AMC 8 · 2023 · #3

Grade 5 rate-ratio

Archetype Arithmetic Expression Decomposition

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 a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation
$(\text{wind chill}) = (\text{air temperature}) - 0.7 \times (\text{wind speed}),$
where temperature is measured in degrees Fahrenheit $(^{\circ}\text{F})$ and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ}\text{F}$ and the wind speed is $18$ mph. Which of the following is closest to the approximate wind chill?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Wind chill is estimated by the formula "(wind chill) = (air temperature) − 0.7 × (wind speed)," where temperature is in $^{\circ}\text{F}$ and wind speed is in mph. Given an air temperature of $36^{\circ}\text{F}$ 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.

Givens: Formula: $\text{wind chill} = \text{air temperature} - 0.7 \times \text{wind speed}$; Air temperature = $36^{\circ}\text{F}$; Wind speed = $18$ mph; Answer choices: (A) 18, (B) 23, (C) 28, (D) 32, (E) 35

Unknowns: The answer choice **closest** to the wind chill produced by the formula

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #3 Eliminate Possibilities

The formula is already given and the quantities carry mixed units (temperature in $^{\circ}\text{F}$, wind speed in mph, the constant $0.7$ acting as "$^{\circ}\text{F}$ lost per mph of wind"). Tool #8 (Analyze the Units) makes the setup transparent: $(\text{mph}) \times (^{\circ}\text{F}/\text{mph}) = {}^{\circ}\text{F}$, so $0.7 \times \text{wind speed}$ 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.

Execute — Answer: B

#8 Analyze the Units 5.OA.A.2 Step 1

Track the units to make sense of the formula.
The constant $0.7$ has units of $^{\circ}\text{F}$ per mph: every $1$ mph of wind subtracts $0.7^{\circ}\text{F}$ from how warm it feels.
So $0.7 \times (\text{wind speed})$ is the number of degrees Fahrenheit shaved off by the wind, and subtracting that from $36^{\circ}\text{F}$ gives the wind chill.
Substitute the given numbers directly.

$$\text{wind chill} = 36 - 0.7 \times 18$$

💡 Translating a stated formula into a numerical expression is exactly the Grade 5 idea of writing simple expressions that record a calculation.

#8 Analyze the Units 5.NBT.B.7 Step 2

By the order of operations, do the multiplication first.
Rewriting $0.7$ as $\tfrac{7}{10}$ makes the arithmetic clean: $0.7 \times 18 = \tfrac{7 \times 18}{10} = \tfrac{126}{10} = 12.6$.
So the $18$ mph wind shaves $12.6^{\circ}\text{F}$ off the air temperature.

$$0.7 \times 18 = 12.6$$

💡 Multiplying a decimal by a whole number is right in the Grade 5 "decimal operations to hundredths" wheelhouse.

#8 Analyze the Units 5.NBT.B.7 Step 3

Now subtract the amount lost from the air temperature to get the exact wind chill: a whole number $36$ minus a decimal $12.6$.

$$36 - 12.6 = 23.4$$

💡 Decimal subtraction to the tenths place is the same Grade 5 decimal-arithmetic standard.

#3 Eliminate Possibilities 5.NBT.A.4 Step 4

The problem asks for the *closest* answer, so compare $23.4$ to each option: $|23.4-18|=5.4$, $|23.4-23|=0.4$, $|23.4-28|=4.6$, $|23.4-32|=8.6$, $|23.4-35|=11.6$.
The smallest gap is to $23$, which is choice (B).

$$|23.4 - 23| = 0.4 \;\Rightarrow\; \textbf{(B)}\ 23$$

💡 Rounding $23.4$ to the nearest whole number is exactly the Grade 5 "round decimals to any place" standard.

[1] #8 5.OA.A.2 Track the units to make sense of the formula. The constant $0.7$ has units of $^

[2] #8 5.NBT.B.7 By the order of operations, do the multiplication first. Rewriting $0.7$ as $\tf

[3] #8 5.NBT.B.7 Now subtract the amount lost from the air temperature to get the exact wind chil

[4] #3 5.NBT.A.4 The problem asks for the *closest* answer, so compare $23.4$ to each option: $|2

Review

Reasonableness: Sanity check: with no wind the wind chill would just be the air temperature, $36^{\circ}\text{F}$. 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 $0.7 \times 18 = 12.6$, leaving $23.4^{\circ}\text{F}$ — 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.

Alternative: Tool #6 (Guess and Check) also works: each answer choice claims a specific drop from $36$. Choice (B) 23 implies a drop of $36-23=13$, choice (C) 28 implies a drop of $8$, and so on. The actual drop is $0.7 \times 18 = 12.6$, which is closest to $13$, so (B) wins immediately — the same answer with slightly different bookkeeping.

CCSS standards used (min grade 5)

5.OA.A.2 Write simple expressions that record calculations with numbers (Translating the verbal wind-chill formula into the numerical expression $36 - 0.7 \times 18$.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Computing the decimal product $0.7 \times 18 = 12.6$ and the decimal subtraction $36 - 12.6 = 23.4$.)
5.NBT.A.4 Round decimals to any place (Rounding $23.4$ to the nearest whole number to match it with the closest answer choice, (B) 23.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 5)

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