Sharing Quantities

The Concept of Parts

If you want to share 42 counters in the ratio 4 : 3:

1. Total Parts: The ratio implies one person gets 4 parts and the other gets 3 parts. $\text{Total Groups} = 4 + 3 = 7$
2. Size of One Part: Divide the total quantity by the number of groups. $\text{Size} = \frac{42}{7} = 6$
3. Calculate Shares:
   • Person A (4 parts): $4 \times 6 = 24$
   • Person B (3 parts): $3 \times 6 = 18$

Check: $24 + 18 = 42$. Correct.

General Formula

To divide quantity $X$ in ratio $m:n$:

$\text{First Part} = \left( \frac{m}{m+n} \right) \times X$

$\text{Second Part} = \left( \frac{n}{m+n} \right) \times X$

Example: Business Profit

Prashanti invested ₹75,000 and Bhuvan invested ₹25,000. Profit is ₹4,000. Share profit based on investment.

1. Ratio: $75000 : 25000 \rightarrow 3 : 1$.
2. Total Parts: $3 + 1 = 4$.
3. Prashanti’s Share: $\frac{3}{4} \times 4000 = 3000$.
4. Bhuvan’s Share: $\frac{1}{4} \times 4000 = 1000$.