Work and Rate Problems - SSAT Middle Level Quantitative

1. A water tank can be filled by two pipes. Pipe A fills the tank in 15 hours when working alone. When both pipes work together, they fill the tank in 6 hours. If Pipe B works alone for 4 hours and then both pipes work together, how much additional time is needed to fill the tank completely?
   Answer: 3⅗ hours Think in rates, the fraction of the tank filled per hour. Pipe A's rate is $\frac{1}{15}$ and the combined rate is $\frac{1}{6}$, so Pipe B's rate is $\frac{1}{6} - \frac{1}{15} = \frac{5}{30} - \frac{2}{30} = \frac{1}{10}$. In 4 hours alone, Pipe B fills $4 \times \frac{1}{10} = \frac{2}{5}$ of the tank, leaving $\frac{3}{5}$. Working together at $\frac{1}{6}$ per hour, the pipes need $\frac{3}{5} \div \frac{1}{6} = \frac{18}{5} = 3\frac{3}{5}$ hours. An answer of 3½ hours treats 3.6 hours as 3 hours 30 minutes, but 0.6 hour is 36 minutes. An answer of 3 hours would fit only if half the tank were left. An answer of 4 hours repeats Pipe B's solo time instead of computing the remaining work.
2. A warehouse uses two conveyor belts to move packages. Belt A moves 45 packages per hour, and Belt B moves 36 packages per hour. Belt A works alone for 3 hours and then breaks down. Belt B then works alone until a total of 531 packages have been moved. How many hours must Belt B operate?
   Answer: 11 hours Break the job into two phases. Phase 1: Belt A works alone for 3 hours at 45 packages per hour, moving $45 \times 3 = 135$ packages. Phase 2: Belt B must move the rest of the 531 packages: $531 - 135 = 396$ packages. At 36 packages per hour, that takes $\frac{396}{36} = 11$ hours. Check: $135 + 36 \times 11 = 135 + 396 = 531$. An answer of 9 hours comes from dividing the remaining 396 packages by Belt A's rate ($396 \div 45 = 8.8$) instead of Belt B's. An answer of 10 hours falls short: $36 \times 10 = 360$ packages, which is 36 too few. Answers of 12 or 13 hours move more packages than needed ($36 \times 12 = 432$). In multi-phase work problems, find what the first worker finished, subtract it from the total, and divide the remainder by the rate of the worker who finishes the job.
3. A swimming pool is being filled and drained simultaneously. The inlet pipe fills the pool at 40 gallons per minute, while the drain removes water at 25 gallons per minute. If the pool starts with 200 gallons and has a capacity of 800 gallons, how long will it take to fill the pool completely?
   Answer: $40$ minutes When you encounter problems involving simultaneous filling and draining, think about the net rate of change. You need to find how much water is actually being added to the pool each minute after accounting for both the inflow and outflow. The inlet adds 40 gallons per minute while the drain removes 25 gallons per minute, giving you a net rate of $40 - 25 = 15$ gallons per minute. This means the pool gains 15 gallons every minute. Since the pool starts with 200 gallons and needs to reach 800 gallons, it needs $800 - 200 = 600$ more gallons. At a net rate of 15 gallons per minute, this will take $600 \div 15 = 40$ minutes. Looking at the wrong answers: Choice A (35 minutes) likely comes from incorrectly calculating the gallons needed or making an arithmetic error. Choice C (45 minutes) might result from using the wrong net rate, perhaps calculating $40 - 25 = 13.33$ through sloppy math. Choice D (50 minutes) could come from dividing 600 by 12 instead of 15, suggesting confusion about the net rate calculation. The correct answer is B: 40 minutes. Study tip: For rate problems involving opposing forces (filling vs. draining, traveling with vs. against wind, etc.), always identify the net effect first. Subtract the opposing rate from the working rate, then use this net rate for your time calculations. This approach prevents the common mistake of using individual rates instead of the combined effect.
4. Two garden hoses can fill a swimming pool. Hose A fills at 15 gallons per minute, and Hose B fills at 20 gallons per minute. If the pool holds 1350 gallons and Hose A runs for 20 minutes before Hose B is turned on, how many additional minutes will it take with both hoses running to fill the pool?
   Answer: $30$ minutes When you encounter a multi-step rate problem like this, break it down into phases and track what happens in each one. First, calculate how much water Hose A adds during its solo 20 minutes: $15 \text{ gallons/minute} \times 20 \text{ minutes} = 300 \text{ gallons}$. This leaves $1350 - 300 = 1050 \text{ gallons}$ still needed. Next, determine the combined rate when both hoses run together: $15 + 20 = 35 \text{ gallons per minute}$. To find how long it takes to fill the remaining 1050 gallons at this combined rate: $\frac{1050 \text{ gallons}}{35 \text{ gallons/minute}} = 30 \text{ minutes}$. This confirms answer B. Now let's see where the wrong answers come from. Choice A (28 minutes) likely results from a calculation error, perhaps incorrectly computing $\frac{1050}{35}$. Choice C (32 minutes) might come from using the wrong remaining volume or making an arithmetic mistake in the division. Choice D (34 minutes) could result from forgetting to add the rates together and instead using just one hose's rate for the second phase. The key strategy for rate problems is to organize your work in clear steps: identify what happens in each time period, calculate any intermediate amounts, then solve for the unknown time using the formula $\text{time} = \frac{\text{amount}}{\text{rate}}$. Always double-check that your rates match the scenario—individual rates for solo work, combined rates when working together.
5. A bakery has two ovens. Oven 1 can bake 24 loaves per hour, and Oven 2 can bake 18 loaves per hour. If they need to bake 306 loaves and Oven 1 operates for 6 hours while Oven 2 operates for the entire time needed, how many total hours must Oven 2 operate?
   Answer: 9 hours Let $t$ be the number of hours Oven 2 operates. Oven 1 runs for 6 hours at 24 loaves per hour, so it bakes $6 \times 24 = 144$ loaves. Oven 2 bakes $18t$ loaves. Together they must make 306 loaves: $144 + 18t = 306$ $18t = 162$, so $t = 9$ hours. Check: $144 + 18 \times 9 = 144 + 162 = 306$. An answer of 8 hours leaves Oven 2 short: $144 + 144 = 288$ loaves. An answer of 10 hours makes $144 + 180 = 324$ loaves, more than needed. Answers of 11 or 12 hours overshoot even more, making 342 or 360 loaves. In problems where workers run for different lengths of time, write each one's output as rate times time and set the sum equal to the total.

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