Average Problems - SSAT Middle Level Quantitative

1. State the formula for the average of $n$ numbers with sum $S$.

   $A=\frac{S}{n}$. The average is computed by dividing the total sum $S$ by the number of elements $n$.

2. State the weighted average formula for values $a,b$ with counts $m,n$.

   $\frac{ma+nb}{m+n}$. The formula weights each value by its count, summing the products and dividing by total count $m+n$.

3. How much does the average change if one value $x$ is increased by $5$ in a set of $n$ numbers?

   Increase of $\frac{5}{n}$. Increasing one value by $5$ raises the total sum by $5$, so the average increases by $5/n$.

4. What is the average of consecutive integers $k,k+1,k+2,k+3,k+4$?

   $k+2$. For five consecutive integers, the average is the middle term, which is $k+2$.

5. What is the average of two numbers $a$ and $b$?

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

6. What is the new average if the average of $10$ numbers is $12$ and one number $22$ is removed?

   $\frac{98}{9}$. Original sum is $10 \times 12 = 120$; subtract $22$ to get $98$, then divide by $9$ for new average.

7. Find the missing number: the average of $6,10,x$ is $9$.

   $x=11$. Set up $(6 + 10 + x)/3 = 9$, solve for $x$ by multiplying both sides by $3$ and subtracting $16$.

8. What is the new average if the average of $8$ numbers is $9$ and a number $17$ is added?

   $\frac{89}{9}$. Original sum is $8 \times 9 = 72$; add $17$ to get $89$, then divide by $9$ for new average.

9. What is the average of $12,18,20$?

   $\frac{50}{3}$. Sum the numbers $12 + 18 + 20 = 50$ and divide by $3$ to find the average.

10. What is the average of $5$ numbers if their sum is $65$?

    $13$. Divide the total sum $65$ by the count $5$ to compute the average.

11. Find the combined average: $4$ students average $80$ and $6$ students average $70$.

    $74$. Compute total sum as $(4 \times 80) + (6 \times 70) = 740$, then divide by $10$ for combined average.

12. State the formula for the sum $S$ of $n$ numbers with average $A$.

    $S=nA$. The sum is obtained by multiplying the average $A$ by the number of elements $n$.

13. What is the average of $6$ numbers if adding $3$ to each number increases the average to $14$?

    $11$. Adding $3$ to each number increases the average by $3$, so subtract $3$ from $14$ to find the original average.

14. How many numbers are in a set with sum $96$ and average $12$?

    $8$. Divide the sum $96$ by the average $12$ to determine the number of elements.

15. Identify the correct expression for the average of $n$ numbers: sum $S$ divided by what quantity?

    Divide by $n$. The average is the total sum $S$ divided by the number of elements $n$.

16. What is the average of $7$ numbers if multiplying each number by $2$ makes the new average $18$?

    $9$. Multiplying each number by $2$ multiplies the average by $2$, so divide $18$ by $2$ to find the original average.

17. Find $x$: the average of $x$ and $14$ is $10$.

    $x=6$. Set up $(x + 14)/2 = 10$, multiply both sides by $2$, and subtract $14$ to solve for $x$.

18. What is the total sum of scores if $5$ tests have an average of $84$?

    $420$. Multiply the average $84$ by the number of tests $5$ to find the total sum.

19. If the average of $4$ numbers is $11$, what is the average after each number is decreased by $2$?

    $9$. Decreasing each number by $2$ decreases the average by $2$, so subtract from $11$.

20. What is the sum of $8$ numbers if their average is $15$?

    $120$. Multiply the average $15$ by the count $8$ to determine the total sum.

21. If the average of $9$ numbers is $6$, what is their sum?

    $54$. Multiply the average $6$ by the count $9$ to compute the total sum.

22. Find the combined average: $3$ items average $10$ and $2$ items average $16$.

    $\frac{62}{5}$. Compute total sum as $(3 \times 10) + (2 \times 16) = 62$, then divide by $5$ for combined average.

23. Find the missing number: the average of $4$ numbers is $7$ and three are $5,6,8$.

    $9$. Calculate required sum as $4 \times 7 = 28$, then subtract known sum $5 + 6 + 8 = 19$ to find the missing value.

24. Find the missing score: average of $5$ tests is $90$; four scores are $88,92,85,95$.

    $90$. Required sum is $5 \times 90 = 450$; subtract known sum $88 + 92 + 85 + 95 = 360$ to find missing score.

25. If $5$ numbers have average $8$, what must the $6$th number be to make the new average $9$?

    $14$. Original sum is $5 \times 8 = 40$; new sum needed is $6 \times 9 = 54$, so subtract to find the sixth number.

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