Calculating Averages - ISEE Middle Level Quantitative Reasoning

1. The mean of $5$ numbers is $6$. Four numbers are $4, 5, 7, 8$. What is the fifth number?

   $6$. Total sum is $6 \times 5 = 30$; subtract sum of known numbers $4+5+7+8=24$ to find the fifth.

2. What is the mean of the data set $-3, -1, 2, 6$?

   $1$. Sum the values $-3+(-1)+2+6=4$ and divide by $4$ for the average.

3. What is the weighted mean of $80$ (weight $2$) and $90$ (weight $3$)?

   $86$. Compute $(80 \times 2 + 90 \times 3)/(2+3)$ to account for the weights.

4. What is the mean of the data set $-2, 0, 6$?

   $\frac{4}{3}$. Sum the values $-2+0+6=4$ and divide by $3$ to calculate the mean.

5. What is the mean of the data set $2, 3, 5, 10$?

   $5$. Sum the values $2+3+5+10=20$ and divide by $4$ to compute the average.

6. What is the mean of the data set $0.2, 0.3, 0.5$?

   $\frac{1}{3}$. Sum the decimals $0.2+0.3+0.5=1.0$ and divide by $3$ to find the mean.

7. State the mean of two numbers $a$ and $b$ using a formula.

   $\frac{a+b}{2}$. The mean of two numbers is their sum divided by $2$.

8. What is the mean of the data set $4, 6, 10$?

   $\frac{20}{3}$. Sum the values $4+6+10=20$ and divide by $3$ to find the average.

9. A set has mean $9$ for $4$ numbers. What value must be added to make the new mean $10$?

   $14$. Original sum is $9 \times 4 = 36$; for new mean $10$ with $5$ numbers, new sum is $50$, so add $14$.

10. Find and correct the mean: For $2, 4, 6, 8$, a student wrote $\frac{2+4+6+8}{3}=7$.

    $\frac{2+4+6+8}{4}=5$. The student divided by $3$ instead of $4$; correct by using the proper count of values.

11. What is the mean of the consecutive integers from $1$ to $9$ inclusive?

    $5$. The mean of consecutive integers from $1$ to $9$ is the middle value, which is $5$.

12. What is the mean of consecutive integers $5, 6, 7, 8, 9$?

    $7$. Sum the integers $5+6+7+8+9=35$ and divide by $5$ for the mean.

13. What is the mean if the sum of $8$ numbers is $60$?

    $\frac{15}{2}$. Divide the total sum $60$ by the number of values $8$ to find the average.

14. What is the mean of the data set $0, 5, 10, 15$?

    $\frac{15}{2}$. Sum the values $0+5+10+15=30$ and divide by $4$ to find the average.

15. Identify the mean of the data set $\frac{1}{2}, \frac{1}{2}, \frac{3}{2}$.

    $\frac{5}{6}$. Sum the fractions $\frac{1}{2}+\frac{1}{2}+\frac{3}{2}=\frac{5}{2}$ and divide by $3$.

16. What is the mean of the data set $7, 7, 7, 7, 7$?

    $7$. When all data points are identical, the mean equals that common value.

17. What is the mean of the data set $1, 2, 3, 4, 10$?

    $4$. Sum the values $1+2+3+4+10=20$ and divide by $5$ for the mean.

18. A set has mean $8$ for $5$ numbers. What is the new mean after removing a value of $3$?

    $\frac{37}{4}$. Original sum is $8 \times 5 = 40$; subtract $3$ for new sum $37$, divide by $4$.

19. What is the mean of the data set $12, 15, 18$?

    $15$. Sum the values $12+15+18=45$ and divide by $3$ to obtain the mean.

20. A set has mean $10$ for $6$ numbers. What is the new mean after adding $16$?

    $\frac{76}{7}$. Original sum is $10 \times 6 = 60$; add $16$ for new sum $76$, divide by $7$.

21. What is the sum of a data set with mean $12$ and $5$ values?

    $60$. Multiply the mean $12$ by the number of values $5$ to obtain the total sum.

22. State the formula for a weighted mean with values $x_i$ and weights $w_i$.

    $\text{weighted mean}=\frac{\sum w_i x_i}{\sum w_i}$. The weighted mean is the sum of each value multiplied by its weight, divided by the total weight.

23. What is the weighted mean of scores $70, 80, 90$ with weights $1, 2, 1$?

    $80$. Calculate $(70 \times 1 + 80 \times 2 + 90 \times 1)/(1+2+1)$ using the given weights.

24. State the formula for the arithmetic mean of $n$ numbers $x_1, x_2, \dots, x_n$.

    $\text{mean}=\frac{x_1+x_2+\cdots+x_n}{n}$. The arithmetic mean is the sum of all values divided by the number of values.

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