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Lesson 3. Identifying fractions

We have been studying how to work with fractions. In particular, we have learned how to add and subtract with fractions:

Lesson 1. Adding fractions.

How to check whether two fractions are equal. Recall that one fraction can be written in various ways (e.g., $1/2 = 2/4$). In this posting, we study how we can identify various different looking fractions are the same. For example, we have
$$\frac{2}{3} = \frac{10}{15}.$$
How do we know? One way to check is to multiply the opposite bottoms up and down. That is, we have
$$\frac{2}{3} = \frac{2 \cdot 15}{3 \cdot 15} = \frac{30}{45}.$$
We also do this for the other fraction $10/15$:
$$\frac{10}{15} = \frac{10 \cdot 3}{15 \cdot 3} = \frac{30}{45}.$$
Therefore, we have
$$\frac{2}{3} = \frac{30}{45} = \frac{10}{15},$$
so ignoring the middle we must have $2/3 = 10/15$. The key here is to multiply opposite bottoms up and down to get the same bottom! Now, notice that because we get the same bottom in this process, all we check to ensure that $2/3 = 10/15$ is the following:
$$2 \cdot 15 = 3 \cdot 10.$$
More generally, to check
$$\frac{A}{B} = \frac{C}{D},$$
all we need to check is $AD = BC$.

Exercise. Suppose that
$$\frac{x + 2}{x} = \frac{3}{4}.$$
Then what is $x$?

Solution. By the given condition, we have $(x+2) \cdot 4 = x \cdot 3$. The left-hand side is $4x + 8$, and the right-hand side is $3x$. This implies that
$$4x + 8 = 3x.$$
Subtracting $3x$ from both sides, we have
$$x + 8 = 0.$$
Subtracting $8$ from both sides, we have $x = -8$, and this is the answer.

Exercise. Assume that
$$\frac{x}{y} = \frac{x + 2}{y + 1}$$
If $x + y = 3$, then what is $x$ and $y$, separately?

Solution. The given condition on the fractions says $x(y + 1) = y(x + 2)$. We can expand the multiplications on both sides to have $xy + x = xy + 2y$. Subtracting $xy$ from both sides, we have $x = 2y$. Plugging this into the equation $x + y = 3$ in $x$, we have $2y + y = 3$, or to put it another way, we have $3y = 3$. Dividing both sides by $3$ yields $y = 1$. Since $x + y = 3$, this implies that $x + 1 = 3$, so subtracting $1$ from both sides gives us $x = 2$. Therefore, the answer is that $x = 2$ and $y = 1$.

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