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GMAT word problems, like the ones from the Official Guide to the GMAT, usually come with explanations. A lot of those explanations start by turning the word problem into equations. Starting with the equations can make an explanation easy to understand: if the equations match up to what the problem says, then the explanation makes sense.
Unfortunately, it can also make an explanation look like a magic trick. When you had to do the problem, how on earth were you supposed to think of the right equation? What makes an equation the right one, anyways?
In simplest terms, an equation is just two pieces of math with an equals sign in between them.
5x + 10y = 500
In GMAT word problems, those two pieces of math have to match up with something in the real world. In fact, they both have to match up with the same real-world thing. The two sides of the equation have to talk about the exact same thing in two different ways.
For example, suppose that a school play makes a total revenue of $500. You can express the revenue using the number 500.
Another way to express the revenue is to split it up by ticket types. For instance, if the only types of tickets sold were children’s tickets and adult tickets, then this is also a good way to express the revenue:
revenue from children’s tickets + revenue from adult tickets
We now have two ways of describing the exact same thing, so we can create a good equation:
revenue from children’s tickets + revenue from adult tickets = 500
Depending on what information the problem gives you, this probably isn’t a very useful equation. Most GMAT equations are more complex. For instance, the problem might tell you that x children’s tickets were sold, and that each one cost $5. y adult tickets were also sold at $10 each. That gives you another way of writing out the total revenue:
5x + 10y
Because 5x + 10y describes the same thing (total revenue) as the number 500, this is a good equation:
5x + 10y = 500
This might seem basic. And it is! But it’s often the most basic things that are the toughest to really understand. When you write an easy equation, it might just seem obvious, and you can’t really explain why you wrote what you wrote. That makes it hard to handle much tougher equations that do require a lot of thought and explanation.
Let’s create an equation from some more complicated text.
Jordan planned to fold exactly 10 paper roses per day between now and his mother’s birthday in order to complete her birthday gift. Instead, he only folded an average of 7 roses per day until the very last day, when he had to fold 43 roses in one day to finish the gift. How many roses did Jordan fold in total?
In order to create an equation, we’ll have to find two different ways of talking about the same value. In this case, the number of roses that Jordan folded would be a good value to work with: it’s right there in the question.
One way to talk about the total number of roses is by looking at Jordan’s original plan. If he planned to fold 10 roses per day, then one way to write the total number of roses is:
Now, let’s find another way to describe the total number of roses. When Jordan actually started folding roses, he did one thing until the last day, and then did something different on the very last day. That gives us a good way to divide up the number of roses:
roses folded before the last day + roses folded on the last day
On the last day, he folded 43 roses. Before the last day, he folded 7 roses per day, or a total of 7(days – 1) roses. So, here’s a second way to write about the total number of roses:
7(days – 1) + 43
Since we now have two ways of talking about the total number of roses, we can put an equals sign between them and create an equation.
10d = 7(d – 1) + 43
If you solve that equation, you’ll find the number of days Jordan spent on the project, which will let you calculate the number of roses. (It’s 120).
Let’s do one more. This time, you’ll need two equations.
Marisha recently completed a 300-mile road trip at an average speed of 50 mph. For the first part of the trip, she drove at a speed of 40 mph. For the second part of the trip, she drove at a speed of 70 mph. How many hours of the trip were spent driving at 70 mph?
We could start by finding two different ways to talk about the total distance, or two different ways to talk about the total time. (We can’t start with the speed, because you can’t just do arithmetic with speeds—going 40 mph and then 70 mph isn’t the same thing as going 110 mph!)
We already know one way to express the total distance: 300 miles.
Another way to express the distance would involve splitting the trip into two parts:
distance of the first part + distance of the second part
We don’t know exactly how long the two parts of the trip were, though, so we’ll need to find a way to describe them in terms of what we do know.
If Marisha drove at 40 mph for the first part of the trip, then the total distance she covered was as follows:
(40 mph)(hours for first part of trip)
And if she drove at 70 mph for the second part of the trip, the distance she covered was as follows:
(70 mph)(hours for second part of trip)
So another way of writing the total distance is like this:
(40 mph)(hours for first part of trip) + (70 mph)(hours for second part of trip)
We now have two different ways of writing the total distance, so we have an equation!
(40 mph)(hours for first part of trip) + (70 mph)(hours for second part of trip) = 300
However, we aren’t quite finished. We have two variables, so we’ll need a second equation. That’s where the total time comes in. One way to express the total time is by just giving the number of hours: 6 hours. The other way is by splitting it up into two parts:
hours for first part of trip + hours for second part of trip = 6
Now we have a system of equations! It looks like this:
40x + 70y = 300
x + y = 6
The first equation gives two ways of talking about the total distance of the trip. The second equation gives two ways of talking about the total time of the trip. By combining them, it’s possible to solve. (The answer to the question is 2 hours.)
Try reframing how you think about equations in GMAT word problems. The right equation is always right for a reason—because both sides of the equation talk about the same real-world quantity. You don’t have to come up with that equation instantly, either. It’s okay to build an equation up from smaller pieces, just like we did here. 📝
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Chelsey Cooley is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170/170 on the GRE. Check out Chelsey’s upcoming GRE prep offerings here.