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Consider the following

An experienced bricklayer can work twice as fast as an apprentice bricklayer. After the bricklayers work together on a job for 6 h, the experienced bricklayer quits. The apprentice requires 12 more hours to finish the job. How long would it take the experienced bricklayer, working alone, to do the job?

Now, it is intuitively obvious that in 6 hours the apprentice does x work, the master does 2x work, and therefore the overall amount of work done is 3*6=18 "work units". Subsequently the apprentice does another 12 "work units" for a total of 30. The apprentice works at a rate of 1/30, thus alone s/he will finish the job in 30 hours whereas the master will only take 15.

How can we write equations for this?

Joe Stavitsky
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3 Answers3

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You have gotten the idea well. First define your variables. If you define $x$ as the work done by the apprentice in one hour and $j$ as the amount of work to do the job, the work done before the experienced bricklayer quits is $6x+6(2x)=18x$ The statement that the apprentice takes another $12$ hours gives us the equation $18x+12x=j$ and we are asked to find $\frac jx$ so we can say $18x+12x=j,\quad 30x=j,\quad \frac jx=30$. If this looks like your solution, that is not a coincidence.

Ross Millikan
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Let x = the total number of hours the experienced bricklayer needs to finish the job alone. Then 2x = the total hours the apprentice bricklayer needs to finish the same job on his own. Then in 6 hrs, the apprentice did 6/2x part of the job, and the experienced one did 6/x job. When the experienced quit, the apprentice had 12/2x part of the job to complete it. So the equation is: 6/x + 6/2x + 12/2x = 1 ==> x = 15 hrs, so 2x = 30 hrs and so the rate of the apprentice is : 1/30.

DeepSea
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The apprentice works at a rate of

$$rt $$

If the experienced worker is $x$ times faster then the experienced worker has a rate of

$$x(rt)$$

We say the experienced worker and the apprentice worked for a time $t_1$. then that means

$$rt_1 + xrt_2$$

work was complete.

We say the apprentice continued to work for a time $t_2$. then that means the total work was

$$rt_1 + xrt_1 + rt_2$$

Now to find the amount of time it takes the experienced worker alone to do the jobe we solve the equation:

$$xrt = rt_1 + xrt_1 + rt_2$$

for the variable t (without subscript)

which yields

$$t = \frac{rt_1 + xrt_1 + rt_2}{xr} =\frac{t_1 + xt_1 + t_2}{x} $$

Sure enough in your example if we substitute values we know x = 2, $t_1$ = 6, $t_2$ = 12

so the time taken is:

$$\frac{t_1 + 2t_1 + t_2}{2} = \frac{6 + 2*6 + 12}{2} = \frac{30}{2} = 15 hours$$

Sidharth Ghoshal
  • 16,771