Fill in the blanks. 46.5 divided by 3.1 equals 465 divided by what equals what.
In this problem, we’ve got a string of three equivalent expressions that all worth the same. The first one is a division, 46.5 divided by 3.1. And, we’re told this is worth the same as another division, 465 divided by something. And then finally, on the end, we’ve got a number missing. This is the answer to the divisions. So, we need to find which numbers belong in the two missing spaces.
Let’s start by looking at the one complete expression that we’ve got. 46.5 divided by 3.1. The way we write fractions is another way of writing a division. 46.5 divided by 3.1. Dividing two decimals is tricky. We’d much rather be working with whole numbers. Our second division shows 465 divided by a number that we don’t know yet. And, this is equal to the first calculation. Look at how the first number in our division has changed. It’s become a whole number. What have we done to get from 46.5 to 465?
The digits in 46.5 have shifted one place to the left. This is what happens when we multiply a number by 10. So, we’ve multiplied the first number in our division by 10. Now, to keep the value of the division the same, and we need to do this because there’s an equal sign in between both divisions, we need to do the same to the second number in our division as we did to the first. This is exactly like fractions when we say we have to do the same to the numerator, the denominator. So if we multiply 3.1 by 10, the digits are going to shift one place to the left. The answer is 31.
Although it looks like we’re working with fractions here, all we’re really doing is working with the numbers in a division. We’ve multiplied the dividend and the divisor, or the first number and the second number, by the same amount, by 10. And because we’ve multiplied by 10, the decimals have turned into whole numbers, much easier to deal with. But because we’ve multiplied by the same amount, the answer to the division, or the quotient, is going to be exactly the same as if the numbers were decimals.
So now, we’ve made the division a lot easier. Let’s calculate the answer. What is 465 divided by 31? Let’s use long division to find the answer. How many 31s are there in 465? We can’t divide the first digit by 31 because it’s too small. So, let’s start by looking at our first two digits. How many 31s are there in 46? We know two 31s are 62, so that’s too many. There’s only one lot of 31 in 46. So, we’ll write one at the top, and we need to find the remainder. One lot of 31 equals 31. So if we take this away from 46, we’ll find out what’s left. Six take away one equals five. And, four take away three equals one. How many 31s are there in 15? 15 is too small. So, we need to bring down another digit to add on the end.
So, our number becomes 155. How many 31s are there in 155? Rather than keep trying different numbers until we find one that fits, we can use our estimation skills here. 31 is very close to 30, and we could call 155 the same as 150. How many 30s fit into 150? It’s the same number of threes that fit into 15. There are five 30s in 150. So, perhaps there are about five 31s in 155. Let’s see. 10 lots of 31 equals 310. And so, five lots of 31 is half of this amount. Half of 300 equals 150. And, half of 10 equals five. 150 plus five equals 155. There are five lots of 31 in 155. So, we can write five at the top. And, let’s prove there’s no remainder.
We’ll take away five lots of 31 which is 155 from whatever we had left, which was 155 itself. And of course, this leaves us with nothing left over. There’s no remainder. And so the answer to our division and our second missing number is 15. 46.5 divided by 3.1 is the same as 465 divided by 31, which is equal to 15. Our two missing numbers are 31 and 15.