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ACT Math : Operations and Fractions

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Example Question #1 : How To Subtract Fractions

Simplify:

$\frac{1}{4}-\frac{x-2}{5}-\frac{2x+7}{20}$

Possible Answers:

$\frac{2-3x}{10}$

$\frac{6+5}{21}$

$\frac{7-12x}{20}$

$\frac{8+3x}{5}$

$\frac{3-3x}{10}$

Correct answer:

$\frac{3-3x}{10}$

Explanation:

First, you must convert your fractions to the common denominator of :

$\frac{5}{20}-\frac{4}{4}*\frac{x-2}{5}-\frac{2x+7}{20}$

Next, do your subtraction:

$\frac{5}{20}-\frac{4x-8}{20}-\frac{2x+7}{20}$

Next, you must be very careful. Notice how you must handle your subtractions in order to maintain the correct distribution of signs:

$\frac{5-(4x-8)-(2x+7)}{20}$

Now, carefully distribute for each group:

$\frac{5-4x+8-2x-7}{20}$

Next, simplify:

$\frac{6-6x}{20}$

Factor out the common in the numerator:

$\frac{2(3-3x)}{20}=\frac{3-3x}{10}$

There is still a common , but that does not help you get your fraction into the form found in the answers.

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Example Question #1 : How To Subtract Fractions

Choose the answer which best solves the equation below:

$\frac{5}{12} - \frac{1}{3} =$

Possible Answers:

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{1}{6}$

$\frac{1}{12}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{12}$

Explanation:

To solve this equation, you must first make sure that both fractions have a common denominator.

In this case the common denominator will be 12:

$\frac{5}{12} - \left(\frac{1}{3} \cdot \frac{4}{4}\right) =$

Then you perform your operation:

$\frac{5}{12} - \left(\frac{1\cdot 4}{3 \cdot 4}\right)$

$=\frac{5}{12} - \left(\frac{4}{12}\right)$

$=\frac{1}{12}$

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Example Question #1 : How To Subtract Fractions

If John has slices of an slice pizza left over, and he eats 3.5 of them, what fraction of the pizza does he have left over?

Possible Answers:

$\frac{7}{18}$

$\frac{17}{36}$

$\frac{17}{24}$

$\frac{15}{24}$

$\frac{9}{18}$

Correct answer:

$\frac{17}{36}$

Explanation:

To find this answer, first you need to set up your equation:

$\frac{12}{18} - \frac{3.5}{18} =$

And you need to get rid of the decimal in the numerator. We can do this by multiplying each fraction by 2/2.

$\left(\frac{12}{18}\cdot \frac{2}{2} \right) - \left(\frac{3.5}{18}\cdot \frac{2}{2} \right) =$

$\frac{24}{36} - \frac{7}{36} =$

Then solve for your answer:

$\frac{17}{36}$

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Example Question #14 : Operations And Fractions

What common number can you add to the numerator and denominator of $\frac{7}{19}$ to get $\frac{1}{2}$ ?

Possible Answers:

Correct answer:

Explanation:

Set up an equation where you add the same unknown number (x) to both the numerator and the denominator of the original fraction, and set the equation equal to $\frac{1}{2}$ .

$\newline \frac{7+x}{19+x}=\frac{1}{2}$

Cross-multiply the fractions to simplify.

$(7+x)\cdot (2)=(19+x)\cdot (1)$

14+2x=19+x

Now, solve for x.

14+2x=19+x

14+x-14=19-14

x=5

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Example Question #15 : Operations And Fractions

Add the following fractions

$\frac{2}{5}+\frac{3}{4}+\frac{1}{12}$

Possible Answers:

$\frac{37}{30}$

$\frac{79}{60}$

$\frac{22}{60}$

$\frac{6}{240}$

$\frac{321}{240}$

Correct answer:

$\frac{37}{30}$

Explanation:

To add fractions you first must find the lowest common denominator. For these fractions it is 60. Then you must multiply the numerator and denominator by the number such that the denominator is equal to the LCD. For example $\frac{2}{5}$ gets multiplied by 12 (on both the numerator and denominator) because 5 times 12 is 60. When you do that you get the expression

$\frac{24}{60}+\frac{45}{60}+\frac{5}{60}$

then you just add the numerators and get

$\frac{74}{60}$

2 goes into both of those numbers so you get

$\frac{37}{30}$

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Example Question #16 : Operations And Fractions

Simplify the expression:

$\frac{1}{2}-\frac{2}{9}+\frac{5}{6}$

Possible Answers:

$\frac{10}{9}$

$\frac{3}{2}$

$\frac{1}{18}$

$\frac{2}{3}$

$\frac{14}{9}$

Correct answer:

$\frac{10}{9}$

Explanation:

First you want to find the least common denominator (in this case it's 18):

$\frac{9*1}{9*2}-\frac{2*2}{2*9}+\frac{1*15}{1*18}$

$\frac{9}{18}-\frac{4}{18}+\frac{15}{18}$

$\frac{9-4+15}{18}$

$\frac{20}{18}=\frac{10}{9}$

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Example Question #1 : Operations

What is the result of adding $20\%$ of $\frac{2}{7}$ to $\frac{1}{4}$ ?

Possible Answers:

$\frac{23}{11}$

$\frac{43}{140}$

$\frac{47}{140}$

$\frac{3}{28}$

$\frac{3}{39}$

Correct answer:

$\frac{43}{140}$

Explanation:

Let us first get our value for the percentage of the first fraction. 20% of 2/7 is found by multiplying 2/7 by 2/10 (or, simplified, 1/5): (2/7) * (1/5) = (2/35)

Our addition is therefore (2/35) + (1/4). There are no common factors, so the least common denominator will be 35 * 4 or 140. Multiply the numerator and denominator of 2/35 by 4/4 and the numerator of 1/4 by 35/35.

This yields:

(8/140) + (35/140) = 43/140, which cannot be reduced.

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Example Question #18 : Operations And Fractions

Which of the following is equal to $\frac{1}{y(y+1)}$ ?

Possible Answers:

$\frac{1}{y}-\frac{1}{y+1}$

$\frac{y}{y-1}$

$\frac{1}{y}+\frac{1}{y+1}$

$\frac{y}{y+1}$

$\frac{1}{y+1}-\frac{1}{y}$

Correct answer:

$\frac{1}{y}-\frac{1}{y+1}$

Explanation:

We have two options here. We can manipulate the answers to work backwards and find a common denominator. This involves simply subtracting or adding two fractions. We can also try to rewrite the numerator by adding and subtracting the value . This serves the purpose of creating a sum in the numerator than can be split into y+1 and . This gives us one of the factors in the denominator in each numerator. When we separate or decompose the fraction, we can divide out by the common factor to re-express this as the difference of two rational expressions.

$\frac{1}{y(y+1)}=\frac{1+y-y}{y(y+1)}=\frac{y+1}{y(y+1)}-\frac{y}{y(y+1)}=\frac{1}{y}-\frac{1}{y+1}$

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ACT Math : Operations and Fractions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Subtract Fractions

Example Question #1 : How To Subtract Fractions

Example Question #1 : How To Subtract Fractions

Example Question #14 : Operations And Fractions

Example Question #15 : Operations And Fractions

Example Question #16 : Operations And Fractions

Example Question #1 : Operations

Example Question #18 : Operations And Fractions

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