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PSAT Math : PSAT Mathematics

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Example Questions

Example Question #53 : Algebraic Fractions

Simplify the expression

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

Possible Answers:

$\small x - 2$

Cannot be simplified

$\small x +3$

$\small 2x$

Correct answer:

$\small x - 2$

Explanation:

The expression's numerator can be factored. Factoring the expression gives,

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

The numerator and denominator can then both be divided by $\small (x+3)$ to give,

$\small \frac{(x-2)(1)}{(1)} = x-2$

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Example Question #54 : Algebraic Fractions

Simplify the expression

$\small \frac{x^3-2x^2+9x-18}{x^2+9}$ .

Possible Answers:

Cannot be simplified

$\small x-2$

x + 2

$\small x - 3$

Correct answer:

$\small x-2$

Explanation:

The expression can be factored to give

$\small \frac{(x-2)(x^2+9)}{(x^2+9)}$ .

$\small (x^2+9)$ can then be divded from the numerator and the denominator to give,

$\small \frac{(x-2)(1)}{(1)} = x -2$

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Example Question #15 : How To Simplify A Fraction

Simplify the expression

$\small \frac{x^2-7x+10}{x^2-9x+20}$ .

Possible Answers:

$\small 2$

$\small 2x$

$\small \frac{x-2}{x-4}$

$\small \frac{x-3}{x-5}$

Correct answer:

$\small \frac{x-2}{x-4}$

Explanation:

The expression must first be factored in order to be solvable. Both the numerator and the denominator can be factored, which would give

$\small \frac{(x-2)(x-5)}{(x-5)(x-4)}$ .

$\small x-5$ can be divided from both the numerator and the denominator to give

$\small \frac{(x-2)(1)}{(1)(x-4)} = \frac{x-2}{x-4}$

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Example Question #55 : Algebraic Fractions

Simplify the expression

$\small \frac{x^3y^{-2}z^2}{xy^3z^2}$ .

Possible Answers:

$\small \frac{x^2z}{y}$

$\small \frac{x^2}{y^5}$

$\small \frac{x^2z^4}{y}$

$\small \frac{x^2}{y}$

Correct answer:

$\small \frac{x^2}{y^5}$

Explanation:

The expression can be rewritten as

$\small x^{(3-1)}y^{(-2-3)}z^{(2-2)} = x^2y^{-5}z^0 = \frac{x^2}{y^5}$

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Example Question #262 : Algebra

A train travels at a constant rate of meters per second. How many kilometers does it travel in minutes? $(1\ km=1000\ m)$

Possible Answers:

Correct answer:

Explanation:

Set up the conversions as fractions and solve:

$\dpi{100} \small \frac{20m}{1sec}\times \frac{60sec^}{1min}\times \frac{1km}{1000m}\times \frac{10min}{1}$

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Example Question #7 : Algebraic Fractions

Simplify. $\frac{4x^{4}z^{3}}{2xz^{2}}$

Possible Answers:

$2x^{4}z^{3}$

$2x^{3}z$

$\frac{2x^{4}z^{3}}{xz}$

Can't be simplified

$4x^{3}z$

Correct answer:

$2x^{3}z$

Explanation:

To simplify exponents which are being divided, subtract the exponents on the bottom from exponents on the top. Remember that only exponents with the same bases can be simplified

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Example Question #2 : How To Simplify A Fraction

Simplify:

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

Possible Answers:

$x-y,\ x+y$

x-y

x+y

2xy

Correct answer:

x-y

Explanation:

x² – y² can be also expressed as (x + y)(x – y).

Therefore, the fraction now can be re-written as (x + y)(x – y)/(x + y).

This simplifies to (x – y).

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Example Question #3 : How To Simplify A Fraction

Simplify:

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

Possible Answers:

$\sqrt{x}-\sqrt{2}$

$\sqrt{x}+\sqrt{2}$

$x-\sqrt{2}$

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

$x+\sqrt{2}$

Correct answer:

$\sqrt{x}-\sqrt{2}$

Explanation:

Notice that the $\sqrt{2}$ term appears frequently. Let's try to factor that out:

$\frac{x\sqrt{2}-2\sqrt{2}}{\sqrt{2x}+2} = \frac{\sqrt{2}*(x-2)}{\sqrt{2}*(\sqrt{x} +\sqrt{2})} = \frac{x-2}{\sqrt{x} +\sqrt{2}}$

Now multiply both the numerator and denominator by the conjugate of the denominator:

$\frac{x-2}{\sqrt{x} +\sqrt{2}} * \frac{\sqrt{x} -\sqrt{2}}{\sqrt{x} -\sqrt{2}} =\frac{(x-2)*(\sqrt{x} -\sqrt{2})}{x-2} =\sqrt{x} -\sqrt{2}$

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Example Question #4 : How To Simplify A Fraction

Simplify:

(2x + 4)/(x + 2)

Possible Answers:

x + 1

x + 4

x + 2

2x + 2

Correct answer:

Explanation:

(2x + 4)/(x + 2)

To simplify you must first factor the top polynomial to 2(x + 2). You may then eliminate the identical (x + 2) from the top and bottom leaving 2.

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Example Question #5 : How To Simplify A Fraction

Simplify the following expression:

$\frac{2x^{4}-32}{2x^{2}-8}$

Possible Answers:

$x^{2}+4$

$2(x^{2}-4)$

$x^{2}-4$

$(x+2)(x-2)$

$2$

Correct answer:

$x^{2}+4$

Explanation:

Factor both the numerator and the denominator:

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

After reducing the fraction, all that remains is:

$(x^{2}+4)$

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PSAT Math : PSAT Mathematics

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #53 : Algebraic Fractions

Example Question #54 : Algebraic Fractions

Example Question #15 : How To Simplify A Fraction

Example Question #55 : Algebraic Fractions

Example Question #262 : Algebra

Example Question #7 : Algebraic Fractions

Example Question #2 : How To Simplify A Fraction

Example Question #3 : How To Simplify A Fraction

Example Question #4 : How To Simplify A Fraction

Example Question #5 : How To Simplify A Fraction

All PSAT Math Resources

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