AMC 10 · 2010 · #2
Grade 6 rate-ratioMakarla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
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: During a $9$-hour work day, Makarla went to two meetings: the first lasted $45$ minutes and the second lasted twice as long. Find what percent of the whole work day she spent in meetings.
Givens: The work day is $9$ hours long; The first meeting took $45$ minutes; The second meeting took twice as long as the first; Answer choices: (A) $15$, (B) $20$, (C) $25$, (D) $30$, (E) $35$
Unknowns: The percent of the $9$-hour work day spent in meetings
Understand
Restated: During a $9$-hour work day, Makarla went to two meetings: the first lasted $45$ minutes and the second lasted twice as long. Find what percent of the whole work day she spent in meetings.
Givens: The work day is $9$ hours long; The first meeting took $45$ minutes; The second meeting took twice as long as the first; Answer choices: (A) $15$, (B) $20$, (C) $25$, (D) $30$, (E) $35$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units
The question stacks a few small tasks: find the second meeting's length, add the two meetings, express the day in the same unit, then turn the ratio into a percent. Tool #7 (Identify Subproblems) keeps each step clean. Tool #8 (Analyze the Units) is the guardrail: the day is in hours and the meetings in minutes, so the one real trap is comparing them before matching units. Convert the day to minutes and the percent falls out.
Execute — Answer: C
4.OA.A.1 Step 1 - Find the second meeting's length.
- It took twice as long as the first, and the first was $45$ minutes, so the second was $2\times 45=90$ minutes.
💡 "Twice as long" just means double it, so the second meeting is two of the first stacked together.
4.NBT.B.4 Step 2 - Add the two meetings to get the total meeting time.
- That is $45+90=135$ minutes.
💡 Total time in meetings is simply the first plus the second, added up.
5.MD.A.1 Step 3 - Put the work day in the same unit as the meetings.
- Since $1$ hour is $60$ minutes, a $9$-hour day is $9\times 60=540$ minutes.
💡 You can only compare minutes to minutes, so trade the hours in for the minutes they are worth.
6.RP.A.3 Step 4 - Compare meeting time to the whole day as a percent.
- The fraction is $\dfrac{135}{540}=\dfrac{1}{4}$, and $\dfrac{1}{4}=25\%$.
- So the answer is (C).
💡 Percent is just the part-over-whole fraction rewritten as a piece of $100$, and a quarter of anything is $25\%$.
4.OA.A.1 Find the second meeting's length. It took twice as long as the first, and the fi 4.NBT.B.4 Add the two meetings to get the total meeting time. That is $45+90=135$ minutes. 5.MD.A.1 Put the work day in the same unit as the meetings. Since $1$ hour is $60$ minute 6.RP.A.3 Compare meeting time to the whole day as a percent. The fraction is $\dfrac{135} Review
Reasonableness: The meetings run $135$ minutes out of a $540$-minute day — a bit more than two hours out of nine, which is roughly a quarter of the day. $25\%$ matches that eyeball estimate, and it lands right in the middle of the choices, so it is the sensible size. Answers like $15\%$ would be far too little for over two hours of meetings.
Alternative: Tool #8 (Analyze the Units) taken further: keep everything in hours instead. The meetings total $135$ minutes $=2.25$ hours, and $\dfrac{2.25}{9}=0.25=25\%$ — the same answer (C), reached by converting the meetings to hours rather than the day to minutes.
CCSS standards used (min grade 6)
4.OA.A.1Interpret a multiplication equation as a comparison (Reading "twice as long" as doubling to get the second meeting's $90$ minutes.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Adding the two meetings, $45+90=135$ minutes, to get total meeting time.)5.MD.A.1Convert among different-sized standard measurement units within a given system (Converting the $9$-hour day into $540$ minutes so it can be compared with the meeting minutes.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Turning the ratio $\tfrac{135}{540}$ into the percent $25\%$.)
⭐ Before you compare two amounts, make sure they are in the same unit — then the percent is just part over whole.
⭐ Before you compare two amounts, make sure they are in the same unit — then the percent is just part over whole.
More like this
Same archetype — closest grade level first.