Competition · AMC preparation · step 4 of 4
AMC 8 · 2018 · #17
Grade 5 rate-ratioPick an answer.
AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We are never told the actual speed in mph or how long the trip takes, so chasing time as a variable is harder than it needs to be. Tool #9 (Easier Related Problem) lets us replace the awkward unit of "time" with a much friendlier unit: "one Bella-step." In the duration of one Bella-step, Bella moves 2.5 ft and Ella — being 5 times faster — moves 5 × 2.5 = 12.5 ft, so the gap between them shrinks by a fixed amount per Bella-step. Tool #8 (Analyze the Units) then keeps the bookkeeping clean: dividing total feet by feet-per-step cancels to leave plain "steps," exactly what the question asks for.
Measure time in Bella-steps
Swap clock-time for an easier unit — one "Bella-step" — during which Bella moves her step length, 2.5 ft.
Recognizing 2 1/2 as the decimal 2.5 is a Grade 4 mixed-number-to-decimal skill.
4.NF.C.6Solve An Easier Related ProblemFind Ella's distance per step
In that same Bella-step, Ella rides five times as far — 5 × 2.5 = 12.5 ft, a "times as fast" comparison.
"N times as fast" is the classic Grade 4 multiplicative-comparison setup: same time, N times the distance.
4.OA.A.2Solve An Easier Related ProblemAdd the two distances
Moving toward each other, the gap shrinks each Bella-step by both distances added: 2.5 + 12.5 = 15 ft.
Adding two decimals to hundredths to get 15.0 ft is the Grade 5 decimal-arithmetic standard.
In the time it takes Bella to take one step, the distance between Bella and Ella shrinks by 15 feet.
▸ Why?
Bella and Ella head straight toward each other, so the ground Bella walks forward and the ground Ella rides forward are two separate stretches of the road between them, and together they are exactly how much the gap closes.
▸ Why?
The road between them splits with no overlap into the stretch Bella has walked, the stretch Ella has ridden, and the gap still left, so the two travelled stretches add back to the amount the gap has lost.
▸ Why?
That closed distance is Bella's 2.5 ft plus Ella's 12.5 ft, which makes 15 ft, and Ella's 12.5 ft is how far she rides during the one span of time Bella spends taking that single step.
▸ Why?
Over that one span of time Ella keeps a steady speed 5 times Bella's, and at a constant speed the distance covered is speed times time, so across the same time interval 5 times the speed lays down 5 times the distance Bella walks.
▸ Why?
Five times Bella's 2.5-ft step means five equal 2.5-ft lengths placed end to end, and 5 groups of 2.5 ft is 12.5 ft.
Divide the gap by 15
Divide total feet by feet-per-step; "ft" cancels, leaving steps: = 704 steps → (A).
Dividing a 4-digit number by a 2-digit divisor is the Grade 5 long-division standard; tracking units confirms the answer is in steps.
5.NBT.B.6Analyze The UnitsThis AMC 8 problem only needs Grade 5 decimal arithmetic and "times as fast" comparison you already know!
- Measure time in Bella-steps
- Find Ella's distance per step
- Add the two distances
- Divide the gap by 15
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