Video: Interpreting and Computing Partitive Division of a Fraction by a Fraction of Different Denominator

Sophia and Liam have walked 13(1/4) km in 2(1/4) hours. As they want to walk for 3 hours, what fraction of their walk have they completed so far? If they keep up the same pace, what distance will they have walked in 3 hours?

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Video Transcript

Sophia and Liam have walked 13 and one-quarter kilometers in two and one-quarter hours. As they want to walk for three hours, what fraction of their walk have they completed so far? Then the second part says if they keep up the same pace, what distance will they have walked in three hours?

Sophia and Liam have walked for two and a quarter hours. And they want to walk for three. To find the fraction of the walk that they’ve completed so far, we’re going to divide the amount they’ve completed by the total. Our instinct might be to write this as two and one-quarter over three. But actually, we know that fraction line means divide. So we’re going to write it as two and one-quarter divided by three, as shown.

Next, we know that to divide when we’re working with mixed numbers, we need to make sure any mixed numbers are written in improper fraction form. Similarly, any integers we write with a denominator of one. Now two times four is eight. And when we add the one, we get nine. So two and one-quarter is equivalent to nine-quarters. We also know that three, since it’s an integer, can be written as three over one. And so the calculation we’re doing is nine over four divided by three over one.

Now, in fact, since three is a factor of nine, we could actually simply divide nine by three. But let’s prove to ourselves that our methods that we have for dividing fractions work when we’re working with integers two. One of those involves multiplying the first fraction by the reciprocal of the second. And so we can say that nine-quarters divided by three over one is actually the same as nine-quarters times one-third. Then we cross cancel. Nine divided by three is three, and three divided by three is one. So we get three-quarters times one over one. And that of course is simply equal to three-quarters. Sophia and Liam have completed three-quarters of the walk so far.

Then the second part says that they’re going to keep up the same pace. So they’re going to have the same average speed. What distance will they have walked in three hours? And so one thing that we could do is use the speed–distance–time formula. Speed is equal to distance divided by time. So we could work out the average speed for the first part of their journey by dividing 13 and a quarter by two and a quarter. Then we can work out the total distance that they’ll walk in three hours by multiplying this speed by the time taken, by three.

But there is in fact another method. We know that they’re going to be walking at the same pace. So the total distance they travel will be directly proportional to the amount of time taken. And so we’re going to divide the distance that they walked in the first part of the journey by the fraction of the walk that they’d completed so far. So that’s 13 and one-quarter divided by three-quarters.

To perform this calculation, we convert 13 and one-quarter into a mixed number. 13 and one-quarter is the same as 53 over four. And that’s because 13 times four is 52. And then we add the numerator one to get 53. So we’re doing fifty-three quarters divided by three-quarters.

Now, of course, since the denominators of these fractions are equal, we simply divide the numerators. And so we get 53 over three. We do of course need to change this back into a mixed number. 53 divided by three is 17 with a remainder of two. Since we’re converting an improper fraction into a mixed number, we know that the denominator remains unchanged. It’s still three. And so we can say 53 over three is equivalent to 17 and two-thirds. And therefore, the total distance they will have walked in three hours, assuming that they maintain the same pace, will be 17 and two-thirds of a kilometer.

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