AMC 10 · 2015 · #21
Grade 6 rate-ratioPick an answer.
AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #4 (Introduce a Variable) names the staircase size n. The hidden idea is that a jumper who needs a final partial jump still spends a whole jump, so the jump counts are the round-up divisions ⌈ n/2 ⌉ and ⌈ n/5 ⌉; Tool #13 (Convert to Algebra) turns the sentence "Dash takes 19 fewer jumps" into the single equation ⌈ n/2 ⌉ - ⌈ n/5 ⌉ = 19. Because the ceilings are awkward to solve head-on, Tool #9 (Solve an Easier Related Problem) first drops the rounding to get the clean rate equation n/2 - n/5 = 19, which locates the answer near n = 63. Tool #6 (Guess and Check) then tests the exact (rounded) counts in a short window around 63, and Tool #2 (Make a Systematic List) collects every n that lands on a difference of exactly 19, since the rounding makes the difference wobble instead of climb smoothly.
Count each jumper's jumps
Cozy climbs 2 per jump, so ⌈ n/2 ⌉ jumps; Dash climbs 5, so ⌈ n/5 ⌉ jumps — round up, a partial jump still counts.
A leftover that is too small for a full stride still forces one extra jump, so you always round the division up.
4.OA.A.3Use Matrix LogicWrite the difference equation
"Dash takes 19 fewer jumps than Cozy" means Cozy's jumps minus Dash's jumps equals 19 — write it straight as an equation in n.
"Fewer" is a subtraction, so the whole story becomes one tidy equation to hunt with.
6.EE.B.6Convert To AlgebraEstimate by ignoring the rounding
Drop the rounding: treat the counts as n/2 and n/5, so the gap is 3n/10. Setting it to 19 gives n ≈ 63 — the answers sit near 63.
Dash saves about 3/10 of a jump per step over Cozy, so 19 saved jumps means roughly 63 steps.
6.RP.A.3Solve An Easier Related ProblemCheck exact counts near 63
Test the exact rounded counts near 63; rounding makes the gap wobble, so it equals 19 only at 63, 64, 66 — 62, 65, and 67 all miss.
Rounding makes the gap jump up by one each time a jumper crosses a multiple, so it can equal 19 at scattered points, not a single block.
4.OA.A.3Guess And CheckSum the valid sizes and add digits
The staircase sizes that work are 63, 64, and 66. Add them to get s, then add the digits of s.
Collect every value the check approved, total them, then add the three digits for the final number.
4.NBT.B.4Make A Systematic ListCount jumps by rounding the division up, turn "19 fewer" into one equation, estimate where the answer lives by ignoring the rounding, then check nearby whole numbers — because the rounding makes the gap wobble, more than one staircase can work.
- Count each jumper's jumps
- Write the difference equation
- Estimate by ignoring the rounding
- Check exact counts near 63
- Sum the valid sizes and add digits