Five ratios are shown in the grid. 15 to 20, six to eight, 48 to 36, 27 to 36, 30 to 40. One of the ratios is not equivalent to three to four. Part a) Identify the ratio that is not equivalent to three to four.
Now, the fact that this question says one of the ratios is not equivalent to three to four means that four of them are, but one of them isn’t. We could go through and check all of the ratios individually. But actually, we don’t need to. And this is because two of the ratios have 36 as one part of the ratio, but they have a different number as the other part.
These two ratios can’t simplify to the same thing, which means it must be one of these ratios that is not equivalent to three to four. So we only have two possibilities to check instead of five.
If we consider the first of these ratios, 48 to 36, we can actually see straightaway that this ratio can’t be equivalent to three to four. This is because the number on the left of the ratio, 48, is greater than the number on the right, 36, whereas in the other ratio, the number on the left, three, is less than the number on the right, four. So these two ratios can’t possibly be equivalent. So this tells us straightaway that the ratio that is not equivalent to three to four is the ratio 48 to 36.
Now, it would be relatively quick to actually check and confirm that the other four ratios do indeed simplify to three to four. For example, for the ratio 30 to 40, we can divide both sides of the ratio by 10. And it does simplify to three to four. You can check this for the other ratios of 15 to 20, six to eight, and 27 to 36.
Part b) Calculate three-fifths divided by one-third.
So in this question, we’re dividing a fraction by another fraction. And to do so, we need to recall the general rule that we used to do this. To divide by a fraction, we flip or invert that fraction. And then instead of dividing, we multiply.
So three-fifths divided by one-third is equivalent to three-fifths multiplied by three over one. Multiplying fractions is much more straightforward. We can multiply the numerators together and multiply the denominators. So we have three multiplied by three over five multiplied by one. Three multiplied by three is nine and five multiplied by one is five. So our answer is nine over five or nine-fifths.
We could also give our answer as a mixed number, in which case it would be one and four-fifths. But as we haven’t been asked to give our answer in a particular format, our answer of nine over five is fine.
Watch out for a common mistake at this stage here, which is to try and cross cancel a factor of three from the three in the numerator and the three in the denominator. Cross cancelling is only a valid method of simplification when we’re multiplying; we can’t use it when we’re dividing.
Part c) Calculate three-fifths plus three fifteenths. Give your answer in its simplest form.
Now, normally, when we’re adding fractions, our first step would be to find a common denominator, which would involve scaling one or both fractions up. However, this question is a little different because the fraction three over 15 can actually be simplified as both the three and the 15 have a common factor of three.
If we divide three by three, we get one and if we divide 15 by three, we get five, which means that we have the same denominator as the first fraction. So three-fifths plus three fifteenths is actually equal to three-fifths plus one-fifth. And we can add these fractions as they both have a denominator of five.
To add these fractions, we keep the denominator the same, but add the numerators. So three-fifths plus one-fifth is equal to four-fifths. You could also answer this question by finding a common denominator for five and 15, in which case, it would be 15 as this is the lowest common multiple of five and 15. However, we’d end up with an answer that was cancelled back down to four-fifths anyway.
So if you can spot that three fifteenths is in fact equivalent to one-fifth, then you’re actually cutting down some of the work and making life easier for yourself.