Competition · AMC preparation · step 4 of 4
AMC 8 · 2024 · #12
Grade 2 arithmeticPick an answer.
AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a multiple-choice problem and the unknown is exactly the number we are asked for (guppies in tank 4). Each answer choice fully determines all four tanks by walking backwards (tank 3 = tank 4 - 3, tank 2 = tank 3 - 2, tank 1 = tank 2 - 1). We can test a single choice, add the four numbers, and see if it equals 90 — Tool #6. Combined with Tool #3 we can eliminate any choice whose total is not 90, and Tool #1 (a quick row of 4 boxes) keeps the four tanks straight.
Chain the four tanks
Draw 4 boxes for the tanks and mark the gaps +1, +2, +3 — knowing any one tank now fixes all four.
Even kindergarten kids can draw boxes and use arrows to show 'a little more' between them.
K.OA.A.1Draw A DiagramGuess the largest choice
Guess the biggest choice for Tank 4, then step backward: Tank 3 = 23, Tank 2 = 21, Tank 1 = 20 — all whole numbers, a good sign.
Subtracting small numbers like 3, 2, and 1 from two-digit numbers is Grade 2 fluent subtraction.
2.NBT.B.5Guess And CheckAdd the four counts
Add the four counts in friendly pairs: 20+21=41 and 23+26=49, so the total is 90 — the guess works.
Adding up to four two-digit numbers using mental grouping is exactly the Grade 2 standard.
Adding the four tank counts produced by the trial gives exactly 90, so that trial satisfies the total-guppy condition.
▸ Why?
The 90 guppies are split among the four tanks with none shared and none left out, so adding the four tank counts must land back on that same 90.
▸ Why?
To total the four counts we lean on the freedom to combine them in any handy order and grouping and still reach the same sum.
▸ Why?
Choosing which two counts to push together first never changes the running total.
▸ Why?
Swapping the order in which the counts are added never changes the total.
Confirm no other choice fits
Because Tank 4 fixes all four tanks, only one choice can total 90 — e.g. (A) 20 gives sum 66, too small. So the biggest choice stands.
Comparing the sum to 90 to decide 'yes' or 'no' is the Grade 1 idea of comparing two-digit numbers.
1.NBT.B.3Eliminate PossibilitiesThis AMC 8 problem only needs Grade 2 two-digit adding and subtracting you already know!
- Chain the four tanks
- Guess the largest choice
- Add the four counts
- Confirm no other choice fits
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