Question 279072
For jobs like this, look at it terms of the whole job being a unitary thing.
Carmen can do the job in 24 minutes, so she does the job at a rate of 1/24 of the total job per minute.
Carmen and Kim working together can do the job in 8 minutes, so their combined rate is 1/8 per minute.
Algebraically, their combined speeds can be shown as:
{{{ 1/24 + 1/x = 1/8 }}}
Multiplying through by 24x we have
{{{x + 24 = 24x/8 = 3x}}}
Subtracting x from both sides
{{{24 = 2x}}}
{{{2x = 24}}}
Dividing both sides by 2
{{{x = 12}}}
Kim can do the job in 12 minutes working alone.
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Checking our work, we were told that combined they did the job in 8 minutes.
In 8 minutes, Carmen can do {{{8*(1/24) = 8/24 = 1/3}}} of the job.
In 8 minutes, Kim can do {{{8*(1/12) = 8/12 = 2/3}}} of the job.
1/3 + 2/3 = 1 job.
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Done