Ratio, Proportion and Averages for CSAT

Prerequisites: Percentages for CSAT · Number System

The Hook: Two Grades of Rice

The Core Concept

1. Ratio — the language of “parts”

A ratio $a : b$ compares two quantities of the same kind: $a : b = \dfrac{a}{b}$. Think of it as parts: in $2 : 3$, the first quantity has 2 parts and the second 3 parts of the same size.

Dividing a quantity. Divide ₹7,200 among A, B and C in the ratio $2 : 3 : 4$. Total parts $= 9$; one part $= 7{,}200 \div 9 = 800$. A $= 1{,}600$, B $= 2{,}400$, C $= 3{,}200$. (Check: sum $= 7{,}200$ ✓.)

Combining two ratios. If $A : B = 3 : 4$ and $B : C = 6 : 5$, make B equal in both. $\text{LCM}(4, 6) = 12$: $A : B = 9 : 12$ and $B : C = 12 : 10$, so $A : B : C = 9 : 12 : 10$.

Comparing ratios. Which is larger, $5 : 7$ or $7 : 10$? Cross-multiply: $5 \times 10 = 50$ and $7 \times 7 = 49$. Since $50 > 49$, $5 : 7$ is larger.

2. Proportion — direct and inverse

• $a : b = c : d$ means $ad = bc$ (product of extremes = product of means).
• Direct proportion: if one doubles, the other doubles (cost and quantity).
• Inverse proportion: if one doubles, the other halves (workers and days for a fixed job; speed and time for a fixed distance — see Time, Speed and Work).

3. Ages

Worked example. The ages of A and B are in the ratio $5 : 3$. After 6 years the ratio will be $7 : 5$. Find their present ages.

Let the ages be $5x$ and $3x$:

$$\frac{5x + 6}{3x + 6} = \frac{7}{5} \Rightarrow 25x + 30 = 21x + 42 \Rightarrow x = 3$$

Ages: 15 and 9. Check: after 6 years, $21 : 15 = 7 : 5$ ✓.

Option-testing shortcut: with numbers in the options, test them in the second condition — often faster than algebra.

4. Mixtures and alligation

When two ingredients of “values” $c$ (cheaper) and $d$ (dearer) are mixed to get a mean value $m$:

$$\boxed{\frac{\text{Quantity of cheaper}}{\text{Quantity of dearer}} = \frac{d - m}{m - c}}$$

graph TD
    C["Cheaper: 40"] --> M["Mean: 45"]
    D["Dearer: 55"] --> M
    M --> R1["Cheaper share = 55 - 45 = 10"]
    M --> R2["Dearer share = 45 - 40 = 5"]
    R1 --> R["Ratio 10 : 5 = 2 : 1"]
    R2 --> R

The “values” can be prices, percentages of concentration, speeds or average marks — anything averaged by weight.

Adding one component. A 60-litre mixture has milk and water in the ratio $2 : 1$ (40 L milk, 20 L water). To make it $1 : 2$, milk stays 40 L, so water must become 80 L — add 60 L of water.

5. Partnership

Profit is shared in proportion to capital × time.

Worked example. A invests ₹30,000 for 12 months and B invests ₹40,000 for 6 months. Profit ₹25,000. Ratio $= 30{,}000 \times 12 : 40{,}000 \times 6 = 3{,}60{,}000 : 2{,}40{,}000 = 3 : 2$. A gets $\dfrac{3}{5} \times 25{,}000 = ₹15{,}000$; B gets ₹10,000.

6. Averages

$$\text{Average} = \frac{\text{Sum}}{\text{Number of items}} \qquad \Longleftrightarrow \qquad \text{Sum} = \text{Average} \times \text{Number}$$

Almost every average problem is solved by converting averages to sums, doing the arithmetic on sums, and converting back.

(a) Deviation method. Marks 72, 75, 78, 81, 69. Assume 75; deviations $-3, 0, +3, +6, -6$ sum to 0, so the average is 75.

(b) Replacement. The average weight of 8 people rises by 2.5 kg when a new person replaces one who weighs 65 kg. New person $= 65 + 8 \times 2.5 = $ 85 kg (the newcomer brings the old weight plus the increase for everyone).

(c) Weighted average. Section A: 30 students averaging 60; section B: 20 students averaging 70. Combined average $= \dfrac{30 \times 60 + 20 \times 70}{50} = \dfrac{1{,}800 + 1{,}400}{50} = 64$. Note that 64 is closer to 60, because section A is larger.

(d) New entry. A batsman averages 40 runs over 10 innings. How many must he score in the 11th to raise the average to 42? Needed total $= 11 \times 42 = 462$; current total $= 400$; required score $= $ 62.

(e) Consecutive and evenly spaced numbers. The average equals the middle term, or the mean of the first and last: average of 1 to $n$ is $\dfrac{n + 1}{2}$.

Memory Anchors

• “Parts first, values later” — find the value of one part, then multiply.
• “Cross the X” for alligation — write the two values on top, the mean in the middle, and subtract diagonally.
• “Average out, sum in” — convert every average to a sum before doing anything.
• “Newcomer = old + n × change” for replacement problems (use minus if the average falls).
• Partnership = money × months.

Confusable pair	Rule
Ratio vs fraction of total	In $2 : 3$, the first is $\tfrac{2}{5}$ of the total, not $\tfrac{2}{3}$
Simple average vs weighted average	Weight by group size when groups differ
Average speed	Not the simple average of two speeds for equal distances — use $\dfrac{2xy}{x + y}$

For Prelims

Speed Strategy

• Use the options. In age and number-ratio questions, test options in the condition you have not yet used.
• Work with sums, not averages, in every multi-step average question.
• Alligation over algebra whenever two things are mixed to get a mean.
• Make the common term equal (LCM) before combining ratios — do not add or multiply them.
• What to skip: three-component mixture questions with repeated replacement are rare and slow; leave them for the end.

Common Traps

Practice

Prelims (CSAT)

Quick Revision

Situation	Approach
Divide in $a : b : c$	One part $= \dfrac{\text{Total}}{a + b + c}$
Combine $A : B$ and $B : C$	Make B equal using LCM
Compare ratios	Cross-multiply
Mixture of two	Alligation: $(d - m) : (m - c)$
Adding one component	Anchor on the unchanged component
Partnership	Capital × time
Average	Sum = average × number
Replacement	New = old ± $n$ × change in average
Groups of unequal size	Weighted average

Key Takeaways

Related Topics

• Percentages for CSAT
• Percentages, Ratio and Averages — quick revision
• Percentages, Ratio and Averages — practice questions
• Time, Speed and Work
• Data Interpretation