## Example Questions

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### 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?

Gary Johnson

Al Gore

Hillary Clinton

Rand Paul

Hillary Clinton

Explanation:

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

### Example Question #1 : 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?

40

36

21

32

52

32

Explanation:

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 #1 : How To Evaluate Algebraic Expressions

If 5x + 30 = 6 – 7x, then x = ?

x = 2

x = –10

x = –37

x = –2

x = –18

x = –2

Explanation:

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 #2 : Evaluating And Simplifying Expressions

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
3/2
-3/2
1/2
-1
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

### Example Question #1 : Evaluating And Simplifying Expressions

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

2

0

–8

Not possible

4

0

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

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

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

52

62

43

49

37

43

Explanation:

To evaluate an expression we make substitutions into the expression

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

### Example Question #2 : How To Evaluate Algebraic Expressions

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

10

24

33

17

23

23

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

### Example Question #21 : 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?

5

7

1

2

Cannot be determined

2

Explanation:

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 #1 : Evaluating Expressions

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

24

23

21

20

22

23

Explanation:

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 : How To Evaluate Algebraic 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%?

60 fl oz

75 fl oz

85 fl oz

40 fl oz

50 fl oz

40 fl oz

Explanation:

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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