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PSAT Math : Rational Expressions

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Example Questions

Example Question #1 : How To Add Rational Expressions With A Common Denominator

Simplify.

$\frac{2x+3}{x+2}+\frac{5x-6}{x+2}$

Possible Answers:

$\frac{7x-3}{x+2}$

$\frac{7x+9}{2x+4}$

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

5x-1

$\frac{5x-1}{2}$

Correct answer:

$\frac{7x-3}{x+2}$

Explanation:

$\frac{2x-3}{x+2}+\frac{5x-6}{x+2}$

Same denominator means you add straight across the numerators, keeping the denominator the same.

$\frac{2x-3+5x-6}{x+2}$

Add like terms.

$\frac{2x+5x+3-6}{x+2}$

Final Answer.

$\frac{7x-3}{x+2}$

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Example Question #1 : How To Add Rational Expressions With A Common Denominator

Simplify.

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

Possible Answers:

$\frac{x-2}{6}$

$\frac{x^{2}-16}{x+4}$

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

$\frac{x^{2}-16}{2x+8}$

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

Correct answer:

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

Explanation:

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

Check for same Denominator

$\frac{x^{2}-4+(-12)}{2x+8}$

Add like terms

$\frac{x^{2}-16}{2x+8}$

Check for GCF or if the expression can be factored

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

After factoring, divide out like terms. x+4

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

Final Answer

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Example Question #51 : Expressions

Simplify the following rational expression: (9x - 2)/(x²) MINUS (6x - 8)/(x²)

Possible Answers:

$\frac{15x+6}{x}$

$\frac{3x+6}{x^{2}}$

$\frac{12x-6}{x}$

3x-10

Correct answer:

$\frac{3x+6}{x^{2}}$

Explanation:

Since both expressions have a common denominator, x², we can just recopy the denominator and focus on the numerators. We get (9x - 2) - (6x - 8). We must distribute the negative sign over the 6x - 8 expression which gives us 9x - 2 - 6x + 8 ( -2 minus a -8 gives a +6 since a negative and negative make a positive). The numerator is therefore 3x + 6.

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Example Question #1 : Expressions

Simplify the expression.

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

Possible Answers:

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

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

$\frac{2x}{3x+6}$

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

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

Correct answer:

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

Explanation:

To add rational expressions, first find the least common denominator. Because the denominator of the first fraction factors to 2(x+2), it is clear that this is the common denominator. Therefore, multiply the numerator and denominator of the second fraction by 2.

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

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

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

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

This is the most simplified version of the rational expression.

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Example Question #1 : How To Evaluate Rational Expressions

If √(ab) = 8, and a²= b, what is a?

Possible Answers:

Correct answer:

Explanation:

If we plug in a² for b in the radical expression, we get √(a³) = 8. This can be rewritten as a^3/2 = 8. Thus, log_a8 = 3/2. Plugging in the answer choices gives 4 as the correct answer.

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Example Question #1 : How To Evaluate Rational Expressions

Function_part1

Possible Answers:

–37/15

37/15

–9/5

–11/5

9/5

Correct answer:

–11/5

Explanation:

Fraction_part2

Fraction_part3

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Example Question #1 : How To Subtract Rational Expressions With Different Denominators

Simplify.

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

Possible Answers:

x+6

3x+3

7x+3

$\frac{x^{2}+7x+3}{x^{2}}$

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

Correct answer:

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

Explanation:

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

Determine an LCD (Least Common Denominator) between and $x^{2}$ .

LCD = $x^{2}$

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

Multiply the top and bottom of the first rational expression by , so that the denominator will be $x^{2}$ .

Distribute the to x+5 .

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

Now you can subtract because both rational expressions have the same denominators.

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

Final Answer.

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

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Example Question #1 : Expressions

Which of the following is equivalent to $\dpi{100} \frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ ? Assume that denominators are always nonzero.

Possible Answers:

$x-t$

$\frac{x}{t}$

$(xt)^{-1}$

$x^{2}-t^{2}$

$t-x$

Correct answer:

$(xt)^{-1}$

Explanation:

We will need to simplify the expression $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We can think of this as a large fraction with a numerator of $\frac{1}{t}-\frac{1}{x}$ and a denominator of $\dpi{100} x-t$ .

In order to simplify the numerator, we will need to combine the two fractions. When adding or subtracting fractions, we must have a common denominator. $\frac{1}{t}$ has a denominator of $\dpi{100} t$ , and $\dpi{100} -\frac{1}{x}$ has a denominator of $\dpi{100} x$ . The least common denominator that these two fractions have in common is $\dpi{100} xt$ . Thus, we are going to write equivalent fractions with denominators of $\dpi{100} xt$ .

In order to convert the fraction $\dpi{100} \frac{1}{t}$ to a denominator with $\dpi{100} xt$ , we will need to multiply the top and bottom by $\dpi{100} x$ .

$\frac{1}{t}=\frac{1\cdot x}{t\cdot x}=\frac{x}{xt}$

Similarly, we will multiply the top and bottom of $\dpi{100} -\frac{1}{x}$ by $\dpi{100} t$ .

$\frac{1}{x}=\frac{1\cdot t}{x\cdot t}=\frac{t}{xt}$

We can now rewrite $\frac{1}{t}-\frac{1}{x}$ as follows:

$\frac{1}{t}-\frac{1}{x}$ = $\frac{x}{xt}-\frac{t}{xt}=\frac{x-t}{xt}$

Let's go back to the original fraction $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We will now rewrite the numerator:

$\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ = $\frac{\frac{x-t}{xt}}{x-t}$

To simplify this further, we can think of $\frac{\frac{x-t}{xt}}{x-t}$ as the same as $\frac{x-t}{xt}\div (x-t)$ . When we divide a fraction by another quantity, this is the same as multiplying the fraction by the reciprocal of that quantity. In other words, $a\div b=a\cdot \frac{1}{b}$ .

$\frac{x-t}{xt}\div (x-t)$ = $\frac{x-t}{xt}\cdot \frac{1}{x-t}$ $=\frac{x-t}{xt(x-t)}$ $= \frac{1}{xt}$

Lastly, we will use the property of exponents which states that, in general, $\frac{1}{a}=a^{-1}$ .

$\frac{1}{xt}=(xt)^{-1}$

The answer is $(xt)^{-1}$ .

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Example Question #1 : Expressions

Simplify (4x)/(x²– 4) * (x + 2)/(x²– 2x)

Possible Answers:

(4x²+ 8x)/(x⁴+ 8x)

x/(x – 2)²

4/(x – 2)²

x/(x + 2)

4/(x + 2)²

Correct answer:

4/(x – 2)²

Explanation:

Factor first. The numerators will not factor, but the first denominator factors to (x – 2)(x + 2) and the second denomintaor factors to x(x – 2). Multiplying fractions does not require common denominators, so now look for common factors to divide out. There is a factor of x and a factor of (x + 2) that both divide out, leaving 4 in the numerator and two factors of (x – 2) in the denominator.

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Example Question #8 : Expressions

what is 6/8 X 20/3

Possible Answers:

9/40

3/20

18/160

120/11

Correct answer: 5

Explanation:

6/8 X 20/3 first step is to reduce 6/8 -> 3/4 (Divide top and bottom by 2)

3/4 X 20/3 (cross-cancel the threes and the 20 reduces to 5 and the 4 reduces to 1)

1/1 X 5/1 = 5

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PSAT Math : Rational Expressions

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : How To Add Rational Expressions With A Common Denominator

Example Question #1 : How To Add Rational Expressions With A Common Denominator

Example Question #51 : Expressions

Example Question #1 : Expressions

Example Question #1 : How To Evaluate Rational Expressions

Example Question #1 : How To Evaluate Rational Expressions

Example Question #1 : How To Subtract Rational Expressions With Different Denominators

Example Question #1 : Expressions

Example Question #1 : Expressions

Example Question #8 : Expressions

All PSAT Math Resources

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