Work and Combined-Rate Problems

0:00 All right, let's jump right into this explainer. If you're gearing up for the ASVAB and the math sections got you feeling a little on edge, hey, take a deep breath, you are in exactly the right place. Today, we're going to totally conquer ASVAB math, and we're going to do it without touching a single calculator. Seriously, how? Well, by mastering a really simple, incredibly friendly concept that I like to call the power of one. Here is our game plan for today. We'll start with work rates and combined teams, then hit shifting teams and opposing rates. After that, sizing, scaling, and estimation, step-by-step worked examples, common traps to avoid, and we'll wrap it all up with a final review and some practice questions. Okay, section one, work rates and combined teams. Let's get our foundational rules down for any combined work problem you're going to face. Use the whole job as the number one. It doesn't matter if the task is building a giant fence, painting a massive house, or filling a gigantic tank. We just treat that entire full task as exactly one.

1:01 We use the formula w equals r times t. That just means work done equals the worker's rate multiplied by their time. By making this whole intimidating job equal to just one, we never ever have to worry about the actual physical size of the wall or the tank. It keeps the math so incredibly simple. Let's actually look at this concept in action. Say you've got a mechanic who can finish a job in five hours and another mechanic who takes 20 hours. Their hourly rates are simply fractions of that whole job, one fifth and one 20th. When they work together as a team, we just add those fractional rates, one fifth plus one 20th. Using a common denominator of 20, that becomes four 20th plus one 20th, which gives us five 20th. Simplify that down, and you get one fourth. Now to find the total combined time, you just invert that one fourth fraction. Literally flip it upside down, and boom, you get exactly four hours total. Shifting teams and opposing rates. We're going to add a little twist here and see what happens when a team changes mid job or when things are actually working against each other.

2:08 Let's patiently break down a partial work scenario. Imagine a worker who needs 12 hours alone, but they start the job completely by themselves and work for four hours before any backup arrives. Step one, find the work they did alone. Four hours out of 12 is four 12ths, which simplifies to one third. Step two, find the remaining work. Well, if one third is done, exactly two thirds of our whole one job is left. Step three, a helper who needs six hours alone finally shows up. Their combined team rate is one 12th plus one sixth, which gives us a shiny new team rate of one fourth. Finally, step four. Divide that remaining work by the new team rate. Two thirds divided by one fourth is exactly the same as two thirds multiplied by four over one, giving us eight thirds hours. You just split the job into stages, and track what gets done in each stage. Now, picture an inlet pipe filling up a water tank while a drain at the very bottom is letting the water out. If a fill pipe adds a positive one third per hour, but a drain removes a negative one sixth per hour, you simply subtract that drain rate from the total.

3:10 One third minus one sixth leaves us with a net rate of positive one sixth. The tank is absolutely still filling up. It just takes six hours with both of those pipes open. Sizing, scaling, and estimation. Let's look at a pretty cool advanced tactic for handling tricky numbers when you don't have that calculator to lean on. When working together, a team is always faster than the fastest individual worker, but they're going to be slower than half of that fastest worker's time. For example, if you have a four-hour worker and a five-hour worker, the fastest is four hours, right? Working together, they obviously have to finish in under four hours. But even if they were both equally that fast, like two four-hour workers, it would still take two hours. Therefore, the combined time absolutely must fall somewhere between two and four hours. This rule is basically a super power. It lets you instantly eyeball and eliminate wrong multiple choice answers without doing a single bit of fractional math. All right, section four. Step-by-step worked examples. It is time to execute our strategy and actually put these formulas to work exactly as you'll see them on the test.

4:17 The good news is the rules do not change, even if we start adding more workers. In example two here, we have three machines that empty a pond in 10, 15, and 30 hours alone. Step one, write their rates as one 10th, one 15th, and one 30th. Step two, grab a common denominator of 30, which gives us three 30ths plus two 30ths plus one 30th. Step three, add them all up for a sum of six 30ths, which beautifully reduces to one fifth. Step four, just invert that fraction to get five hours. Look at that. We just took three super intimidating fractions and turned them into one incredibly simple answer. Now, here's another classic ASVAB setup in example three, finding an unknown worker. We know the team rate is four hours, which means one fourth of the job per hour. We know one solo worker takes five hours, meaning one fifth. So how do we find the unknown partners time? We just subtract the solo rate from the total team rate. Step three shows one fourth minus one fifth. With a common denominator of 20, that gives us a result of one 20th.

5:18 Invert that result and we instantly see the unknown worker takes exactly 20 hours alone. Common traps to avoid. We've built an awesome mathematical foundation so far, but now we really need to flag the specific pitfalls the ASVAB uses to try and trip you up. Okay, let's run through these traps so you don't fall for them. First, adding times instead of rates. Two workers taking three and six hours do not need nine hours together. Guys, a team is always faster. Second trap, averaging the times. No, also completely wrong. Third, forgetting to invert at the end. If your combined rate is one fourth, the time is four hours, not a quarter of an hour. Fourth, adding a drain rate instead of subtracting it. Always remember drains work against you. And fifth, giving the wrong elapsed time. On those shifting team problems we talked about, carefully read whether they're asking for the total elapsed time from the very beginning or just the time after the extra help showed up. Final review and practice questions. We are going to lock in all this knowledge right now with two exact practice questions from our material.

6:26 Let's run the gauntlet. Okay, question one. A landscaper can mow a large park alone in eight hours. After working for three hours, what fraction of the park is still unmowed? A is five eighths, B is three eighths, C is one eighth, D is three fifths. I invite you to pause the explainer right here. Really try solving this partial work question without your calculator. Then hit play when you're ready. All right, let's see how you did. The correct answer is A, five eighths. Why? Well, each hour, the landscaper mows one eighth of the park. So three hours of work covers three eighths of the park. But because the question specifically asks what is still unmowed, we take our whole one park and subtract the three eighths that's already finished. That leaves exactly five eighths remaining. Here is our final challenge. Question two. One clerk files a batch in four hours, a second clerk files it in 12 hours. Working together, how long will it take them? A is eight hours, B is 16 hours, C is three hours, D is one third of an hour.

7:27 This is the ultimate classic combined rate question. Go ahead, pause the explainer one last time to test your skills. And we're back. The correct answer is C, three hours. Let's break it down. Step one, add the hourly rates one fourth plus one 12th. Step two, convert to a common denominator of 12, making it three 12ths plus one 12th. Step three, the sum is four 12ths, which simplifies perfectly down to one third. Step four, we invert that one third fraction to get a simple solid three hours. Do you see how smoothly one fourth plus one 12th becomes three hours when we just follow those steps? And that brings us to the end. The power of one means that every huge intimidating task is just one single job, waiting to be conquered one simple fraction at a time. No matter how complex the ASVAB math looks, treating the whole job as one makes you completely unstoppable. And you can do it entirely without a calculator. So I'll leave you with this. What other complex overwhelming challenges in your life could you conquer just by breaking them down into simpler fractions?

8:30 Think about it. Thanks for joining me on this explainer and keep practicing.