Competition · AMC preparation · step 4 of 4
AMC 8 · 2006 · #15
Grade 6 rate-ratioPick an answer.
AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The rates are in seconds per page, and the condition is equal total time in seconds. Tool #8 (Analyze the Units) makes the setup almost automatic: (sec/page) × (pages) = sec, so equal time means 30 × (Chandra's pages) = 45 × (Bob's pages). That immediately exposes a 3:2 page-count ratio without heavy algebra. Tool #13 (Convert to Algebra) gives a single name x to Chandra's page count so we can write the equation, but the work that finishes the problem is the ratio, not symbol manipulation.
Name the unknown
Let x be Chandra's page count; Bob reads the rest, 760 - x pages.
Naming the unknown with a single letter is the Grade 6 "use variables to represent numbers" move.
6.EE.B.6Convert To AlgebraWrite both reading times
Equal time means rate × pages match on both sides: 30x = 45(760 - x).
Tracking the unit "seconds per page" makes the equation feel inevitable: pages cancel, and what is left on both sides is the time in seconds.
6.RP.A.3Analyze The UnitsRead it as a ratio
Chandra is faster, so page counts flip to the inverse of the speeds — the ratio 3:2.
Equal time with different rates is the classic "faster reader gets more pages" trade — the ratio 45:30 flips to a 3:2 page split.
Because Chandra and Bob read for the same amount of time, their page counts land in the ratio 3 to 2 — the inverse of their 30- and 45-second per-page rates.
▸ Why?
Equal reading time means their two totals of seconds are the same number, and each total is the per-page rate multiplied by the pages read, so 30 times Chandra's pages equals 45 times Bob's pages.
▸ Why?
Reading every page at a fixed 30 or 45 seconds adds that same amount once for each page, so a total time is just the per-page rate counted once per page.
▸ Why?
Each reader's total time equals the one shared reading time, so the two totals must equal each other.
▸ Why?
When the products 30 times Chandra's pages and 45 times Bob's pages are equal, the page counts must trade off against the rates, so Chandra's pages to Bob's pages stand in the ratio 45 to 30, which is the ratio 3 to 2.
▸ Why?
Dividing the equal products by the two rates undoes the multiplication and leaves Chandra's pages to Bob's pages equal to 45 to 30.
▸ Why?
Writing the ratio 45 to 30 as the fraction 45 over 30, both its top and bottom are divisible by 15, and dividing top and bottom by that shared 15 turns it into 3 over 2 without changing the fraction's value.
Split 760 in ratio 3 to 2
Split 760 into 5 shares of 152; Chandra's 3 shares give 456.
Splitting a total in a given ratio is a Grade 6 rate-reasoning skill: count the shares, find one share, multiply.
6.RP.A.3Analyze The UnitsWhen two readers spend equal time but read at different speeds, the faster reader handles more pages — and the seconds-per-page numbers tell you exactly how to split the total in a Grade 6 ratio.
- Name the unknown
- Write both reading times
- Read it as a ratio
- Split 760 in ratio 3 to 2
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