PSAT Math - How to evaluate algebraic expressions

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.

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.

The symbol $\square$ is defined as follows:

$c\square d =(2c)^{d}$

Which of the following is equivalent to $4\square3$ ?

Explanation

Symbols may appear scary, but these problems are often easier than they first appear. Begin by taking the numbers you are given and replacing the letters in the equation with these numbers.

$c\square d = (2c)^{d}$

$4\square 3 = (2(4))^{3}$

We can now solve the equation. Be sure to perform the operations within the parentheses (PEMDAS) first.

$4\square 3=(2(4))^{^{3}}$

$4\square 3=(8)^{3}$

$4\square3 = 512$

The correct answer is 512.

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.

Fahrenheit temperature can be converted to its Celsius equivalent using the formula

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

Celsius temperature can be converted to its Fahrenheit equivalent using the formula

$F = \frac{9}{5} C + 32$ .

Mercury boils at $357 ^{\circ } C$ . To the nearest degree, convert this to its equivalent Fahrenheit temperature.

$675^{\circ } F$

$700^{\circ } F$

$181^{\circ } F$

$166^{\circ } F$

None of the other responses is correct.

Explanation

The formula to use is

$F = \frac{9}{5} C + 32$ .

Substitute 357 for :

$F = \frac{9}{5} (357) + 32 = 642.6 + 32 = 674.6$

To the nearest degree, this is $675^{\circ } F$ .

IF 5x3 = 40, then what is the value of 12x – (x/2)?

Explanation

Use the first equation to solve for x, then plug into the 2nd equation to find a value.

5x3 = 40

x3 = 8

x = 2

12(2) – (2/2) = 24 – 1 = 23

If $m > n$ , which of the following has to be true?

$\frac{m}{2} > \frac{n}{2}$

$m^{2} > n^{2}$

$mn > 0$

$\left | m \right | > \left | n \right |$

$mn > -mn$

Explanation

Plug in numbers for each alternative. If both sides of the inequality $\frac{m}{2} > \frac{n}{2}$ are multiplied by 2, the result is the original inequality, $m > n$ . The other options fail (if confused, try plugging in $m$ as a positive, $n$ as a negative).

How to evaluate algebraic expressions

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