SAT Math: Finding Your Lucky Number

Those wacky people at College Board sure love a good riddle. Often in the SAT Math sections, you'll come across a word problem asking you to find some number by analzing a given set of clues. To make things even more fun, these questions are often found in the Student-Produced Response section—no multiple choice!

Here's a look at one of the more difficult “number-finding” questions you might find on the SAT:

A four-digit integer, DEFG, in which D, E, F, and G each represent a different digit, is formed according to the following rules:

I. G – D = E + F
II. E = F + 2
III. D = E + 3

What is the number DEFG?

Of course, this question would be pretty simple if it were multiple-choice: You could just look at the five answer choices and see which one fit all three of these rules. The SAT masterminds know this, and they're not about to let you off so easily. What should you do?

If you remember your system of equations, you know that you need equal numbers of variables and equations in order to completely solve them. In this example, we have 4 variables and only 3 equations. This means we need some other piece of information. How about the fact that the variables represent digits in a number? Therefore, each variable must be a number between 0 and 9. That’s good for us—it means there is a limited number of values that each digit can be.

We can start by figuring out which are the smallest and largest digits. Let’s look at equation I:

G – D = E + F.

Let's put this equation in terms of only one variable. If we rearrange and add D to both sides, we can see that:

G = D + E + F

G is the sum of all three other digits, making it biggest digit in the group. That means that D, E, or F must be the smallest. Based on equation II, F must be smaller than E. Based on equation III, E must be smaller than D. If F is smaller than E, and E is smaller than D, then F must be the smallest digit.

At this point, we could mess around with the equations some more, but that might waste time. Instead, let’s do some guess-and-check. We know F is the smallest digit—it’s probably something like 0, 1, or 2. Let’s say it is 1. Now plug that value into equation II:

E = F + 2

E = 1 + 2 = 3

If F equals 1, E would equal 3. Now, plug these values into equation III:

D = E + 3

D = 3 + 3 = 6

If F equals 1, D would equal 6. Remember, based on equation I, G is the sum of all 3 other digits:

G = D + E + F

G = 6 + 3 + 1 = 10

This says G equals 10…but that’s not possible! Each variable has to be between 0 and 9. Therefore, F cannot be 1. What else do we now know? D, E, and F were too big. F can’t be greater than 1, because this would make G even larger—and it’s already too big. That means F must equal 0. Let’s try that, but plugging it into equation II:

E = F + 2

E = 0 + 2 = 2

E equals 2. On to equation III:

D = E + 3

D = 2 + 3 = 5

D equals 5. Last but not least, solve for G:

G = 5 + 2 + 0 = 7.

G equals 7—and that works! This makes our number DEFG equal to 5207. Double-check that answer with your three equations, and go pencil the answer onto your answer sheet.

Here's a quick review of how we solved that problem:

• Understand the question. Find and organize the information you need to solve the question. Can you write an equation? If so, do it! In this case you couldn’t, but you remembered the important fact that each digit could only be from 0 to 9.
• If you’re stuck, it’s okay to guess and check. With this example, we knew we had limited options for each digit (they could only be numbers between 0 and 9). Once we organized the digits from smallest to largest, we could make an educated guess about where to start.
• Double-check. This is an especially important step when you have a Student-Produced Response question. It’s easier to make a mistake when you’re finding your own answer.

Try some additional number-finding questions using these flashcards.

Got a math problem you can’t solve? Drop it in the comments or email it to testpreptutor@sparknotes.com.

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