AMC 8 · 2023 · #3

Grade 5 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

formula-substitutionfraction-decimal-conversionmulti-digit-arithmetic dimensional-analysisidentify-subproblems ↑ Prerequisites: order-of-operationsmulti-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

Wind chill is estimated by the formula "(wind chill) = (air temperature) − 0.7 × (wind speed)," where temperature is in ^°F and wind speed is in mph. Given an air temperature of 36^°F and a wind speed of 18 mph, find which of (A) 18, (B) 23, (C) 28, (D) 32, (E) 35 is closest to the wind chill.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

The formula is already given and the quantities carry mixed units (temperature in ^°F, wind speed in mph, the constant 0.7 acting as "^°F lost per mph of wind"). Tool #8 (Analyze the Units) makes the setup transparent: (mph) × (^°F/mph) = {}^°F, so 0.7 × wind speed is just the number of degrees you subtract from the air temperature. After computing, Tool #3 (Eliminate Possibilities) handles the "closest" comparison against the five choices. No need for algebra (#13) — direct substitution and arithmetic are enough.

1STEP 1

Substitute the given values into the formula, giving 36 - 0.7 × 18, since 0.7 × (wind speed) is the ^°F the wind shaves off.

wind chill = 36 - 0.7 × 18

2STEP 2

Order of operations first: 0.7 × 18 = $\frac{7}{10}$ × 18 = 12.6, the ^°F the 18 mph wind removes.

0.7 × 18 = 12.6

3STEP 3

Subtract the loss from the air temperature: 36 - 12.6 = 23.4, the exact wind chill.

36 - 12.6 = 23.4

4STEP 4

The problem wants the closest value: 23.4 sits only 0.4 from (B), far nearer than any other choice.

|23.4 - 23| = 0.4 → (B) 23

Answer

Sanity check: with no wind the wind chill would just be the air temperature, 36^°F. Wind makes it feel colder, and 18 mph is a fairly stiff breeze, so a drop of around 12 to 13 degrees feels right. The formula gives a drop of 0.7 × 18 = 12.6, leaving 23.4^°F — well below 36 but still well above freezing, a believable wind-chill reading. Among the choices, only 23 is essentially the same as 23.4, so (B) is correct.

💡Key takeaway

This AMC 8 problem only needs Grade 5 decimal multiplication, decimal subtraction, and rounding that you already know!

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