Help with a Tricky Number Problem for AMC10 Prep

Question

I need to find two whole numbers that together add up to 17,402. One of these numbers is a multiple of 10, and if you take away the last digit (which is a zero), you get the other number. I've tried a few things, but I'm not sure how to figure out their difference. The answer choices are:

(A) 10,272 (B) 11,700 (C) 13,362 (D) 14,238 (E) 15,426

Could you show me how to work this out?

Answer 1

Let's denote the smaller number as 'a'. The larger number, being a multiple of 10, is then 10a.

The sum of the two numbers is 10a + a = 11×a = 17,402'. To find a, we divide 17,402 by 11 to get a = 1,582.

The difference between the two numbers is 10a - a = 9a. Calculating this gives us 9×1,582 = 14,238. So, the answer is '14,238', which is option (D).

Answer 2

Let x = the smaller number

Then 10x - the larger number

If you add them, you get 17402.

So x + 10x = 17402

11x = 17402 Divide by 11

x = 1582 This is the smaller number.

10x = 15820. This is the larger number.

Check. 1582 + 15820 really does equal 17402.

If you really need to know their difference, then subtract them 15820 - 1582 = 14238.