Problem
Recover a missing count from the mean
A table lists test scores and how many students got each. Five scored 10, two scored 15, and the 20-point cell is blank. The class average is 16 points. Find how many students scored 20.
Reasoning · Grade 5-2 Averages
Your answer
Watch out Multiplying the average by the class size fails: the class size is unknown too.
How to solve
Strategy Organize Information in More Ways — The missing count sits on both sides of the average — it changes the total points AND it changes the number of students — so chasing it straight through the division is awkward. The fix is to re-record every score not as itself but as its distance from 16, the average. Then the average becomes a balance point: everything sitting below 16 has to be balanced exactly by everything sitting above 16. The 10s and 15s are all below and are fully known, so they give a fixed amount to balance, and each 20 supplies a fixed amount of the balancing. Watching the units (points and students) keeps the setup honest, and a direct recomputation of the average at the end confirms the count.
1STEP 1
Read the table and name what is missing
What is missing is how many scored 20.
class size = 5 + 2 + (students who scored 20)
A frequency table is just a shorthand list: '5' under '10' means the score 10 appears 5 times. Writing the class size as 7 plus the missing number is the reminder that the blank affects the head count too, which is the mistake this problem is built to catch.
6.SP.B.5Analyze The Units2STEP 2
Write down what the average promises
The average is total points over students.
(total points) = 16 points/student × (number of students)
The average is one number standing in for the whole data set — the level everything would sit at after fair sharing. Reading it that way turns it from a formula into something that can be pushed around like any total.
6.SP.A.3Analyze The Units3STEP 3
Re-measure every score from the average instead of from zero
Re-measure each score from the average.
16 - 10 = 6 below, 16 - 15 = 1 below, 20 - 16 = 4 above
Sharing equally means the students above the average give away exactly what the students below the average need. Measuring from 16 makes that give-and-take visible, and it makes the unknown count appear in only one place instead of two.
6.SP.A.3Organize Information In More WaysMeasuring every score from the average instead of from zero makes the unknown count appear in only one place.
Why?
The average is the level everything sits at after fair sharing, so what the scores above give away exactly matches what the scores below need.
🧱The mean is the total shared out equallyThe average is the total divided by how many there are, so the total is the average times the count — and everything above that level gives away exactly what everything below it needs.
Why?
Sliding every score down by the same average leaves all the gaps between them untouched, so nothing is lost by re-measuring.
🧱Shift both by the same amount and the difference holdsSlide two points the same distance along the line and the gap between them never changes.
4STEP 4
Add up everything that is below the average
The shortfall totals 32 points.
5 × 6 = 30, 2 × 1 = 2, 30 + 2 = 32 points below
This part is entirely known — no blank cell is involved — so the whole shortfall can be computed with two multiplications and one addition before anything about the 20s is decided.
4.OA.A.3Organize Information In More Ways5STEP 5
See how many 20s it takes to make up that shortfall
Each 20 adds 4, so 8 students.
32 ÷ 4 = 8
'How many 4s make 32?' is a division within 100 — the balance idea has reduced the whole problem to one basic fact, with no equation to rearrange.
4.OA.A.3Organize Information In More Ways6STEP 6
Check by rebuilding the average from scratch
Rebuilding the average gives 16.
10 × 5 + 15 × 2 + 20 × 8 = 50 + 30 + 160 = 240, 240 ÷ 15 = 16
Recomputing the average straight from the finished table is the surest check there is, because it uses the original definition rather than the shortcut that produced the answer.
6.SP.B.5Guess And CheckAnswer
8 students
32 ÷ 4 = 8
The answer is a count of students, which is what was asked, and 8 is a whole number — a fractional answer would have meant a mistake. Its size makes sense too: the average 16 is closer to 20 than it is to 10, so the 20-point group has to be the biggest one, and 8 does outnumber the 5 + 2 = 7 students below the average. Testing nearby counts shows how tightly 8 is pinned: with 5 students at 20 the class average would be 180 divided by 12 = 15, too low, and with 10 students at 20 it would be 280 divided by 17, about 16.5, too high. Only 8 lands exactly on 16, and the direct recomputation 240 divided by 15 = 16 confirms it.
Takeaway
Measure every score from the average instead of from zero — then the points below and the points above must balance, and the blank in the table falls right out.
- Read the table and name what is missing
- Write down what the average promises
- Re-measure every score from the average instead of from zero
- Add up everything that is below the average
- See how many 20s it takes to make up that shortfall
- Check by rebuilding the average from scratch