AMC 10 · 2015 · #21
Grade 6 number-theoryPick an answer.
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
Each jump count is a rounding up.
A leftover that is too small for a full stride still forces one extra jump, so you always round the division up.
A leftover too small for a full stride still forces one extra jump, so the division is always rounded up.
▸ Why?
Dividing leaves a remainder smaller than one stride, and that leftover still has to be covered.
▸ Why?
One fewer jump would fall short of the end, so the next whole number is the smallest that works.
Write the difference equation
The condition is one plain difference.
"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
Ignoring the rounding locates the answer.
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
Only three sizes survive the exact check.
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
Their digit total is 13, choice (D).
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