AMC 10 · 2014 · #3

Grade 6 rate-ratio

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

linear-equations-one-varfraction-arithmetic convert-to-algebra ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 2 insights

Problem

Randy's trip is split into three parts. The first one-third of the whole trip is on gravel, the next 20 miles are on pavement, and the last one-fifth of the whole trip is on dirt. Find the total length of the trip in miles.

Pick an answer.

(A)

(B)

$\frac{400}{11}$

(C)

$\frac{75}{2}$

(D)

(E)

$\frac{300}{7}$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The whole trip is the one number everything else is measured against, so Tool #4 (Introduce a Variable) names it x and turns "one-third" and "one-fifth" into x/3 and x/5. Tool #7 (Identify Subproblems) splits the job cleanly: first combine the two fraction parts, then see what fraction the pavement must fill. Tool #13 (Convert to Algebra) writes the sentence "the parts add to the whole" as one equation, so the 20 miles pins down x instead of floating free.

1STEP 1

Name the whole trip

Let x be the whole trip in miles: gravel and dirt become x/3 and x/5, while pavement stays the given 20 miles.

gravel = x/3, pavement = 20, dirt = x/5

2STEP 2

Combine the two fraction parts

Over the common denominator 15, 1/3 = 5/15 and 1/5 = 3/15, so the two fractions cover 8/15 of the trip.

1/3 + 1/5 = 5/15 + 3/15 = 8/15

3STEP 3

Find the pavement's share

The whole trip is 15/15, so pavement fills the rest: 15/15 - 8/15 = 7/15, and that share is the 20 miles.

1 - 8/15 = 7/15, so 7/15x = 20

4STEP 4

Solve for the total

Multiply both sides of 7/15 · x = 20 by the reciprocal 15/7: x = 20 · 15/7 = 300/7 miles, choice (E).

x = 20 · 15/7 = 300/7 → (E)

Answer

300/7

Plug x = 300/7 back in: gravel = 1/3·300/7 = 100/7 and dirt = 1/5·300/7 = 60/7. Their sum is 160/7, and adding the 20 = 140/7 miles of pavement gives 300/7, exactly the whole trip. The value 300/7 ≈ 42.9 miles is a bit larger than 40, which fits: the pavement is only 7/15 (under half) of the trip, so the whole trip must be more than twice 20.

💡Key takeaway

The 20 miles is whatever fraction of the trip the two fractions leave over, so find that leftover fraction first.

Name the whole trip
Combine the two fraction parts
Find the pavement's share
Solve for the total

Name the whole trip

Combine the two fraction parts

Find the pavement's share

Solve for the total

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