AMC 8 · 2019 · #16

Grade 6 rate-ratioalgebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

ratelinear-equations-one-varfraction-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: ratelinear-equations-one-var

📏 Medium solution 💡 3 insights

Problem

Qiang has already driven 15 miles at 30 mph. He now wants to keep going at 55 mph for some extra distance d so that, when you average over the whole trip, the overall speed comes out to exactly 50 mph. How many extra miles d does he need to drive?

Pick an answer.

(A)

(B)

(C)

(D)

110

(E)

135

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

Since this is multiple choice and each candidate d gives a clean overall-average-speed check, Tool #6 (Guess and Check) on the five choices is the fastest honest path — far simpler than setting up and solving the rational equation $\frac{15+d}{0.5 + \frac{d}{55}}$ = 50 with algebra. Tool #3 (Eliminate Possibilities) is the natural companion for any AMC multiple-choice question — we keep ruling out choices until one survives. Tool #8 (Analyze the Units) is the bookkeeping that keeps miles, mph, and hours consistent so the comparison to 50 mph is meaningful.

1STEP 1

The first leg is fixed: 15 mi ÷ 30 mph = 0.5 hr, so any candidate makes total distance 15 + d over total time 0.5 + $\frac{d}{55}$ hr.

t₁ =

\frac{15 mi}{30 mph}

= 0.5 hr, avg speed =

\frac{15 + d}{0.5 + \frac{d}{55}}

2STEP 2

Test the middle choice d = 90: total 105 mi over 0.5 + $\frac{90}{55}$ ≈ 2.14 hr gives avg ≈ 49.2 mph — just under 50, so we need a bigger d.

\frac{105}{2.136}

≈ 49.2 mph ＜ 50

3STEP 3

Since average speed grows with d, choices 45 and 62 (both below 90) fail even harder — eliminate them, leaving only 110 and 135.

d = 45, 62, 90 all give avg ＜ 50 mph → ruled out

4STEP 4

Test d = 110: total 125 mi over 0.5 + $\frac{110}{55}$ = 2.5 hr gives 125 ÷ 2.5 = exactly 50 mph — a perfect hit.

\frac{125}{2.5}

= 50 mph ✓

5STEP 5

Check d = 135: it overshoots to avg ≈ 50.8 mph above 50, so 110 is the unique answer.

\frac{150}{2.955}

≈ 50.8 mph ＞ 50 → (D)

Answer

110

A common wrong intuition is to average 30 and 55 to get 42.5 and panic that no d can reach 50. But d only changes the second leg, and Qiang spends much more time on the slow leg (0.5 hr) than the fast leg if d is small. To pull the time-weighted average up to 50, the fast leg has to dominate the trip — needing 2 full hours at 55 mph (i.e. 110 miles) versus just 0.5 hr at 30 mph. The ratio of fast-time to slow-time being 4{:}1 matches the way 50 sits between 30 and 55 (50 is 20 above 30 but only 5 below 55, i.e. 4{:}1 — the same ratio). That cross-check confirms (D) 110.

💡Key takeaway

This AMC 8 problem only needs Grade 6 rate reasoning — total distance divided by total time — that you already know!

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