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ACT Math : ACT Math

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Example Questions

Example Question #1 : Find The Sum Or Difference Of Two Matrices

If $\begin{bmatrix} a&b\\ c&d \end{bmatrix} + \begin{bmatrix} 3&14\\ 12&4 \end{bmatrix} = \begin{bmatrix} 19&2\\ 4&5 \end{bmatrix}$ , what is a+b+c+d ?

Possible Answers:

Correct answer:

Explanation:

You can treat matrices just like you treat other members of an equation. Therefore, you can subtract the matrix

$\begin{bmatrix} 3&14\\ 12&4 \end{bmatrix}$

from both sides of the equation. This gives you:

$\begin{bmatrix} a&b\\ c&d \end{bmatrix}=\begin{bmatrix} 19&2\\ 4&5 \end{bmatrix}-\begin{bmatrix} 3&14\\ 12&4 \end{bmatrix}$

Now, matrix subtraction is simple. You merely subtract each element, matching the correlative spaces with each other:

$\begin{bmatrix} a&b\\ c&d \end{bmatrix} = \begin{bmatrix} 19-3&2-14\\ 4-12&5-4 \end{bmatrix}$

Then, you simplify:

$\begin{bmatrix} 19-3&2-14\\ 4-12&5-4 \end{bmatrix} = \begin{bmatrix} 16&-12\\ -8&1 \end{bmatrix}$

Therefore, a+b+c+d = 16-12-8+1=-3

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Example Question #2581 : Act Math

$\begin{bmatrix} 0&5 \\ 3&4 \end{bmatrix} - \begin{bmatrix} 2&-1 \\ 4& 3 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} -2&4 \\ -1&1 \end{bmatrix}$

$\begin{bmatrix} -4&6 \\ 1&-1 \end{bmatrix}$

$\begin{bmatrix} -2&6 \\ 1&-1 \end{bmatrix}$

$\begin{bmatrix} -2&6 \\ -1& 1 \end{bmatrix}$

$\begin{bmatrix} -2&4 \\ -1& 1 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} -2&6 \\ -1& 1 \end{bmatrix}$

Explanation:

When subtracting matrices, subtracting component-wise.

$\begin{bmatrix} 0&5 \\ 3&4 \end{bmatrix} - \begin{bmatrix} 2&-1 \\ 4& 3 \end{bmatrix}=$

$\begin{bmatrix} 0-2&5-(-1)) \\ 3-4& 4-3 \end{bmatrix}=$

$\begin{bmatrix} -2&6 \\ -1&1 \end{bmatrix}$

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Example Question #2581 : Act Math

If $\begin{bmatrix} 12& 14\\ -4 & -2 \\ 3 & 3 \end{bmatrix}- \begin{bmatrix} 9& -1\\ 4 & -5 \\ 6 & -9 \end{bmatrix}=\begin{bmatrix} a&b\\c&d\\e&f\end{bmatrix}$ , then what is the value of a+f ?

Possible Answers:

Correct answer:

Explanation:

Match each term from the first matrix with the corresponding number from the second matrix and subtract.

$\begin{bmatrix} 12 -9 & 14 - (-1) \\ -4 -4 & -2-(-5) \\ 3-6 & 3-(-9) \end{bmatrix}$

Simplify.

$\begin{bmatrix} 12 -9 & 14 - (-1) \\ -4 -4 & -2-(-5) \\ 3-6 & 3-(-9) \end{bmatrix} = \begin{bmatrix} 3& 15 \\ -8 & 3 \\ -3 & 12 \end{bmatrix}$

Each of the numbers in the matrix solution now corresponds to the letters, a through f. Add a (upper left) and f (lower right).

a+f=3+12=15

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Example Question #1 : How To Find Combinations With A Matrix

Matthew, Emily, Cynthia, John, and Stephanie are all running for student council. If three equivalent positions are available in this year's election, how many different leadership boards are possible?

Possible Answers:

125

Correct answer:

Explanation:

One possible approach is to enumerate all possibilities using initials for names (ie. MEC, MEJ, MES, etc). If we recognize that order does not matter in this problem, we can also enter 5 nCr 3 into our calculators, and that will give us the answer of .

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Example Question #31 : Matrices

Simplify.

$3\cdot \begin{bmatrix} 2 & 6 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix}6 & 2 \end{bmatrix}$

$\begin{bmatrix}6 & 6 \end{bmatrix}$

$\begin{bmatrix}18 & 6 \end{bmatrix}$

$\begin{bmatrix}6 & 18 \end{bmatrix}$

$\begin{bmatrix}2 & 18 \end{bmatrix}$

Correct answer:

$\begin{bmatrix}6 & 18 \end{bmatrix}$

Explanation:

When we are asked to simplify a matrix that is being multiplied by a constant we simply multiply each component of the matrix with the scalar factor that is on the outside of the matrix.

Therefore,

$3\cdot \begin{bmatrix} 2 & 6 \end{bmatrix} \rightarrow \begin{bmatrix} 3\cdot 2 & 3\cdot 6 \end{bmatrix}=\begin{bmatrix} 6 & 18 \end{bmatrix}$

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Example Question #32 : Matrices

At a sale, a necklace that was originally $\$120$ is marked ten percent off. Cara has a coupon that will allow her to get ten percent off the discounted price. How much will Cara pay for the necklace if she uses the coupon?

Possible Answers:

$\$96$

$\$97.20$

$\$96.60$

$\$100$

$\$100.40$

Correct answer:

$\$97.20$

Explanation:

Taking ten percent off of $\$120$ gives us $\$108$ ,

$120\cdot (1-0.10)=120\cdot 0.90=108$ .

Taking ten percent off of this amount gives us $\$97.20$

$108 \cdot (1-0.10)=108\cdot (0.90)=97.20$ .

NOTE: This problem cannot be solved by taking twenty percent off of $\$120$ .

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Example Question #1 : Foil

For all x, (4x – 3)² =

Possible Answers:

12x²+ 24x – 9

16x²– 9

16x²+ 24x + 9

16x²+ 9

16x²– 24x + 9

Correct answer:

16x²– 24x + 9

Explanation:

To solve this problem, you should FOIL: (4x – 3)(4x – 3) = 16x²– 12x – 12x + 9 = 16x²– 24x + 9.

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Example Question #1 : Distributive Property

Which of the following is equivalent to (2g – 3h)²?

Possible Answers:

4g²– 6gh + 9h²

4g²– 12gh + 9h²

4g²– 12gh + 3h²

4g²+ 9h²

g²– 12gh + 9h²

Correct answer:

4g²– 12gh + 9h²

Explanation:

Use FOIL: (2g – 3h)(2g – 3h) = 4g²– 6gh – 6gh + 9h²= 4g²– 12gh + 9h²

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Example Question #2 : How To Use Foil With The Distributive Property

Use FOIL on the following expression:

x(x + 1)(x – 1)

Possible Answers:

x²– x

x – 1

x³– x²

x³– x

Correct answer:

x³– x

Explanation:

FOIL (First, Outside, Inside, Last): (x + 1)(x – 1) which is (x²– 1) then multiply it by x, which is (x³– x)

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Example Question #3 : Distributive Property

Multiply: (4x + 3)(2x + 4)

Possible Answers:

30x + 12

3x² + 12x – 12

8x² + 22x +12

6x² + 16x + 24

8x² + 34

Correct answer:

8x² + 22x +12

Explanation:

To solve you must use FOIL (first outer inner last)

Multiply 4x and 2x to get 8x²

Multiply 4x and 4 to get 16x

Multiply 3 and 2x to get 6x

Multiply 3 and 4 to get 12

Add the common terms and the awnser is 8x² + 22x + 12

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ACT Math : ACT Math

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Find The Sum Or Difference Of Two Matrices

Example Question #2581 : Act Math

Example Question #2581 : Act Math

Example Question #1 : How To Find Combinations With A Matrix

Example Question #31 : Matrices

Example Question #32 : Matrices

Example Question #1 : Foil

Example Question #1 : Distributive Property

Example Question #2 : How To Use Foil With The Distributive Property

Example Question #3 : Distributive Property

All ACT Math Resources

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