2.04 Fractions of a quantity

Evaluate $24\times\frac{7}{40}$24×740​.

Think: Using the same reasoning as above, we can represent $24\times\frac{7}{40}$24×740​ as $\frac{7}{40}$740​ of $24$24.

It is also the same as $\frac{7\times24}{40}$7×2440​. We can evaluate the multiplication in the numerator and then simplify the resulting fraction. Or, we can cancel common factors in the numerator and denominator before evaluating the multiplication.

Do: It is easier to cancel the common factors first. But what are the common factors?

Notice that $24=3\times8$24=3×8. Also, $40=5\times8$40=5×8. So $8$8 is a common factor. If we put these into our original fraction, we get $\frac{7\times3\times8}{5\times8}$7×3×85×8​.

When we cancel the $8$8s from the numerator and denominator, we are left with $\frac{7\times3}{5}$7×35​. Evaluating the multiplication in the numerator gives us $\frac{21}{5}$215​. Since there are no common factors between $21$21 and $5$5, this fraction is fully simplified.

So $24\times\frac{7}{40}=\frac{21}{5}$24×740​=215​.

If we evaluate the multiplication in the numerator first, we should get the same answer. Using the vertical algorithm, or a calculator we can see that $24\times7=168$24×7=168.

We know that $40=5\times8$40=5×8, so we can try dividing $168$168 by $8$8 and we get $21$21.

So we have $\frac{21\times8}{5\times8}$21×85×8​ and after cancelling the $8$8s we get $\frac{21}{5}$215​.

Reflect: Both of these methods gave the same answer, but which one was better?

The benefit of cancelling first is that we work with smaller numbers, which is easier to do in our heads. However, there may not always be common factors that we can cancel in which case we will have to use the other method.

The benefit of multiplying first is that it will always work. However, sometimes it will result in very big numbers which we still have to simplify later. This may require the use of a calculator or vertical multiplication algorithm.

The best method to use will depend on the situation, so we should practice both.

Find $\frac{3}{4}$34​ of $\$5$$5 in cents.

Think: We want to approach this question the same way as the previous one. However, we will also need to convert the dollars into cents.

We can use the fact that $\$1=100$$1=100c.

Do: First we can write $\$5$$5 in cents. This means we are finding $\frac{3}{4}$34​ of $5\times100$5×100. We could evaluate this multiplication now, but it will be easier to do it later.

Now we have $\frac{3}{4}\times5\times100$34​×5×100. We can move all of the factors into the numerator to get $\frac{3\times5\times100}{4}$3×5×1004​.

$3,4,$3,4, and $5$5 have no common factors. However, we can rewrite $100$100 as $4\times25$4×25. This gives us $\frac{3\times5\times4\times25}{4}$3×5×4×254​. And cancelling the $4$4s gives us $3\times5\times25$3×5×25.

We can evaluate the final multiplications as we normally would for whole numbers, which gives us $375$375 cents.