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ACT Math : Fractions

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Example Question #21 : Fractions

A stadium holds a total of 13,758 seats. Of that total, 6,732 seats are full with spectators. How many seats, to the nearest percent, are currently full?

Possible Answers:

$49\%$

$39\%$

$51\%$

$43\%$

$44\%$

Correct answer:

$49\%$

Explanation:

To answer this question, we must find the percentage of the amount of seats filled up within the stadium. To find a percentage, there are two steps.

First, you divide the part of the total number by the total number itself (in this case the amount of seats filled divided by the total number of seats). So, for this data:

$\frac{part}{whole} = \frac{6,732}{13,758} = 0.489$

The second step is to multiply your answer by 100 so that it will now be represented as a percentage. So for this data:

$\frac{6,732}{13,758} = 0.489 \cdot 100 = 48.9 %$

The question asked us to round our answer to the nearest whole percent. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

$48.9% \rightarrow 49%$ $48.9 \rightarrow 49$

So our answer for the amount of seats filled within the stadium is $49\%$ .

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Example Question #1 : Complex Fractions

Which of the following is equal to $\frac{\left(7+\frac{1}{x}\right)}{\left(14+\frac{2}{x}\right)}$ ?

Possible Answers:

$\frac{1}{2}$

$\frac{x}{2}$

$\frac{1}{2x}$

$\frac{7x}{2}$

x^2+14x+2

Correct answer:

$\frac{1}{2}$

Explanation:

First we must take the numerator of our whole problem. There is a fraction in the numerator with as the denominator. Therefore, we multiply the numerator of our whole problem by $\frac{x}{x}$ , giving us $7+\frac{1}{x}=7x+1$ .

Now we look at the denominator of the whole problem, and we see that there is another fraction present with as a denominator. So now, we multiply the denominator by $\frac{x}{x}$ , giving us $14+\frac{2}{x}=14x+2$ .

Our fraction should now read $\frac{(7x+1)}{(14x+2)}$ . Now, we can factor our denominator, making the fraction $\frac{(7x+1)}{(2(7x+1))}$ .

Finally, we cancel out 7x+1 from the top and the bottom, giving us $\frac{1}{2}$ .

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Example Question #1 : Complex Fractions

Simplify: $\frac{3}{\frac{3}{2}}$

Possible Answers:

$\frac{9}{2}$

$\frac{3}{2}$

Correct answer:

Explanation:

Rewrite $\frac{3}{\frac{3}{2}}$ into the following form:

$3\div \frac{3}{2}$

Change the division sign to a multiplication sign by flipping the 2nd term and simplify.

$3\cdot \frac{2}{3}=2$

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Example Question #1 : Complex Fractions

Evaluate: $\frac{\frac{1}{x}}{x^2}$

Possible Answers:

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

$\frac{1}{x}$

$\frac{1}{x^3}$

$\frac{1}{2x^2}$

Correct answer:

$\frac{1}{x^3}$

Explanation:

The expression $\frac{\frac{1}{x}}{x^2}$ can be rewritten as:

$\frac{1}{x}\div{x^2}$

Change the division sign to a multiplication sign and take the reciprocal of the second term. Evaluate.

$\frac{1}{x}\cdot\frac{1}{x^2}=\frac{1}{x^3}$

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Example Question #4 : Complex Fractions

Simplify: $\frac{\frac{1}{6}}{3} + \frac{\frac{1}{3}}{6}$

Possible Answers:

$\frac{2}{9}$

$\frac{1}{4}$

$\frac{1}{9}$

$\frac{5}{2}$

Correct answer:

$\frac{1}{9}$

Explanation:

The expression $\frac{\frac{1}{6}}{3} + \frac{\frac{1}{3}}{6}$ can be simplified as follows:

We can simplify each fraction by multiplying the numerator by the reciprocal of the denominator.

$\frac{1}{6}\div3 \rightarrow \frac{1}{6}\cdot \frac{1}{3}=\frac{1}{9}$

$\frac{1}{3}\div 6 \rightarrow \frac{1}{3}\cdot \frac{1}{6}=\frac{1}{9}$

From here we add our two new fractions together and simplify.

$=\frac{1}{18}+\frac{1}{18}=\frac{2}{18}=\frac{1}{9}$

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Example Question #5 : Complex Fractions

Simplify the following:

$\frac{7+\frac{1}{3}}{\frac{3}{5}}$

Possible Answers:

$\frac{13}{7}$

$\frac{28}{5}$

$\frac{7}{10}$

$\frac{110}{9}$

$\frac{25}{4}$

Correct answer:

$\frac{110}{9}$

Explanation:

Begin by simplifying your numerator. Thus, find the common denominator:

$\frac{\frac{21}{3}+\frac{1}{3}}{\frac{3}{5}}$

Next, combine the fractions in the numerator:

$\frac{\frac{22}{3}}{\frac{3}{5}}$

Next, remember that to divide fractions, you multiply the numerator by the reciprocal of the denominator:

$\frac{22}{3}\times \frac{5}{3}$

Since nothing needs to be simplified, this is just:

$\frac{22}{3}\times \frac{5}{3}=\frac{110}{9}$

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Example Question #6 : Complex Fractions

Simplify,

$\frac{\frac{ab}{c} + \frac{c}{a}}{\frac{b}{c}} - a$

Possible Answers:

$\frac{b}{ac}$

$\frac{a^2bc + bc^2}{a}$

$\frac{c^2}{ab}$

$\frac{ba^2 + c}{ab}$

$\frac{ab + c - a}{c^2}$

Correct answer:

$\frac{c^2}{ab}$

Explanation:

Convert the numerators and denominators into single fractions, then simplify.

Start by finding the lowest common denominator in both the numerator and denominator of the complex fraction.

$\frac{\frac{ab}{c} + \frac{c}{a}}{\frac{b}{c}} - a$

$\frac{\frac{ab}{c}\left ( \frac{a}{a} \right ) + \frac{c}{a}\left ( \frac{c}{c}\right )}{\frac{b}{c}} - a$

$\frac{\frac{a^2b}{ac} + \frac{c^2}{ac}}{\frac{b}{c}} - a$

Add fractions with like denominators.

$\frac{\frac{a^2b + c^2}{ac}}{\frac{b}{c}} - a$

Simplify. Divide complex fractions by multiplying the numerator by the reciprocal of the denominator.

$\frac{\frac{a^2b + c^2}{ac}}{\frac{b}{c}} - a = \frac{a^2b + c^2}{ac}\cdot \frac{c}{b} - a = \frac{a^2b + c^2}{ab} - a$

Solve.

$\frac{a^2b + c^2}{ab} - a\left ( \frac{ab}{ab} \right ) = \frac{a^2b + c^2}{ab} - \frac{a^2b}{ab} = \frac{c^2}{ab}$

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Example Question #7 : Complex Fractions

Subtract: $\frac{3}{2}-\frac{2}{3}-\frac{4}{5}$

Possible Answers:

$\frac{1}{6}$

$-\frac{4}{5}$

$-\frac{1}{2}$

$\frac{1}{30}$

Correct answer:

$\frac{1}{30}$

Explanation:

The least common multiple can be found by multiplying the denominators: 2, 3, and 5. The common denominator of these numbers is 30. Multiply the numerator with what was multiplied to the denominator of each term, and then solve.

$\frac{3}{2}-\frac{2}{3}-\frac{4}{5}$

$\frac{3(3*5)}{2(3*5)}-\frac{2(2*5)}{3(2*5)}-\frac{4(2*3)}{5(2*3)}$

$\frac{45}{30}-\frac{20}{30}-\frac{24}{30}=\frac{1}{30}$

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Example Question #1 : Complex Fractions

What is $\frac{\frac{4}{9}}{2} - \frac{\frac{5}{12}}3{}$ ?

Possible Answers:

$\frac{1}{4}$

$\frac{5}{36}$

$\frac{3}{18}$

$\frac{1}{12}$

$\frac{1}{36}$

Correct answer:

$\frac{1}{12}$

Explanation:

First, simplify both sides. $\frac{\frac{4}{9}}{2}$ becomes $\frac{4}{18}$ and $\frac{\frac{5}{12}}{3}$ becomes $\frac{5}{36}$ . The LCF between $\frac{4}{18}$ and $\frac{5}{36}$ is 36. Thus, $\frac{8}{36} - \frac{5}{36} = \frac{3}{36}.$ This simplifies to $\frac{1}{12}$ .

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Example Question #1 : How To Subtract Complex Fractions

Simplify:

$\frac{4}{\frac{4-12}{6}}-\frac{\frac{3}{4}}{5}$

Possible Answers:

$\frac{-15}{4}$

$\frac{-3}{7}$

$\frac{-63}{20}$

$\frac{40}{3}$

$\frac{3}{7}$

Correct answer:

$\frac{-63}{20}$

Explanation:

Begin by simplifying the denominator of the first fraction:

$\frac{4}{\frac{4-12}{6}}-\frac{\frac{3}{4}}{5}=\frac{4}{\frac{-8}{6}}-\frac{\frac{3}{4}}{5}$

Now, remember that division of fractions is done by multiplying the numerator by the reciprocal of the denominator. Thus:

$\frac{4}{\frac{-8}{6}}-\frac{\frac{3}{4}}{5}=4*\frac{-6}{8}-\frac{3}{4}*\frac{1}{5}$

Simplify a bit:

$\frac{-3}{1}-\frac{3}{20}=\frac{-60}{20}-\frac{3}{20}=\frac{-63}{20}$

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ACT Math : Fractions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #21 : Fractions

Example Question #1 : Complex Fractions

Example Question #1 : Complex Fractions

Example Question #1 : Complex Fractions

Example Question #4 : Complex Fractions

Example Question #5 : Complex Fractions

Example Question #6 : Complex Fractions

Example Question #7 : Complex Fractions

Example Question #1 : Complex Fractions

Example Question #1 : How To Subtract Complex Fractions

All ACT Math Resources

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