Team A has 4 girls and 7 boys. Team A takes 5 days to paint 4 rooms. Team B has 7 girls and 10 boys. Team B takes 4 days to paint 5 rooms. Team C has 8 girls and 5 boys. How long will it take team C to paint 6 rooms? (Assume every girl works at a constant rate, every boy works at a constant rate, and every room is identical)
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Wait, so we have to figure out the rates from the two teams? I set up equations but got stuck. Can someone explain the next step?
Let g = girl's rate, b = boy's rate. Team A: 5(4g+7b)=4, so 4g+7b=4/5. Team B: 4(7g+10b)=5, so 7g+10b=5/4. Solve for g and b.
I did that but my numbers looked messy. Maybe I made an arithmetic error. Thanks!
This one took me longer than expected. The fractions are annoying. But once you get g and b, Team C is straightforward.
Is there a faster way? I feel like solving two equations with fractions is time-consuming.
You could scale to avoid fractions, but it's still two equations. Maybe not worth it.
I got g=0.05 and b=0.1? Then Team C rate = 8*0.05+5*0.1=0.9 rooms/day, so 6/0.9=6.67 days, which isn't an option. I'm confused.
Check your solving. From 4g+7b=0.8 and 7g+10b=1.25. Multiply first by 7: 28g+49b=5.6; second by 4: 28g+40b=5.0. Subtract: 9b=0.6 -> b=0.0666..., then g=(0.8-7b)/4=0.08333...
Oh, I see my mistake. Thanks!
What it tests
Your ability to combine individual work rates, usually by working with the fraction of a job completed per unit of time.
Common trap
Adding the times instead of the rates, or forgetting to take the reciprocal when combining rates.