Prove that the following rational numbers are equal: (i) and (ii) and (iii) and (iv) and

Understand the Question Two rational numbers are equal if they reduce to the same simplest form (lowest terms). To convert a fraction to simplest form, divide both the numerator and the denominator by their Greatest Common Divisor (GCD). If both simplified fractions are identical, the rational numbers are proved equal. Part (i) Prove that and are equal. Step 1 · Simplify and Compare The first rational number is already in simplest form: For the second rational number, the . Dividing numerator and denominator by : Since both simplify to , the numbers are equal. Answer > (i) Part (ii) Prove that and are equal. Step 1 · Simplify and Compare The first rational number is already in simplest form: For the second rational number, the . Dividing numerator and denominator by : Since both simplify to , the numbers are equal. Answer > (ii) Part (iii) Prove that and are equal. Step 1 · Simplify and Compare The first rational number is already in simplest form: For the second rational number, the . Dividing numerator and denominator by : Since both simplify to , the numbers are equal. Answer > (iii) Part (iv) Prove that and are equal. Step 1 · Simplify and Compare Simplifying by dividing the numerator by the denominator: The second number is already . Since both simplify to , the numbers are equal. Answer > (iv) Common Mistakes Unequal Division: Dividing only the numerator or only the denominator by the common factor instead of both. Sign Errors: Dropping the negative sign while simplifying negative rational numbers like .

The World of Numbers | Exercise 3.3

Question 1

Prove that the following rational numbers are equal:

(i) $\dfrac{2}{3}$ and $\dfrac{4}{6}$

(ii) $\dfrac{5}{4}$ and $\dfrac{10}{8}$

(iii) $-\dfrac{3}{5}$ and $-\dfrac{6}{10}$

(iv) $\dfrac{9}{3}$ and $3$

Solution

Understand the Question

Two rational numbers are equal if they reduce to the same simplest form (lowest terms).
To convert a fraction to simplest form, divide both the numerator and the denominator by their Greatest Common Divisor (GCD).
If both simplified fractions are identical, the rational numbers are proved equal.

(i) Prove that $\dfrac{2}{3}$ and $\dfrac{4}{6}$ are equal.

Step 1 · Simplify and Compare

The first rational number is already in simplest form: $\dfrac{2}{3}$

For the second rational number, the $\text{GCD}(4, 6) = 2$ . Dividing numerator and denominator by $2$ :

\begin{aligned} \dfrac{4}{6} &= \dfrac{4 \div 2}{6 \div 2} \\[0.6em] &= \dfrac{2}{3} \end{aligned}

Since both simplify to $\dfrac{2}{3}$ , the numbers are equal.

Answer

(i) $\dfrac{2}{3} = \dfrac{4}{6}$

(ii) Prove that $\dfrac{5}{4}$ and $\dfrac{10}{8}$ are equal.

Step 1 · Simplify and Compare

The first rational number is already in simplest form: $\dfrac{5}{4}$

For the second rational number, the $\text{GCD}(10, 8) = 2$ . Dividing numerator and denominator by $2$ :

\begin{aligned} \dfrac{10}{8} &= \dfrac{10 \div 2}{8 \div 2} \\[0.6em] &= \dfrac{5}{4} \end{aligned}

Since both simplify to $\dfrac{5}{4}$ , the numbers are equal.

Answer

(ii) $\dfrac{5}{4} = \dfrac{10}{8}$

(iii) Prove that $-\dfrac{3}{5}$ and $-\dfrac{6}{10}$ are equal.

Step 1 · Simplify and Compare

The first rational number is already in simplest form: $-\dfrac{3}{5}$

For the second rational number, the $\text{GCD}(6, 10) = 2$ . Dividing numerator and denominator by $2$ :

\begin{aligned} -\dfrac{6}{10} &= -\dfrac{6 \div 2}{10 \div 2} \\[0.6em] &= -\dfrac{3}{5} \end{aligned}

Since both simplify to $-\dfrac{3}{5}$ , the numbers are equal.

Answer

(iii) $-\dfrac{3}{5} = -\dfrac{6}{10}$

(iv) Prove that $\dfrac{9}{3}$ and $3$ are equal.

Step 1 · Simplify and Compare

Simplifying $\dfrac{9}{3}$ by dividing the numerator by the denominator:

\begin{aligned} \dfrac{9}{3} &= 3 \end{aligned}

The second number is already $3$ .

Since both simplify to $3$ , the numbers are equal.

Answer

(iv) $\dfrac{9}{3} = 3$

Common Mistakes

Unequal Division: Dividing only the numerator or only the denominator by the common factor instead of both.
Sign Errors: Dropping the negative sign while simplifying negative rational numbers like $-\dfrac{6}{10}$ .

Question 1

More questions in Exercise 3.3