Problem
Work rates add, times do not
One person folds 240 paper boats alone in 1 hour. The other folds the same 240 in 40 minutes. Together they fold at their usual steady speeds. Find how many minutes the 240 boats take.
Reasoning · Grade 6-1 Ratio and Rate (2)
Your answer
Watch out Averaging the times is wrong — working together is faster, so add the rates.
How to solve
Strategy Analyze the Units — The tempting move is to combine the two times, 60 minutes and 40 minutes, somehow — but minutes are the wrong thing to add. Look at the units instead: what actually piles up when two people work side by side is boats per minute. So I break the question into three small subproblems: how many boats per minute does Emma fold, how many does Jacob fold, and how many do the two of them fold together. Once I have the combined boats per minute, the time is just 240 boats shared out at that speed. A one-minute picture makes the idea obvious: in a single minute Emma's pile and Jacob's pile sit next to each other, and the two piles simply add.
1STEP 1
Put both times in the same unit
Turn the hour into 60 minutes.
1 hour = 60 minutes
You can never compare or combine two speeds while one is measured in hours and the other in minutes. Matching the units first is the cheapest way to avoid a wrong answer.
6.RP.A.3Analyze The Units2STEP 2
Find Emma's rate in boats per minute
One folds 4 boats a minute.
240 ÷ 60 = 4 boats per minute
A unit rate answers 'how much in ONE minute?'. Dividing the total by the number of minutes is exactly the sharing-equally division a young solver already knows.
6.RP.A.2Analyze The Units3STEP 3
Find Jacob's rate in boats per minute
The other folds 6 boats a minute.
240 ÷ 40 = 6 boats per minute
Less time for the same job means a bigger one-minute amount. Seeing 6 come out larger than 4 is a free check that the division was set up the right way round.
6.RP.A.2Analyze The Units4STEP 4
Add the rates, not the times
Together that is 10 a minute.
4 + 6 = 10 boats per minute together
Boats per minute is a count of boats, and counts of things made in the same minute simply add up. Minutes, by contrast, are not being stacked — the two people spend the same minutes, they do not spend separate ones.
6.RP.A.3Draw A DiagramWorking together, the two rates add, even though the two times certainly do not.
Why?
A rate says how much of the job each person finishes in one minute, and that amount does not change because someone else is working too.
🧱A bigger unit is a fixed count of smaller unitsA bigger unit always equals the same fixed number of smaller units (1 hour = 60 minutes, 1 km = 1000 m), so a measure can be renamed between them.
Why?
The job done in one minute is the two shares put together with no gap and no overlap, so the shares simply add.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
5STEP 5
Divide the job by the combined rate
Dividing 240 gives 24 minutes.
240 ÷ 10 = 24 minutes
Once the speed is a single number, 'how long?' is one division: total amount divided by amount per minute. The units say it too — boats divided by boats-per-minute leaves minutes.
6.RP.A.3Identify Subproblems6STEP 6
Check the same answer with the fraction-of-a-job method
Taking the job as 1 also gives 24 minutes.
1/60 + 1/40 = 2/120 + 3/120 = 5/120 = 1/24, 1 ÷ 1/24 = 24
Calling the job 1 turns each worker into a fraction of a job per minute, and those fractions add for exactly the same reason the boat counts did. Dividing 1 by 1/24 asks 'how many twenty-fourths in a whole?', which is 24.
6.NS.A.1Solve An Easier Related ProblemAnswer
24 minutes
240 ÷ 10 = 24
The units come out right: 240 boats divided by 10 boats per minute leaves minutes, and the answer is a time. The size is right too. Two people working together must finish faster than either one alone, so the answer has to be below 40 minutes — the faster person's time — and 24 is. It also has to be more than half of 40, namely 20 minutes, because Emma is slower than Jacob and so cannot be as much help as a second Jacob would be; 24 sits between 20 and 40, exactly where it should. As a direct check, in 24 minutes Emma folds 4 x 24 = 96 boats and Jacob folds 6 x 24 = 144 boats, and 96 + 144 = 240. Notice what would have gone wrong with the tempting shortcuts: adding the times gives 100 minutes and averaging them gives 50 minutes, and both are slower than Jacob working alone, which is impossible.
Takeaway
When two people work together, add how much they do each minute — never add the times they take alone.
- Put both times in the same unit
- Find Emma's rate in boats per minute
- Find Jacob's rate in boats per minute
- Add the rates, not the times
- Divide the job by the combined rate
- Check the same answer with the fraction-of-a-job method