AMC 8 · 2024 · #10

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

ratemulti-digit-arithmeticfraction-decimal-conversion dimensional-analysis ↑ Prerequisites: multi-digit-arithmeticfraction-decimal-conversion

📏 Medium solution 💡 3 insights

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Problem

In January 1980 the CO₂ level at Mauna Loa was 338 ppm, and it has gone up by about 1.515 ppm every year since. We need to estimate the CO₂ level in January 2030 and round to the nearest whole number.

Pick an answer.

(A)

399

(B)

414

(C)

420

(D)

444

(E)

459

AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

This is a rate problem: ppm per year times years gives ppm of increase, which then adds to a starting ppm. Tool #8 (Analyze Units) keeps the recipe honest — "years × (ppm/year) = ppm," and "ppm + ppm = ppm." The only tricky arithmetic is 1.515 × 50, where Tool #9 (Easier Related Problem) lets us replace it with the friendlier 1.5 × 50 = 75 and then add a small correction. Finally, since the question is multiple-choice, Tool #3 (Eliminate Possibilities) confirms the answer against the five choices.

1STEP 1

Subtract the years: 2030 − 1980 = 50 years — years minus years is years, so that's how long CO₂ has been climbing.

2030 - 1980 = 50 years

2STEP 2

Multiply rate by time: 1.515 × 50. Split as 1.5×50 = 75 plus 0.015×50 = 0.75, giving a total increase of 75.75 ppm.

1.515 × 50 = (1.5 × 50) + (0.015 × 50) = 75 + 0.75 = 75.75 ppm

3STEP 3

Add the increase onto the 1980 base (ppm + ppm = ppm): 338 + 75.75 = 413.75 ppm, the expected 2030 level.

338 + 75.75 = 413.75 ppm

4STEP 4

The tenths digit 7 rounds up: 413.75 ≈ 414, matching (B); the other choices need a far larger or smaller rate, so eliminate them.

413.75 ≈ 414 → (B)

Answer

414

Sanity check the size: in 50 years, CO₂ goes up by about 1.5 ppm/year × 50 = 75 ppm, so the 2030 level should be roughly 338 + 75 ≈ 413 ppm. Our answer 414 matches that ballpark almost exactly, and the small extra 0.75 from the 0.015 correction explains the rounding up to 414. Units are right (ppm), and the answer sits squarely between the too-low (A) 399 and too-high (C) 420 options.

💡Key takeaway

This AMC 8 problem only needs Grade 5 decimal arithmetic and rounding you already know!

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