Expressions - SAT Math

Card 0 of 20

Question

Simplify the following rational expression: (9x - 2)/(x2) MINUS (6x - 8)/(x2)

Answer

Since both expressions have a common denominator, x2, we can just recopy the denominator and focus on the numerators. We get (9x - 2) - (6x - 8). We must distribute the negative sign over the 6x - 8 expression which gives us 9x - 2 - 6x + 8 ( -2 minus a -8 gives a +6 since a negative and negative make a positive). The numerator is therefore 3x + 6.

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Question

Simplify the following rational expression:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

Answer

Since both fractions in the expression have a common denominator of $x^{2}$ , we can combine like terms into a single numerator over the denominator:

$\frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}$

$=\frac{(7x-18)+(6x-14)}{x^{2}}$

$=\frac{13x-32}{x^{2}}$

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Question

Simplify the following rational expression:

$\frac{5x-5}{2x^{2}} + \frac{7x+9}{2x^{2}}$

Answer

Since both rational terms in the expression have the common denominator $2x^{2}$ , combine the numerators and simplify like terms:

$\frac{5x-5}{2x^{2}} + \frac{7x+9}{2x^{2}}$

$=\frac{(5x-5)+(7x+9)}{2x^{2}}$

$=\frac{12x+4}{2x^{2}}$

$=\frac{6x+2}{x^2}$

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Question

Simplify the following expression:

$\frac{10x-9}{x^{3}}+\frac{11x+12}{x^{3}}$

Answer

Since both terms in the expression have the common denominator $x^{3}$ , combine the fractions and simplify the numerators:

$\frac{10x-9}{x^{3}}+\frac{11x+12}{x^{3}}$

$=\frac{(10x-9)+(11x+12)}{x^{3}}$

$=\frac{21x+3}{x^{3}}$

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Question

Simplify the expression.

$\frac{x}{2x+4}+\frac{x}{x+2}$

Answer

To add rational expressions, first find the least common denominator. Because the denominator of the first fraction factors to 2(x+2), it is clear that this is the common denominator. Therefore, multiply the numerator and denominator of the second fraction by 2.

$\frac{x}{2x+4}+\frac{x}{x+2}$

$\frac{x}{2x+4}+\frac{(2)x}{(2)(x+2)}$

$\frac{x}{2x+4}+\frac{2x}{2x+4}$

$\frac{3x}{2x+4}$

This is the most simplified version of the rational expression.

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Question

Simplify the following:

$\frac{x+3}{x-2}+\frac{x-2}{x+3}$

Answer

To simplify the following, a common denominator must be achieved. In this case, the first term must be multiplied by (x+2) in both the numerator and denominator and likewise with the second term with (x-3).

$\frac{(x+3)(x+3)}{(x-2)(x+3)}+\frac{(x-2)(x-2)}{(x+3)(x-2)}$

$\frac{(x+3)^2+(x-2)^2}{(x+3)(x-2)}$

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Question

Which of the following is equivalent to $\dpi{100} \frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ ? Assume that denominators are always nonzero.

Answer

We will need to simplify the expression $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We can think of this as a large fraction with a numerator of $\frac{1}{t}-\frac{1}{x}$ and a denominator of $\dpi{100} x-t$ .

In order to simplify the numerator, we will need to combine the two fractions. When adding or subtracting fractions, we must have a common denominator. $\frac{1}{t}$ has a denominator of $\dpi{100} t$ , and $\dpi{100} -\frac{1}{x}$ has a denominator of $\dpi{100} x$ . The least common denominator that these two fractions have in common is $\dpi{100} xt$ . Thus, we are going to write equivalent fractions with denominators of $\dpi{100} xt$ .

In order to convert the fraction $\dpi{100} \frac{1}{t}$ to a denominator with $\dpi{100} xt$ , we will need to multiply the top and bottom by $\dpi{100} x$ .

$\frac{1}{t}=\frac{1\cdot x}{t\cdot x}=\frac{x}{xt}$

Similarly, we will multiply the top and bottom of $\dpi{100} -\frac{1}{x}$ by $\dpi{100} t$ .

$\frac{1}{x}=\frac{1\cdot t}{x\cdot t}=\frac{t}{xt}$

We can now rewrite $\frac{1}{t}-\frac{1}{x}$ as follows:

$\frac{1}{t}-\frac{1}{x}$ = $\frac{x}{xt}-\frac{t}{xt}=\frac{x-t}{xt}$

Let's go back to the original fraction $\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ . We will now rewrite the numerator:

$\frac{(\frac{1}{t}-\frac{1}{x})}{x-t}$ = $\frac{\frac{x-t}{xt}}{x-t}$

To simplify this further, we can think of $\frac{\frac{x-t}{xt}}{x-t}$ as the same as $\frac{x-t}{xt}\div (x-t)$ . When we divide a fraction by another quantity, this is the same as multiplying the fraction by the reciprocal of that quantity. In other words, $a\div b=a\cdot \frac{1}{b}$ .

$\frac{x-t}{xt}\div (x-t)$ = $\frac{x-t}{xt}\cdot \frac{1}{x-t}$ $=\frac{x-t}{xt(x-t)}$ $= \frac{1}{xt}$

Lastly, we will use the property of exponents which states that, in general, $\frac{1}{a}=a^{-1}$ .

$\frac{1}{xt}=(xt)^{-1}$

The answer is $(xt)^{-1}$ .

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Question

Tim is two years older than his twin sisters, Rachel and Claire. The sum of their ages is 65. How old is Tim?

Answer

The answer is 23.

Since Rachel and Claire are twins they are the same age. We will use the variable r to represent both Rachel and Claire's ages.

From the question we can form two equations. They are:

t = r + 2 and 65 = t + 2r

lets plug the first equation into the second to solve for r.

65 = (r + 2) + 2r

65 = 3r +2

63 = 3r

r = 21 This means Rachel and Claire are 21 years old. Plug this into the equation so

t = 23 Tim is 23 years old.

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Question

If f(x) = 5x/a, and g(x) = ax/10, what is f(2) + g(5)?

Answer

(20+a2) / 2a is the answer. Find a common denominator to add the 2 fractions after plugging in the values. So we get 10/a + a/2 after plugging in the values. The lowest common denominator would be 2a. Mutiply 10/a by 2/2, and multiply a/2 by a/a. The new equation appears as 20/2a + a2/2a. Add them together to get (20 + a2)/2a.

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Question

(√(4) + √(16))2 = ?

Answer

If we use order of operations (PEMDAS), the answer is 36.

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Question

x * y = –a, a – x = 2a. What is y?

Answer

Use the second equation to find x in terms of a. Plug it back in the second equation, that will give you 1 = y.

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Question

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?

Answer

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.

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Question

Half of one hundred divided by five and multiplied by one-tenth is __________.

Answer

Let's take this step by step. "Half of one hundred" is 100/2 = 50. Then "half of one hundred divided by five" is 50/5 = 10. "Multiplied by one-tenth" really is the same as dividing by ten, so the last step gives us 10/10 = 1.

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Question

Let x&y be defined as (x – y)xy . What is the value of –1_&_2?

Answer

We are told that x&y = (x – y)xy .

–1&2 = (–1 – 2)(–1)(2) = (–3)–2

To simplify this, we can make use of the property of exponents which states that a– b = 1/(ab ).

(–3)–2 = 1/(–3)2 = 1/9

The answer is 1/9.

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Question

If 18 – w is 8 less than 32, what is the value of $\dpi{100} \small -\frac{1}{3}w$ ?

Answer

We need to rewrite this problem in mathematic terms.

8 less than 32 can be written as 32 – 8

so we will get the equation

18 – w = 32 – 8.

Now, we need to solve for w.

–w = 32 – 8 – 18

–w = 6

w = –6

Find the value of the given expression, $\dpi{100} \small -\frac{1}{3}w$ , by plugging in –6 for w.

$\dpi{100} \small -\left (\frac{1}{3} \right )\left ( -6 \right )=2$

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Question

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?

Answer

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.

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Question

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

Answer

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.

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Question

If x = 2 and y = 3, then evaluate 2(x – 3) + 5y2

Answer

To evaluate an expression we make substitutions into the expression

2(x – 3) + 5y2 becomes 2(2 – 3) + 5 * 32 = –2 + 45 = 43

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Question

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

Answer

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

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Question

A rowing team paddles upstream at a rate of 10 miles every 2 hours and downstream at a rate of 27 miles every 3 hours. Assuming they are paddling at the same rate up and downstream, what is the speed of the water?

Answer

Upstream: p – w = (10/2) or p – w = 5 miles/hour

Downstream: p + w = (27/3) or p + w = 9 miles/hour

Then we add the two equations together to cancel out the w's. After adding we see

2p = 14

p = 7 miles/hour where p is the rate of the paddling. We plug p into the equation to find

w = 2 miles/hour where w is the rate of the stream's water.

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