Expressions - ACT Math

Card 0 of 981

Question

Find the product of and .

Answer

Solve the first equation for .

Solve the second equation for .

The final step is to multiply and .

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Question

Amy buys concert tickets for herself and her friends. She initially buys them at \$40/ticket. Weeks later, her other friends ask her to buy them tickets, but the prices have increased to \$54. Amy buys 7 tickets total and spends \$350. How many tickets has she paid \$40 on?

Answer

Amy has bought 7 tickets, x of them at \$40/ticket, and the remaining 7-x at \$54/ticket. She spends at total of

$\ \ \$350 = 40x + (54)(7-x) = 40x + 378 – 54x = 378 – 14x\rightarrow \ 14x = 378-350 = 28 \rightarrow x =2.$

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Question

The following table shows the temperature of a cup of coffee at different times

Time 1:09 1:11 1:13 1:15 1:17

Temperature (ºF) 187.1 184.4 181.7 179.0 176.3

If this trend continues, what will the temperature of the coffee at minute 1:25?

Answer

The table shows that for every two minutes, the temperature of the coffee lowers 2.7ºF. At 1:25, 16 minutes, or eight 2-minute intervals have passed, and the temperature of the coffee has lowered by 8*2.7ºF, reaching a temperature of 165.5ºF.

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Question

What is

$\frac{3}{\sqrt{6}}$ ?

Answer

To find an equivalency we must rationalize the denominator.

To rationalize the denominator multiply the numerator and denominator by the denominator.

$\frac{3}{\sqrt{6}}$ $\frac{\sqrt{6}}{\sqrt{6}}=$

$\frac{3\sqrt{6}}{6}=$

To simplify completely, factor out a three from the numerator and denominator resulting in the final solution.

$\frac{\sqrt{6}}{2}$

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Question

Simplify the following expression:

$\frac{5}{7}+\frac{11}{14}$

Answer

In order to add fractions, we must first make sure they have the same denominator.

So, we multiply $\frac{5}{7}$ by $\frac{2}{2}$ and get the following:

$\frac{5}{7}\left(\frac{2}{2}\right)+\frac{11}{14}$

$\frac{10}{14}+\frac{11}{14}$

Then, we add across the numerators and simplify:

$\frac{10}{14}+\frac{11}{14}=\frac{21}{14}=\frac{3}{2}$

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Question

Simplify the following:

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

Answer

To simplify the following, a common denominator must be achieved. In this case, the first term must be multiplied by (x+2) in both the numerator and denominator and likewise with the second term with (x-3).

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

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

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Question

Simplify the following $\frac{3x}{x-3} + \frac{7}{x-4}$

Answer

Find the least common denominator between x-3 and x-4, which is (x-3)(x-4). Therefore, you have $\frac{3x(x-4)}{(x-3)(x-4)} + \frac{7(x-3)}{(x-4)(x-3)}$ . Multiplying the terms out equals $\frac{3x^{2}-12x+ 7x-21}{x^{2}-7x+12}$ . Combining like terms results in $\frac{3x^{2}-5x-21}{x^{2}-7x+12}$ .

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Question

Select the expression that is equivalent to

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

Answer

To add the two fractions, a common denominator must be found. With one-term denominators, it is easier to simply find the least common denominator between them and multiply each side to obtain it.

In this case, the least common denominator between and is . So the first fraction needs to be multiplied by and the second by :

$\frac{x^2 + 3x - 2}{3x} + \frac{x-3}{x^2} = \frac{x^2 + 3x - 2}{3x}* \left(\frac{x}{x}\right) + \frac{x-3}{x^2} * \left(\frac{3}{3}\right)$

$\frac{x^3 + 3x^2 - 2x}{3x^2} + \frac{3x-9}{3x^2}$

Now, we can add straight across, remembering to combine terms where we can.

$\frac{x^3 + 3x^2 - 2x}{3x^2} + \frac{3x-9}{3x^2} = \frac{x^3 +3x^2 -2x +3x - 9}{3x^2} = \frac{x^3 + 3x^2 +x -9}{3x^2}$

So, our simplified answer is $\frac{x^3 + 3x^2 +x -9}{3x^2}$

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Question

Combine the following two expressions if possible.

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

Answer

For binomial expressions, it is often faster to simply FOIL them together to find a common trinomial than it is to look for individual least common denominators. Let's do that here:

$\frac{x+3}{x+7} + \frac{x^2}{4-x} = \frac{x+3}{x+7} * \frac{(4-x)}{(4-x)} + \frac{x^2}{4-x} * \frac{(x+7)}{(x+7)}$

FOIL and simplify.

$\frac{-x^2 + x + 12}{-x^2 -3x+28} + \frac{x^3 +7x^2}{-x^2 -3x+28}$

Combine numerators.

$\frac{-x^2 + x + 12}{-x^2 -3x+28} + \frac{x^3 +7x^2}{-x^2 -3x+28} = \frac{x^3 +6x^2 + x + 12}{-x^2-3x+28}$

Thus, our answer is $\frac{x^3 +6x^2 + x + 12}{-x^2-3x+28}$

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Question

Find the product of and .

Answer

Solve the first equation for .

Solve the second equation for .

The final step is to multiply and .

Compare your answer with the correct one above

Question

Amy buys concert tickets for herself and her friends. She initially buys them at \$40/ticket. Weeks later, her other friends ask her to buy them tickets, but the prices have increased to \$54. Amy buys 7 tickets total and spends \$350. How many tickets has she paid \$40 on?

Answer

Amy has bought 7 tickets, x of them at \$40/ticket, and the remaining 7-x at \$54/ticket. She spends at total of

$\ \ \$350 = 40x + (54)(7-x) = 40x + 378 – 54x = 378 – 14x\rightarrow \ 14x = 378-350 = 28 \rightarrow x =2.$

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Question

The following table shows the temperature of a cup of coffee at different times

Time 1:09 1:11 1:13 1:15 1:17

Temperature (ºF) 187.1 184.4 181.7 179.0 176.3

If this trend continues, what will the temperature of the coffee at minute 1:25?

Answer

The table shows that for every two minutes, the temperature of the coffee lowers 2.7ºF. At 1:25, 16 minutes, or eight 2-minute intervals have passed, and the temperature of the coffee has lowered by 8*2.7ºF, reaching a temperature of 165.5ºF.

Compare your answer with the correct one above

Question

What is

$\frac{3}{\sqrt{6}}$ ?

Answer

To find an equivalency we must rationalize the denominator.

To rationalize the denominator multiply the numerator and denominator by the denominator.

$\frac{3}{\sqrt{6}}$ $\frac{\sqrt{6}}{\sqrt{6}}=$

$\frac{3\sqrt{6}}{6}=$

To simplify completely, factor out a three from the numerator and denominator resulting in the final solution.

$\frac{\sqrt{6}}{2}$

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Question

Simplify the following expression:

$\frac{5}{7}+\frac{11}{14}$

Answer

In order to add fractions, we must first make sure they have the same denominator.

So, we multiply $\frac{5}{7}$ by $\frac{2}{2}$ and get the following:

$\frac{5}{7}\left(\frac{2}{2}\right)+\frac{11}{14}$

$\frac{10}{14}+\frac{11}{14}$

Then, we add across the numerators and simplify:

$\frac{10}{14}+\frac{11}{14}=\frac{21}{14}=\frac{3}{2}$

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Question

Simplify the following:

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

Answer

To simplify the following, a common denominator must be achieved. In this case, the first term must be multiplied by (x+2) in both the numerator and denominator and likewise with the second term with (x-3).

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

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

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Question

Simplify the following $\frac{3x}{x-3} + \frac{7}{x-4}$

Answer

Find the least common denominator between x-3 and x-4, which is (x-3)(x-4). Therefore, you have $\frac{3x(x-4)}{(x-3)(x-4)} + \frac{7(x-3)}{(x-4)(x-3)}$ . Multiplying the terms out equals $\frac{3x^{2}-12x+ 7x-21}{x^{2}-7x+12}$ . Combining like terms results in $\frac{3x^{2}-5x-21}{x^{2}-7x+12}$ .

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Question

Select the expression that is equivalent to

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

Answer

To add the two fractions, a common denominator must be found. With one-term denominators, it is easier to simply find the least common denominator between them and multiply each side to obtain it.

In this case, the least common denominator between and is . So the first fraction needs to be multiplied by and the second by :

$\frac{x^2 + 3x - 2}{3x} + \frac{x-3}{x^2} = \frac{x^2 + 3x - 2}{3x}* \left(\frac{x}{x}\right) + \frac{x-3}{x^2} * \left(\frac{3}{3}\right)$

$\frac{x^3 + 3x^2 - 2x}{3x^2} + \frac{3x-9}{3x^2}$

Now, we can add straight across, remembering to combine terms where we can.

$\frac{x^3 + 3x^2 - 2x}{3x^2} + \frac{3x-9}{3x^2} = \frac{x^3 +3x^2 -2x +3x - 9}{3x^2} = \frac{x^3 + 3x^2 +x -9}{3x^2}$

So, our simplified answer is $\frac{x^3 + 3x^2 +x -9}{3x^2}$

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Question

Combine the following two expressions if possible.

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

Answer

For binomial expressions, it is often faster to simply FOIL them together to find a common trinomial than it is to look for individual least common denominators. Let's do that here:

$\frac{x+3}{x+7} + \frac{x^2}{4-x} = \frac{x+3}{x+7} * \frac{(4-x)}{(4-x)} + \frac{x^2}{4-x} * \frac{(x+7)}{(x+7)}$

FOIL and simplify.

$\frac{-x^2 + x + 12}{-x^2 -3x+28} + \frac{x^3 +7x^2}{-x^2 -3x+28}$

Combine numerators.

$\frac{-x^2 + x + 12}{-x^2 -3x+28} + \frac{x^3 +7x^2}{-x^2 -3x+28} = \frac{x^3 +6x^2 + x + 12}{-x^2-3x+28}$

Thus, our answer is $\frac{x^3 +6x^2 + x + 12}{-x^2-3x+28}$

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Question

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

Answer

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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Question

Simplify the following rational expression:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

Answer

Since both fractions in the expression have a common denominator of $x^{2}$ , we can combine like terms into a single numerator over the denominator:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

$=\frac{(7x-18)+(6x-14)}{x^{2}}$

$=\frac{13x-32}{x^{2}}$

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