2. Rational numbers

Lesson

Fractions describe parts of a whole, but they can also describe parts of a quantity.

Find $\frac{1}{12}$112 of $36$36.

Let's start by drawing a grid of $36$36:

To find one twelfth, we split this grid into $12$12 equal parts.

Looking at the pieces, each piece has $3$3 squares. So $\frac{1}{12}$112 of $36$36 is $3$3.

We can also work this out using arithmetic. We know that $\frac{1}{12}$112 of $36$36 can be written using multiplication, $\frac{1}{12}\times36$112×36. This is the same as $1\times\frac{36}{12}$1×3612 and $\frac{1\times36}{12}$1×3612.

The third expression is the most useful. First, if we evaluate the multiplication in the numerator we get $\frac{36}{12}$3612. Then we can cancel the greatest common factor from the numerator and denominator. In this case it is $12$12. This gives us $\frac{3}{1}$31 which is the same as $3$3.

We can check this answer by multiplying back. $12\times3=36$12×3=36, so we know that $3$3 is $\frac{1}{12}$112 of $36$36.

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 canceling 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 canceling 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 canceling 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.

Summary

Finding the fraction of a quantity is the same as multiplying a whole number by a fraction.

To multiply a whole number by a fraction, multiply the whole number by the numerator.

It is often easier to cancel common factors in the numerator and denominator before evaluating the multiplication.

Evaluate $\frac{2}{5}\times35$25×35.

Find $3$3 groups of $\frac{4}{5}$45.

We've seen how to multiply whole numbers by fractions. Can we use the same techniques to multiply fractions by fractions?

Evaluate $\frac{2}{3}\times\frac{4}{5}$23×45.

Finding $\frac{2}{3}\times\frac{4}{5}$23×45 is the same as finding $\frac{2}{3}$23 of $\frac{4}{5}$45. We can find this by starting with a diagram of $\frac{4}{5}$45:

Then we can split each of these fifths into thirds:

Notice that the circles is now divided into fifteenths (that is, $3\times5$3×5) and twelve parts have been shaded ($3\times4$3×4).

Now we can shade in two thirds of each of the original pieces:

And we finish with eight fifteenths. So $\frac{2}{3}\times\frac{4}{5}=\frac{8}{15}$23×45=815.

Each of these steps we've done before. We can think of $\frac{2}{3}$23 of $\frac{4}{5}$45 as $\frac{2}{3}$23 of $\frac{12}{15}$1215 (since $\frac{4}{5}$45 and $\frac{12}{15}$1215 are equivalent fractions). Since $\frac{12}{15}$1215 is $12$12 fifteenths we then want to find $\frac{2}{3}$23 of $12$12, and this is the number of fifteenths we are left with.

This suggests another method for multiplying fractions. By equivalent fractions, $\frac{2}{3}\times\frac{4}{5}=\frac{2}{3}\times\frac{4\times3}{5\times3}$23×45=23×4×35×3.

Since this is $\frac{2}{3}\times4\times3$23×4×3 fifteenths, we are multiplying a fraction by a whole number, so we can write $\frac{2}{3}\times4\times3=\frac{2\times4\times3}{3}$23×4×3=2×4×33.

If we cancel the common factor of $3$3, we get $2\times4$2×4 fifteenths which is $\frac{8}{15}$815.

So $\frac{2}{3}\times\frac{4}{5}=\frac{2\times4}{3\times5}$23×45=2×43×5.

We can generalize this method to any fractions. So whenever we want to multiply two fractions, we can multiply the numerators and the denominators separately. Sometimes we might have to simplify the resulting fraction afterwards.

Let's use this method from now on.

Evaluate $\frac{6}{11}\times\frac{5}{9}$611×59.

**Think:** First we can write this as one fraction by multiplying the numerators and the denominators separately.

**Do:** Writing the multiplication as one fraction gives us $\frac{6\times5}{11\times9}$6×511×9.

Evaluating the numerator gives $6\times5=30$6×5=30 and evaluating the denominator gives $11\times9=99$11×9=99.

So $\frac{6}{11}\times\frac{5}{9}=\frac{30}{99}$611×59=3099. Since $3$3 is a common factor of $30$30 and $99$99 we can also simplify the fraction further. Canceling $3$3 gives $\frac{11}{30}$1130.

**Reflect:** We could also cancel the common factors a step earlier. Sometimes this makes the multiplication easier to evaluate.

Evaluate $\frac{3}{7}\times\frac{7}{5}$37×75.

**Think:** First we can write this as one fraction by multiplying the numerators and the denominators separately. Instead of evaluating the multiplications, we can look for common factors to cancel first.

**Do:** Writing the multiplication as one fraction gives us $\frac{3\times7}{7\times5}$3×77×5.

We can see that $7$7 is a factor in both the numerator and denominator. Canceling this gives us $\frac{3}{5}$35.

So $\frac{3}{7}\times\frac{7}{5}=\frac{3}{7}$37×75=37.

**Reflect:** In this case, it was definitely quicker to cancel the common factors first, so it is worth checking for common factors.

Summary

To multiply two fractions, multiply the numerators and the denominators separately.

Evaluate $\frac{3}{5}\times\frac{4}{7}$35×47.

Evaluate $\frac{5}{3}\times\frac{21}{2}$53×212.

Evaluate $\frac{3}{77}\times\frac{35}{8}$377×358.