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ACT Math : Algebra

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Example Questions

Example Question #1 : How To Divide Monomial Quotients

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:

3yp

3ax

2yp

12xp

2xy

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:

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

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

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

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

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

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:

6x+8

8x+6

$8x^{2}+6x$

$6x^{2}+8x$

$8x^{2}+6$

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 #591 : Algebra

Multiply:

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

Possible Answers:

20x+11

$20x^{2}+28x$

$9x^{2}+28$

$9x^{2}+28x$

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:

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

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

None of the other answers

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

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

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:

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

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

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

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

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

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 #592 : Algebra

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:

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

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

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

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

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

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:

$135.00

$281.25

$178.50

$301.75

$215.25

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 : How To Use The Direct Variation Formula

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:

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

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

d=2v-6

d=2v+6

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

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

An instrument reads two values, and daily. The values directly vary with respect to each other. If on Monday the value of was and was , which of the following could be the values for and on Wednesday?

Possible Answers:

x = 45; y = 5

x = 25; y = 5

x = 35; y = 3

x = 80; y = 10

x = 40; y = 20

Correct answer:

x = 25; y = 5

Explanation:

Direct variation means that any pairing of the related values will always have the same ratio, thus we know that for any other values and , those values will be equal according to the following equation:

$\frac{x}{y} = \frac{a}{b}$

Thus, for our information, we know:

$\frac{40}{8} = \frac{a}{b}$

This means that the new values of and , when divided must be equal to . Therefore, the only possible answer is x = 25; y = 5

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ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Divide Monomial Quotients

Example Question #1 : Monomials

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

Example Question #591 : Algebra

Example Question #2 : Variables

Example Question #1 : How To Multiply Monomial Quotients

Example Question #592 : Algebra

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

Example Question #2 : How To Use The Direct Variation Formula

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

All ACT Math Resources

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