Question 6

We can find a rational number between any two rational numbers using different methods.

Step 1 — Mean Method

This is a very common method. We take the average of the two numbers. Let the two rational numbers be $a$ and $b$. Their average is found by adding them. Then we divide the sum by 2. This gives us a rational number. It lies strictly between $a$ and $b$.

Let's find a number between $1/4$ and $1/2$. Here, $a = 1/4$ and $b = 1/2$. We add $a$ and $b$ together.

$= \frac{1}{4} + \frac{1}{2}$

$= \frac{1}{4} + \frac{2}{4}$

$= \frac{3}{4}$

Then we divide the sum by 2.

$= \frac{3}{4} \div 2$

$= \frac{3}{4} \times \frac{1}{2}$

$\boxed{\frac{3}{8}}$

Step 2 — Common Denominator Method

This method uses equivalent fractions. First, we make the denominators the same. Let's use $1/4$ and $1/2$ again. The common denominator for 4 and 2 is 4.

$\frac{1}{4} = \frac{1}{4}$

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$

Now we need numbers between $1/4$ and $2/4$. There is no integer numerator between 1 and 2. We can multiply both by 10/10. This creates more space between them.

$\frac{1}{4} = \frac{1 \times 10}{4 \times 10} = \frac{10}{40}$

$\frac{2}{4} = \frac{2 \times 10}{4 \times 10} = \frac{20}{40}$

Now we have $10/40$ and $20/40$. We can pick any fraction in between. For example, $11/40$ is a good choice.

$\boxed{\frac{11}{40}}$

Step 3 — Decimal Expansion Method

This method uses decimal forms. We convert the rational numbers to decimals. Let's use $1/4$ and $1/2$ one more time.

$\frac{1}{4} = 0.25$

$\frac{1}{2} = 0.5$

Now we can choose any decimal number. It must lie between 0.25 and 0.5. For instance, 0.3 is a rational number. It is between the two given numbers. We can also choose 0.4 or 0.45.

$\boxed{0.3}$

Answer

(i) Mean Method: We find the average of the two rational numbers. (ii) Common Denominator Method: We convert numbers to equivalent fractions with a larger common denominator. (iii) Decimal Expansion Method: We convert numbers to decimals and choose a decimal between them.