Operations and Fractions - ACT Math

Card 0 of 18

Question

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

Answer

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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Question

Add the following fractions

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

Answer

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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Question

Simplify the expression:

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

Answer

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

$\frac{91}{92}-\frac{22}{29}+\frac{115}{118}$

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

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

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

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Question

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

Answer

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 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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Question

$\dpi{100} \small \frac{1}{3}\div \frac{3}{5} =$

Answer

Cross multiply or multiply using the reciprocal of the second fraction.

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Question

Simplify the following expression:

$\frac{2}{3}\div\frac{5}{6}$

Answer

$\frac{2}{3}\div\frac{5}{6}=\frac{2}{3}\times\frac{6}{5}=\frac{12}{15}=\frac{4}{5}$

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Question

Evaluate the following:

$7\frac{1}{3}\div6\frac{2}{3}$

Answer

Start by converting 71/3 to 22/3, and 6 2/3 to 20/3. We then multiply 22/3 by the reciprocal of 20/3, 3/20, and you get 66/60. This simplifies to 11/10.

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Question

Solve.

$\small \frac{2}{5}\div \frac{1}{25}=$

Answer

Remember, to divide a number by a fraction, multiply the number by the reciprocal of the fraction. In this case, $\small \small \frac{2}{5}\div \frac{1}{25}=\frac{2}{5}\times \frac{25}{1}=\frac{50}{5}=10$

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Question

Simplfiy the following expression;

$\frac{2}{3}\times\frac{6}{8}\times\frac{4}{12}$

Answer

Multiply the numerators 2 x 6 x 4 = 48. Then multiply the denominators 3 x 8 x 12 = 288. The answer is 48/288. To simplify, divide both numerator and denominator by 48 to get 1/6.

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Question

Evaluate –3–2 * 2–3.

Answer

Because the exponents are negative, we can convert –3–2 to 1/9 and 2–3 to 1/8. We then multiply straight across the top and the bottom, giving you 1/72.

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Question

What value of $s$ makes the equation $\frac{7}{s}=\frac{8}{10}$ true?

Answer

We can simply cross multiply to obtain $70=8s$ and divide by 8 to solve for $s$ .

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Question

Simplify the following into one fraction

$\frac{2}{7}\cdot\frac{12}{5}\cdot\frac{25}{6}$

Answer

To multiply fractions you multiply the entire numerator and the entire denominator together. However, before we do that we can cancel anything from the denominator with anything in the numerator.

Six cancels with 12

$\frac{2}{7}\cdot\frac{\textbf{12}}{5}\cdot\frac{25}{\textbf{6}}=\frac{2}{7}\cdot\frac{2}{5}\cdot25$

5 cancels with 25

$\frac{2}{7}\cdot\frac{2}{\textbf5}\cdot\textbf{25}=\frac{2}{7}\cdot2\cdot5$

multiply it all out and get

$\frac{20}{7}$

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Question

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

Answer

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 $\frac{1}{2}$ .

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

Cross-multiply the fractions to simplify.

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

Now, solve for x.

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Question

Solve for x where

$4\frac{1}{3}=x+1\frac{1}{2}$

Answer

To solve this, subtract 1 1/2 from both sides. Convert to common denominators.

4 1/3 – 1 1/2 = 4 2/6 – 1 3/6.

In order to subtract, you'll want to "borrow" from the 4 2/6. Rewrite 4 2/6 as 3 8/6 and then subtract 1 3/6 from this. Your solution is 2 5/6. Most calculators will also do these calculations for you.

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Question

Choose the answer which best solves the equation below:

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

Answer

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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Question

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

Answer

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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Question

Solve for :

$\frac{7}{12}x+\frac{5}{4}=\frac{1}{2}+\frac{4}{3}x$

Answer

$\frac{7}{12}x+\frac{5}{4}=\frac{1}{2}+\frac{4}{3}x$

Begin by isolating your variable:

$\frac{7}{12}x-\frac{4}{3}x=\frac{1}{2}-\frac{5}{4}$

Next, you need to find the common denominator. For the left side of your equation, it is . For the right, it is . This means that you need to rewrite as follows:

$\frac{7}{12}x-\frac{4}{3}\frac{4}{4}x=\frac{2}{4}-\frac{5}{4}$

Now, simplify and combine terms:

$\frac{7}{12}x-\frac{16}{12}x=-\frac{3}{4}$

$-\frac{9}{12}x=-\frac{3}{4}$

You can further simplify the left side:

$-\frac{3}{4}x=-\frac{3}{4}$

Next, multiply both sides by $-\frac{4}{3}$ . This gives you:

$x=-\frac{3}{4}-\frac{4}{3}=1$

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Question

Simplify:

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

Answer

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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