Madison made a list of all the numbers between 176 and 231 that have the digit five. How many numbers did Madison have on her list?
The numbers between 176 and 231 are all three-digit numbers. This means they each contain a number of hundreds, tens, and ones. We’re told that Madison makes a list of all the numbers between 176 and 231 that have the digit five. Before we can think about what numbers these are and how many there are of them, we need to think about where the digit five is going to appear. Is it possible that we may have some numbers with the digit five in the hundreds place? No. The number that Madison starts from has one hundred. And the number that she counts up to has two hundreds. The numbers in between will either have a one or a two in the hundreds place, not a five.
What about the tens place? Will Madison need to look for any numbers where the five represents five tens or 50? Again, the answer is no. Madison starts counting at 176. And the 150 numbers come before this, and she ends counting on 231. And the 250 numbers come after this. There are no numbers in between with a digit five in the tens place.
And so we can say that all of the numbers that Madison finds have the digit five in the ones place. How many are there? Let’s start by finding all the multiples of 10 in between our two numbers. 180, 190, 200, 210, 220, and 230. The numbers that have five ones will be the numbers halfway between these multiples of 10. We can’t include 175 because we have to start with 176, but halfway between 180 and 190 is 185. Halfway between 190 and 200 is 195. Halfway between 200 and 210 is 205. Halfway between 210 and 220 is 215. And halfway between 220 and 230 is 225.
We can’t go as far as 235 because we have to stop at 231. The numbers that Madison has on her list are 185, 195, 205, 215, and 225. And so the number of numbers that Madison has on her list is five.