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SAT Math : Algebra

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Example Questions

Example Question #501 : Problem Solving

The sum of the number of pennies and nickels equals 70 and the total dollar amount of the change equals $1.50. How many nickels and pennies are there?

Possible Answers:

$20\ pennies,\ 50\ nickels$ $\displaystyle 20\ pennies,\ 50\ nickels$

$50\ pennies,\ 20\ nickels$ $\displaystyle 50\ pennies,\ 20\ nickels$

$60\ pennies,\ 10\ nickels$ $\displaystyle 60\ pennies,\ 10\ nickels$

$None\ of\ the\ answers$ $\displaystyle None\ of\ the\ answers$

$10\ pennies,\ 60\ nickels$ $\displaystyle 10\ pennies,\ 60\ nickels$

Correct answer:

$50\ pennies,\ 20\ nickels$ $\displaystyle 50\ pennies,\ 20\ nickels$

Explanation:

Assume there are x pennies. Hence the number of nickels is $70-x$ $\displaystyle 70-x$ .

Now you have to set up an equation for the dollar value of the pennies and nickels which will be $x+5(70-x)=150$ $\displaystyle x+5(70-x)=150$ .

Now solving for $x$ $\displaystyle x$ results in $x=50$ $\displaystyle x=50$ (number of pennies). Hence the number of nickels will be $70-50=20$ $\displaystyle 70-50=20$ .

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Example Question #23 : How To Simplify An Expression

If $\displaystyle log 2 = 0.301$ , then solve $\log \ 32$ $\displaystyle \log \ 32$

Possible Answers:

1.505 $\displaystyle 1.505$

1.431 $\displaystyle 1.431$

1.209 $\displaystyle 1.209$

1.301 $\displaystyle 1.301$

1.612 $\displaystyle 1.612$

Correct answer:

1.505 $\displaystyle 1.505$

Explanation:

To solve this problem we use the logarithm rule $log (x^{n}) = n log(x)$ $\displaystyle log (x^{n}) = n log(x)$

$log(32) = log (2^{5}) = 5 log(2)$ $\displaystyle log(32) = log (2^{5}) = 5 log(2)$ or 1.505 $\displaystyle 1.505$

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Example Question #791 : Algebra

$a=\frac{x^2-y^2}{x-y}$ $\displaystyle a=\frac{x^2-y^2}{x-y}$

If both $\displaystyle x$ and $\displaystyle y$ are positive, what is the simplest form of $\displaystyle a$ ?

Possible Answers:

$x+y$ $\displaystyle x+y$

$xy$ $\displaystyle xy$

$x^2-y^2-1$ $\displaystyle x^2-y^2-1$

$1$ $\displaystyle 1$

$x-y$ $\displaystyle x-y$

Correct answer:

$x+y$ $\displaystyle x+y$

Explanation:

$x^2-y^2$ $\displaystyle x^2-y^2$ can also be expressed as $(x-y)(x+y))$ $\displaystyle (x-y)(x+y))$

$a=\frac{(x-y)(x+y)}{x-y}=x+y$ $\displaystyle a=\frac{(x-y)(x+y)}{x-y}=x+y$

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Example Question #792 : Algebra

Which of the following does not simplify to $\displaystyle x$ ?

Possible Answers:

$\frac{x(4x)}{4x}$ $\displaystyle \frac{x(4x)}{4x}$

(3-3)x $\displaystyle (3-3)x$

$\displaystyle 5x-(6x-2x)$

All of these simplify to $\displaystyle x$

$\displaystyle (x - 1)(x + 2) - x^2 + 2$

Correct answer:

(3-3)x $\displaystyle (3-3)x$

Explanation:

5x – (6x – 2x) = 5x – (4x) = x

(x – 1)(x + 2) - x² + 2 = x² + x – 2 – x² + 2 = x

x(4x)/(4x) = x

(3 – 3)x = 0x = 0

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Example Question #793 : Algebra

Edward is $\displaystyle e$ years old. He is 5 years younger than his sister Francine. In terms of $\displaystyle e$ , how old will Francine be in 2 years?

Possible Answers:

e+3 $\displaystyle e+3$

e+7 $\displaystyle e+7$

e+4 $\displaystyle e+4$

e-3 $\displaystyle e-3$

e-7 $\displaystyle e-7$

Correct answer:

e+7 $\displaystyle e+7$

Explanation:

Let f = Francine's age now.

e = f – 5

f = e + 5

In 2 years, Francine will be f + 2. Use our previous calculation to substitute.

f + 2 = (e + 5) + 2 = e + 7

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Example Question #402 : Gre Quantitative Reasoning

a # b = (a * b) + a

What is 3 # (4 # 1)?

Possible Answers:

Correct answer:

Explanation:

Work from the "inside" outward. Therefore, first solve 4 # 1 by replacing a with 4 and b with 1:

4 # 1 = (4 * 1) + 4 = 4 + 4 = 8

That means: 3 # (4 # 1) = 3 # 8. Solve this now:

3 # 8 = (3 * 8) + 3 = 24 + 3 = 27

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Example Question #1 : Simplifying Expressions

Simplify the result of the following steps, to be completed in order:

1. Add 7x to 3y

2. Multiply the sum by 4

3. Add x to the product

4. Subtract x – y from the sum

Possible Answers:

28x + 13y

28x + 12y

28x + 11y

29x + 13y

28x – 13y

Correct answer:

28x + 13y

Explanation:

Step 1: 7x + 3y

Step 2: 4 * (7x + 3y) = 28x + 12y

Step 3: 28x + 12y + x = 29x + 12y

Step 4: 29x + 12y – (x – y) = 29x + 12y – x + y = 28x + 13y

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Example Question #1 : Simplifying Expressions

Which is the greater quantity: the median of 5 positive sequential integers or the mean of 5 positive sequential integers?

Possible Answers:

The relationship cannot be determined

The median is greater

The quantities are equal

The mean is greater

Correct answer:

The quantities are equal

Explanation:

If the first integer is $\dpi{100} \small n$ $\displaystyle \dpi{100} \small n$ , then $\dpi{100} \small n+(n+1)+(n+2)+(n+3)+(n+4)=5n+10$

$\dpi{100} \small \frac{5n+10}{5}=n+2$

This is the same as the median.

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Example Question #2 : How To Simplify An Expression

You are told that $\dpi{100} \small x$ $\displaystyle \dpi{100} \small x$ can be determined from the expression:

Capture7

Determine whether the absolute value of $\dpi{100} \small x$ $\displaystyle \dpi{100} \small x$ is greater than or less than 2.

Possible Answers:

$\dpi{100} \small |x|>2$ $\displaystyle \dpi{100} \small |x|>2$

The relationship cannot be determined from the information given.

The quantities are equal

$\dpi{100} \small |x|<2$ $\displaystyle \dpi{100} \small |x|< 2$

Correct answer:

$\dpi{100} \small |x|>2$ $\displaystyle \dpi{100} \small |x|>2$

Explanation:

The expression is simplified as follows:

Capture8

Since $\dpi{100} \small 2^{4}=16$ $\displaystyle \dpi{100} \small 2^{4}=16$ the value of $\dpi{100} \small x$ $\displaystyle \dpi{100} \small x$ must be slightly greater for it to be 17 when raised to the 4th power.

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Example Question #3 : Simplifying Expressions

Which best describes the relationship between (x+y)^3 $\displaystyle (x+y)^3$ and x^3+y^3 $\displaystyle x^3+y^3$ if $x,y\neq 0$ $\displaystyle x,y\neq 0$ ?

Possible Answers:

The relationship cannot be determined from the information given.

$\small (x+y)^{3}>x^{3}+y^{3}$ $\displaystyle \small (x+y)^{3}>x^{3}+y^{3}$

$\small (x+y)^{3}=x^{3}+y^{3}$ $\displaystyle \small (x+y)^{3}=x^{3}+y^{3}$

$\displaystyle (x+y)^3< x^3 + y^3$

Correct answer:

The relationship cannot be determined from the information given.

Explanation:

Use substitution to determine the relationship.

For example, we could plug in x=2 $\displaystyle x=2$ and y=3 $\displaystyle y=3$ .

$\displaystyle (x+y)^3=(2+3)^3=5^3=125$

$\displaystyle x^3+y^3=2^3+3^3=4+9=13$

So far it looks like the first expression is greater, but it's a good idea to try other values of x and y to be sure. This time, we'll try some negative values, say, x=-4 $\displaystyle x=-4$ and y=-3 $\displaystyle y=-3$ .

$\displaystyle (x+y)^3=(-4+-3)^3=-7^3=-343$

$\displaystyle x^3+y^3=-4^3+-3^3=-64+-27=-91$

This time the first quantity is smaller. Therefore the relationship cannot be determined from the information given.

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SAT Math : Algebra

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #501 : Problem Solving

Example Question #23 : How To Simplify An Expression

Example Question #791 : Algebra

Example Question #792 : Algebra

Example Question #793 : Algebra

Example Question #402 : Gre Quantitative Reasoning

Example Question #1 : Simplifying Expressions

Example Question #1 : Simplifying Expressions

Example Question #2 : How To Simplify An Expression

Example Question #3 : Simplifying Expressions

All SAT Math Resources

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