In the previous posts, Part 1, Part 2, and Part 3, we have learned how to solve digit problems involving 2-digit numbers. In this post, we are going to discuss digit problems involving 3-digit numbers. Let’s have the following problem.

**Problem**

The hundreds digit of a 3-digit number is twice its units digit and its tens digit is 1 more than its units digit. If 297 is subtracted from the number, then its digits are reversed. What is the number?

**Solution and Explanation**

Before we solve the problem, let’s recall that in the first two parts of this series that a number with tens digit and units digit can be represented by and when we reverse its digits can be represented by . For example, the number with and can be represented as and when we reverse the digits, it becomes .

In the same way, if we let be the hundreds digit of a number, be the tens digit, and be the units digit, then we can represent the number as

and with its digits reversed as

.

In the first sentence of the problem above, it says that the hundreds digit is twice the units digit. Therefore,

(*).

In the second sentence, it says that if 297 is subtracted from the number, then the digits is reversed. Putting this in equation, we have

**number – 297 = number with digits reversed**.

That is,

.

Simplifying, we have

.

Dividing both sides by 99, we have

.

But from (*), h = 2u. So, substituting, we have

.

So, the units digit is equal to .

Now, the tens digit is 1 more than the units digit, so it’s .

The hundreds digit is twice the units digit, so it’s .

Therefore, the number is .

Check: Let’s try to subtract and see if the digits are reversed.

.

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