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Lesson 4. Multiplying fractions

So far, we have been studying some essential properties of fractions. In this lesson, we will learn how to multiply two fractions.

How to multiply fractions. Multiplying two fractions is much simpler than adding them. That is, you multiply tops with tops and bottoms with bottoms:
$$\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}.$$
Exercise. Compute
$$\frac{2}{100} \cdot \frac{5}{100}.$$
Solution. We have
$$\frac{2}{100} \cdot \frac{5}{100} = \frac{2 \cdot 5}{100 \cdot 100} = \frac{10}{10000} = \frac{1}{1000}.$$
Note that the last equal sign can be checked by reviewing this previous lesson.

Don't be confused between addition and multiplication. We have
$$\frac{2}{100} + \frac{5}{100} = \frac{2 + 5}{100} = \frac{7}{100} = \frac{70}{1000},$$
while
$$\frac{2}{100} \cdot \frac{5}{100} = \frac{2 \cdot 5}{100 \cdot 100} = \frac{10}{10000} = \frac{1}{1000},$$
so they are not equal. The first is bigger than the second (because they have the same bottom).

Identification of fractions using multiplication. Note that in the above explanation, we have used
$$\frac{7}{100} = \frac{70}{1000}.$$
This can be explained by what we learned in this previous lesson, but we can also see this as using multiplication of fractions.

Step 1. Multiplying $1$ does not change anything:
$$\frac{7}{100} = \frac{7}{100} \cdot 1.$$
Step 2. $1$ can be rewritten as $10/10$ (because $10$ divided by $10$ is $1$):
$$1 = \frac{10}{10}.$$
Step 3. Replace $1$ with $10/10$ in Step 1:
$$\frac{7}{100} = \frac{7}{100} \cdot \frac{10}{10}.$$
Step 4. Use multiplication of fractions:
$$\frac{7}{100} = \frac{7}{100} \cdot \frac{10}{10} = \frac{7 \cdot 10}{100 \cdot 10} = \frac{70}{1000}.$$

Exercise. There is a cake and you want to cut it into equally sized pieces so that two pieces of them is a third of the whole cake. How many pieces do you need to cut the cake into?

Solution. Let $x$ be the number of (equally sized) pieces you cut the cake into. The proportion of each piece is $1/x$. Since two of these is $1/3$, you get the following equation:
$$2 \cdot \frac{1}{x} = \frac{1}{3}.$$
Since $2$ divided by $1$ is $2$, we have $2 = 2/1$, so replacing $2$ with $2/1$ on the left-hand side of the above equation, we have
$$2 \cdot \frac{1}{x} = \frac{2}{1} \cdot \frac{1}{x} = \frac{2 \cdot 1}{1 \cdot x} = \frac{2}{x}.$$
This must be equal to $1/3$ because of the previous equation, so we get the equation
$$\frac{2}{x} = \frac{1}{3}.$$
Recalling what we learned about how to identify fractions, this gives us $2 \cdot 3 = x \cdot 1$. This means that $6 = x$, or to put it another way, we get $x = 6$, which is the answer: you need to cut the cake into $6$ equally sized pieces.

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