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Sophia and Liam have walked 13 and one-quarter kilometers in two and one-quarter hours.
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As they want to walk for three hours, what fraction of their walk have they completed so far?
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Then the second part says if they keep up the same pace, what distance will they have walked in three hours?
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Sophia and Liam have walked for two and a quarter hours.
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And they want to walk for three.
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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.
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Our instinct might be to write this as two and one-quarter over three.
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But actually, we know that fraction line means divide.
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So we’re going to write it as two and one-quarter divided by three, as shown.
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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.
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Similarly, any integers we write with a denominator of one.
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Now two times four is eight.
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And when we add the one, we get nine.
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So two and one-quarter is equivalent to nine-quarters.
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We also know that three, since it’s an integer, can be written as three over one.
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And so the calculation we’re doing is nine over four divided by three over one.
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Now, in fact, since three is a factor of nine, we could actually simply divide nine by three.
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But let’s prove to ourselves that our methods that we have for dividing fractions work when we’re working with integers two.
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One of those involves multiplying the first fraction by the reciprocal of the second.
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And so we can say that nine-quarters divided by three over one is actually the same as nine-quarters times one-third.
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Then we cross cancel.
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Nine divided by three is three, and three divided by three is one.
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So we get three-quarters times one over one.
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And that of course is simply equal to three-quarters.
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Sophia and Liam have completed three-quarters of the walk so far.
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Then the second part says that they’re going to keep up the same pace.
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So they’re going to have the same average speed.
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What distance will they have walked in three hours?
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And so one thing that we could do is use the speed–distance–time formula.
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Speed is equal to distance divided by time.
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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.
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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.
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But there is in fact another method.
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We know that they’re going to be walking at the same pace.
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So the total distance they travel will be directly proportional to the amount of time taken.
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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.
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So that’s 13 and one-quarter divided by three-quarters.
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To perform this calculation, we convert 13 and one-quarter into a mixed number.
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13 and one-quarter is the same as 53 over four.
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And that’s because 13 times four is 52.
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And then we add the numerator one to get 53.
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So we’re doing fifty-three quarters divided by three-quarters.
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Now, of course, since the denominators of these fractions are equal, we simply divide the numerators.
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And so we get 53 over three.
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We do of course need to change this back into a mixed number.
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53 divided by three is 17 with a remainder of two.
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Since we’re converting an improper fraction into a mixed number, we know that the denominator remains unchanged.
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It’s still three.
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And so we can say 53 over three is equivalent to 17 and two-thirds.
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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.