AMC 10 · 2021 · #5
Easy mode Grade 5Telephone poles stand in a line along a road, spaced evenly apart.
Elmer the emu goes from one pole to the pole right next to it in 44 equal steps. Oscar the ostrich covers that same distance in 12 equal jumps.
The 41st pole in the line is exactly one mile (5280 feet) from the first pole.
How many feet longer is one of Oscar's jumps than one of Elmer's steps?
Pick an answer.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Telephone poles stand along a road at equal spacing. Elmer the emu crosses the gap between two neighboring poles in 44 equal strides; Oscar the ostrich crosses that same gap in 12 equal leaps. From the 1st pole to the 41st pole is exactly one mile, or 5280 feet. Find how many feet longer one leap is than one stride.
Givens: The poles are evenly spaced, so every gap between neighboring poles is the same length.; Elmer takes 44 equal strides to cross one gap.; Oscar takes 12 equal leaps to cross that same one gap.; The distance from the 1st pole to the 41st pole is 5280 feet.
Unknowns: The length of one of Oscar's leaps minus the length of one of Elmer's strides, in feet.
Understand
Restated: Telephone poles stand along a road at equal spacing. Elmer the emu crosses the gap between two neighboring poles in 44 equal strides; Oscar the ostrich crosses that same gap in 12 equal leaps. From the 1st pole to the 41st pole is exactly one mile, or 5280 feet. Find how many feet longer one leap is than one stride.
Givens: The poles are evenly spaced, so every gap between neighboring poles is the same length.; Elmer takes 44 equal strides to cross one gap.; Oscar takes 12 equal leaps to cross that same one gap.; The distance from the 1st pole to the 41st pole is 5280 feet.
Plan
Primary tool: #8 Analyze the Units
Secondary: #1 Draw a Diagram, #7 Identify Subproblems
Every number here is either a distance (feet) or a count of steps, and what the question wants is feet per stride and feet per leap. Watching the units tells you exactly which division to do: feet divided by number of steps gives feet per step. The one place units cannot help is the pole count, so a quick sketch settles how many gaps 41 poles create before any dividing starts.
Execute — Answer: B
4.OA.A.3 Step 1 Count gaps, not poles
- Sketch a few poles in a row.
- Two poles make 1 gap, three poles make 2 gaps, four poles make 3 gaps.
- Each new pole adds exactly one new gap, so the number of gaps is always one less than the number of poles.
- From the 1st pole to the 41st pole there are 41 poles, hence 40 gaps, not 41.
💡 The first pole starts the line without creating a gap behind it, so poles always outnumber gaps by exactly one.
5.NBT.B.6 Step 2 Find one gap in feet
- The 40 gaps are all the same length and together they make up the full 5280 feet.
- Split the mile into 40 equal pieces to get the length of a single gap.
💡 Equal spacing means the mile is cut into 40 identical pieces, so one piece is the mile divided by 40.
5.NBT.B.6 Step 3 Divide the same 132 feet two ways
- Both animals cross one gap, so both cover 132 feet.
- Elmer does it in 44 equal strides, so one stride is 132 feet divided by 44 steps.
- Oscar does it in 12 equal leaps, so one leap is 132 feet divided by 12 steps.
- In each case the units are feet divided by steps, which gives feet per step.
💡 One fixed distance shared among more steps means shorter steps, so the same 132 feet gives a small stride and a big leap.
4.NBT.B.4 Step 4 Subtract to compare
- The question asks how much longer a leap is than a stride, which is a subtraction, not a ratio.
- One leap is 11 feet and one stride is 3 feet, so the leap is 11 - 3 = 8 feet longer.
- That is choice (B).
- A common slip is to stop at Oscar's leap length of 11 feet and pick choice (D) without subtracting.
💡 "How much longer" always means the difference of the two lengths, so the last move is a subtraction.
4.OA.A.3 Sketch a few poles in a row. Two poles make 1 gap, three poles make 2 gaps, four 5.NBT.B.6 The 40 gaps are all the same length and together they make up the full 5280 feet 5.NBT.B.6 Both animals cross one gap, so both cover 132 feet. Elmer does it in 44 equal st 4.NBT.B.4 The question asks how much longer a leap is than a stride, which is a subtractio Review
Reasonableness: Check the two step counts against one gap: 44 strides of 3 feet give $44 \times 3 = 132$ feet, and 12 leaps of 11 feet give $12 \times 11 = 132$ feet, so both animals really do cover the same gap. The size also fits common sense: Oscar needs $\frac{44}{12} = \frac{11}{3}$ times fewer steps than Elmer, so his leap should be $\frac{11}{3}$ times a stride, and indeed $3 \times \frac{11}{3} = 11$. An 11-foot ostrich leap and a 3-foot emu stride are believable for large birds, and the 8-foot difference is one of the offered choices.
Alternative: Work with whole-mile totals instead of one gap. Across the 40 gaps Elmer takes $40 \times 44 = 1760$ strides and Oscar takes $40 \times 12 = 480$ leaps, and both cover 5280 feet. Then one stride is $\frac{5280}{1760} = 3$ feet and one leap is $\frac{5280}{480} = 11$ feet, giving the same difference of 8 feet. This version skips finding the 132-foot gap length entirely.
CCSS standards used (min grade 5)
4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Turning 41 evenly spaced poles into 40 equal gaps before any arithmetic begins.)5.NBT.B.6Find whole-number quotients with up to four-digit dividends and two-digit divisors (Dividing 5280 by 40 for the gap length, then 132 by 44 and by 12 for one stride and one leap.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Taking 11 - 3 to answer how much longer a leap is than a stride.)
⭐ Count the gaps between the poles, not the poles, and then let the units guide you: feet divided by number of steps gives the length of one step.
⭐ Count the gaps between the poles, not the poles, and then let the units guide you: feet divided by number of steps gives the length of one step.
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