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Lesson 1. Adding fractions

The notion of fractions is a fundamental concept in math. As soon as we learn how to count $1, 2, 3, \dots$, the next natural step is to understand the notion of proportion. Generally, speaking, the following three words have the same meaning:

  1. fraction,
  2. proportion,
  3. ratio.
More precisely, proportion/ratio of $A$ to $B$ means $A/B = \frac{A}{B}$.


Exercise. What is the ratio of $2$ to $4$?

Solution. Ratio of $2$ to $4$ is $2/4$, so the answer is $2/4$.

Perhaps, some people would argue that the answer to the above exercise is $1/2$. That's also true. In fact, we have
$$\frac{1}{2} = \frac{2}{4}.$$
Think about it. Having one piece out of two equal-size pieces of a cake is the same as having two pieces out of four equal-size pieces of the cake. This means that there are multiple ways to write a single fraction. The best way to see this is the trick of multiplying $1$. What do I mean by this?

Step 1. Note that multiplying $1$ does not change anything:
$$\frac{1}{2} \cdot 1 = \frac{1}{2}.$$
Step 2. Note that $1$ can be written as follows:
$$1 = \frac{2}{2}.$$
Step 3. Replace $1$ in the equality given in Step 1 by $2/2$:
$$\frac{1}{2} \cdot \frac{2}{2} = \frac{1}{2}.$$
Step 4. Multiply tops out and bottoms out:
$$\frac{1 \cdot 2}{2 \cdot 2} = \frac{1}{2}.$$
Step 5. Get the answer:
$$\frac{2}{4} = \frac{1}{2}.$$

Adding fractions: multiply the other bottom up and down. First of all, when the bottoms are the same, you keep one bottom and add the tops. For example, we have
$$\frac{2}{6} + \frac{3}{6} = \frac{2 + 3}{6} = \frac{5}{6}.$$
To add two fractions with different bottoms, you need to multiply the other bottom up and down. For example, consider the addition $1/2 + 1/3$. The fraction $1/2$ thinks that the other bottom is $3$. If we multiply $3$ up and down to $1/2$, we get
$$\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}.$$
Now, the fraction $1/2$ thinks that the other bottom is $2$. If we multiply $2$ up and down to $1/3$, we get
$$\frac{1}{3} = \frac{1 \cdot 2}{3 \cdot 2} = \frac{2}{6}.$$
Therefore, we get
$$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{3 + 2}{6} = \frac{5}{6}.$$

Exercise. Compute the following addition:
$$\frac{3}{25} + \frac{9}{4}.$$

Solution. Again, for each fraction, multiply the other bottom to the top and the bottom. We get
$$\frac{3}{25} = \frac{3 \cdot 4}{25 \cdot 4} = \frac{12}{100},$$
and
$$\frac{9}{4} = \frac{9 \cdot 25}{4 \cdot 25} = \frac{225}{100}.$$
Therefore, we have
$$\frac{3}{25} + \frac{9}{4} = \frac{12}{100} + \frac{225}{100} = \frac{12 + 225}{100} = \frac{237}{100},$$
which is the answer.

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