AMC 8 · 2011 · #10

Grade 6 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

linear-equations-one-varrateunit-conversion convert-to-algebraidentify-subproblems ↑ Prerequisites: ratemulti-digit-arithmetic

📏 Medium solution 💡 3 insights

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Problem

A Gotham City taxi charges

2.40 for the first

\frac{1}{2}

mile</span>, then <span class="mk-c">

0.20 for each additional 0.1 mile. You have

10 total</span> and want to leave a <span class="mk-c">

2 tip. How many miles can you ride?

Pick an answer.

(A)

3.0

(B)

3.25

(C)

3.3

(D)

3.5

(E)

3.75

AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The $10 has to do three different jobs — tip, flat fare for the first 0.5 mile, and the per-0.1-mile charge after that — so Tool #7 (Identify Subproblems) cleanly peels them off one at a time: take out the tip, take out the flat fare, then ask "how far does the leftover money buy?" Tool #8 (Analyze the Units) turns the awkward$ 0.20 per 0.1 mile rate into the friendlier $2 per mile, which makes the final division a one-line decimal computation rather than an algebra problem.

1STEP 1

Set aside the tip first: you hand over the $2 at the end, leaving only $ 8 for the meter.

10.00 -

2.00 = $8.00 for the meter

2STEP 2

Pay the flat fare for the first $\frac{1}{2}$ mile: subtract $2.40, leaving $ 5.60 for the extra distance.

8.00 -

2.40 = $5.60 left for extra distance

3STEP 3

Convert the rate: $0.20 per 0.1 mile is the same as $ 2 per mile — a friendlier unit.

0.20/(0.1 mile) = (

0.20 × 10)/(0.1 × 10 mile) = $2.00/(1 mile)

4STEP 4

At $2 per mile, the leftover$ 5.60 buys 2.8 extra miles (5.60 ÷ 2).

5.60/(

2.00 / mile) = 2.8 miles of extra distance

5STEP 5

Add the first 0.5 mile back to the 2.8 extra: 0.5 + 2.8 = 3.3 miles — choice (C).

0.5 + 2.8 = 3.3 miles → (C)

Answer

3.3

Check the dollars: first 0.5 mile costs

2.40, the extra 2.8 miles cost 2.8 ×

2.00 =

5.60, and the tip is

2.00. Total = 2.40 + 5.60 + 2.00 =

10.00 exactly — every dollar is accounted for, so 3.3 miles is right. A quick sanity check on size: at

2 per mile after the first half, an extra $5.60 should buy under 3 extra miles, so the total ride should be a bit under 3.5 miles. 3.3 fits perfectly.

💡Key takeaway

Big AMC 8 word problems often shrink down to Grade 6 rate reasoning — once you split off the tip and the flat fare, only a tidy decimal division is left.

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