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SAT Math : How to find arithmetic mean

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Example Questions

Example Question #31 : Arithmetic Mean

Find the arithmetic mean of the following set of numbers:

$\dpi{100} \small 6,8,12,16,18$

Possible Answers:

$\dpi{100} \small 16$

$\dpi{100} \small 8$

$\dpi{100} \small 6$

$\dpi{100} \small 18$

$\dpi{100} \small 12$

Correct answer:

$\dpi{100} \small 12$

Explanation:

The mean is adding all the numbers then dividing by the number of items.

In this case we have, $\dpi{100} \small x-b,x-y,x,x+b,x+y$

where $\dpi{100} \small x=12, b= 6, y = 4$

When this happens, the average is $\dpi{100} \small x$

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Example Question #61 : Data Analysis

$\small \textup{The average (arithmetic mean) of}\:x,\,y,\,\textup{and}\,z\,\textup{is 8.}$

$\small \textup{If}\,y=5x\,\textup{and}\,z=6,\,\textup{what is the value of}\,x?$

Possible Answers:

$\small 6$

$\small 5$

$\small 8$

$\small 15$

$\small 3$

Correct answer:

$\small 3$

Explanation:

Arithmetic mean: $\small \left ( x+y+z\right )\div3=8$ given: $\small z=6\;\;y=5x$

Solve: $\small \left ( x+5x+6\right )\div3=8$

$\small 6x+6=24$

$\small 6x=18$

$\small x=3$

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Example Question #31 : Arithmetic Mean

A class consists of 10 students. The teacher wrote the test and expects the class to have an average score of at least 80. The average score of 9 people was calculated and it was 78. What score does the last person have to get in order to reach the teacher's standard?

Possible Answers:

$98$

$82$

$94$

$100$

$88$

Correct answer:

$98$

Explanation:

The total score of those 9 students is $9\cdot 78=702$ .

The standard score for the entire class was $10\cdot 80=800$ .

Therefore, the last person needs a score of $800-702=98$ .

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Example Question #31 : Arithmetic Mean

What is the average (mean) of

$2x+7, 3x-10, and\ -14x+12$

Possible Answers:

4x+7

-3(x+1)

3(x+1)

-3(x-1)

3(x-1)

Correct answer:

-3(x-1)

Explanation:

By adding all of the expressions we get -9x+9

which averages to -3x+3 ,

which will simplify to -3(x-1) .

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Example Question #31 : How To Find Arithmetic Mean

The mean of a series of 6 numbers (11, 12, 14, 17, 18, x) is 15. Solve for x.

Possible Answers:

Correct answer:

Explanation:

If the mean of 6 numbers is 15, that implies that the sum of the numbers is equal to 90 (6 x 15).

Therefore, $\small 90=(11+12+14+17+18+x)$ or $\small 90=(72+x)$ $\small 90-72=x=18$

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Example Question #33 : Arithmetic Mean

A police officer walked into a room full of suspects and turn out their pockets. The chart below shows the number of coins each man had.

Screen_shot_2013-09-04_at_10.20.59_am

The police claim the man with the amount of money closest to the arithmetic mean of the group is guilty. Who is it?

Possible Answers:

Suspect 1

Suspect 4

Suspect 3

Suspect 2

Suspect 6

Correct answer:

Suspect 2

Explanation:

Screen_shot_2013-09-04_at_10.26.20_am

Summing total money and dividing by the number of suspects gives us an average of approximately $2.73. By comparing it to the amounts held by the suspects, we can see that Suspect #2 is guilty.

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Example Question #32 : Statistics

The average temperature for 6 days in July is $87^\circ F$ . If on the seventh day, the temperature is $54^\circ F$ , what is the average temperature over those seven days?

Possible Answers:

$82.3^\circ F$

$70.5^\circ F$

There is not enough information to answer this question.

None of the available answers.

$76.0^\circ F$

Correct answer:

$82.3^\circ F$

Explanation:

In order to calculate mean, we can act as though the temperature was exactly $87^\circ F$ for the first six days and $54^\circ F$ on the seventh day. The mean can be expressed:

$\frac{(6\times 87)+54}{7}=\frac{522+54}{7}=\frac{576}{7}\approx 82.3^\circ F$

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Example Question #34 : Arithmetic Mean

If the average of $b + 2$ is equal to the average of $b + 4 + 6$ . What is the value of $b$ ?

Possible Answers:

Correct answer:

Explanation:

$\frac{b+2}{2}=\frac{b+4+6}{3}$

$3b+6=2b+8+12$

$b=14$

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Example Question #32 : How To Find Arithmetic Mean

The test average for a class of 13 students is 87%. If two students take make-up tests the next day and score 97% and 56% respectively, what is the new average?

Possible Answers:

$84.8\%$

$85.6\%$

The average does not change

$80\%$

$94.5\%$

Correct answer:

$85.6\%$

Explanation:

In order to calculate the new average, we can act as though each of the thirteen students who originally took the test each earned 87%:

$\frac{(13\times87)+97+56}{15}=85.6\%$

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Example Question #62 : Data Analysis

Tom makes a bet with his friend Bob. Bob wagers $100, but Tom decides to make him a deal. If Bob wins, Tom will pay him $100. If Tom wins, Bob will instead pay $1 for the first day, then every day for 10 days he must match the total amount paid so far.

For the first day, Bob will pay $1. The second day, he will pay an additional dollar, since he has only paid $1 thus far. The third day he will pay $2, the fourth $4, and so on.

If Bob has a 75% chance to win, which of the following is the closest to the amount of money Tom will make on average with this bet?

Possible Answers:

$\$ 100$

$\$0$

$\$50$

- $\$25$

$\$25$

Correct answer:

$\$50$

Explanation:

We are asked to find out how much Tom would win on average. Since Bob has a 75% chance to win, Tom must have a 25% chance to win. The amount Tom will lose is given to us: $100. We now need to find how much he will win.

The first day, Bob will only pay $1. Similarly for the second day. The third day Bob will now have to pay $2. The fourth day, $4. We notice that each day we will be doubling how much he pays (since he is matching how much was paid previously). Thus he is paying:

$1+2^0+2^1+2^2+2^3+2^4+...+2^{8} = 1+1+2+4+8+16+...+256$

Now, there are a couple ways to deal with this series. There are only 10 terms, so we can simply add them up to get our answer. Another method would be looking at it as a geometric series. If you recall, the sum of a geometric series (series that have terms differentiated by a constant multiplier) is given by: $a\cdot \frac{1-r^{n}}{1-r}$ where is the initial value, is the ratio between any 2 terms, and is the number of terms. Here we just have a=1 , since the first term is a 1. Each term is half the previous term, so r=2 , and we have 9 terms ( $2^{0}$ to $2^{8}$ ) so n=9 .

Hence this would be: $a \cdot \frac{1-r^{n}}{1-r}(1) = (1)\cdot \frac{1-(2)^{(9)}}{1-(2)}$

and our answer would be $1+ \frac{-511}{-1} = 1+511 = 512$

We now have that Bob will have to pay $512 if Tom wins, while Tom will have to pay $100 if Bob wins. We finally take into account the odds of winning. 25% of the time, Tom will win $512. 75% of the time he will lose $100. We can more easily interpret this with fractions: 1/4 chance to win, 3/4 chance to lose.

Now we simply keep in mind how averages work: The numerator is going to be the sum of all the cases, while the denominator is the number of cases. In this situation, we have 4 cases. 3 of them Tom loses, 1 he wins. Thus we have:

$\frac{512 - 100 - 100 - 100}{4} = \frac{512-300}{4} = \frac{212}{4} = 53$

So on average, Tom will make $53 with this bet. The closest answer to this is $50.

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SAT Math : How to find arithmetic mean

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #31 : Arithmetic Mean

Example Question #61 : Data Analysis

Example Question #31 : Arithmetic Mean

Example Question #31 : Arithmetic Mean

Example Question #31 : How To Find Arithmetic Mean

Example Question #33 : Arithmetic Mean

Example Question #32 : Statistics

Example Question #34 : Arithmetic Mean

Example Question #32 : How To Find Arithmetic Mean

Example Question #62 : Data Analysis

All SAT Math Resources

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