AMC 8 · 2006 · #16

Grade 6 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation
ratelinear-equations-one-varlcmfraction-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: ratelinear-equations-one-varlcm
📏 Medium solution 💡 3 insights
Problem
A 760-page novel is split among Alice, Bob, and Chandra so that all three read for the same total time. Alice needs 20 seconds per page, Bob needs 45 seconds per page, and Chandra needs 30 seconds per page. Find that shared reading time, in seconds.

Pick an answer.

(A)
6400
(B)
6600
(C)
6800
(D)
7000
(E)
7200

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Set Up an Equation

The three readers all share one unknown — the time T each spends reading. Tool #4 (Introduce a Variable) names that unknown so we can describe each person's contribution in pages. Tool #3 (Set Up an Equation) ties everything together: the three page counts must add to 760. Once T appears in a single linear equation, solving for it is straightforward.

1STEP 1

Name the shared time: let T be the seconds each person reads.

T = seconds each person reads
2STEP 2

Invert "seconds per page" to pages: in T seconds each person reads T divided by their per-page time.

Alice: T20\frac{T}{20} pages, Bob: T45\frac{T}{45} pages, Chandra: T30\frac{T}{30} pages
3STEP 3

Parts make the whole: the three page counts add up to 760.

T20\frac{T}{20} + T45\frac{T}{45} + T30\frac{T}{30} = 760
4STEP 4

Clear denominators: the LCM of 20, 45, 30 is 180, so multiplying through gives 9T+4T+6T = 19T.

180 · T20\frac{T}{20} + 180 · T45\frac{T}{45} + 180 · T30\frac{T}{30} = 180 · 760 ; 9T + 4T + 6T = 180 · 760 ; 19T = 180 · 760
5STEP 5

Solve: since 760 = 19 · 40, the 19 cancels, giving T = 180 · 40 = 7200.

T = 180⋅76019\frac{180 \cdot 760}{19} = 180 · 76019\frac{760}{19} = 180 · 40 = 7200 → (E)
Answer
7200
Plug T = 7200 back in. Alice reads 720020\frac{7200}{20} = 360 pages, Bob reads 720045\frac{7200}{45} = 160 pages, Chandra reads 720030\frac{7200}{30} = 240 pages. Total: 360 + 160 + 240 = 760 pages — exactly the book length. The fastest reader (Alice) handles the most pages and the slowest (Bob) the fewest, which fits the equal-time setup.
💡Key takeaway

When the same time hides behind different speeds, name it once and let one equation do the work.

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