Competition · AMC preparation · step 4 of 4
AMC 8 · 2010 · #5
Grade 6 arithmeticPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Every length in this problem is a vertical distance, so the entire problem is a single one-dimensional sum: stool + Alice + overhead reach = floor-to-bulb height. Tool #8 (Analyze the Units) is the gating move because the data is mixed in meters and centimeters; converting everything to cm first makes the arithmetic trivial. Tool #7 (Identify Subproblems) then splits the calculation into two small targets — (a) the bulb's height above the floor, and (b) Alice's reach without the stool — so the final equation reduces to one subtraction.
Convert meters to centimeters
Put everything in one unit: the 2.4 m ceiling is 240 cm and Alice's 1.5 m height is 150 cm; 10 cm and 46 cm already fit.
Converting m to cm inside the metric system is exactly the Grade 5 "convert standard measurement units" standard.
5.MD.A.1Analyze The UnitsFind the bulb's height
Subproblem (a): the bulb hangs 10 cm below the 240 cm ceiling, so its height above the floor is 230 cm.
Reading "10 cm below the ceiling" as a subtraction is the standard Grade 4 distance word-problem move.
4.MD.A.2Identify SubproblemsFind Alice's reach
Subproblem (b): Alice's floor reach is height plus overhead, 150 cm + 46 cm = 196 cm.
Stacking two vertical lengths (her body, then her arm above her head) is a Grade 4 length word-problem skill.
4.MD.A.2Identify SubproblemsSet up the reach equation
"Just reaches" means stool + reach = bulb height, so s + 196 = 230.
Translating "just reaches" into a one-variable equation is Grade 6 expression-and-equation work.
When Alice stands on the stool and just touches the bulb, the stool height s satisfies s + 196 = 230: her fingertip height while on the stool equals the bulb's height above the floor.
▸ Why?
The left side, s + 196, is her fingertip height while standing on the stool: the stool raises her whole 196 cm floor reach straight up, so her reach on the stool is the stool height stacked on top of that reach.
▸ Why?
The floor-to-fingertip distance is one vertical line cut into two touching pieces — floor up to the stool top (s), then the stool top up to her fingertips (her 196 cm reach) — with no gap or overlap, so the pieces add back to the whole.
▸ Why?
The right side, 230, is the bulb's height above the floor: the full 240 cm from floor to ceiling is the bulb's height plus the 10 cm gap from the bulb up to the ceiling, so the bulb's height is what is left below that gap.
▸ Why?
The floor-to-ceiling height splits into two touching pieces — floor up to the bulb, then the bulb up to the ceiling (10 cm) — with no gap or overlap, so those pieces add back to the whole 240 cm and the lower piece is 230.
▸ Why?
The two sides are set equal because "just reaches" means her fingertips stop exactly at the bulb — her reach height and the bulb's height name one and the same height, so the expression for one may be set equal to the value of the other.
▸ Why?
Her stool-top reach equals the fingertip height, and the bulb also sits at that fingertip height of 230 cm; two measures equal to the same height are equal to each other.
Solve for the stool height
Subtract 196 from both sides: s = 230 - 196 = 34 cm, choice (B).
Inverse operations on a one-step equation give the stool height directly.
6.EE.B.7Analyze The UnitsOnce every length is in the same unit, this AMC 8 problem is just one Grade 6 equation: s + 196 = 230.
- Convert meters to centimeters
- Find the bulb's height
- Find Alice's reach
- Set up the reach equation
- Solve for the stool height
A parent dashboard for the family lives at sensimlab.com.