WRT Practice PS Set – 3
Work & Rate: Level 3
1.
Tom can do a job in 6 hours, Jerry in 8 hours, and Spike in 12 hours. How long will it take if they all work together?
Solution (B):
Combined Rate = \( \frac{1}{6} + \frac{1}{8} + \frac{1}{12} \).
LCD = 24. \( \frac{4}{24} + \frac{3}{24} + \frac{2}{24} = \frac{9}{24} = \frac{3}{8} \).
Time = \( 8/3 \approx 2.66 \) hours.
2.
Pipe A fills a tank in 4 hours, and Pipe B empties it in 6 hours. If both are open, how long to fill the tank?
Solution (D):
Rate A = \(1/4\) (positive). Rate B = \(1/6\) (negative).
Net Rate = \( \frac{1}{4} – \frac{1}{6} = \frac{3}{12} – \frac{2}{12} = \frac{1}{12} \).
Time = 12 hours.
3.
A can complete 1/3 of a job in 4 days. B can complete 1/2 of the same job in 5 days. If they work together, how many days to complete the full job?
Solution (C):
A’s full time = \(4 \times 3 = 12\) days. Rate A = \(1/12\).
B’s full time = \(5 \times 2 = 10\) days. Rate B = \(1/10\).
Combined Rate = \( \frac{1}{12} + \frac{1}{10} = \frac{5}{60} + \frac{6}{60} = \frac{11}{60} \).
Time = \( 60/11 \approx 5.45 \) days.
4.
Working together, X and Y can finish a job in 12 days. X is twice as fast as Y. How long does it take X alone to do the job?
Solution (A):
Let Rate Y = \(r\). Then Rate X = \(2r\).
Total Rate = \(3r\).
Time = \( \frac{1}{3r} = 12 \implies 3r = \frac{1}{12} \implies r = \frac{1}{36} \).
Rate X = \(2r = \frac{2}{36} = \frac{1}{18}\).
X takes 18 days.
5.
Adam works at a rate of 50 tiles/hour. Brianna works at 55 tiles/hour. If the job is 1400 tiles, how long do they take together?
Solution (B):
Combined Rate = \(50 + 55 = 105\) tiles/hour.
Time = \( \frac{1400}{105} = \frac{40}{3} = 13 \frac{1}{3} \) hours.
\( \frac{1}{3} \) hour = 20 mins. Total = 13 hrs 20 mins.
6.
Three pumps X, Y, and Z can fill a tank in 4, 5, and 20 hours respectively. If they all run together, how long does it take?
Solution (E):
Rate = \( \frac{1}{4} + \frac{1}{5} + \frac{1}{20} \).
LCD = 20. \( \frac{5}{20} + \frac{4}{20} + \frac{1}{20} = \frac{10}{20} = \frac{1}{2} \).
Time = 2 hours.
7.
Machine A produces \(w\) widgets in 5 minutes. Machine B produces \(w\) widgets in 10 minutes. How many minutes to produce \(3w\) widgets together?
Solution (D):
Rate A = \(w/5\). Rate B = \(w/10\).
Combined = \( \frac{2w}{10} + \frac{w}{10} = \frac{3w}{10} \) widgets/min.
We need \(3w\) widgets.
Time = \( \frac{3w}{3w/10} = 10 \) minutes.
8.
A pool fills in 6 hours with the inlet pipe. The drain empties it in 8 hours. If both are open, and the pool is already half full, how long until it is full?
Solution (C):
Net rate = \( \frac{1}{6} – \frac{1}{8} = \frac{4}{24} – \frac{3}{24} = \frac{1}{24} \) pool/hour.
We need to fill the remaining \(1/2\) pool.
Time = \( \frac{1/2}{1/24} = 12 \) hours.
9.
Team A takes 4 days to build a structure. Team B takes 6 days. Team C takes 12 days. If they work together for 1 day, what fraction of the structure is left?
Solution (A):
Combined Rate = \( \frac{1}{4} + \frac{1}{6} + \frac{1}{12} = \frac{3}{12} + \frac{2}{12} + \frac{1}{12} = \frac{6}{12} = \frac{1}{2} \).
In 1 day, they finish 1/2.
Remaining = \( 1 – 1/2 = 1/2 \).
10.
It takes 5 men 10 days to paint a house. How many men are needed to paint the house in 2 days?
Solution (B):
Total work = \( 5 \times 10 = 50 \) man-days.
\( M \times 2 = 50 \implies M = 25 \).
Score: 0 / 0