Question 1

We will simplify each rational number to its lowest terms.

Step 1 — Prove equality for (i)

Let's simplify the first rational number. The number is $\frac{2}{3}$. It is already in its simplest form.

$\boxed{\frac{2}{3}}$

Now, let's simplify the second rational number. We find the greatest common divisor (GCD). The GCD of 4 and 6 is 2. We divide both numerator and denominator by 2.

$\frac{4}{6}$

$= \frac{4 \div 2}{6 \div 2}$

$\boxed{\frac{2}{3}}$

Both numbers simplify to $\frac{2}{3}$. Thus, they are equal.

Step 2 — Prove equality for (ii)

Let's simplify the first rational number. The number is $\frac{5}{4}$. It is already in its simplest form.

$\boxed{\frac{5}{4}}$

Now, let's simplify the second rational number. We find the greatest common divisor (GCD). The GCD of 10 and 8 is 2. We divide both numerator and denominator by 2.

$\frac{10}{8}$

$= \frac{10 \div 2}{8 \div 2}$

$\boxed{\frac{5}{4}}$

Both numbers simplify to $\frac{5}{4}$. Thus, they are equal.

Step 3 — Prove equality for (iii)

Let's simplify the first rational number. The number is $-\frac{3}{5}$. It is already in its simplest form.

$\boxed{-\frac{3}{5}}$

Now, let's simplify the second rational number. We find the greatest common divisor (GCD). The GCD of 6 and 10 is 2. We divide both numerator and denominator by 2.

$-\frac{6}{10}$

$= -\frac{6 \div 2}{10 \div 2}$

$\boxed{-\frac{3}{5}}$

Both numbers simplify to $-\frac{3}{5}$. Thus, they are equal.

Step 4 — Prove equality for (iv)

Let's simplify the first rational number. The number is $\frac{9}{3}$. We divide the numerator by the denominator.

$\frac{9}{3}$

$\boxed{3}$

Now, let's look at the second number. The number is 3. It is already in its simplest form.

$\boxed{3}$

Both numbers simplify to 3. Thus, they are equal.

Answer

(i) $\frac{2}{3}$ and $\frac{4}{6}$ are equal because both simplify to $\frac{2}{3}$. (ii) $\frac{5}{4}$ and $\frac{10}{8}$ are equal because both simplify to $\frac{5}{4}$. (iii) $-\frac{3}{5}$ and $-\frac{6}{10}$ are equal because both simplify to $-\frac{3}{5}$. (iv) $\frac{9}{3}$ and $3$ are equal because both simplify to $3$.