Which of the following is equivalent to eight 10ths? Seven-eights, 12 15ths, 15 12ths, or 20 eights.
In a moment, we’re going to calculate the answer to this question. We can also find the answer by using reasoning. We’re asked to find which of the four fractions is equivalent to or has the same value as the fraction eight 10ths.
Now, what can we say about the fraction eight 10ths? Well, one thing we can say is that it’s worth less than one. It’s a proper fraction. We know this because the top number — the numerator — is less than the denominator, which is the bottom number. So in other words, if we have 10 equal parts and we want to show eight of them, then eight out of 10 or eight 10ths is less than a whole.
Now, what do we notice about our possible answers? Well, in two of our answers, the numerator is actually larger than the denominator. 15 is greater than 12 and 20 is greater than eight. These are improper fractions. This means that they have a value greater than one. So they’re not going to be equivalent to eight 10ths. So we don’t need to do any calculation. We just need to use reasoning. And we can see that these last two fractions are not going to be the same as eight 10ths. The answer we’re looking for is going to be either seven-eights or 12 15ths.
And, normally, when we’re looking for an equivalent fraction, we might write out the fraction and we might multiply the denominator and the numerator by the same amount. And then, we could find another fraction that was worth the same. This also works with division. We could divide the denominator and the numerator by the same amount. But this method isn’t going to work with the two possible answers that we have. For example, there’s no number that we could divide the denominator the numerator by to give us seven-eights or multiply. And the same is true of 12 15ths. There’s no whole number that we could divide or multiply by to give us 12 15ths. So what can we do?
Let’s simplify eight 10ths. Perhaps by making the fraction as simple as possible, we could then find a fraction that we can multiply the denominator and the numerator by to make out the seven-eights or 12 15ths. So what’s the largest whole number that we can divide eight and 10 by? Well, apart from the number one, it’s the only number we can divide eight and 10 by and that’s two. Let’s divide the denominator first. 10 divided by two equals five and eight divided by two equals four. And if we look at the diagram, we can see that this statement is true. Eight 10ths equals four-fifths.
Now which of our two possible answers could be equal to four-fifths? Do you think it may be the one that has a multiple of five as the denominator and a multiple of four as the numerator? Let’s see whether four-fifths is equal to 12 15ths. As we’ve just noticed, 15 is a multiple of five. But what do we multiply five by to get 15? In other words, what do we multiply the denominator by to get the denominator in 12 15ths? Five times three equals 15.
Now, remember, if we want to make an equivalent fraction, we need to do the same to the numerator as we’ve done to the denominator. So does four times three equal 12? Yes, it does. Four-fifths is equal to 12 15ths. And so, that means eight 10ths must be also equal to 12 15ths. And if we look at the diagram, we’ll go through each of these three fractions. Four-fifths is the same as eight 10ths, which is also exactly the same as 12 15ths.
The fraction that’s equivalent to eight 10ths is 12 15ths.