Bunuel wrote:
New homeowners Jo and Colin are painting their basement. Working alone, Jo could paint the entire basement in seven hours; working alone, Colin could paint it in six. They work together at their separate and constant rates for two hours; then Colin goes to work at his office while Jo continues to paint. How long in hours and minutes will it take for the basement to be completely painted?
A. 3:00
B. 3:30
C. 4:00
D. 4:18
E. 4:40
1) First segment of work: together
Jo's rate = 6 (or \(\frac{1}{6}\), see end of post) and Colin's rate = 7 (or \(\frac{1}{7}\))
Combined rate**: \(\frac{(a+b)}{ab}\) = \(\frac{(6+7)}{(6*7)}\) =\(\frac{13}{42}\)
Two hours together,
\(rt=W\):
\(\frac{13}{42}\) * 2 = \(\frac{26}{42}\) or \(\frac{13}{21}\) of work is finished.
Work remaining: \(\frac{8}{21}\)
2) Second segment of work, Jo alone.
\(\frac{W}{r} = t\) for Jo to finish
(8/21)/(1/7) = \(\frac{8}{21}\)*\(\frac{7}{1}\) = \(\frac{8}{3}\) hrs. Jo alone = 2\(\frac{2}{3}\) hrs
3) Total time: 2 + 2\(\frac{2}{3}\) = 4\(\frac{2}{3}\) hrs is 4 hrs 40 minutes, or 4:40
Answer E
**
The formula is ubiquitous here. I haven't seen it explained often, but I am not adept at the search function. Just in case, the formula comes from adding rates:\(\frac{1}{6}\) +\(\frac{1}{7}\) = \(\frac{13}{42}\) (add 6 and 7 for numerator, multiply 6 and 7 for denominator).
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