Q: Which of the following expressions is equivalent to $$ \frac{\frac{x+2}{8}}{\frac{y-1}{9}} $$?

$$\frac{x+2}{8} \times \frac{9}{y-1}$$

Q: Which of the following is NOT equivalent to $$ \frac{\frac{x+2}{8}}{\frac{y-1}{9}} $$?

$$\frac{x+2}{y-1} \times \frac{9}{8}$$

Q: Given the expression $$ \frac{\frac{m+1}{n}}{\frac{p-2}{q}} $$, which of the following is equivalent?

$$\frac{m+1}{n} \times \frac{q}{p-2}$$

Question 1

Simplify the following complex fraction: 
$$
\frac{\frac{2}{5-x} + \frac{3}{x-5}}{\frac{2}{x} + \frac{3}{x-5}}
$$

Accepted Answer

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

Question 2

Which of the following expressions is equivalent to 
$$
\frac{\frac{x+2}{8}}{\frac{y-1}{9}}
$$?

Accepted Answer

$$\frac{x+2}{8} 	imes \frac{9}{y-1}$$

Question 3

If $$x = 4$$ and $$y = 2$$, what is the value of 
$$
\frac{\frac{x+2}{8}}{\frac{y-1}{9}}
$$?

Accepted Answer

$$\frac{6}{8} \div \frac{1}{9} = \frac{27}{4}$$

Question 4

Which of the following is NOT equivalent to 
$$
\frac{\frac{x+2}{8}}{\frac{y-1}{9}}
$$?

Accepted Answer

$$\frac{x+2}{y-1} 	imes \frac{9}{8}$$

Question 5

A chemist is mixing two solutions. The first solution is represented by $$\frac{2}{5-x}$$ liters and the second by $$\frac{3}{x-5}$$ liters. If the total mixture is divided by $$\frac{2}{x} + \frac{3}{x-5}$$, what is the simplified form of the resulting expression?

Accepted Answer

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

Question 6

Which of the following is the correct first step to simplify the complex fraction 
$$
\frac{\frac{a}{b} + \frac{c}{d}}{\frac{e}{f} + \frac{g}{h}}
$$?

Accepted Answer

Find a common denominator for both the numerator and denominator.

Question 7

A medical dosage formula is given by 
$$
\frac{\frac{d}{w}}{\frac{t}{h}}
$$, where $$d is $$the drug amount, $$w is $$weight, $$t is $$time, and $$h is $$hours. What is the simplified form?

Accepted Answer

$$\frac{d h}{w t}$$

Question 8

Which of the following is a common mistake when simplifying 
$$
\frac{\frac{a}{b}}{\frac{c}{d}}
$$?

Accepted Answer

Multiplying $$a by$$ $$c$$ and $$b by$$ $$d$$ instead of multiplying by the reciprocal

Question 9

If $$x = 5$$, what is the value of the denominator in the complex fraction 
$$
\frac{\frac{2}{5-x} + \frac{3}{x-5}}{\frac{2}{x} + \frac{3}{x-5}}
$$?

Accepted Answer

Undefined

Question 10

A recipe calls for a ratio of ingredients given by 
$$
\frac{\frac{a+2}{b}}{\frac{c-1}{d}}
. If$$ $$a=4$$, $$b=2$$, $$c=5$$, and $$d=3$$, what is the simplified value?

Accepted Answer

$$\frac{6}{2} \div \frac{4}{3} = \frac{9}{2}$$

Question 11

Which of the following best describes the process of simplifying a complex fraction?

Accepted Answer

Multiply the numerator and denominator by the least common denominator (LCD) of all fractions involved.

Question 12

Given the expression 
$$
\frac{\frac{m+1}{n}}{\frac{p-2}{q}}
$$, which of the following is equivalent?

Accepted Answer

$$\frac{m+1}{n} 	imes \frac{q}{p-2}$$

Question 13

If a student simplifies 
$$
\frac{\frac{a}{b} + \frac{c}{d}}{\frac{e}{f} + \frac{g}{h}}
 by $$only adding the numerators and denominators, what mistake have they made?

Accepted Answer

They ignored the need for common denominators.

Question 14

Which of the following is the correct way to rewrite 
$$
\frac{\frac{x+2}{8}}{\frac{y-1}{9}}
 as a $$single fraction?

Accepted Answer

$$\frac{(x+2) \cdot 9}{8(y-1)}$$

Question 15

A student claims that 
$$
\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d}
. Is $$this correct?

Accepted Answer

Yes, because dividing by a fraction is the same as multiplying by its reciprocal.

Intermediate Algebra Study Guide: Complex Fractions & Equivalent Expressions

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