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Phyllis: 6
Taneisha: 2
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
66% (02:38) correct
34%
(02:32)
wrong
based on 1041
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Phyllis, Shania, and Taneisha are each working at a constant rate to fill envelopes for a local city council letter mailing informational campaign. Shania would take 3 hours to complete the job alone, and Taneisha can complete the job faster than Phyllis.
If Phyllis, Shania, and Taneisha, working together, can complete the job in 1 hour, select for Phyllis the time in hours she would take to complete the job alone, and select for Taneisha the time in hours she would take to complete the job alone that are jointly consistent with the information provided. Make only two selections, one in each column.
If Phyllis, Shania, and Taneisha, working together, can complete the job in 1 hour, select for Phyllis the time in hours she would take to complete the job alone, and select for Taneisha the time in hours she would take to complete the job alone that are jointly consistent with the information provided. Make only two selections, one in each column.
| Phyllis | Taneisha | |
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 6 | ||
| 9 |
ShowHide Answer
Official Answer
Phyllis: 6
Taneisha: 2
Bunuel
Official Solution:
Since it takes Shania 3 hours to complete the job alone, in 1 hour she'd complete \(\frac{1}{3}\) of the job, leaving \(\frac{2}{3}\) of the job to be completed by Phyllis and Taneisha. One being faster than the other implies that the faster one, Taneisha, can complete the job in less than 3 hours, and the slower one, Phyllis, can complete the job in more than 3 hours.
Among the options, the times less than 3 hours are 1 hour and 2 hours. If Taneisha took 1 hour, she would be able to finish the entire job alone in 1 hour, which cannot be true since the problem states all three together take 1 hour. Therefore, the only consistent option is that Taneisha takes 2 hours to complete the job alone.
Assuming Phyllis takes \(p\) hours to complete the job alone, we'd have (Taneisha's rate) + (Phyllis's rate) \(= \frac{1}{2} + \frac{1}{p} = \frac{2}{3}\), which gives \(p = 6\) hours.
Correct answer:
Phyllis "6"
Taneisha "2"
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06 Jan 2025, 13:04
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1. Let p, s, and t be the working rates of Phyllis, Shania, and Taneisha, respectively. The amount of work is W.
2. The question asks us to find \(\frac{W}{p}\) and \(\frac{W}{t}\). We can label them as x and y, respectively. Remember that out of the answer options, they're natural numbers.
3. It is known that \(s = \frac{W}{3}\), \(t > p\), and \(p + s + t = W\). The first equation can be used in the third: \(p + t = \frac{2}{3}W\).
4. Replacing it using x and y, we have: \(\frac{W}{x} + \frac{W}{y} = \frac{2}{3}W \rightarrow \frac{1}{x} + \frac{1}{y} = \frac{2}{3}\) and \(\frac{W}{x} < \frac{W}{y} \rightarrow \frac{1}{x} < \frac{1}{y} \rightarrow x > y\).
5. \(\frac{1}{x} + \frac{1}{y} = \frac{2}{3} \rightarrow \frac{1}{y} = \frac{2}{3} - \frac{1}{x} = \frac{2x - 3}{3x} \rightarrow y = \frac{3x}{2x - 3}\). Then, \(x > y \rightarrow x > \frac{3x}{2x - 3} \rightarrow 2x^2 - 3x = 3x \rightarrow x > 3\).
6. x can be any answer option that is larger than 3 - 4, 6, or 9. However, \(\frac{3x}{2x - 3}\) must also be an answer choice. So, let's test out each case:
- x = 4. \(\frac{3x}{2x - 3} = \frac{3 * 4}{2 * 4 - 3} = \frac{12}{5}\), this doesn't work.
- x = 6. \(\frac{3x}{2x - 3} = \frac{3 * 6}{2 * 6 - 3} = \frac{18}{9} = 2\), this works.
- x = 9. \(\frac{3x}{2x - 3} = \frac{3 * 9}{2 * 9 - 3} = \frac{27}{15}\), this doesn't work.
7. Our answer will be: Phyllis - 6 and Taneisha - 2.
2. The question asks us to find \(\frac{W}{p}\) and \(\frac{W}{t}\). We can label them as x and y, respectively. Remember that out of the answer options, they're natural numbers.
3. It is known that \(s = \frac{W}{3}\), \(t > p\), and \(p + s + t = W\). The first equation can be used in the third: \(p + t = \frac{2}{3}W\).
4. Replacing it using x and y, we have: \(\frac{W}{x} + \frac{W}{y} = \frac{2}{3}W \rightarrow \frac{1}{x} + \frac{1}{y} = \frac{2}{3}\) and \(\frac{W}{x} < \frac{W}{y} \rightarrow \frac{1}{x} < \frac{1}{y} \rightarrow x > y\).
5. \(\frac{1}{x} + \frac{1}{y} = \frac{2}{3} \rightarrow \frac{1}{y} = \frac{2}{3} - \frac{1}{x} = \frac{2x - 3}{3x} \rightarrow y = \frac{3x}{2x - 3}\). Then, \(x > y \rightarrow x > \frac{3x}{2x - 3} \rightarrow 2x^2 - 3x = 3x \rightarrow x > 3\).
6. x can be any answer option that is larger than 3 - 4, 6, or 9. However, \(\frac{3x}{2x - 3}\) must also be an answer choice. So, let's test out each case:
- x = 4. \(\frac{3x}{2x - 3} = \frac{3 * 4}{2 * 4 - 3} = \frac{12}{5}\), this doesn't work.
- x = 6. \(\frac{3x}{2x - 3} = \frac{3 * 6}{2 * 6 - 3} = \frac{18}{9} = 2\), this works.
- x = 9. \(\frac{3x}{2x - 3} = \frac{3 * 9}{2 * 9 - 3} = \frac{27}{15}\), this doesn't work.
7. Our answer will be: Phyllis - 6 and Taneisha - 2.
