2. Ratio, Proportion and Partnership Without Slow Algebra

Why This Chapter Matters

Ratio and partnership questions look different on the surface, but most reduce to equal parts, proportional change, or capital multiplied by time. This chapter also feeds directly into salary, age, mixture, and DI comparisons.

Core Ideas

• Ratios only make sense when both quantities are of the same kind and written in the same unit.
• A ratio like $3:5$ describes parts, not actual values. Multiply both parts by the same number to reach real quantities.
• Direct proportion means both quantities move together; inverse proportion means one rises while the other falls.
• Compound ratios combine more than one comparison, and linked ratios must be aligned through the common term before merging them.
• In partnership, profit share depends on $\text{capital}\times \text{time}$, not capital alone.
• Whenever totals are given, convert ratios to unit parts before solving for individual values.

High-Value Formulas

How To Approach Questions

1. Write the given comparison as parts.
2. If two ratios share one term, equalise that common term before combining them.
3. Translate totals, differences, or percentage conditions into one linear relation.
4. In partnership questions, build a contribution table before calculating the ratio of profits.
5. If time or investment changes midway, split the contribution into separate intervals.

Worked Examples

Example 1

Prompt: The ratio of boys to girls is $3:5$ and the total is $64$. Find the number of girls.

Approach: Total parts $=3+5=8$. Each part $=64÷8=8$. Girls $=5\times 8=40$.

Example 2

Prompt: A invests $6000$ for $12$ months and B invests $8000$ for $9$ months. Find the profit ratio.

Approach: Compare $6000\times 12:8000\times 9=72000:72000=1:1$. Both receive equal profit shares.

Example 3

Prompt: A starts with $\text{Rs }2000$, B joins after $3$ months with $\text{Rs }4000$, and C invests $\text{Rs }10000$ for only $2$ months. Find the profit ratio at year end.

Approach: Use weighted capital: $2000\times 12:4000\times 9:10000\times 2=24:36:20=6:9:5$.

Common Mistakes

• Comparing quantities before converting them to the same unit.
• Using only the investment amount in partnership instead of amount multiplied by time.
• Joining two ratios without matching the common middle term.
• Forgetting to add all ratio parts before finding the unit value.
• Treating a ratio as a percentage directly.

Quick Revision

For ratio chapters, think in parts; for partnership chapters, think in weighted parts over time.