5  Ratio and Proportion

5.1 Introduction

Ratios and proportions form the language of comparison in quantitative aptitude.
- Ratios compare quantities of the same kind (e.g., boys:girls, price:cost).
- Proportion expresses equality of ratios (a:b = c:d).
Mastery of these concepts is essential for word problems, mixtures, partnerships, speed–time–distance, and data interpretation.

5.2 Core Definitions

  • Ratio (a:b):
    \(a:b=\frac{a}{b}\), where \(b\neq0\).
  • Proportion:
    \(a:b=c:d \;\Rightarrow\; \frac{a}{b}=\frac{c}{d}\).
  • Continued Proportion:
    \(a:b=b:c\); here \(b^2=ac\).
  • Mean Proportion:
    Mean of \(a\) and \(b\) is \(\sqrt{ab}\).

5.3 Properties of Ratios

  1. Scaling: \(a:b = ka:kb\) (for \(k>0\)).
  2. Inversion: \(a:b = 1/b:1/a\).
  3. Equality: If \(a:b=c:d\), then \(a:c=b:d\).
  4. Compounded ratio: \((a:b)\times(c:d)=(ac:bd)\).
  5. Duplicate, triplicate ratios: \((a:b)^2=(a^2:b^2)\), etc.

5.4 Types of Proportion

  • Direct proportion: \(a\propto b \;\Rightarrow\; \frac{a_1}{a_2}=\frac{b_1}{b_2}\).
  • Inverse proportion: \(a\propto 1/b \;\Rightarrow\; a_1b_1=a_2b_2\).
  • Partnership proportion: Profits divide in ratio of capitals × time.
  • Compound proportion: Combination of more than one ratio.

5.5 Rule of Three (Simple Proportion)

If \(a:b=c:x\), then
\[ x=\frac{b\times c}{a} \]

This is the “unitary method” shortcut.

Example: If 5 pens cost 50, cost of 8 pens = \((50\times8)/5=80\).

5.6 Mixtures and Ratio Applications

  • Mixing quantities: If two ingredients are mixed in ratio \(a:b\), their share % are \(\frac{a}{a+b}\times100\) and \(\frac{b}{a+b}\times100\).
  • Alligation rule: Used to find mixing ratios when average value is given.

Example: Mix milk at 20/litre with milk at 30/litre to get 26/litre.
Cheaper:Dearer = (30-26):(26-20)=4:6=2:3.

5.7 Variation Problems

  • Direct variation: \(y=kx\)
  • Inverse variation: \(y=k/x\)
  • Joint variation: \(y=kxz\)

Example: Work varies directly with men and time, inversely with days.

5.8 Partnership (Business Ratios)

If A invests \(C_A\) for \(T_A\) months, and B invests \(C_B\) for \(T_B\) months, then
Profit share ratio = \(C_A\times T_A : C_B\times T_B\).

Example: A invests 2000 for 6 months, B invests 3000 for 4 months.
Profit ratio = \(2000\times6 : 3000\times4 = 12000:12000=1:1\).

5.9 Solved Examples

  1. Simplify ratio of 50 paise : 2 rupees = 50:200 = 1:4.
  2. Divide 400 in ratio 2:3:5 = 80:120:200.
  3. If a:b=2:3 and b:c=4:5, then a:b:c=8:12:15.
  4. If 12 workers finish a job in 15 days, how many days for 20 workers?
    Inverse proportion: \(12\times15=20\times x\), \(x=9\) days.
  5. The mean proportion between 9 and 25 = \(\sqrt{9\times25}=15\).
  6. Two numbers are in ratio 5:7, sum=96. Numbers=40, 56.
  7. A:B=3:5, B:C=2:3. Find A:B:C. Solution=6:10:15.
  8. A mixture of milk and water is 3:1. Add 5 L water to make it 2:1. Find milk.
    Let milk=3x, water=x. Adding 5 → ratio=(3x):(x+5)=2:1 ⇒3x=2x+10 ⇒x=10. Milk=30 L.

5.10 Common Traps

  • Mixing direct and inverse variation.
  • Forgetting to scale ratios to whole numbers.
  • Wrong base in partnership (time neglected).
  • Confusing mean with arithmetic mean instead of geometric mean.
  • Misapplying alligation (swap differences!).

5.11 Practice Set – Level 1

  1. Simplify 48:60.
  2. Divide 600 in ratio 1:2:3.
  3. a:b=2:5, b:c=3:4. Find a:b:c.
  4. If 15 men do a work in 25 days, how many days for 25 men?
  5. Find mean proportion between 12 and 27.

5.12 Practice Set – Level 2

  1. Two numbers are in ratio 7:9, difference=16. Find numbers.
  2. A:B=5:7, B:C=6:11. Find A:B:C.
  3. Mixture of alcohol:water=7:3. Add 10 L water to make ratio 7:5. Find alcohol.
  4. A invests 3000 for 10 months, B invests 5000 for 6 months. Profit share?
  5. If 8 workers finish a task in 20 days, how many workers for 10 days?

5.13 Practice Set – Level 3

  1. A:B=4:5, B:C=5:6, C:D=6:7. Find A:B:C:D.
  2. In a partnership, A invests 2400 for 8 months, B invests 3600 for 6 months, C invests 4800 for 4 months. Divide 7200 profit.
  3. By selling mixture of tea worth 20/kg and 30/kg at 26/kg, find ratio of mixing.
  4. If a:b=2:3 and (a+b):(b+c)=5:7, find a:b:c.
  5. If x varies jointly as y and z, and inversely as t, then x=48 when y=4, z=6, t=2. Find x when y=8, z=5, t=4.

5.14 Answer Key (Outline)

Level 1: 1) 4:5; 2) 100,200,300; 3) 6:15:20; 4) 15 days; 5) 18.
Level 2: 6) 56,72; 7) 30:36:66; 8) 70 L; 9) 5:5=1:1; 10) 16.
Level 3: 11) 120:150:180:210; 12) 1920,2160,3120; 13) 2:3; 14) 2:3:4; 15) 48.

5.15 Summary

  • Ratios express relative comparison; proportions equate ratios.
  • Properties: scaling, inversion, compounding.
  • Direct, inverse, joint variations model most problems.
  • Partnership shares = capital × time.
  • Alligation is a shortcut for mixtures.
  • Always reduce to smallest whole numbers.