Ratios

• How ratios express part-to-part and part-to-whole relationships for Class 12 quant reasoning.
• To simplify, compound, and scale ratios in mixture, partnership, and speed problems.
• To convert ratios to fractions and percentages for comparison across different totals.
• To solve exam questions on dividing quantities in a given ratio and finding missing terms.

Level 1 — Foundations

Verbal: Ratio a : b compares two quantities in same units. a : b = ma : mb for any m ≠ 0.

Forms:

Dividing quantity Q in ratio m : n: First part = Q × m/(m+n); second = Q × n/(m+n).

Compound ratio: (a:b) and (b:c) linked — make b equal (LCM technique) to merge to a:b:c.

Units: Convert to same unit before ratio (km vs m trap).

Level 2 — Exam depth

Partnership ratio: Capital × time invested → profit share ratio.

Speed ratio: Distance same → speed ratio inverse of time ratio.

Continued proportion: a/b = b/c → b² = ac (mean proportional).

Exam shortcut: If ratio 3:5 and total 40, parts 3+5=8; one part=5; answers = 15 and 25.

Ratio change problems: Add equal amount to both → new ratio; set algebra or part scaling.

Divide amount in ratio with total

Divide ₹7200 in ratio 5 : 7 : 8.
Sum parts = 20. One part = 7200/20 = 360.
Shares: 5×360=**1800**; 7×360=**2520**; 8×360=**2880**.
Check sum = 7200 ✓.

Merge two ratios to three-way

A:B = 2:3, B:C = 4:5. Equalise B: A:B = 8:12, B:C = 12:15 → **A:B:C = 8:12:15**.

• Simplify 45:60:75.
• Total 90; ratio 2:3 — find smaller share.
• If a:b=3:4 and b:c=2:1, find a:c.
• Stretch: Profit split: A invests 5000 for 6 mo, B 8000 for 4 mo — ratio?

Revision tip: Revisit adjacent topics in Quantitative Reasoning before mixed practice on Ratios.

Open the Practice tab for graded questions on Ratios.

Convert ratios to fractions of total immediately when a sum is given — avoids setting up unnecessary algebra. For partnership problems, write capital×time lines before merging ratios. Check unit consistency (months vs years) before simplifying. Keep ratio answers in simplest integer form unless the question specifies decimals.

Ratio methods feed ages problems (present age ratios with future shifts) and mixture/alligation at higher levels. In pie charts, sector ratios translate to angles — practise converting 2:3:5 ratio pies to degrees without a calculator. Partnership splits combine ratio with percentages of profit — write profit as total, then apply ratio shares. Speed–time–distance problems use inverse ratio when distance is constant — link ranking puzzles with rate comparisons mentally.

Keep a dedicated notebook spread for this topic: one page for methods, one for worked mistakes, and one for mixed drill from the Practice tab. Review weekly by explaining the core idea aloud in under sixty seconds without notes.