AMC 8 · 2010 · #5

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

unit-conversionmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: unit-conversionmulti-digit-arithmetic

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Problem

Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Alice must change a light bulb that hangs $10$ cm below a ceiling $2.4$ m above the floor. She is $1.5$ m tall and can reach $46$ cm above her head. Standing on a stool, she can just barely touch the bulb. How tall is the stool, in centimeters?

Givens: Ceiling height $= 2.4$ m above the floor; Light bulb hangs $10$ cm below the ceiling; Alice's height $= 1.5$ m; Alice's reach above her head $= 46$ cm; Standing on the stool, Alice just reaches the bulb; Answer choices: (A) $32$, (B) $34$, (C) $36$, (D) $38$, (E) $40$ (cm)

Unknowns: The height of the stool in centimeters

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

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.

Execute — Answer: B

#8 Analyze the Units 5.MD.A.1 Step 1

Convert the meter measurements to centimeters so every length uses the same unit.
Using $1$ m $= 100$ cm, the ceiling becomes $240$ cm and Alice's height becomes $150$ cm; the $10$ cm and $46$ cm values are already in centimeters.

$$2.4 \text{ m} \times 100 \tfrac{\text{cm}}{\text{m}} = 240 \text{ cm}, \quad 1.5 \text{ m} \times 100 \tfrac{\text{cm}}{\text{m}} = 150 \text{ cm}$$

💡 Converting m to cm inside the metric system is exactly the Grade 5 "convert standard measurement units" standard.

#7 Identify Subproblems 4.MD.A.2 Step 2

Find subproblem (a): the bulb's height above the floor.
The bulb hangs $10$ cm below the ceiling, so subtract that from the ceiling height.

$$240 \text{ cm} - 10 \text{ cm} = 230 \text{ cm}$$

💡 Reading "$10$ cm below the ceiling" as a subtraction is the standard Grade 4 distance word-problem move.

#7 Identify Subproblems 4.MD.A.2 Step 3

Find subproblem (b): Alice's reach without the stool.
Add her height to the distance she can reach above her head.

$$150 \text{ cm} + 46 \text{ cm} = 196 \text{ cm}$$

💡 Stacking two vertical lengths (her body, then her arm above her head) is a Grade 4 length word-problem skill.

#7 Identify Subproblems 6.EE.B.7 Step 4

Set up the "just reaches" equation.
Let $s$ be the stool height in centimeters.
Standing on the stool, Alice's reach is $s + 196$ cm, and this must equal the bulb's height $230$ cm.

$$s + 196 = 230$$

💡 Translating "just reaches" into a one-variable equation is Grade 6 expression-and-equation work.

#8 Analyze the Units 6.EE.B.7 Step 5

Solve for $s$ by subtracting $196$ from both sides.

$$s = 230 - 196 = 34 \text{ cm} \;\Rightarrow\; \textbf{(B)}$$

💡 Inverse operations on a one-step equation give the stool height directly.

[1] #8 5.MD.A.1 Convert the meter measurements to centimeters so every length uses the same unit

[2] #7 4.MD.A.2 Find subproblem (a): the bulb's height above the floor. The bulb hangs $10$ cm b

[3] #7 4.MD.A.2 Find subproblem (b): Alice's reach without the stool. Add her height to the dist

[4] #7 6.EE.B.7 Set up the "just reaches" equation. Let $s$ be the stool height in centimeters.

[5] #8 6.EE.B.7 Solve for $s$ by subtracting $196$ from both sides.

Review

Reasonableness: A $34$ cm stool is about $13$ inches tall — a short kitchen step stool, which fits the everyday setting. Sanity check the totals: $34 + 150 + 46 = 230$ cm, and $240 - 10 = 230$ cm, so reach-on-stool exactly matches bulb height. The other choices either leave Alice short of the bulb ($32$ cm gives $228 < 230$) or overshoot it ($36$ cm gives $232 > 230$), so $34$ is the unique fit.

Alternative: Tool #6 (Guess and Check) on the choices: for each candidate $s$, compute $s + 196$ and compare to $230$. (A) $32 + 196 = 228$; (B) $34 + 196 = 230$ ✓; (C) $36 + 196 = 232$; (D) $38 + 196 = 234$; (E) $40 + 196 = 236$. Only (B) gives the exact reach.

CCSS standards used (min grade 6)

5.MD.A.1 Convert among different-sized standard measurement units within a given system (Converting the ceiling height ($2.4$ m to $240$ cm) and Alice's height ($1.5$ m to $150$ cm) so every length is in centimeters.)
4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money (Combining the given lengths into the bulb's floor height ($240 - 10 = 230$ cm) and Alice's standing reach ($150 + 46 = 196$ cm).)
6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations of the form $x + p = q$ (Modeling "just reaches" as $s + 196 = 230$ and solving the one-step linear equation to get $s = 34$ cm.)

⭐ Once every length is in the same unit, this AMC 8 problem is just one Grade 6 equation: $s + 196 = 230$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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