Benjamin bought a villa for 467334 dollars. He paid for it in 18 equal installments. Find the value of each installment.
The first sentence in our problem tells us that Benjamin bought a villa for the price of 467334 dollars. But like a lot of expensive things that we need to buy, he didn’t pay for this all in one go. We’re told that he paid for the villa in 18 equal installments. In other words, that large amount of money was split up over time into 18 equal pieces. And the problem asked us to find the value of each installment. This is a division problem. We need to divide 467334 by 18. These are some large numbers to be dividing. How can we find the answer? Well, we’re going to have to use long division. And to do so, we’re going to have to write out the calculation so that we can use this written method.
So how many 18s are there in 467334? We can’t divide one digit on its own by 18. So let’s start by looking at the first two digits of our number. How many 18s are there in 46? In maths, they’re often methods and little tricks and ways of working that we can use. And these depend on the numbers in each calculation. In this division, we’re dividing by 18. And if we look at the number 18, we can say to ourselves that this is very close to the number 20. So in this particular calculation, we can use estimation to help. So rather than going through all the multiples of 18 to try and see how close we can get to 46, we can think to ourselves, how many 20s are there in 46? And then we can use this to get an idea of how many 18s there might be.
So how many 20s are there in 46? Well, we know two lots of 20 is 40. So let’s see what two 18s might be. Eight twos are 16. And one lot of two is two plus the one we’ve exchanged, 36. 36 is 10 less than 46. So it’s as close as we’re going to get. There are two lots of 18 in 46 then. And if we subtract two lots of 18 which we said was 36, the remainder is 10. 10 is too small to be dividing by 18. So once again, we’re going to have to include another digit. The next digit along is a seven. So if we bring that down, our number becomes 107. Again, let’s not start by asking how many 18s there are in 107. Let’s think to ourselves how many 20s there might be.
We know that five 20s make 100. So let’s see what five lots of 18 equals. Eight fives are 40. One lot of five is five plus the four that we’ve exchanged gives us an answer of 90. If we add another lot of 18, it’s going to take us more than 107. So there must be only five lots of 18 in 107. Five lots of 18, as we’ve just said, equal 90. And again, we can subtract to find the remainder. The difference between 107 and 90 is 17. There are no 18s in 17. So we need to include another digit. This time the next digit along is a three. How many 18s are there in 173? How many 20s are there in 173? We know there are five 20s in 100, six 20s in 120, seven in 140, and then eight in 160. If eight 20s make 160, let’s see how close we can get with eight 18s. Eight eights are 64. One eight is eight plus the six we’ve exchanged is 14.
Now, 144 looks quite a lot less than 173. Maybe we can fit another 18 in. If we add another 18 to 144, we can find out what nine 18s might be. Four plus eight equals 12. Four tens plus one ten equal five tens plus the one we’ve exchanged equals six tens. And then we have 100 in the hundreds place. Nine 18s are 162. And so we can fit nine 18s in 173. Now, we need to subtract to find the remainder. Nine 18s are 162. And if we subtract each column, we can see that we’re left with 11. There are no 18s in 11. So we need to include another digit. The number 11 now becomes 113.
How many 18s are there in 113? This time, we don’t need to use estimation to help. We can see from our calculations that five lots of 18 is 90. But it looks like we can fit one more lot of 18. To find six lots of 18, we need to add 18 to 90. Of course, the answer is 108. So there must be six lots of 18 in 113. Let’s find the remainder. Six lots of 18 are 108. We know the difference between eight and 13 is five. So the difference between 108 and 113 is still five. And there’s one last digit we need to include.
How many 18s are there in 54? We can use one of our earlier facts here to help us. We know that two 18s are 36. This means that if we add one more lot of 18, we’ll find three lots of 18. Six plus eight equals 14. And three tens plus one ten plus the one we’ve exchanged equals five tens. There are three 18s in 54 exactly, as though we’ve ended with no remainder. Three 18s are 54. And if we subtract 54 from 54, there’s nothing left. Our five-digit answer is at the top of the calculation. The large numbers in this calculation meant that we had to use a written method like long division to find the answer. If Benjamin bought a villa for 467334 dollars and paid for it in 18 equal installments, then we can find the value of each installment by dividing 467334 by 18. The value of each installment is 25963 dollars.