AMC 10 · 2009 · #3

Grade 4 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation
fraction-arithmeticratio-proportionequal-spacing identify-subproblemsphysical-representation ↑ Prerequisites: fraction-arithmetic
📏 Medium solution 💡 2 insights
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Problem
A trip runs between two given fractions on the number line. Find the point reached after covering one third of it.

Pick an answer.

(A)
$\frac {1}{3}$
(B)
$\frac {5}{12}$
(C)
$\frac {1}{2}$
(D)
$\frac {7}{12}$
(E)
$\frac {2}{3}$
How to solve
Strategy Draw a Diagram

Everything hard about this problem is in reading the phrase, not in the arithmetic, so Tool #1 (Draw a Diagram) comes first: a number line makes the start, the finish, and the direction visible all at once, and it is the direction that four of the five answer choices attack. Tool #15 (Organize Information in More Ways) then does the real work — rewrite both ends in twelfths instead of fourths, chosen so that a trip which must be cut into three equal parts cuts on whole marks. Tool #7 (Identify Subproblems) splits the job into measure-the-trip and then cut-it-into-thirds, and it forces the trisection to be shown rather than eyeballed: the marks are third-marks only because the gaps between them come out equal. Finally Tool #11 (Work Backwards) checks the candidate against the definition from the other end — measure the two pieces it creates and confirm they land in a 1:2 ratio.

1STEP 1

Draw the trip and its direction

The trip runs from the start toward the finish, in that direction.

start = 1/4 ⟶ finish = 3/4, target = start + 1/3(finish - start)
2STEP 2

Switch the ruler to twelfths

A common denominator puts both ends on the same ruler.

1/4 = (1 · 3)/(4 · 3) = 3/12, 3/4 = (3 · 3)/(4 · 3) = 9/12
3STEP 3

Cut the trip into three equal pieces

The trip cuts into three equal jumps.

9/12 - 3/12 = 6/12, 6/12 = 2/12 + 2/12 + 2/12
4STEP 4

Take one jump from the start

One jump from the start lands on 5/12.

3/12 + 2/12 = 5/12
5STEP 5

Check the 1-to-2 split

The one-to-two split checks out, so the answer is 5/12, choice (B).

5/12 - 3/12 = 2/12, 9/12 - 5/12 = 4/12, 2/12 : 4/12 = 1 : 2 → (B)
Answer
5/12
Put every choice on the same twelfths ruler: (A) 4/12, (B) 5/12, (C) 6/12, (D) 7/12, (E) 8/12, against a start of 3/12 and a finish of 9/12. Since 1/3 ＜ 1/2, the point must sit strictly between the start and the midpoint 6/12, which already kills (C), (D) and (E). Of the two survivors, (A) sits only 1/12 past the start — one sixth of the 6/12 trip, not one third — while (B) sits 2/12 past, exactly one third. Each wrong choice is a named misreading: (A) treats "one third" as the number itself, (C) is halfway rather than a third of the way, (D) is two thirds of the way (equivalently, one third of the way if the trip is run backwards from 3/4), and (E) is the number two thirds.
💡Key takeaway

Rewrite both ends in twelfths so the trip cuts evenly into three, then count one jump from the end the problem starts at — that lands you on 5/12.

  • Draw the trip and its direction
  • Switch the ruler to twelfths
  • Cut the trip into three equal pieces
  • Take one jump from the start
  • Check the 1-to-2 split

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