AMC 8 · 2023 · #11

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

rateunit-conversionestimation dimensional-analysisestimation ↑ Prerequisites: rateunit-conversion

📏 Medium solution 💡 3 insights

Problem

NASA's Perseverance Rover flew 292,526,838 miles to Mars in about 6.5 months. Of the five answer choices, which one is closest to the rover's average speed in miles per hour?

Pick an answer.

(A)

$6{,}000$

(B)

$12{,}000$

(C)

$60{,}000$

(D)

$120{,}000$

(E)

$600{,}000$

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The numbers are huge (292,526,838 miles) and the answer choices are spread far apart (each is 2{×} or 10{×} the previous), so we don't need an exact answer. Tool #9 (Easier Related Problem) says: replace ugly numbers with nearby round numbers and compute the easier version. Tool #8 (Analyze the Units) keeps us honest about miles vs. months vs. hours so we don't divide the wrong things. Tool #7 (Subproblems) splits the work into two small jobs — convert 6.5 months to hours, then divide distance by hours.

1STEP 1

Round 292,526,838 up to a friendly 300,000,000 miles; the choices are far apart, so this won't flip the answer (Tool #9).

292,526,838 mi ≈ 300,000,000 mi

2STEP 2

Convert 6.5 months to hours so units come out mi/hr: 6.5 × 30 × 24 = 4680 hours (Tools #7, #8).

6.5 mo× 30 day/mo× 24 hr/day=6.5× 720=4680 hr

3STEP 3

Round 4680 up to 5000 hours, which divides cleanly into 300,000,000 — again choosing a number we can divide in our head (Tool #9).

4680 hr ≈ 5000 hr

4STEP 4

Divide rounded distance by rounded time (Tool #7): 300,000,000 ÷ 5000 = 60,000 mi/hr — the units cancel to miles per hour (Tool #8).

300,000,000 mi/5000 hr=

\frac{300,000,000}{5000}

mi/hr=60,000 mi/hr

5STEP 5

60,000 matches choice (C) exactly; neighbors 12,000 and 120,000 are off by ≥ 2×, so rounding can't move it. The answer is 60,000.

60,000 mi/hr → (C) 60,000

Answer

60,000

Sanity-check the magnitude. About 300,000,000 miles over about 6.5× 30=195 days is roughly 1,500,000 miles per day. Dividing by 24 gives about 62,500 miles per hour, very close to our 60,000. The two neighboring choices, 12,000 (a factor of 5 smaller) and 120,000 (a factor of 2 bigger), are both far enough away that no reasonable rounding flips the answer. (C) is consistent.

💡Key takeaway

This AMC 8 problem only needs Grade 6 unit-rate (miles-per-hour) reasoning you already know!

More like this