Competition · AMC preparation · step 4 of 4
AMC 8 · 2005 · #5
Grade 4 arithmeticPick an answer.
AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #7 (Identify Subproblems) splits "minimum packs for 90 cans" into a short chain: peel off as many 24-packs as fit, then peel off 12-packs from what's left, then finish with 6-packs. Each sub-problem is one division-with-remainder, and stacking the largest packs first keeps the pack count as small as possible. Tool #3 (Eliminate Possibilities) then verifies the minimum: the smaller answer choice (A) 4 packs is impossible, so the candidate from Tool #7 really is the minimum.
Take out the 24-packs
Start biggest: 90 ÷ 24 = 3 packs (72 cans), leaving 18 cans.
Division-with-remainder is the Grade 4 "how many groups fit, and what is left over" move.
4.OA.A.3Identify SubproblemsTake out the 12-packs
Next size: 18 ÷ 12 = 1 pack (12 cans), leaving 6 cans.
Same move, smaller numbers: one 12-pack covers all the 12s in 18, leaving a multiple of 6.
4.OA.A.3Identify SubproblemsTake out the last 6-pack
Last 6 cans = one 6-pack, 0 left — total is exactly 90.
One 6-pack closes out the leftover and lands the total at exactly 90.
3.OA.A.3Identify SubproblemsAdd the packs
Add the pack counts: 3 + 1 + 1 = 5 packs.
Combine the three subproblem counts to get the candidate minimum.
3.OA.A.3Identify SubproblemsRule out the smaller choice
Check (A) 4 fails: every pack is a multiple of 6, and no mix of 4 packs hits 90, so 5 is truly the minimum.
Pack totals are all multiples of 6, so we can search small cases by hand and confirm 4 packs cannot reach 90.
No group of four packs chosen from sizes 6, 12, and 24 can total exactly 90 cans.
▸ Why?
The cans from four packs are just the four sizes added together, so the biggest total four packs can reach is 24 + 24 + 24 + 24 = 96 cans; hitting exactly 90 would mean trimming precisely 6 cans off that 96.
▸ Why?
But you cannot shave just 6 cans off four 24-packs: the gentlest change is swapping one 24-pack for a 12-pack, and since 24 is four sixes and 12 is two sixes, that swap drops the total by 12 at once — from 96 straight down to 84 — leaping right over 90, and every other swap removes even more.
When you want the fewest packs that total an exact amount, start with the biggest pack and peel off as many as fit, then move to the next size. That "biggest first" plan turns this AMC 8 problem into three short Grade 4 division-with-remainder steps.
- Take out the 24-packs
- Take out the 12-packs
- Take out the last 6-pack
- Add the packs
- Rule out the smaller choice
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