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A factory uses two machines to produce widgets. The first can produce
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04 Aug 2025, 13:46
04 Aug 2025, 13:46
Expert Reply
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Question Stats:
50% (00:49) correct
50% (01:37) wrong based on 2 sessions
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A factory uses two machines to produce widgets. The first can produce $n$ widgets in 20 minutes. The second can produce $n$ widgets in 30 minutes. If the machines are both operating consistently at the specified rates, how many minutes will it take to produce a total of $\(3 n\)$ widgets?
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36
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Re: A factory uses two machines to produce widgets. The first can produce
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14 Aug 2025, 22:25
14 Aug 2025, 22:25
Expert Reply
Given:
- Machine 1 produces $n$ widgets in 20 minutes.
- Machine 2 produces $n$ widgets in 30 minutes.
- Both machines work simultaneously.
- We want to find how many minutes it takes for the two machines together to produce $3 n$ widgets.
Step 1: Find rates of production for each machine
- Rate of Machine 1 :
$$
\(\text { Rate }_1=\frac{n \text { widgets }}{20 \text { minutes }}=\frac{n}{20} \text { widgets per minute }\)
$$
- Rate of Machine 2 :
$$
\(\text { Rate }_2=\frac{n \text { widgets }}{30 \text { minutes }}=\frac{n}{30} \text { widgets per minute }\)
$$
Step 2: Find combined rate of production
$$
\(\text { Combined Rate }=\text { Rate }_1+\text { Rate }_2=\frac{n}{20}+\frac{n}{30}=n\left(\frac{1}{20}+\frac{1}{30}\right)\)
$$
Find common denominator:
$$
\(\frac{1}{20}+\frac{1}{30}=\frac{3}{60}+\frac{2}{60}=\frac{5}{60}=\frac{1}{12}\)
$$
So,
$$
\(\text { Combined Rate }=n \times \frac{1}{12}=\frac{n}{12} \text { widgets per minute }\)
$$
Step 3: Find time to produce $3 n$ widgets at combined rate
$$
\(\text { Time }=\frac{\text { Total widgets }}{\text { Rate }}=\frac{3 n}{\frac{n}{12}}=3 n \times \frac{12}{n}=3 \times 12=36 \text { minutes }\)
$$
Final answer:
It will take 36 minutes to produce $3 n$ widgets.
36
- Machine 1 produces $n$ widgets in 20 minutes.
- Machine 2 produces $n$ widgets in 30 minutes.
- Both machines work simultaneously.
- We want to find how many minutes it takes for the two machines together to produce $3 n$ widgets.
Step 1: Find rates of production for each machine
- Rate of Machine 1 :
$$
\(\text { Rate }_1=\frac{n \text { widgets }}{20 \text { minutes }}=\frac{n}{20} \text { widgets per minute }\)
$$
- Rate of Machine 2 :
$$
\(\text { Rate }_2=\frac{n \text { widgets }}{30 \text { minutes }}=\frac{n}{30} \text { widgets per minute }\)
$$
Step 2: Find combined rate of production
$$
\(\text { Combined Rate }=\text { Rate }_1+\text { Rate }_2=\frac{n}{20}+\frac{n}{30}=n\left(\frac{1}{20}+\frac{1}{30}\right)\)
$$
Find common denominator:
$$
\(\frac{1}{20}+\frac{1}{30}=\frac{3}{60}+\frac{2}{60}=\frac{5}{60}=\frac{1}{12}\)
$$
So,
$$
\(\text { Combined Rate }=n \times \frac{1}{12}=\frac{n}{12} \text { widgets per minute }\)
$$
Step 3: Find time to produce $3 n$ widgets at combined rate
$$
\(\text { Time }=\frac{\text { Total widgets }}{\text { Rate }}=\frac{3 n}{\frac{n}{12}}=3 n \times \frac{12}{n}=3 \times 12=36 \text { minutes }\)
$$
Final answer:
It will take 36 minutes to produce $3 n$ widgets.
36
gmatclubot
Re: A factory uses two machines to produce widgets. The first can produce [#permalink]
14 Aug 2025, 22:25
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