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Person A makes 300 cakes in 2 hours. Person B makes 300 cakes in 2 hours. Person C makes 300 cakes in 3 hours

How long does it take all 3 working together to make 300 cakes. (Assume no synergies and rates are additive. So Person A and Person B would make 600 cakes in 2 hours together).

So I wasn't sure what to do here so I tried to figure out the per hour rate of each.

Person A makes 150 cakes in an hour. Person B makes 150 cakes in an hour. Person C makes 100 cakes in an hour.

So that wasn't getting me anywhere. All I know is that in an hour, they all make 150+150+100 = 400 cakes. So, to make 300 cakes, they need less than 1 hour.

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  • $\begingroup$ Common-denominator $6$ hours, then $R_A=\frac {900\text {cakes}}{6\text{hours}}$, $R_B=\frac {900\text {cakes}}{6\text{hours}}$, $R_C=\frac {600\text {cakes}}{6\text{hours}}$ and $R_A+R_B+R_C=\frac {2400\text {cakes}}{6\text{hours}}=\frac {100\text{cakes}}{0.25\text{hours}}$, multiply by $3$ to get $300$ cakes in... $\endgroup$ – abiessu Jan 30 '14 at 20:27
  • $\begingroup$ You should have kept going with, to paraphrase you, "they make 400 cakes per hour". $\endgroup$ – David Mitra Jan 30 '14 at 20:30
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Let's back up and go back to your observation, and complete the thought:

If together they are able to make $400$ cakes in an hour, then to make $300$ cakes takes them $$\dfrac{300}{400} \text{hours} = 45\text{ min}$$

We can also setup the problem as one of ratios: $$\dfrac{400}{1 \text{hour}} = \frac {300}{x\text{ hours}}$$ and then solve for $x$

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Another way to tackle it is, you can think of work problems like distance problems. Just like ratetime=distance, ratetime=work

When you have several people working together, you add their rates together. You did a great job of finding this, so we know that they make 400cakes/hour (which is their rate together) to find out how long it takes them to make 300 cakes we use the work equation and solve for t (time) 400c/h*t=300

so like was said before, t=3/4 of an hour or 45 minutes

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