In previous posts, we’ve discussed how to solve systems of equations. But often, you will not be given a simply list of equations. Rather, you will get a chunk of text asking you to find some particular value. For example,

Example 1
Tom and Liz are baking cookies. Tom can bake 10 cookies in an hour and Liz can bake 5 cookies in an hour. Working together, how long will it take them to bake 40 cookies?

Example 2
Dembe is reading a long book. He finishes a quarter of it, then puts it down to take a walk. After he comes back, he reads another 50 pages. Now, he has finished 30% of the book. How long is the book?

Example 3
Raymond is trying to calculate how much he needs to invest today in order to have $2,000,000 in an account in 18 years. He can guarantee an annual return of 15% on any funds invested. How much does he need to invest?

The key to such problems lies in translating the text to mathematical equations. As we will see, solving the actual equations is generally not the hard part; the trick lies in the translation.

The problems above are each examples of different kinds of problems. The first problem is an example of a rate problem, the second one is an example of a ratio problem and the last one is a compound interest problem.

Solving Example 1


Let’s look at our first problem. To determine how long it takes the pair to bake 40 cookies, we would try first figuring out how quickly they bake cookies. Let’s try to figure out how many cookies they would bake in an hour. Well, during that hour, Tom would bake 10 cookies and Liz would bake 5 cookies. So together, they would bake 15 cookies.

Now, we ask: if someone could bake 15 cookies in an hour, how long would it take them to bake 40 cookies? To answer this, we use an old equation:

    \[d = rt\]

where d is the total number of cookies baked, r is the number of cookies baked each hour, and t is the amount of hours spent baking cookies. Turning to our problem, we know that their combined rate is 15 cookies per hour, and we want to know how long it takes to bake 40 cookies. Therefore, we set up the following equation:

    \[40 = 15t\]

and solve for t. We learn, by dividing both sides by 15, t = \frac{8}{3}. So it takes them \frac{8}{3} hours to do so (or, in other words, 2 hours and 40 minutes).

This kind of problem often involves some kind of task (eg baking cookies) or distance (eg how far one runs). The key is to use what you know about the outcome (d), rate (r), or time (t) to solve for the unknown quantity. In this case, the problem gave us the rate and outcome, and we solved for the time it took to get that outcome.

Solving Example 2


The key here is setting up the right equation. Let’s use our variables. Let p = the number of pages in Dembe’s book. We know that, after reading .25p + 50 pages, Dembe has finished .3p pages (i.e. 30% of the book). Thus, we get

    \[.25p + 50 = .3p\]

Now that we have this equation, it is easy to solve for the value of p. We subtract to get:

    \[50 = .05p\]

    \[p = 1000\]

Thus, we conclude that Dembe’s book was 1,000 pages long.

Solving Example 3


Again, let’s use our variables. Let x = the amount of money Raymond invests today. Then, in 18 years, with an annual rate of 15%, Raymond will have x(1.15)^18 in that account. We want to make that amount equal to 1,000,000. Thus, we get the equation:

    \[x(1.15)^{18} = 1,000,000\]

And now, we can solve for x by dividing by 1.15^{18}:

    \[x \approx 80805.\]

Thus, Raymond needs to invest about $80,805 today.

Now there is no cookie-cutter recipe that will handle all word problems. But in general, thinking about what the words mean and translating them into mathematical formulas carefully will work.

Practice Problems:

  1. Felipe left for his morning jog with a water bottle that was three-quarters full. On his route, he passed by a high-tech water fountain and added precisely 300 milliliters of water to his bottle (the bottle still was not full). Then, he continued on his run. At the end of it, he drank two-thirds of the water then in his bottle before heading home. Let w be the maximum capacity of Felipe’s water bottle (in milliliters). Write an expression for how much water was left in his water bottle when he reached his home. If he had exactly 200 milliliters of water when he reached home, what was the capacity of his water bottle?
  2. The kindergarteners are trying to decide which shape is best. Every kindergartener likes only one shape. 3/5 of the class is tall and the rest are short. Of the tall students, 1/6 like circles, 2/3 like squares, and everyone else likes trapezoids. Suppose 1/2 of the class likes circles. What proportion of the short students like circles?

  3. Evan is a distance runner. He runs at a constant rate of 10 miles per hour for as many hours as you like. How long will it take him to finish a marathon (26.2 miles)?


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