Word Problems at a Glance

Marty, our resident baker, can frost 2 cupcakes per minute. He has baked both chocolate and vanilla cupcakes, but he also loves to decorate his cupcakes using sprinkles, yum. How long will it take him to frost 55 cupcakes?

Don't worry, Shmooper, we'll take this problem step-by-step, nice and slow, and get those cupcakes in the end.

Step 1: First we carefully read over the problem. What did you think of the sentence about the types of cupcakes and sprinkles? Did you stop and think, "What the heck does this have to do with the problem?" If so, good for you. This is exactly what we meant by irrelevant information; disregard that sentence entirely.

Step 2: What's our unknown in this situation? We've been given a whole lot of details about Marty, even a bunch we don't want, but the only thing we don't know is how long (i.e., how much time) it will take to frost those delicious cupcakes.

Unknown = time it takes to frost 55 cupcakes

Step 3: Let's write out what we want to know and how we're going to find it. We know how many cupcakes we want frosted total, and we know how many can be frosted in one minute. We just need how many minutes it will take. A wordy equation that describes the situation is:

Number of cupcakes frosted = Cupcake frosting rate × Total time spent frosting cupcakes

Step 4: We generally support treating everyone as an individual and not making assumptions, but in math it's okay to assign labels:

Number of cupcakes frosted = 55 cupcakes

Total time spent frosting cupcakes = t minutes

Step 5: Now we use insert our labels into the word equation to get an honest-to-Betsy algebraic equation.

55 = 2t

Step 6: We know what to do with poor, defenseless equation sitting all alone. Show no mercy and solve it.

Step 7: But wait, what was the question again? We need to make sure that we're answering it properly.

Okay, t is the time it takes to frost all the cupcakes, so it will take Marty 27.5 minutes to frost 55 cupcakes. And we will be waiting eagerly for him to finish.

This one's about figuring out a grade for a class; it doesn't get more real-world—or meta—than that. Mae is taking a class in which her grade is determined by her scores on four exams. She has taken three exams so far, and she got scores of 92, 84, and 88 out of 100 points. She wants to average 90% in the class. What is the minimum score she can get on her last exam to still get a 90% in the class?

Step 1: Reading the problem through, pretty much all the information given here is relevant. No words are wasted here.

Step 2: We know the scores for three of the four exams, and Mae is shooting for a total average of 90%. We need to know what score on the last exam will give an average of exactly 90%; anything above that on the test will be gravy. Pro tip, though: don't put gravy on your test.

Unknown = score on the last exam

Step 3: The average of the four tests can be written as:

Step 4: We can't make a proper equation without numbers. We need to be careful here, though. We've been talking about the average for the class as a percent, but the test scores were given to us as the number of points. We either need to convert the average to points, or the exam scores to percents.

Let's put everything into percents, then convert those percents to decimals so we can work with 'em.

Average = 90%, which is the same as 0.9.

Score 1 = 92% = 0.92

Score 2 = 84% = 0.84

Score 3 = 88% = 0.88

Score 4 = x

Step 5:

Step 6: Now that we're looking at this equation, we don't like all of those decimals sitting around. If we multiply everything by 100, that will get rid of them.

That's better. Now to solve.

360 = 264 + 100x

96 = 100x

x = 0.96

Step 7: In Step 4, we labeled x as the percentage of the points that Mae needed to score on the last exam to get a 90% in the class. That's not what the original problem was asking for, though. Putting down x = 0.96 as your answer would be like saying "The French Revolution" after someone asks you what you had for lunch. That is not what you ate; answer the real question, please.

There are 100 points on the last test, so Mae needs to score a 96 to finish the class with at least a 90%. Harsh. Better start studying now, Mae.

Through painstaking research, Gary the deranged hamster farmer has determined that a single hamster running on its wheel can power a light bulb for 30 minutes. Gary wants to replace as much of his electricity as possible with hamster power. He needs to replace at least 5 light bulbs in his house, but if he replaces 10 or more, the energy company will figure out his scheme and send Animal Control after him. If Gary keeps his lights on for 8 hours a day, how many hamsters can he use?

Step 1: There are several key facts we need to pull out of the question.

First, that there's a relationship between hamsters and the amount of time a light bulb is in use: it takes 2 hamsters to run a light bulb for 1 hour.

Second, light bulbs stay on for 8 hours a day. That means that, for instance, 2 light bulbs will be used for 16 hours total, and so on.

Third, Gary will replace "at least" 5 light bulbs (so that includes 5), but won't do 10 or more (so 9 is good, but 10 is not).

Fourth, Gary somehow uses hamsters to farm. How? We don't know. Why? We don't really care. It's not relevant to the problem.

Step 2: How many hamsters does it take to run 5 light bulbs for a day? How many hamsters does it take to run 7? What about 9? We want to know them all. Not 10, though; for some reason, we're helping Gary evade the law with his crazy animal labor scheme, so we're not going to let him get caught.

Unknown = number of hamsters to run the light bulbs for a day

Step 3: We know a lot about the light bulbs: how many Gary can use, and how long each one runs in a day. We also know how long a hamster can run a light bulb. We just want to know how many hamsters are needed.

We're going to need to use 2 equations to get this done. One to find how many hours a day of light bulb use there are, and the other to find how many hamsters it will take to create that much light.

Number of light bulb hours = Number of light bulbs × Hours used each day

Number of hamsters = Number of light bulb hours × Hamsters needed per hour

Step 4: Let's fill in all of the labels we just made. For our light bulbs, though, we have a minimum number and a maximum number we can have, so we'll need to look at both.

Number of light bulbs (minimum) = 5 light bulbs

Number of light bulbs (maximum) = 9 light bulbs

Number light bulb hours = t hours

Number of hamsters = h hamsters

Step 5: We've done the hard work already; now we just plug in our labels and get ready to solve up a storm.

h (min and max) = 2t

Step 6: Time to solve for the minimum and maximum number of hamsters needed to pull off this scheme.

h (minimum) = 2(40) = 80

h (maximum) = 2(72) = 144

Step 7: Gary can use between 80 and 144 hamsters to power his light bulbs each day. In math-speak, we can show this as:

80 ≤ h ≤ 144 hamsters

That is really messed up; guess Gary really is deranged.

The inequality signs are inclusive because he can use "at least" 5 light bulbs (80 hamsters worth of light) and up to 9 light bulbs (but not more). He can't replace 9.5 light bulbs, so that's why we went up to 9 and not 10.