Expert: Josh - 9/10/2008

Question
It is greater than 43 and less than 52. If you add the digits, the sum is 8. What is the number?

It is less than 60 and greater than 55. If you add the digits, the sum is 13. What is the number?

Answer
Shaquan,

These questions test your understanding of the metric system.

In the first question, since the number (let's call this X) is bounded between 43 and 52, it must be a double digit number.

Recall that a number, for example, 962 may be written as 900+60+2 and this is equivalent to 9*100 + 66*10 + 2*1. In the metric system, a double digit X may be represented literally by the string "ab", or mathematically, as X=a*10+b.

Let's take a moment to digest this. Consider the number 49. Suppose we let X=49. We can then write this as X=a*10+b, where a=4 and b=9.

Now, we interpret the information which we have been given.
The part where it says "add the digits, the sum is 8" means the sum of digit "a" and "b" must equal 8.

Thus, we have the unknown number X=a*10+b and a+b=8. The first equation is the consequence of number representation in the metric system. What to do next? To find out what X is, we need to work out the value of both digits "a" and "b".

There are some constraints on the values that a and b can take. Since the unknown double digit number is between 43 and 52, the digit "a" must be EITHER 4 or 5.

Rewriting a+b=8 as b=8-a, we have two possible solutions.

case 1: If a=5, b=8-a=3.
case 2: If a=4, then b=8-a=4.

Case 1 is not an acceptable solution because "ab"=53 is outside the bound given by 43 and 52. So, we conclude the answer must be (from case 2) "ab"=44.

I'll leave the second question for you to attempt.