ACT Average Problems: Find a Missing Value from the Mean

To find one missing value when you know the arithmetic mean, multiply the mean by the number of values to recover the total. Then subtract the sum of the known values. Count the missing value as part of the group.

$$\text{missing value}=(\text{mean}\times\text{count})-\text{known total}$$

ACT includes averages among the essential math skills used in its multi-step problems, according to its official exam content guide. The questions below are original Kibo practice examples, not official ACT items.

Why rebuilding the total works

The arithmetic mean is the sum of the values divided by how many values there are. OpenStax’s explanation of measures of center gives this definition. Here, “average” means arithmetic mean.

$$\text{mean}=\frac{\text{total}}{\text{count}}$$

Multiplying both sides by the count reverses that division and gives the required total. The count must be positive, and a given mean must be exact to determine an exact missing value. A rounded mean may leave more than one possible answer.

1. Count the whole group. Include the unknown value and check whether the question adds or removes anything.
2. Recover the required total. Multiply that group’s count by its mean.
3. Subtract what you know. Add the known values, then subtract their sum from the required total.
4. Check the original mean. Add every value, including your answer, and divide by the original count.

Example: one number is missing

Five values have a mean of $18$. Four of them are $12, 16, 20, 23$. What is the fifth value?

All five values must add to $5\times18=90$. The known values add to $12+16+20+23=71$. The missing value is therefore $90-71=19$.

$$\frac{12+16+20+23+19}{5}=\frac{90}{5}=18$$

The calculation $71\div4=17.75$ finds the mean of the four known values. Using that as the fifth value would leave the overall mean at $17.75$, which misses the required $18$.

You can also write $\frac{71+x}{5}=18$, then multiply both sides by $5$ before isolating $x$. Our guide to solving linear equations explains the same balance-preserving operations.

Example: adding a value changes the count

Four notebooks have a mean of $26$ used pages. A fifth notebook brings the mean to $28$ used pages. How many used pages are in the fifth notebook?

The original total is $4\times26=104$ pages. The new total is $5\times28=140$ pages. Subtracting totals gives $140-104=36$ used pages in the added notebook.

$$\frac{104+36}{5}=28$$

Subtracting the means gives $28-26=2$, which measures the increase in the average. It does not tell you how many pages the new notebook contributes. Each mean describes a different-sized group.

Example: removing a value changes the count too

Six measurements have a mean of $14$. After one measurement is removed, the remaining five have a mean of $12$. What measurement was removed?

The original total is $6\times14=84$. The remaining total is $5\times12=60$. The removed measurement is $84-60=24$. Check: $(84-24)\div5=12$.

Removing $24$, a value above the old mean of $14$, makes the mean fall. This direction check can catch a subtraction mistake, though you still need the calculation for the exact value.

Common mistakes to catch before choosing an answer

• Dividing too early. Dividing the known sum by the known count describes only the known values. Recover the whole group’s total before solving for its missing contribution.
• Keeping the old count. After an addition or removal, label the old and new counts separately. Multiply each mean by the count it belongs to.
• Subtracting two averages. Compare the two totals to find the added or removed value.
• Assuming two unknowns must be equal. A mean can determine their sum without determining either value separately. Another condition is needed to separate them.
• Ignoring the context. A negative or fractional answer can be valid for measurements, but a count of objects must satisfy the conditions in the question.

Try five original practice questions

For each question, write the required total before calculating the answer. Treat every stated mean as exact, and keep the solutions covered until you have checked your work.

1. The mean of $8, 11, 15, 19, x$ is $14$. Find $x$.
2. Four lengths have a mean of $6.5$ centimeters. Three lengths are $4.2$, $7.1$, and $8.3$ centimeters. Find the fourth length.
3. Five boxes contain a mean of $12$ pencils. Adding a sixth box raises the mean to $15$ pencils. How many pencils are in the sixth box?
4. Eight values have a mean of $21$. Removing one value makes the remaining mean $20$. What value was removed?
5. The mean of $5, 7, 10, a, b$ is $9$. What is $a+b$? Can either individual value be determined from this information alone?

Answers and checks

1. $x=17$. The required total is $5\times14=70$, and the known total is $8+11+15+19=53$. Subtract: $70-53=17$. Check: $(53+17)\div5=14$.
2. $6.4$ centimeters. The required total is $4\times6.5=26$; the known lengths add to $19.6$. Subtract: $26-19.6=6.4$. Check: $(19.6+6.4)\div4=6.5$.
3. $30$ pencils. Subtract the old total from the new total: $6\times15-5\times12=90-60=30$. Check: $(60+30)\div6=15$.
4. $28$. Subtract the remaining total from the original total: $8\times21-7\times20=168-140=28$. Check: $(168-28)\div7=20$.
5. $a+b=23$, since $5\times9-(5+7+10)=45-22=23$. Check: $(22+23)\div5=9$. Neither individual value is fixed: both $a=11, b=12$ and $a=10, b=13$ fit the information.

Use the mistake to choose your next practice step

If a question went wrong, note whether you used the wrong count, treated the mean as the total, or made an arithmetic error. Our ACT and SAT error-log template can help you turn that observation into a specific next step. Rework one missed question with the solution covered, then try a fresh problem.

Try a free ACT practice question with Kibo when you are ready to continue.

Prepared with AI assistance. These original examples were checked by separate calculations; no human review is claimed. Sources checked October 7, 2026. ACT is a registered trademark of ACT, Inc., which is not affiliated with or endorsing this guide.