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.