David and Isabella share 30
cupcakes in the ratio two to three. How many do they each get?
One way we have to answer this
question is the add-divide-multiply method. But there is another method which
is equally valid. And that involves thinking about
bar modeling and fractions. The first step is the same as the
add-divide-multiply method. We look at our ratio. That’s the ratio two to three. We then add the numbers in our
ratio. Two plus three is equal to five,
meaning we’re looking at a total of five parts in this question. And so a bar model will be made up
of five parts, as shown.
Remember, order is important. So David gets two of these
parts. That’s represented by the yellow
bits. And Isabella gets three. That’s represented by the pink
parts. Now, we can think about this in
terms of fractions. And we can see that the portion of
the bar that I’ve shaded yellow represents two-fifths of the whole, whereas the
portion that I’ve shaded pink represents three-fifths of the total.
We can extend this into the number
of cupcakes and say that, well, if there are 30 cupcakes, David gets two-fifths of
these. Similarly, Isabella will get
three-fifths of these. By recalling that “of” can quite
regularly be interchanged with the multiplication symbol, we see that we can answer
this question either by finding two-fifths of 30 and three-fifths of 30 or by
multiplying two-fifths by 30 and three-fifths by 30.
Let’s multiply two-fifths by
30. We’ll write 30 as a fraction. It’s 30 over one. Next, we cross cancel. We divide the denominator of our
first fraction by five and the numerator of our second by five. Two times six gives us 12, and one
times one gives us one. But 12 ones is simply 12. And so we see that David gets 12 of
To find the number that Isabella
gets, we could subtract this from 30. But let’s multiply again. We do three-fifths times 30 over
one and, once again, divide through by five. This gives us three times six,
which is 18 over one, or simply 18. David gets 12 cupcakes, and
Isabella gets 18.
Note that, of course, we can check
our answer by checking that 12 and 18 do add up to 30, which they do. Note also that at this stage, we
could have found the value of one-fifth by dividing 30 by five. Then whatever value we get, we
double to find the value of two-fifths. Either method is perfectly