Competition · AMC preparation · step 4 of 4
AMC 8 · 2022 · #6
Grade 3 arithmeticalgebraPick an answer.
AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a multiple-choice problem where each candidate for the smallest number is a small whole number (4, 5, 6, 7, 8). For each candidate x, we can compute the largest number (4x) and then check whether 15 sits exactly in the middle of x and 4x (equal gaps). Tool #6 (Guess and Check) tests each candidate directly, and Tool #3 (Eliminate Possibilities) lets us cross off the ones that fail. This avoids reaching for algebra (Tool #13) when the bounded answer set makes plug-and-check the fastest, most age-appropriate path.
Unpack equally spaced
"Equally spaced" means 15 sits exactly halfway between the smallest and the largest, so the two gaps must be equal.
Equal gaps is a one-line rule — once you see it, the rest is just arithmetic checking on each choice.
Because the three numbers are equally spaced, the middle number sits exactly halfway between the smallest and the largest: the gap from the smallest to the middle equals the gap from the middle to the largest.
▸ Why?
The whole distance from the smallest number to the largest is split by the middle number into two touching pieces — smallest-to-middle and middle-to-largest — that leave no gap and no overlap, so those two pieces add back to the whole distance.
▸ Why?
"Equally spaced" says each step to the next number has the same length, so those two pieces are equal; and the length of each piece is a difference — middle minus smallest, and largest minus middle — because subtracting recovers the very amount you would add to the smaller number to reach the larger one.
Test x equals 4
Test x = 4: largest = 4 × 4 = 16, but gaps 15 - 4 = 11 and 16 - 15 = 1 differ, so (A) fails.
Multiplying 4 × 4 and subtracting are Grade 3 multiplication / word-problem skills.
3.OA.A.3Eliminate PossibilitiesTest x equals 5
Test x = 5: largest = 4 × 5 = 20, but gaps 15 - 5 = 10 and 20 - 15 = 5 differ, so (B) fails.
Same Grade 3 multiply-and-subtract check, just on the next candidate.
3.OA.A.3Eliminate PossibilitiesTest x equals 6
Test x = 6: largest = 4 × 6 = 24, and gaps 15 - 6 = 9 and 24 - 15 = 9 match — this works!
When both differences come out the same, the "equally spaced" rule is satisfied — we've found the smallest number.
3.OA.A.3Guess And CheckRule out the rest
Check the rest: x = 7 gives gaps 8 and 13, x = 8 gives 7 and 17 — both unequal, so only (C) 6 survives.
Eliminating the rest of the choices makes sure (C) is the only valid answer — no second contestant slipped through.
3.OA.D.8Eliminate PossibilitiesThis AMC 8 problem only needs Grade 3 multiplication and subtraction you already know — just try each answer choice and see which one makes equal gaps!
- Unpack equally spaced
- Test x equals 4
- Test x equals 5
- Test x equals 6
- Rule out the rest
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