AMC 8 · 2024 · #12

Grade 2 arithmetic

Archetype Small-Domain Constrained Search

sequences-arithmeticlinear-equations-one-var systematic-enumeration ↑ Prerequisites: multi-digit-arithmeticmental-arithmetic

📏 Medium solution 💡 3 insights

📘 View easy version →

Problem

Rohan keeps 90 guppies in 4 fish tanks.

There is 1 more guppy in the 2nd tank than in the 1st tank.
There are 2 more guppies in the 3rd tank than in the 2nd tank.
There are 3 more guppies in the 4th tank than in the 3rd tank.

How many guppies are in the 4th tank?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Rohan has 4 fish tanks holding 90 guppies in total. Each tank has a few more guppies than the one before: tank 2 has 1 more than tank 1, tank 3 has 2 more than tank 2, and tank 4 has 3 more than tank 3. We need the number of guppies in tank 4.

Givens: Total guppies across all 4 tanks: 90; Tank 2 has 1 more guppy than tank 1; Tank 3 has 2 more guppies than tank 2; Tank 4 has 3 more guppies than tank 3; Answer choices: (A) 20, (B) 21, (C) 23, (D) 24, (E) 26

Unknowns: The number of guppies in tank 4

Understand

Plan

Primary tool: #6 Guess and Check

Secondary: #3 Eliminate Possibilities, #1 Draw a Diagram

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.

Execute — Answer: E

#1 Draw a Diagram K.OA.A.1 Step 1

Draw four boxes in a row labeled Tank 1, Tank 2, Tank 3, Tank 4.
Mark the gaps between the boxes with +1, +2, +3 to record how much bigger each next tank is.
This picture makes it obvious that once we know any one tank, we know all four.

$$\boxed{T_1}\ \xrightarrow{+1}\ \boxed{T_2}\ \xrightarrow{+2}\ \boxed{T_3}\ \xrightarrow{+3}\ \boxed{T_4}$$

💡 Even kindergarten kids can draw boxes and use arrows to show 'a little more' between them.

#6 Guess and Check 2.NBT.B.5 Step 2

Guess the largest answer choice first: try (E) Tank 4 = 26.
Walk backwards through the picture: Tank 3 = 26 - 3 = 23, Tank 2 = 23 - 2 = 21, Tank 1 = 21 - 1 = 20.
All four counts are whole numbers, which is a good sign.

$$T_4=26,\ T_3=23,\ T_2=21,\ T_1=20$$

💡 Subtracting small numbers like 3, 2, and 1 from two-digit numbers is Grade 2 fluent subtraction.

#6 Guess and Check 2.NBT.B.6 Step 3

Add the four tank counts and check whether they total 90.
Group friendly pairs: $20+21=41$ and $23+26=49$, then $41+49=90$.
The total matches, so the guess works.

$$20+21+23+26 = 41+49 = 90\ \checkmark$$

💡 Adding up to four two-digit numbers using mental grouping is exactly the Grade 2 standard.

#3 Eliminate Possibilities 1.NBT.B.3 Step 4

Since choice (E) already satisfies the total of 90 and the four-tank rule, every other choice must fail (the four tank counts are forced once Tank 4 is fixed, so only one choice can give the right total).
For example, choice (A) 20 would give tanks 20, 17, 15, 14 with sum 66 — too small.
So we can eliminate the others and confirm (E).

$$\text{Choice (E)}\colon 20+21+23+26=90\ \Rightarrow\ \textbf{(E)}$$

💡 Comparing the sum to 90 to decide 'yes' or 'no' is the Grade 1 idea of comparing two-digit numbers.

[1] #1 K.OA.A.1 Draw four boxes in a row labeled Tank 1, Tank 2, Tank 3, Tank 4. Mark the gaps b

[2] #6 2.NBT.B.5 Guess the largest answer choice first: try (E) Tank 4 = 26. Walk backwards throu

[3] #6 2.NBT.B.6 Add the four tank counts and check whether they total 90. Group friendly pairs:

[4] #3 1.NBT.B.3 Since choice (E) already satisfies the total of 90 and the four-tank rule, every

Review

Reasonableness: Check the magnitude: if all four tanks held the same amount, each would hold $90 \div 4 = 22.5$. Tank 4 should be a bit bigger than 22.5 because it has the most extras (+1+2+3 over Tank 1). Our answer 26 is just a few above 22.5 — that fits perfectly. The four numbers (20, 21, 23, 26) are all positive whole numbers and add to 90, matching every condition in the problem.

Alternative: An alternative is Tool #13 (Convert to Algebra): let $x$ be Tank 1, then Tank 2 $= x+1$, Tank 3 $= x+3$, Tank 4 $= x+6$. The sum gives $4x+10=90$, so $x=20$ and Tank 4 $= 26$. Same answer, but Guess-and-Check is faster for an elementary student and avoids any algebra setup.

CCSS standards used (min grade 2)

K.OA.A.1 Represent addition and subtraction with objects, fingers, or drawings (Drawing the four tanks as boxes with +1, +2, +3 arrows between them to model the relationships.)
1.NBT.B.3 Compare two two-digit numbers using symbols (Comparing the sum of the guessed counts to 90 to decide whether the guess is correct.)
2.NBT.B.5 Fluently add and subtract within 100 (Walking backwards from Tank 4 to Tank 1 by subtracting 3, 2, and 1 from two-digit numbers.)
2.NBT.B.6 Add up to four two-digit numbers using strategies (Adding the four tank counts 20 + 21 + 23 + 26 to check the total equals 90.)

⭐ This AMC 8 problem only needs Grade 2 two-digit adding and subtracting you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 2)

More like this