SAT Math - How to evaluate algebraic expressions

If , , and , then what is the value of $\frac{d}{b}$ ?

$\frac{1}{6}$

$\frac{3}{2}$

None of the given answers.

Explanation

If ,

then

$d=\frac{c}{3}$ .

Therefore,

$\frac{d}{b}=\frac{\frac{c}{3}}{2c}=\frac{c}{3}\cdot\frac{1}{2c}=\frac{c}{6c}=\frac{1}{6}$

x is 75% of y, which is 8 times the amount of 150% of z. What is x in terms of z?

None of the other answers

7.5z

12z

Explanation

Just take this step by step. First write out each equation:

x = 0.75y

y = 8(1.5z)

First, simplify the second equation:

y = 12z

Then, substitute y from this equation into the first:

x = 0.75(12z) = 9z

If x and y are integers such that x > y > 0 and _x_2 + _y_2 = 100

Which of the following can be the value of x + y?

I. 10

II. 12

III. 14

IV. 16

V. 18

Explanation

Note that x must be greater than y and that y must be greater than 0. This means that x and y are different, positive integers. In addition, the sum, _x_2 + _y_2 must equal to 100. If we list squares beginning from the square of the first integer greater than 0 (12) up to the square of the greatest integer less than 100 (92) we will get:

1, 4, 9, 16, 25, 36, 49, 64, 81

We must observe that the only two numbers that will add up to 100 are 36 and 64.

Remember that x > y > 0 and that _x_2 + _y_2 =100.

This means that x must be $\dpi{100} \small \sqrt{64}$ and y must be $\dpi{100} \small \sqrt{36}$

When we solve for x and y we get:

x = 8

and y = 6.

Therefore, x + y can only be 14.

High and low

Above is a graph which gives the high and low temperatures, in degrees Fahrenheit, over a one week period for Jefferson City. Temperature given in degrees Fahrenheit can be converted to the Celsius scale using the following formula, where and are the temperature expressed in degrees Celsius and degrees Fahrenheit, respectively:

$C = \frac{5}{9} (F-32)$

On how many days of the week shown did the temperature get below $-10 ^{\circ } C$ ?

Two

One

None

Three

Four

Explanation

Convert $-10 ^{\circ } C$ to Fahrenheit by substituting for in the given formula, and solving for :

$C = \frac{5}{9} (F-32)$

$-10 = \frac{5}{9} (F-32)$

$\frac{9} {5} \cdot (-10) = \frac{9} {5} \cdot \frac{5}{9} (F-32)$

$-10 ^{\circ } C$ is equivalent to $14 ^{\circ } F$ .

Below is the line graph, with a horizontal line drawn at the $14 ^{\circ } F$ point. High and low 3

The low temperature is below $14 ^{\circ } F$ on two different days (Wednesday and Thursday, both $13 ^{\circ } F$ ).

represents a positive quantity and represents a negative quantity.

$x^{2} = 60$

$y^{2} = 15$

Evaluate $x^{3}+3x^{2}y+3xy^{2}+ y^{3}$ .

$15\sqrt{15}$

$-15\sqrt{15}$

$405\sqrt{15}$

$-405\sqrt{15}$

Explanation

$x^{3}+3x^{2}y+3xy^{2}+ y^{3}$ can be recognized as the cube of the binomial , so

$x^{3}+3x^{2}y+3xy^{2}+ y^{3} = (x+y)^{3}$

$x^{2} = 60$ ,

so, taking the square root of both sides,

$x = \pm \sqrt{60}$

By the Product of Radicals rule, simplify:

$x = \pm \sqrt{4} \cdot \sqrt{15}$

$= \pm 2\sqrt{15}$

Since represents a positive quantity, we choose

$x = 2\sqrt{15}$

$y^{2} = 15$ ,

so, taking the square root of both sides,

$y = \pm \sqrt{15}$

Since represents a negative quantity.

$y = - \sqrt{15}$

Substituting:

$x^{3}+3x^{2}y+3xy^{2}+ y^{3}$

$= (x+y)^{3}$

$=\left [ (2 \sqrt{15}) + \left ( -\sqrt{15}\right ) \right ] ^{3}$

$= ( \sqrt{15}) ^{3}$

$= ( \sqrt{15}) ^{2} \cdot \sqrt{15}$

$= 15\sqrt{15}$

If ab - bc + d = d2 - c2, then what is the value of a when b is two, c is negative one, and d is zero?

-1/2

-1

-3/2

1/2

3/2

Explanation

ab - bc + d = d2 - c2

We need to substitute values in for b, c, and d, and then solve the equation for a.

a(2) - 2(-1) + 0 = 02 - (-1)2

2a +2 + 0 = 0 - (1)

2a + 2 = -1

2a = -3

a = -3/2

The answer is -3/2.

High and low 2

Above is a graph which gives the high and low temperatures, in degrees Celsius, over a one week period for Washington City. Temperature given in degrees Celsius can be converted to the Fahrenheit scale using the following formula, where and are the temperature expressed in degrees Celsius and degrees Fahrenheit, respectively:

In degrees Fahrenheit, what was the highest temperature of the week shown (Nearest whole degree)?

$79 ^{\circ }F$

$74 ^{\circ }F$

$69 ^{\circ }F$

$64 ^{\circ }F$

$84^{\circ }F$

Explanation

The highest temperature during the week was $26 ^{\circ } C$ , which, as can be seen in the graph, was reached on Tuesday and Friday. Setting in the given formula:

$F =1.8 \cdot 26 + 32$

Rounded to the nearest whole number, this is $79 ^{\circ }F$ .

The projected sales of a movie are given by the function s(p) = 3000/(2_p_ + a) where s is the number of movies sold, in thousands; p is the price per movie, in dollars; and a is a constant. If according to projections, 75,000 cartidges will be sold at $15 each, how many movies are sold at $20 each?

20,000

50,000

60,000

150,000

200,000

Explanation

You set up the equation to solve for a.

75 = 3000/(2(15) + a)

a = 10

You then set up the formula again for each movie costing $20, s(20)= 3000/(2(20) + 10), and solve for x, the number sold, giving you 60.

Twenty percent of a number, , is four greater than the product of that number and six. Which of the following algebraic equations could be used to find ?

Explanation

The "is" in the question means "equal," so whatever comes before "is" must be equal to whatever comes after. We will find an expression for the information before "is" and an expression for the information after "is," and then we will set these two expressions equal.

Twenty percent of a number can be represented as 0.2n, because 20% expressed as a decimal is 0.2, and because twenty percent "of" a number means the product of that number and twenty percent.

Four greater than the product of a number and six means that we must first find the product of that number and six, and then increase this value by 4.

The product of a number and six means that we must multiply this number by six, which can be represented by 6n. Increasing 6n by 4 can be modeled by the expression 6n + 4, or 4 + 6n (because of the commutative property of addition).

Setting the two expressions equal gives us 0.2n = 4 + 6n .

If 11x + 4 = 19x – 12, then what is 2x – 4?

–8

Not possible

Explanation

First solve for x. The first equation would simplify as:

16 = 8x

x = 2

If we plug x = 2 into the second expression:

2(2) – 4 = 0

0 is the correct answer.

How to evaluate algebraic expressions

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