AMC 10 · 2017 · #1
Grade 8 rate-ratioPick an answer.
Both collections grow by a fixed number of books each month, so a single variable — the number of months from now — pins down both totals at the same instant (Tool #4). Tool #8 (Analyze the Units) converts each "books per month" rate into books added over those months, and Tool #13 (Convert to Algebra) turns the sentence "twice as many" into one linear equation and solves it. The trap worth guarding against is direction: the person with fewer books is the one being doubled, so the comparison must be read carefully before any solving starts.
Turn each rate into books added
The monthly rate times the months gives the books added.
A per-month rate is a multiplier: let it run for x months and it contributes rate times x books.
6.RP.A.3Analyze The UnitsWrite both totals with one variable
Both totals are written with the same single letter.
Each total is starting amount plus rate times time, and both are read off the same calendar.
6.EE.B.6Introduce A VariableTranslate "twice as many"
Twice multiplies Kymbrea's side, not LaShawn's.
In "A is twice B", the 2 attaches to B — the collection being compared against, not the one doing the overtaking.
In the phrase one is twice the other, the doubling attaches to the collection being compared against.
▸ Why?
Twice a quantity is that quantity repeated two times, which is exactly what multiplying records.
▸ Why?
Both totals are read off the same calendar, so setting them equal after doubling is a legitimate comparison.
Solve the linear equation
Tidying up gives 25 months.
With the variable on both sides, sweep the x terms to one side and the plain numbers to the other; what remains reads off the month.
8.EE.C.7Convert To AlgebraWrite each growing total as starting amount plus rate times months, then check which collection the word "twice" actually doubles before you solve.
- Turn each rate into books added
- Write both totals with one variable
- Translate "twice as many"
- Solve the linear equation