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## Example Questions

### Example Question #1 : How To Evaluate Algebraic Expressions

A total of 150 million votes were tallied in a presidential election. Votes were cast for either Hillary Clinton, Rand Paul, Al Gore, or Gary Johnson. If Clinton received 3 times the number of votes as Johnson, Paul received 30% of the vote, and Gore receieved 30 million total votes, who received the most votes in the election?

**Possible Answers:**

Gary Johnson

Rand Paul

Al Gore

Hillary Clinton

**Correct answer:**

Hillary Clinton

There are a few ways to do this problem, but we will focus on the total number of votes method as follows. First, let Clinton = C, Gore = G, Paul = P, and Johnson = J. We know C + G + P + J = 150 million. We also know that C = 3J. Paul received 30% of the vote which is 150,000,000 * .3 = 45 million votes. Gore received 30 million votes. We can now create an equation with individual totals and substitute 3J for Clinton's vote total:

3J + 30 million + 45 million + J = 150 million

4J = 75 million

J = 18.75 million

Then C = 3J = 56.25 million. So Clinton received 56.25 million votes, Paul received 45 million votes, Gore received 30 million votes, and Johnson received 18.75 million votes. The correct answer is Hillary Clinton.

### Example Question #2 : Evaluating Expressions

Justin makes 61.9% of his free throws. During the season he had 84 free throw attempts. How many of Jason’s shots did *not* go in?

**Possible Answers:**

52

32

40

36

21

**Correct answer:**

32

Find how many free throws Justin made: 84 x 0.619 = 51.99. Since the problem talks free throws, we round to 52 shots went in. To calculate shots missed:

84 – 52 = 32.

### Example Question #2 : Evaluating And Simplifying Expressions

If 5x + 30 = 6 – 7x, then x = ?

**Possible Answers:**

x = 2

x = –18

x = –37

x = –10

x = –2

**Correct answer:**

x = –2

Combine like terms by subtracting 6 from both sides so: 5x + 24 = –7x. Then subtract 5x from both sides: 24 = –12x. Divide both sides by –12 and x = –2.

### Example Question #4 : Evaluating Expressions

If ab - bc + d = d^{2} - c^{2}, then what is the value of a when b is two, c is negative one, and d is zero?

**Possible Answers:**

**Correct answer:**-3/2

ab - bc + d = d^{2} - c^{2}

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

a(2) - 2(-1) + 0 = 0^{2} - (-1)^{2}

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

2a + 2 = -1

2a = -3

a = -3/2

The answer is -3/2.

### Example Question #3 : Evaluating And Simplifying Expressions

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

**Possible Answers:**

2

4

Not possible

0

–8

**Correct answer:**

0

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.

### Example Question #4 : Evaluating And Simplifying Expressions

If x = 2 and y = 3, then evaluate 2(x – 3) + 5y^{2}

**Possible Answers:**

37

43

62

49

52

**Correct answer:**

43

To evaluate an expression we make substitutions into the expression

2(x – 3) + 5y^{2} becomes 2(2 – 3) + 5 * 3^{2} = –2 + 45 = 43

### Example Question #3 : Evaluating And Simplifying Expressions

IF 5x^{3} = 40, then what is the value of 12x – (x/2)?

**Possible Answers:**

10

17

33

24

23

**Correct answer:**

23

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

5x^{3} = 40

x^{3} = 8

x = 2

12(2) – (2/2) = 24 – 1 = 23

### Example Question #4 : Evaluating And Simplifying Expressions

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?

**Possible Answers:**

2

5

7

Cannot be determined

1

**Correct answer:**

2

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.

### Example Question #5 : Evaluating And Simplifying Expressions

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

**Possible Answers:**

22

24

20

23

21

**Correct answer:**

23

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.

### Example Question #1 : Evaluating Expressions

Drink A is 20% water by weight, and drink B is 35% water by weight. How many fluid ounces of drink A must be added to 80 oz of drink B to have a drink whose final proportion of water is 30%?

**Possible Answers:**

40 fl oz

60 fl oz

85 fl oz

75 fl oz

50 fl oz

**Correct answer:**

40 fl oz

It's easiest if we convert all percentages to actual oz of water for each step here. As such, 35% of the 80 oz of drink B would have 0.35(80) = 28 oz of water in it.

We can set up an equation that similarly converts each "percentage of a fixed weight of liquid" to ensure that our final weight is equivalent to 30% to the sum of drink A and B. On the left side, the fixed values are the percentages of each drink individually, and on the right side is what the question *requires *as a fixed percentage of the final weight:

0.2(A) + 0.35(80) = 0.3(A + 80)

0.2A + 28 = 0.3A + 24

**A = 40 **

Solving for A, we get 40 oz of A that must be poured into B. You may plug this back into the equation to check it.

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