Ratio

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Ratio — Quick Facts

Definition: A ratio $a:b$ compares two quantities. It answers: “how many times larger is $a$ than $b$?”

Expressing Ratios:

• $a:b$ (colon notation)
• $a/b$ (fraction)
• “$a$ to $b$” (words)

Equivalent Ratios: $\frac{a}{b} = \frac{ka}{kb}$ for any non-zero $k$

Types of Ratios:

• Part-to-part: $A:B$ (comparing two parts of the same whole)
• Part-to-whole: $A:(A+B)$ (comparing one part to the total)
• Compound ratio: $a:b$ combined with $c:d$ gives $ac:bd$

Dividing a Quantity by Ratio:

If $x:y:z = a:b:c$ and total is $T$: $$x = \frac{a}{a+b+c} \times T, \quad y = \frac{b}{a+b+c} \times T, \quad z = \frac{c}{a+b+c} \times T$$

⚡ CAT Exam Tip: When a ratio changes, set up equations. “If the ratio of boys to girls changes from 3:2 to 4:3” means something was added or removed — write the change explicitly.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Ratio — Study Guide

Worked Example 1:

Divide ₹540 in the ratio 2:3:4.

Total parts = $2 + 3 + 4 = 9$ First share = $\frac{2}{9} \times 540 = ₹120$ Second share = $\frac{3}{9} \times 540 = ₹180$ Third share = $\frac{4}{9} \times 540 = ₹240$

Worked Example 2:

The ratio of incomes of two persons is 5:3 and their expenditures are in the ratio 7:4. If each saves ₹2000, find their incomes.

Let incomes be $5x$ and $3x$. Expenditures be $7y$ and $4y$.

$5x - 7y = 2000$ $3x - 4y = 2000$

Multiply second by 5, first by 3: $15x - 21y = 6000$ $15x - 20y = 10000$

Subtract: $-y = -4000 \Rightarrow y = 4000$ Then $x = 4000$

Incomes: $5 \times 4000 = ₹20,000$ and $3 \times 4000 = ₹12,000$

Continued Ratio:

Express $a:b:c$ as $a:b$ and $b:c$. Given $a:b = 3:4$ and $b:c = 5:6$, find $a:b:c$.

Make $b$ common: LCM of 4 and 5 = 20

$a:b = 3:4 = 15:20$ $b:c = 5:6 = 20:24$

Therefore $a:b:c = 15:20:24$

Ratio of Areas: If two similar shapes have sides in ratio $a:b$, their areas are in ratio $a^2:b^2$.

If two similar shapes have volumes in ratio $a^3:b^3$, their linear ratio is $\sqrt[3]{a^3}:\sqrt[3]{b^3} = a:b$.

⚡ Common Student Mistake: Forgetting that in ratio problems, all parts must be expressed in the same units before forming the ratio.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Ratio — Comprehensive Notes

Inverse Ratio:

If $a:b$ is the ratio, the inverse ratio is $b:a$. The ratio $a:b$ is inverse to $c:d$ if $a \times b = c \times d$ or $a/b = d/c$.

Example: If $a:b = 3:4$, the inverse ratio is $4:3$.

Proportion:

Four quantities $a, b, c, d$ are in proportion if $a:b = c:d$ or $a/b = c/d$.

• Mean proportional: $x$ is the mean proportional between $a$ and $b$ if $a:x = x:b$, so $x^2 = ab$, thus $x = \sqrt{ab}$
• Third proportional: $c$ is the third proportional to $a$ and $b$ if $a:b = b:c$
• Fourth proportional: $d$ is the fourth proportional to $a, b, c$ if $a:b = c:d$

Example: Find the fourth proportional to 3, 5, and 9.

$\frac{3}{5} = \frac{9}{d} \Rightarrow 3d = 45 \Rightarrow d = 15$

Changing Ratios:

Example: The ratio of boys to girls in a school was 4:5. After 20 new boys joined, the ratio became 6:5. Find the original numbers.

Original: boys $= 4x$, girls $= 5x$

After adding 20 boys: $\frac{4x + 20}{5x} = \frac{6}{5}$ Cross multiply: $5(4x + 20) = 6(5x)$ $20x + 100 = 30x$ $100 = 10x \Rightarrow x = 10$

Original boys = $40$, original girls = $50$

Check: After adding 20 boys → 60 boys, 50 girls, ratio = 6:5 ✓

Business-Related Ratios:

Profit and Loss Ratios:

• Cost Price (CP) and Selling Price (SP)
• Profit = SP - CP
• Loss = CP - SP
• Profit% = $\frac{\text{Profit}}{\text{CP}} \times 100$
• Loss% = $\frac{\text{Loss}}{\text{CP}} \times 100$

Example: If profit on an article is 20% of the cost price and another article is sold at the same profit percentage, find ratio of profits given their costs are in ratio 3:5.

Profit on first = $\frac{20}{100} \times 3x = 0.6x$ Profit on second = $\frac{20}{100} \times 5x = x$

Ratio = $0.6x : x = 3:5$

Alligation:

To find the ratio in which two ingredients at prices $p_1$ and $p_2$ must be mixed to get a mixture at price $p$:

$$\text{Ratio} = \left|\frac{p_2 - p}{p - p_1}\right|$$

Example: Mix 40/kg rice and 60/kg rice to get 50/kg mixture.

$\text{Ratio} = \frac{60 - 50}{50 - 40} = \frac{10}{10} = 1:1$

JAMB Pattern Analysis (CAT 2015-2024):

• 2015: Basic division in given ratio
• 2017: Continued ratio
• 2019: Income-expenditure ratio problems
• 2021: Changing ratio problems
• 2023: Alligation and mixture ratio
• 2024: Ratio with percentage profit/loss

⚡ Exam Strategy: When a problem mentions “ratio changes,” always write down the original and final ratios explicitly and set up an equation. The key is identifying what changed (added, removed, or transferred) between the two ratios.