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ACT Math : Variables

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Example Questions

Example Question #1 : Variables

Mike wants to sell candy bars for a $50\%$ profit. If he sells each bar for $\$1.20$ , how much did each bar cost him?

Possible Answers:

$\$0.60$

$\$0.80$

$\$1.00$

$\$0.75$

Correct answer:

$\$0.80$

Explanation:

In order to solve this problem, set up the following equation:

$\frac{1.20}{x}=\frac{150\%}{100\%}$

Cross multiply:

150x = 120

Divide:

$\frac{150x}{150} = \frac{120}{150} = 0.80$

The original cost of the of each candy bar is $\$0.80$

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Example Question #2 : Variables

Choose the answer that is the simplest form of the following expression of monomial quotients:

$\frac{12x^3y^2p}{4ab}\div\frac{3x^3y}{2ab}$

Possible Answers:

2yp

3yp

2xy

12xp

3ax

Correct answer:

2yp

Explanation:

$\frac{12x^3y^2p}{4ab}\div\frac{3x^3y}{2ab}$

To divide monomial quotients, simply invert the divisor and multiply:

$\frac{12x^3y^2p}{4ab} *\frac{2ab}{3x^3y} = \frac{24x^3y^2pab}{12abx^3y}$

Then, reduce:

$\frac{24yp}{12} = 2yp$

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Example Question #1 : Monomials

Choose the answer that is the simplest form of the following expression of monomial quotients:

$\frac{10m^2np}{3xy}\div\frac{3p^2}{2xym}$

Possible Answers:

$\frac{9p}{20m^3n}$

$\frac{20x^3}{9y}$

$\frac{20n^3}{9m}$

$\frac{20m^3n}{9p}$

$\frac{20p^3n}{9m}$

Correct answer:

$\frac{20m^3n}{9p}$

Explanation:

$\frac{10m^2np}{3xy}\div\frac{3p^2}{2xym}$

To find your answer, you have to invert the divisor and multiply across:

$\frac{10m^2np}{3xy} * \frac{2xym}{3p^2} = \frac{20m^3npxy}{9p^2xy}$

Then, reduce:

$\frac{20m^3n}{9p}$

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Example Question #1 : How To Multiply A Monomial By A Polynomial

Multiply: $2x\ast(4x+3)$

Possible Answers:

$8x^{2}+6x$

8x+6

$8x^{2}+6$

6x+8

$6x^{2}+8x$

Correct answer:

$8x^{2}+6x$

Explanation:

To solve you must multiply by both terms in (4x+3)

$2x\ast4x=8x^{2}$

$2x\ast3=6x$

$8x^{2}+6x$

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Example Question #1 : Variables

Multiply:

$4x\cdot(5x+7)$

Possible Answers:

$20x^{2}+28x$

$9x^{2}+28x$

$9x^{2}+28$

20x+11

Correct answer:

$20x^{2}+28x$

Explanation:

Multiply by both terms in (5x+7)

$4x\cdot5x=20x^{2}$

$4x\cdot7=28x$

$20x^{2}+28x$

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Example Question #2 : Variables

Multiply $\left (7x^{2}-12x+4 \right ) \cdot 3x$

Possible Answers:

$21x^{3}-36x^{2}+12x$

$7x^{3}-12x^{2}+4x$

$21x^{3}+36x^{2}+12x$

None of the other answers

$21x^{2}-36x+12$

Correct answer:

$21x^{3}-36x^{2}+12x$

Explanation:

When multiplying a polynomial by a monomial, each term in the polynomial gets multiplied by the monomial. Calculate each term one at a time, then add the results to get the final answer. In this case, we start by multiplying $7x^{2}\cdot3x$ . $7\cdot3=21$ and $x^{2}\cdot x=x^{3}$ , thus we get $21x^{3}$ . For the second term of the polynomial, we multiply $-12\cdot3=-36$ and $x\cdot x= x^{2}$ , resulting in $-36x^{2}$ . Finally, we multiply $4\cdot3 = 12$ and $1\cdot x = x$ , resulting in 12x . Adding the three terms that we just found, we come to the answer of $21x^{3}-36x^{2}+12x$ .

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Example Question #1 : How To Multiply Monomial Quotients

Choose the answer that is the best solution to the following expression of monomial quotients:

$\frac{4x^2p^3}{3mn^2} * \frac{2mn}{12xy}$

Possible Answers:

$\frac{8xp^3}{36ny}$

$\frac{2xp}{9n^2y}$

$\frac{4xp^3}{5ny}$

$\frac{4xp}{11n^2y}$

$\frac{2xp^3}{9ny}$

Correct answer:

$\frac{2xp^3}{9ny}$

Explanation:

$\frac{4x^2p^3}{3mn^2} * \frac{2mn}{12xy}$

To multiply monomial quotients, treat them as you would any other fraction. Combine like terms wherever possible:

$\frac{8x^2p^3mn}{36mn^2xy}$

Then, you need to reduce:

$\frac{2xp^3}{9ny}$

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Example Question #4 : Variables

Choose the answer that is the simplest form of the following expression of monomial quotients:

$\frac{2x^3y^4}{10z^3} * \frac{4z^2p}{10xy}$

Possible Answers:

$\frac{2p}{25z}$

$\frac{2x^2y^3p}{25z}$

$\frac{8x^2y^3p}{25z^2}$

$\frac{2xyp}{15z}$

$\frac{13x^2y^3p}{100z}$

Correct answer:

$\frac{2x^2y^3p}{25z}$

Explanation:

$\frac{2x^3y^4}{10z^3} * \frac{4z^2p}{10xy}$

To simplify, first multiply across:

$\frac{8x^3y^4z^2p}{100z^3xy}$

Then, reduce:

$\frac{2x^2y^3p}{25z}$

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Example Question #1 : How To Use The Direct Variation Formula

The price of silver varies directly as the square of the mass. If 3.6 g of silver is worth $64.80, what is the value of 7.5 g of silver?

Possible Answers:

$178.50

$301.75

$135.00

$215.25

$281.25

Correct answer:

$281.25

Explanation:

This is a direct variation problem of the form y = kx² The first set of data 3.6 g and $64.80 is used to calculate the proportionality constant, k. So 64.80 = k(3.6)² and solving the equation gives k = 5.

Now we move to the new data, 7.5 g and we get y = 5(7.5)² to yield an answer of $218.25.

$135.00 is the answer obtained if using proportions. This is an error because it does not take into consideration the squared elements of the problem.

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Example Question #2 : Variables

The diameter of a specific brand of candy wrapper is 6cm longer than half the volume of the candy itself. Find the expression for the diameter, , in terms of the volume, .

Possible Answers:

$d=\frac{1}{2}v-6$

$d=\frac{4}{3}v^2$

d=2v-6

$d=\frac{1}{2}v+6$

d=2v+6

Correct answer:

$d=\frac{1}{2}v+6$

Explanation:

The question asks for an equation that can relate and to each other, based on the information given. We are told that half the volume + 6cm determines the total diameter.

This gives us:

$d=\frac{1}{2}v+6$

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ACT Math : Variables

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Variables

Example Question #2 : Variables

Example Question #1 : Monomials

Example Question #1 : How To Multiply A Monomial By A Polynomial

Example Question #1 : Variables

Example Question #2 : Variables

Example Question #1 : How To Multiply Monomial Quotients

Example Question #4 : Variables

Example Question #1 : How To Use The Direct Variation Formula

Example Question #2 : Variables

All ACT Math Resources

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