Problem Solving

Two women Ruth and Ursula are working on an embroidery design. If Ursula works alone, she needs eight more hours to complete the design than if they both worked together. If Ruth works alone, she would need 4.5 hours more to complete the design than if they both worked together. What time would it take Ruth alone to complete the design?

(A)10.5 hours
(B)12.5 hours
(C)14.5 hours
(D)18.5 hours
(E)None of the above

OA - A

I would appreciate a direct solution to this problem using equations only, as I've already figured out the backdoor method to solving this problem.

knight247 wrote:Two women Ruth and Ursula are working on an embroidery design. If Ursula works alone, she needs eight more hours to complete the design than if they both worked together. If Ruth works alone, she would need 4.5 hours more to complete the design than if they both worked together. What time would it take Ruth alone to complete the design?

(A)10.5 hours
(B)12.5 hours
(C)14.5 hours
(D)18.5 hours
(E)None of the above

OA - A

I would appreciate a direct solution to this problem using equations only, as I've already figured out the backdoor method to solving this problem.

Let the job = 1.

Times:
Let t = the time for Ruth and Ursula working together.
Thus:
Time for Ruth alone = t + 4.5.
Time for Ursula alone = t + 8.

Rates:
Rate for Ruth and Ursula together = 1/t.
Rate for Ruth alone = 1/(t+4.5) = 2/(2t+9).
Rate for Ursula alone = 1/(t+8).

Since Ruth's rate + Ursula's rate = their combined rate, we get:
2/(2t+9) + 1/(t+8) = 1/t.

Putting the lefthand side over a common denominator, we get:
(2t+16) + (2t+9) / (2t+9)(t+8) = 1/t

(4t+25) / (2t² + 25t + 72) = 1/t.

Cross-multiplying, we get:
4t² + 25t = 2t² + 25t + 72

2t² = 72

t² = 36

t=6.

Thus, the time Ruth alone = t + 4.5 = 6 + 4.5 = 10.5 hours.

The correct answer is A.

knight247 wrote:Two women Ruth and Ursula are working on an embroidery design. If Ursula works alone, she needs eight more hours to complete the design than if they both worked together. If Ruth works alone, she would need 4.5 hours more to complete the design than if they both worked together. What time would it take Ruth alone to complete the design?

(A)10.5 hours
(B)12.5 hours
(C)14.5 hours
(D)18.5 hours
(E)None of the above

OA - A

An alternate approach is to plug in an easy value for the amount of work and to use the answer choices, which represent R's time alone.

It is given that R's time alone is 4.5 hours more than the time for R and U together, while U's time alone is 8 more hours.
Thus, the answer choices imply the following times:

A: R = 10.5 hours, R+U = 10.5-4.5 = 6 hours, U = 6+8 = 14 hours.

B: R = 12.5 hours, R+U = 12.5-4.5 = 8 hours, U = 8+8 = 16 hours.

C: R = 14.5 hours, R+U = 14.5-4.5 = 10 hours, U = 10+8 = 18 hours.

D: R = 18.5 hours, R+U = 18.5-4.5 = 14 hours, U = 14+8 = 22 hours.

On the GMAT, rates tend to be INTEGER values.
Thus, the amount of work should be a multiple both OF R'S TIME ALONE and OF THE OTHER TIMES in the answer choice.
Taking multiples of R's possible times, we get:
A: 10.5, 21, 42...
B: 12.5, 25, 50, 75...
C: 14.5, 29, 58, 87...
D: 18.5, 37, 74, 111....

Only A offers an easy value (42) that is a multiple of the other times in the answer choice (6 and 14).
Thus, the correct answer almost certainly is A.

Answer choice A: R = 10.5 hours, R+U = 10.5-4.5 = 6 hours, U = 6+8 = 14 hours
Let the job = 42 units.
R's rate = 42/10.5 = 4 units per hour.
U's rate = 42/14 = 3 units per hour.
According to the times above, R and U working together should take 6 hours.
Since the combined rate for R and U = 4+3 = 7 units per hour, the time for R and U to produce 42 units = 42/7 = 6 hours.
Success!

The correct answer is A.