Question 3

Step 1 · Compare Quotient to Dividend for Different Divisors

Let $\text{Dividend} = D$, $\text{Divisor} = d$, and $\text{Quotient} = q$, where $q = D \div d$.

Case 1: Divisor is equal to $1$ ($d = 1$) $5 \div 1 = 5$ $\dfrac{3}{5} \div 1 = \dfrac{3}{5}$ When the divisor is $1$, the quotient is equal to the dividend.

Case 2: Divisor is greater than $1$ ($d > 1$) $10 \div 2 = 5 \quad (5 < 10)$ $\begin{aligned} \dfrac{1}{5} \div 2 &= \dfrac{1}{5} \times \dfrac{1}{2} \\[0.6em] &= \dfrac{1}{10} \quad \left(\dfrac{1}{10} < \dfrac{1}{5}\right) \end{aligned}$ When the divisor is greater than $1$, the quotient is less than the dividend.

Case 3: Divisor is between $0$ and $1$ ($0 < d < 1$) $\begin{aligned} 10 \div \dfrac{1}{2} &= 10 \times \dfrac{2}{1} \\[0.6em] &= 20 \quad (20 > 10) \end{aligned}$ $\begin{aligned} \dfrac{1}{2} \div \dfrac{1}{3} &= \dfrac{1}{2} \times \dfrac{3}{1} \\[0.6em] &= \dfrac{3}{2} = 1\dfrac{1}{2} \quad \left(1\dfrac{1}{2} > \dfrac{1}{2}\right) \end{aligned}$ When the divisor is between $0$ and $1$, the quotient is greater than the dividend.

Step 1 · Compare Divisor and Quotient Across Examples

Test the relationship between divisor and quotient across different examples:

• $10 \div 2 = 5 \implies \text{Divisor } (2) < \text{Quotient } (5)$
• $10 \div 5 = 2 \implies \text{Divisor } (5) > \text{Quotient } (2)$
• $4 \div 2 = 2 \implies \text{Divisor } (2) = \text{Quotient } (2)$
• $1 \div \dfrac{1}{2} = 2 \implies \text{Divisor } \left(\dfrac{1}{2}\right) < \text{Quotient } (2)$
• $\dfrac{1}{4} \div \dfrac{1}{2} = \dfrac{1}{2} \implies \text{Divisor } \left(\dfrac{1}{2}\right) = \text{Quotient } \left(\dfrac{1}{2}\right)$

The divisor can be less than, greater than, or equal to the quotient depending entirely on the dividend.