1.84 divided by what equals four and three-fifths.
Just like any other division where we have a missing divisor, we can rearrange the numbers in this division to help us find the missing divisor. If we divide 1.84 by four and three-fifths this will give us our missing number. So to solve the problem, it seems like we have to divide a decimal by a mixed number. Wouldn’t it be easier if both of these were decimals? Well, to help us, we can convert four and three-fifths into a decimal. If we multiply the numerator and the denominator in three-fifths by two, we can find an equivalent fraction. That’s a number of tenths, six-tenths. This means that four and three-fifths is actually the same as four and six-tenths. And we know where the tenth place is a decimal. So four and six-tenths will be four with a six in the tenths place, 4.6.
Our division now involves two decimal numbers. Now, won’t this be easy if they were both whole numbers? If we multiply 1.84 by 100, the digits are going to shift two places to the left. So 1.84 becomes 184. Our decimal has become a whole number. For the answer to our division to stay the same, we need to also multiply 4.6 by 100 too. So the digits in 4.6 are also going to move two places to the left, 460. Now, we have a division involving two whole numbers. We’re dividing by a three-digit number here. So we’re going to need to use long division to find the answer. But if we look carefully at this division, it’s a little bit unusual.
How many lots of 460 are there in 184? The dividend or the first number in our division is smaller than the divisor of a number we’re dividing by. This means that we can predict that the answer to our division or the quotient is going to be less than one. And it might look like we’ve got a bit of a problem here. If we look at all three digits that we have in 184, when we ask ourselves how many lots of 460 are there in 184, there aren’t any. 184 is too small. So the first thing we can do is to write zero at the top. And like we would do with any other long division, we can say that there’s a remainder. Zero lots of 460 are zero, and if we subtract zero from 184, we still have that 184 as a remainder. And the reason why we do this is it allows us to do what we normally do with long division, and that’s to bring down another digit to help.
At the moment, there are no other digits. But what we can do is to move into the decimal places. And 184 is exactly the same as 184.0. But if we write the number as a decimal like this, we’ve got a zero that we can bring down to help us. So our number 184 becomes 1840. Before we start dividing, because we’ve moved into the decimal places, it’s worth just drawing the decimal point in the answer. This way, we know our answer’s going to be a decimal, too. Now, we can ask ourselves how many 460s are there in 1840? What facts can we work out about multiples of 460 to help us?
Well, we know what one lot of 460 is worth. And we also know that if we double 460, we can find out what two lots of 460 are worth. Double 60 is 120, and double 400 is 800 plus the 100 in 120 takes us to 900. Two lots of 460 are 920. There are nine hundreds in 920. And if we look at the number we’re dividing by, we can see that there are 18 hundreds. 18 is double nine. Let’s try doubling 920 to see what four lots of 460 are worth. Double 20 is 40. And double 900 is 18 hundreds or 1800. This is the exact answer we’re looking for. There are four lots of 460 in 1840. So we can write four as part of our answer at the top. This goes in the tens place. We said four lots of 460 or 1840. And if we subtract this from the number we had to start with, we can show that there’s nothing left over. It divides perfectly.
So we’ve reached the end of our calculation, and our prediction that the answer was going to be a decimal less than one was correct. We’ve used long division to find that 184 divided by 460 equals 0.4. Or if we go back to our decimals, we can also say that 1.84 divided by 4.6 is equal to 0.4. Or we could change 4.6 back to the mixed number and say that 1.84 divided by four and three-fifths equals 0.4. And if we swap the divisor and the quotient around, we know now that 1.84 divided by 0.4 equals four and three-fifths. This is the same as our original calculation. And the missing number that we found is 0.4.