Problem
Move the mean by adding a value
The first 7 math tests average 80 points. One more test makes 8 in all. The average of all 8 must be 2 points higher. Find the score needed on the 8th test.
Reasoning · Grade 5-2 Averages
Your answer
Watch out Scoring 82 seems enough, but the score must also lift the other seven.
How to solve
Strategy Work Backwards — The problem hands me the finished state — an average of 82 over 8 tests — and asks for the ingredient that produces it, which is exactly the situation Work Backwards is for. An average is not something I can add or subtract directly, but the total behind it is, so I turn each average into its total: the 7 old tests must sum to 80 x 7, the 8 finished tests must sum to 82 x 8, and the new score is simply the difference. Then I draw the eight scores as bars levelled off at 82, which shows the same answer from a different angle: the new score has to cover its own place at 82 and also hand out the 2 extra points that each of the 7 earlier tests needs to reach 82. Finally a couple of guesses checked against the target average confirm that 96 is the only score that works.
1STEP 1
Turn the old average back into a total
The first seven total 560 points.
80 × 7 = 560
The average is a single number standing in for the whole set, so multiplying it back by how many scores there are recovers the real total that the average was hiding.
6.SP.A.3Work Backwards2STEP 2
Work out the total the 8 tests must reach
The eight must total 656 points.
80+2 = 82, 82 × 8 = 656
Reading the target average backwards into a target total turns a question about a share-out into a question about a plain sum, which is much easier to handle.
6.SP.B.5Work Backwards3STEP 3
Subtract to find the new score
Subtracting gives 96 points.
656 - 560 = 96
Once both averages are written as totals, the unknown score is just the gap between them — a single subtraction with no averaging left to do.
4.OA.A.3Work Backwards4STEP 4
See it as levelling bars: 82 for itself plus 2 for each old test
It is 82 plus 14 to lift the other seven.
2 × 7 = 14, 82 + 14 = 96
A picture of bars being levelled makes visible why the new score has to overshoot the target average: the extra it carries is exactly what lifts every earlier test.
6.SP.A.3Draw A DiagramThe new score has to cover its own place at the new average plus a lift of 2 for each of the seven old tests.
Why?
Raising the average means every test now sits two higher, and the only place that extra height can come from is the new score.
🧱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?
What the old tests gain the new score must give up, so the two changes balance and the total stays honest.
🧱What one part gains the other losesMove some amount from one addend to the other and the total does not budge, so you may make one number round for free.
5STEP 5
Check the answer, and check that nearby scores fail
Recomputing gives an average of 82.
(560+96) ÷ 8 = 656 ÷ 8 = 82
Checking one score on each side of 96 shows the average climbs steadily as the score climbs, so there is no second answer hiding nearby.
6.SP.B.5Guess And CheckAnswer
96 points
656 − 560 = 96
The answer is a test score, so points are the right unit, and 96 is a believable score on a test where 80 is typical. The size makes sense too: to pull an average up at all the new score must beat the old average of 80, and 96 does. It must also beat the new average of 82, because it has to lift the other 7 tests as well, and 96 beats 82 by 14 — exactly the 2 points each that the 7 earlier tests need. The final check divides evenly: 656 divided by 8 = 82 with nothing left over, as an exact average must.
Takeaway
To raise an average, the new score must pay for its own place plus a little extra for every score already there.
- Turn the old average back into a total
- Work out the total the 8 tests must reach
- Subtract to find the new score
- See it as levelling bars: 82 for itself plus 2 for each old test
- Check the answer, and check that nearby scores fail