Why we add 1 to denominator while trying to find rational numbers between two rational numbers?

Question

As I read my book, they added 1 to denominator and multiplied with it, there's no explanation given, so please help me. I have inserted the image, please see the example four and please explain it to me.

Answer 1

The example is asking for six rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$. Since there aren't any integers between $3$ and $4$, you're going to have to change the denominators of the fractions such that there are six integers in between the two numerators.

In other words, we need the numerator of the second fraction to be the seventh lowest integer greater than the numerator of the first fraction, so that there are six integers in between ($6$ integers in between plus an integer at the end: $6+1=7$).

We can set this up as an algebra problem:

$$3x+7=4x$$

$$x=7$$

This is why the fractions are multiplied by $\frac{7}{7}$.

Answer 2

If you wanted six rational numbers between $0$ and $1$ then a possibility might be $\frac17$, $\frac27$, $\frac37$, $\frac47$, $\frac57$, $\frac67$. Note the $6+1=7$ in the denominators so you can have six simple distinct numerators and equally spaced results: you need one more space than the number of intermediate values.

You could express these as $\frac67 \times 0+\frac17 \times 1$, $\frac57 \times 0+\frac27 \times 1$, $\frac47 \times 0+\frac37 \times 1$, $\frac37 \times 0+\frac47 \times 1$, $\frac27 \times 0+\frac57 \times 1$, $\frac17 \times 0+\frac67 \times 1$, which are obviously distinct, rational and interpolated between $0$ and $1$.

Similarly if you wanted six rational numbers between $\frac35$ and $\frac45$ then a possibility might be $\frac67 \times \frac35 +\frac17\times \frac45 =\frac{22}{35}$, $\frac57 \times \frac35 +\frac27\times \frac45 =\frac{23}{35}$, $\frac47 \times \frac35 +\frac37\times \frac45 =\frac{24}{35}$, $\frac37 \times \frac35 +\frac47\times \frac45 =\frac{25}{35}$, $\frac27 \times \frac35 +\frac57\times \frac45 =\frac{26}{35}$, $\frac17 \times \frac35 +\frac67\times \frac45 =\frac{26}{35}$, which are obviously distinct, rational and interpolated between $\frac35$ and $\frac45$. Essentially this is what happened in the original example.