The ratio of handbags to backpacks in a cloakroom is two to three and a half. Part a) What fraction of the bags in the cloakroom are backpacks? Part b) The cloakroom has 341 bags in total. Assuming the cloakroom contains only handbags and backpacks, how many handbags are in the cloakroom?
So if we take a look at our ratio. Our ratio is in the form where one of the parts, so the right-hand side, has a fraction contains, we got two to three and a half. And what we’re gonna do is use a property of ratios to help us solve this problem. Well let’s imagine that I had red counters and blue counters and they were in the ratio one to two. Then, the fraction of red counters would be one out of three or one-third. And that’s because there’re three parts in total and one of those parts is red.
Now, if I scaled my ratio up by multiplying both sides by the same values, so in this case two, I will get two to four. So therefore, if I wanted to find the fraction of red this time, it would be two out of six because two parts are red and they’re six in total. However, if we simplify this fraction by dividing the numerator and denominator by the same values, so our factors of both two and six, so we divide it by two, we’ll get one over three or one-third again. So you can see the fraction hasn’t changed even though we scaled up the ratio.
So once more, if we scale it up again, this time we’re gonna multiply both sides by five, we’d have the ratio 10 to 20. Well, this time the fraction of red will be 10 out of 30 because there 10 parts are red and there are 30 parts in total. Well, again, if we simplify this by dividing the numerator and denominator by a common factor, so 10, we’ll get one over three or a third. So we can see that whatever we do with scaling of our ratio, so if we multiply both sides by the same thing, the fractions will not change; they will stay the same.
So therefore, in our question, if we got handbags to backpacks and we have the ratio two to three and a half, what we can do is scale this up to remove the fraction. And to do that, what we can do is multiply both sides by two. So we have two multiplied by two, we get four. Three and a half multiplied by two gives us seven. So we’ve now got the ratio four to seven. So therefore, we can say the fraction of backpacks is gonna be equal to seven and that’s because seven is the number of parts that are backpacks over 11 and that’s because 11 is the total number of parts cause it’s four add seven. So therefore, we can confirm that the fraction that are backpacks is seven 11s or seven over 11.
Okay, so now, let’s move on to part b. Well, in part b, what we want to do is find out how many handbags are in the cloakroom. Well, we know there are 341 bags in total. And we also know from part a that seven 11s or seven over 11 are backpacks. Well, first of all, we need to work out what fraction are gonna be handbags then. And this is going to be four 11s, so four over 11. And that’s cause you do one minus seven over 11. And we can either do this in a calculator. Or if we take a look and think about one, well one is 11 over 11 or 11 11ths. So we’ve got 11 11ths minus seven 11ths. We’ve got the same denominator. So we just subtract the numerator. So 11 minus seven gives us four. So four 11ths is the fraction of handbags in the cloakroom.
So therefore, to work out this problem, what we want to do is find four 11ths of 341. Well, when we say four 11ths of, what we mean when we’re dealing with fractions is four 11ths multiplied by 341. And when we calculate this, we’re gonna get 1364 over 11. That’s cause we can do four multiplied by 341 which is 1364 and then this is just over 11. And we can do that because if we think about multiplying a fraction by a number, well it’s the same as multiplying a fraction by 341 over one. And when we multiply fractions, we multiply the numerators and the denominators.
There is an alternative way we can work out. And that is by dividing by the denominator and then multiplying by the numerator. So we do 341 divided by 11 and then multiply it by four. That would be the alternative way to work this out. But we’ve got 1364 over 11. So now we need to divide this. So therefore, we get a value of 124. So we know there are 124 handbags in the cloakroom. But we can double-check this using the other method that I mentioned.
Well, if we do 341 divided by 11, this is 31. So therefore, we’re gonna have 31 multiplied by four. Well, 31 multiplied by four is 124, which then matches the answer that we got using the other method. So therefore, we can confirm that the number of handbags in the cloakroom is 124.