AMC 10 · 2010 · #10
Grade 7 rate-ratioShelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?
Pick an answer.
AMC 10 2010 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.
Toolkit + CCSS Solution
Understand
Restated: Shelby's scooter goes $30$ mph in dry weather and $20$ mph in rain. She drove in the sun, then in the rain, covering $16$ miles in a total of $40$ minutes. Find how many of those minutes were driven in the rain.
Givens: Dry speed is $30$ miles per hour; Rain speed is $20$ miles per hour; Total distance is $16$ miles; Total time is $40$ minutes; Answer choices: (A) $18$, (B) $21$, (C) $24$, (D) $27$, (E) $30$ minutes
Unknowns: The number of minutes Shelby drove in the rain
Understand
Restated: Shelby's scooter goes $30$ mph in dry weather and $20$ mph in rain. She drove in the sun, then in the rain, covering $16$ miles in a total of $40$ minutes. Find how many of those minutes were driven in the rain.
Givens: Dry speed is $30$ miles per hour; Rain speed is $20$ miles per hour; Total distance is $16$ miles; Total time is $40$ minutes; Answer choices: (A) $18$, (B) $21$, (C) $24$, (D) $27$, (E) $30$ minutes
Plan
Primary tool: #4 Introduce a Variable
Secondary: #8 Analyze the Units
Two unknown time chunks share one total time and one total distance, so Tool #4 (Introduce a Variable) is the natural fit: name the rain time, write the sun time as "what's left," and turn the distance fact into one equation. Tool #8 (Analyze the Units) keeps the arithmetic honest — speeds are per hour but the answer is wanted in minutes, so units must be converted at the start and again at the end.
Execute — Answer: C
6.RP.A.3 Step 1 Match the units
- The speeds are given per hour, so the total time must be in hours before it can be multiplied by a speed.
- Convert $40$ minutes to hours by dividing by $60$.
💡 Distance $=$ speed $\times$ time only works when the time unit matches the speed's unit, so switch minutes into hours first.
7.EE.B.4 Step 2 Name the unknown and set up
- Let $r$ be the time driven in the rain, in hours.
- The sun time is whatever is left of the $\tfrac{2}{3}$ hour, so it is $\tfrac{2}{3} - r$.
- Each distance is speed times time, and the two distances add to $16$ miles.
💡 One variable is enough because the sun time is forced to be the leftover once the rain time is chosen.
7.EE.B.4 Step 3 Solve for the rain time
Distribute the $30$, combine the $r$ terms, then isolate $r$.
💡 Peeling operations off one at a time strips the equation down to the value of $r$.
6.RP.A.3 Step 4 Convert back to minutes
The question asks for minutes, so change $\tfrac{2}{5}$ hour back into minutes by multiplying by $60$.
💡 Return to the unit the question actually asked for — the answer is $24$ minutes, choice (C).
6.RP.A.3 The speeds are given per hour, so the total time must be in hours before it can 7.EE.B.4 Let $r$ be the time driven in the rain, in hours. The sun time is whatever is le 7.EE.B.4 Distribute the $30$, combine the $r$ terms, then isolate $r$. 6.RP.A.3 The question asks for minutes, so change $\tfrac{2}{5}$ hour back into minutes b Review
Reasonableness: Check both totals. Rain time $= \tfrac{2}{5}$ hr, so sun time $= \tfrac{2}{3} - \tfrac{2}{5} = \tfrac{10}{15} - \tfrac{6}{15} = \tfrac{4}{15}$ hr. Distances: sun $= 30 \times \tfrac{4}{15} = 8$ miles, rain $= 20 \times \tfrac{2}{5} = 8$ miles, total $8 + 8 = 16$ miles. Times in minutes: sun $= \tfrac{4}{15}\times 60 = 16$ min, rain $= 24$ min, total $40$ min. Both the distance and the time come out exactly right, and $24$ minutes is choice (C).
Alternative: Tool #6 (Guess and Check) on the answer choices. Try (C) $24$ min of rain $= 0.4$ hr, leaving $16$ min $\approx 0.2\overline{6}$ hr of sun. Distance $= 30(0.2\overline{6}) + 20(0.4) = 8 + 8 = 16$ miles — an exact match on the first sensible guess, confirming (C) without solving an equation.
CCSS standards used (min grade 7)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Converting $40$ minutes to $\tfrac{2}{3}$ hour so the time unit matches the miles-per-hour speeds, and converting the answer $\tfrac{2}{5}$ hour back to $24$ minutes.)7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Letting $r$ be the rain time, writing the sun time as $\tfrac{2}{3} - r$, building the distance equation $30(\tfrac{2}{3} - r) + 20r = 16$, and solving it for $r = \tfrac{2}{5}$ hour.)
⭐ Name the rain time as one variable, make the sun time the leftover, and add up the distances — the whole race problem shrinks to a single Grade 7 equation.
⭐ Name the rain time as one variable, make the sun time the leftover, and add up the distances — the whole race problem shrinks to a single Grade 7 equation.
More like this
Same archetype — closest grade level first.