}$ $\displaystyle = \frac{13}{4}\cdot \frac{4}{11}$

$= \frac{13}{1}\cdot \frac{1}{11}$ $\displaystyle = \frac{13}{1}\cdot \frac{1}{11}$

$= \frac{13}{11}$ $\displaystyle = \frac{13}{11}$

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Example Question #1 : Complex Fractions

Solve:

$76-\frac{75}{6}=$ $\displaystyle 76-\frac{75}{6}=$

Possible Answers:

$\frac{127}{2}$ $\displaystyle \frac{127}{2}$

$\displaystyle 75$

$\displaystyle 74$

$\frac{1}{6}$ $\displaystyle \frac{1}{6}$

$\frac{7}{6}$ $\displaystyle \frac{7}{6}$

Correct answer:

$\frac{127}{2}$ $\displaystyle \frac{127}{2}$

Explanation:

First reduce the fraction. We can divide both the numerator and the denominator by 3.

$\frac{75\div 3}{6\div 3}=\frac{25}{2}$ $\displaystyle \frac{75\div 3}{6\div 3}=\frac{25}{2}$

Now our expression looks like this:

$76-\frac{25}{2}$ $\displaystyle 76-\frac{25}{2}$

When you add or subtract fractions, you need to have the same denominator. The lowest common deonminator here is 2. So we need to multiply and solve:

$\left ( \frac{2}{2}\right)\cdot76-\frac{25}{2}=\frac{152}{2}-\frac{25}{2}=\frac{127}{2}$ $\displaystyle \left ( \frac{2}{2}\right)\cdot76-\frac{25}{2}=\frac{152}{2}-\frac{25}{2}=\frac{127}{2}$

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PSAT Math : Complex Fractions

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

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Example Question #1 : Complex Fractions

Example Question #82 : Fractions

Example Question #1 : Complex Fractions

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