AMC 8 · 2006 · #15
Grade 6 rate-ratioProblem
Problems 14, 15 and 16 involve Mrs. Reed's English assignment.
A Novel Assignment
The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds.
Chandra and Bob, who each have a copy of the book, decide that they can save time by "team reading" the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Chandra and Bob split a $760$-page novel so they spend equal reading time. Chandra reads pages $1$ through some page $x$ at $30$ seconds per page; Bob reads pages $x+1$ through $760$ at $45$ seconds per page. Find the last page $x$ that Chandra reads.
Givens: Total pages: $760$; Chandra's rate: $30$ seconds per page; Bob's rate: $45$ seconds per page; Chandra reads from page $1$ to page $x$, so she reads $x$ pages; Bob reads from page $x+1$ to page $760$, so he reads $760 - x$ pages; Answer choices: (A) $425$, (B) $444$, (C) $456$, (D) $484$, (E) $506$
Unknowns: The last page number $x$ that Chandra reads
Understand
Restated: Chandra and Bob split a $760$-page novel so they spend equal reading time. Chandra reads pages $1$ through some page $x$ at $30$ seconds per page; Bob reads pages $x+1$ through $760$ at $45$ seconds per page. Find the last page $x$ that Chandra reads.
Givens: Total pages: $760$; Chandra's rate: $30$ seconds per page; Bob's rate: $45$ seconds per page; Chandra reads from page $1$ to page $x$, so she reads $x$ pages; Bob reads from page $x+1$ to page $760$, so he reads $760 - x$ pages; Answer choices: (A) $425$, (B) $444$, (C) $456$, (D) $484$, (E) $506$
Plan
Primary tool: #8 Analyze the Units
Secondary: #13 Convert to Algebra
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: $(\text{sec/page}) \times (\text{pages}) = \text{sec}$, so equal time means $30 \times (\text{Chandra's pages}) = 45 \times (\text{Bob's pages})$. 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.
Execute — Answer: C
6.EE.B.6 Step 1 - Name the unknown and the partner count.
- Let $x$ be the number of pages Chandra reads.
- Bob then reads the rest, $760 - x$ pages.
💡 Naming the unknown with a single letter is the Grade 6 "use variables to represent numbers" move.
6.RP.A.3 Step 2 - Write each reader's total time by tracking units.
- Seconds per page times pages gives seconds — that is just rate $\times$ amount.
- The equal-time condition then gives one clean equation.
💡 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.1 Step 3 - Read the equation as a ratio.
- The product $\text{rate} \times \text{pages}$ being equal means the page counts are in the inverse ratio of the rates.
- Chandra is faster ($30 < 45$), so she reads more pages.
💡 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.
6.RP.A.3 Step 4 - Split $760$ pages in the ratio $3:2$.
- The two parts have $3 + 2 = 5$ shares total, so each share is $760 \div 5 = 152$ pages.
- Chandra gets $3$ shares.
💡 Splitting a total in a given ratio is a Grade 6 rate-reasoning skill: count the shares, find one share, multiply.
6.EE.B.6 Name the unknown and the partner count. Let $x$ be the number of pages Chandra r 6.RP.A.3 Write each reader's total time by tracking units. Seconds per page times pages g 6.RP.A.1 Read the equation as a ratio. The product $\text{rate} \times \text{pages}$ bein 6.RP.A.3 Split $760$ pages in the ratio $3:2$. The two parts have $3 + 2 = 5$ shares tota Review
Reasonableness: Check the time: Chandra reads $456$ pages at $30$ sec/page for $456 \times 30 = 13{,}680$ seconds. Bob reads $760 - 456 = 304$ pages at $45$ sec/page for $304 \times 45 = 13{,}680$ seconds. The two totals match, so the page split is correct. The answer also passes a sanity check: Chandra is $1.5\times$ as fast as Bob ($45/30 = 1.5$), and indeed $456 / 304 = 1.5$ — she read $1.5$ times as many pages, just as the equal-time condition requires.
Alternative: Tool #13 (Convert to Algebra) without the ratio shortcut: solve $30x = 45(760 - x)$ directly. Expand to $30x = 34{,}200 - 45x$, add $45x$ to both sides to get $75x = 34{,}200$, then divide: $x = 34{,}200 / 75 = 456$. Same answer (C), more arithmetic — the ratio path is cleaner once you see the unit cancellation.
CCSS standards used (min grade 6)
6.EE.B.6Use variables to represent numbers and write expressions when solving real-world problems (Letting $x$ stand for Chandra's page count so Bob's count $760 - x$ and both reading times can be written as expressions.)6.RP.A.1Understand the concept of a ratio and use ratio language to describe a ratio relationship (Reading $30x = 45(760 - x)$ as the page-count ratio $x : (760 - x) = 45 : 30 = 3 : 2$.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Setting up time $=$ (seconds per page) $\times$ (pages) and splitting $760$ pages in the $3 : 2$ ratio to get $x = \tfrac{3}{5} \times 760 = 456$.)
⭐ When 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.
⭐ When 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.