AMC 10 · 2008 · #18

Grade 8 rate-ratio

Archetype Meeting Time with Piecewise Rates └ Sub-archetype Constant-Rate Closing Speed

ratelinear-equations-one-varfraction-arithmetic convert-to-algebra ↑ Prerequisites: rate

📏 Short solution 💡 2 insights

Problem

Bricklayer Brenda takes $9$ hours to build a chimney alone, and bricklayer Brandon takes $10$ hours to build it alone. When they work together, they talk a lot, and their combined output decreases by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chimney?

Pick an answer.

(A)

$\ 500$

(B)

$\ 900$

(C)

$\ 950$

(D)

$\ 1000$

(E)

$\ 1900$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: Brenda can build the whole chimney in 9 hours; Brandon can build it in 10 hours. When they work together their combined speed drops by 10 bricks each hour. Together they finish the chimney in 5 hours. Find how many bricks the chimney has.

Givens: Brenda alone finishes the chimney in 9 hours; Brandon alone finishes the chimney in 10 hours; Working together, their combined rate is 10 bricks per hour less than the sum of their solo rates; Working together they finish in 5 hours

Unknowns: The total number of bricks in the chimney

Understand

Plan

Primary tool: #4 Introduce a Variable

Secondary: #8 Analyze the Units, #13 Convert to Algebra

The total number of bricks is unknown, yet every rate depends on it. Naming that total with one letter turns 'per hour' language into exact fractions. Tracking the units (bricks per hour times hours equals bricks) tells you exactly what equation to write, and the whole word problem then collapses into a single linear equation to solve.

Execute — Answer: B

#4 Introduce a Variable 6.RP.A.3 Step 1

Name the total and write each rate

Let N be the number of bricks in the chimney.
Brenda finishes the same N bricks in 9 hours, so she lays N/9 bricks per hour.
Brandon finishes N bricks in 10 hours, so he lays N/10 bricks per hour.

$$\text{Brenda: } \frac{N}{9}\ \text{bricks/hr}, \qquad \text{Brandon: } \frac{N}{10}\ \text{bricks/hr}$$

💡 One fixed job divided by the time it takes is a speed, so the same unknown total controls both rates.

#8 Analyze the Units 7.EE.B.4 Step 2

Build the combined-rate equation

Together their rate is the sum of the two rates minus the 10 bricks per hour they lose to talking: N/9 + N/10 - 10 bricks per hour.
Rate times time equals work, and in 5 hours they finish the whole chimney, which is N bricks.
So 5 times that combined rate equals N.

$$5\left(\frac{N}{9} + \frac{N}{10} - 10\right) = N$$

💡 Bricks-per-hour multiplied by hours gives bricks, so the five-hour output must equal the one whole chimney.

#13 Convert to Algebra 8.EE.C.7 Step 3

Clear fractions and solve

Distribute the 5, then multiply every term by 18 (the least common multiple of 9 and 10 scaled to clear both denominators) to remove fractions.
This leaves N on both sides; collecting them gives the total directly.

$$\frac{5N}{9} + \frac{5N}{10} - 50 = N \;\Rightarrow\; 10N + 9N - 900 = 18N \;\Rightarrow\; 19N - 900 = 18N \;\Rightarrow\; N = 900$$

💡 Clearing denominators turns the messy fraction equation into plain whole numbers with the unknown on both sides, so one subtraction isolates it. The chimney has 900 bricks, choice (B).

[1] #4 6.RP.A.3 Let N be the number of bricks in the chimney. Brenda finishes the same N bricks

[2] #8 7.EE.B.4 Together their rate is the sum of the two rates minus the 10 bricks per hour the

[3] #13 8.EE.C.7 Distribute the 5, then multiply every term by 18 (the least common multiple of 9

Review

Reasonableness: Check N = 900 against the story. Brenda lays 900/9 = 100 bricks/hr, Brandon lays 900/10 = 90 bricks/hr; summed that is 190, minus the 10 lost to talking gives 180 bricks/hr. Over 5 hours that is 180 x 5 = 900 bricks, exactly one chimney. It also passes a sanity filter: without the talking penalty their combined rate would be 190 bricks/hr, and 5 hours near that rate should land just under 1000, so 900 fits and the far-apart choices 500 and 1900 do not.

Alternative: Instead of one variable, reason in fraction-of-job terms: in 5 hours Brenda alone does 5/9 of the chimney and Brandon does 5/10 = 1/2, together 5/9 + 1/2 = 19/18 of a chimney. That is 1/18 of a chimney too much, and that surplus is exactly the work the talking penalty removed: 10 bricks/hr x 5 hr = 50 bricks equal 1/18 of the chimney, so the chimney is 50 x 18 = 900 bricks.

CCSS standards used (min grade 8)

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Turning each bricklayer's solo time into a bricks-per-hour rate expressed with the unknown total)
7.EE.B.4 Use variables to represent quantities and construct simple equations (Writing the word-problem condition (5 hours of combined work equals the whole chimney) as an equation)
8.EE.C.7 Solve linear equations in one variable (Clearing fractions and collecting the unknown from both sides to solve for the number of bricks)

⭐ Give the unknown total a name, turn every 'per hour' into a fraction of that name, and rate times time equals work does the rest.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Name the total and write each rate

Build the combined-rate equation

Clear fractions and solve

Review

CCSS standards used (min grade 8)

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