Question 3

We will explore how division changes a number based on the divisor's value.

Step 1 — Understanding Division Terms

Let us define the parts of a division problem. We have a dividend, a divisor, and a quotient. Dividend divided by divisor equals quotient.

$\text{Dividend} \div \text{Divisor} = \text{Quotient}$

Let us use an example. Consider $6 \div 2 = 3$. Here, 6 is the dividend. 2 is the divisor. 3 is the quotient.

Step 2 — Quotient Compared to Dividend

We want to compare the quotient with the dividend. Let the dividend be 'D'. Let the divisor be 'd'. Let the quotient be 'q'. So, $q = D \div d$. We are comparing 'q' with 'D'.

• Case 1: Divisor is 1

  Let us try dividing a number by 1. Suppose the dividend is 5. The divisor is 1. The quotient is $5 \div 1$.

  $5 \div 1$ $= 5$

  The quotient is 5. The dividend was 5. So, the quotient is equal to the dividend.

  $\boxed{\text{When the divisor is 1, the quotient is equal to the dividend.}}$

  Let us try another example. Suppose the dividend is $\frac{3}{5}$. The divisor is 1. The quotient is $\frac{3}{5} \div 1$.

  $\frac{3}{5} \div 1$ $= \frac{3}{5}$

  The quotient is $\frac{3}{5}$. The dividend was $\frac{3}{5}$. The quotient is equal to the dividend.

• Case 2: Divisor is greater than 1

  Let us try dividing by a number greater than 1. Suppose the dividend is 10. The divisor is 2. The quotient is $10 \div 2$.

  $10 \div 2$ $= 5$

  The quotient is 5. The dividend was 10. We see that 5 is less than 10. So, the quotient is less than the dividend.

  $\boxed{\text{When the divisor is greater than 1, the quotient is less than the dividend.}}$

  Let us try an example with fractions. Suppose the dividend is $\frac{1}{5}$. The divisor is 2. The quotient is $\frac{1}{5} \div 2$.

  $\frac{1}{5} \div 2$ $= \frac{1}{5} \times \frac{1}{2}$ $= \frac{1}{10}$

  The quotient is $\frac{1}{10}$. The dividend was $\frac{1}{5}$. We know that $\frac{1}{10}$ is less than $\frac{1}{5}$. So, the quotient is less than the dividend.

• Case 3: Divisor is between 0 and 1

  Let us try dividing by a number between 0 and 1. This means the divisor is a proper fraction. Suppose the dividend is 10. The divisor is $\frac{1}{2}$. The quotient is $10 \div \frac{1}{2}$.

  $10 \div \frac{1}{2}$ $= 10 \times \frac{2}{1}$ $= 20$

  The quotient is 20. The dividend was 10. We see that 20 is greater than 10. So, the quotient is greater than the dividend.

  $\boxed{\text{When the divisor is between 0 and 1, the quotient is greater than the dividend.}}$

  Let us try an example with fractions. Suppose the dividend is $\frac{1}{2}$. The divisor is $\frac{1}{3}$. The quotient is $\frac{1}{2} \div \frac{1}{3}$.

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

  The quotient is $1\frac{1}{2}$. The dividend was $\frac{1}{2}$. We know that $1\frac{1}{2}$ is greater than $\frac{1}{2}$. So, the quotient is greater than the dividend.

Step 3 — Divisor Compared to Quotient

Now, let us look at the relationship between the divisor and the quotient. We want to see if there is a simple rule.

• Example 1: Dividend = 10, Divisor = 2. Quotient = $10 \div 2 = 5$. Here, Divisor (2) is less than Quotient (5).

• Example 2: Dividend = 10, Divisor = 5. Quotient = $10 \div 5 = 2$. Here, Divisor (5) is greater than Quotient (2).

• Example 3: Dividend = 4, Divisor = 2. Quotient = $4 \div 2 = 2$. Here, Divisor (2) is equal to Quotient (2).

• Example 4: Dividend = 1, Divisor = $\frac{1}{2}$. Quotient = $1 \div \frac{1}{2} = 2$. Here, Divisor ($\frac{1}{2}$) is less than Quotient (2).

• Example 5: Dividend = $\frac{1}{4}$, Divisor = $\frac{1}{2}$. Quotient = $\frac{1}{4} \div \frac{1}{2} = \frac{1}{2}$. Here, Divisor ($\frac{1}{2}$) is equal to Quotient ($\frac{1}{2}$).

From these examples, we see that the relationship changes. Sometimes the divisor is less than the quotient. Sometimes it is greater. Sometimes they are equal. This relationship depends on the dividend too. So, there is no similar direct relationship.

$\boxed{\text{There is no similar direct relationship between the divisor and the quotient.}}$

Answer

(i) When the divisor is between 0 and 1, the quotient is greater than the dividend. (ii) When the divisor is greater than 1, the quotient is less than the dividend. (iii) When the divisor is 1, the quotient is equal to the dividend. (iv) There is no similar direct relationship between the divisor and the quotient.