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
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.

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.

How to solve
Strategy Introduce a Variable

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.

1STEP 1

Name the total and write each rate

Let N be the chimney's brick count. Total divided by time gives the solo rates: Brenda N/9, Brandon N/10 bricks per hour.

Brenda: N/9 bricks/hr, Brandon: N/10 bricks/hr
2STEP 2

Build the combined-rate equation

Talking costs them 10 an hour, so together they lay N/9 + N/10 - 10, and 5 hours of that is the whole chimney: 5(N/9 + N/10 - 10) = N.

5(N/9 + N/10 - 10) = N
3STEP 3

Clear fractions and solve

Distribute the 5 and multiply through by 18 to clear the fractions: 19N - 900 = 18N, so N = 900 bricks, choice (B).

5N/9 + 5N/10 - 50 = N → 10N + 9N - 900 = 18N → 19N - 900 = 18N → N = 900
Answer
900
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.
💡Key takeaway

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.

  • Name the total and write each rate
  • Build the combined-rate equation
  • Clear fractions and solve

More like this

Same archetype — closest grade level first.