Question Video: Dividing Two Fractions Mathematics

Calculate 4/5 ÷ 3/4.


Video Transcript

Calculate four-fifths divided by three-quarters.

One method we have to divide fractions is to create equivalent fractions with a common denominator and then simply divide the numerators. Now, whilst we don’t need to find the least common denominator, it does ensure that there’ll be minimal simplifying at the other end. So what is the common denominator of our two fractions? Well, the least common multiple of five and four is 20. So that’s the denominator we’re going to use. So how do we convert from fifths into twentieths? Well, to get from five to 20, we multiply by four. To create an equivalent fraction, we must do the same to the numerator. So four times four is 16, meaning four-fifths is equivalent to sixteen twentieths.

We repeat this process for three-quarters. To turn quarters into twentieths, we multiply by five. And so we have to do the same to our numerator. We’re going to multiply three by five, which is 15, meaning three-quarters is equivalent. It’s the same as fifteen twentieths. So we’re dividing sixteen twentieths by fifteen twentieths. Once their denominators are the same, we’re able simply to divide the numerators. So we’re going to divide 16 by 15. Of course, the fraction line actually means divide. So 16 divided by 15 can be written as 16 over 15.

And since this is an improper fraction — in other words, the numerator is greater than the denominator — we’re going to convert it into a mixed number. We ask ourselves how many 15s make 16? Well, it’s one remainder one. This means the integer part is one and the numerator of our fraction is also one. The denominator remains unchanged, so 16 over 15 is equivalent to one and one fifteenth. And so four-fifths divided by three-quarters is one and one fifteenth. But this isn’t the only method we have to divide fractions.

To divide by a fraction, we can multiply by the reciprocal of that fraction. So the first fraction remains unchanged. And instead of dividing, we’re timesing. Reciprocal means one over. But if we already have it in fraction form, we simply invert it. We switch the numerators and denominators. So the reciprocal of three-quarters is four-thirds. And the sum becomes four-fifths times four-thirds.

But why does this work? Well, what we’re doing is dividing by the numerator of our second fraction to get a unit fraction and then multiplying by the denominator to get the whole. And the beauty of this method is we already know how to multiply fractions. We simply multiply their numerators and then separately multiply their denominators. So this becomes four times four over five times three, which is sixteen fifteenths. Once again, we know that this becomes one and one fifteenth. So we have our alternative method for dividing four-fifths by three-quarters. Now, both of these methods are equally valid.

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