Ratio and Proportion: quick revision, 15 bites

Ratio Compares, Proportion Equates

Two ideas power half of arithmetic.

Ratio is a **comparison**; proportion is an **equality of two ratios**. Every proportion problem ultimately reduces to a ratio problem.

Takeaway. Master ratios first — proportion is just two ratios set equal.

The Product of Extremes = Product of Means

One cross-multiplication solves most proportions.

In any proportion $a:b :: c:d$, the product of the **extremes** equals the product of the **means**. This single identity is the engine behind almost every proportion problem.

Takeaway. Cross-multiply whenever a proportion has one unknown.

Ratio Has No Units

Ratios are pure numbers — no units.

Takeaway. Convert to the same unit first; then only multiply/divide — never add/subtract.

The 4-Step Method for Any Ratio Problem

Four steps crack any ratio question.

Takeaway. The multiplier $k$ converts abstract ratios into solvable equations.

Dividing a Quantity in a Given Ratio

Split any amount using one formula.

For ratio $a:b$, the total is split into $a+b$ equal parts. Each part $= \frac{\text{Total}}{a+b}$.

Takeaway. Total parts = sum of ratio terms; each share = its term × one part.

Direct vs Inverse Proportion

Two proportions — opposite behaviours.

Takeaway. Direct: ratio stays equal. Inverse: product stays equal.

Never Add the Same Number to Both Terms

Adding to both terms breaks the ratio.

Takeaway. Ratio is preserved by multiplication/division, never by addition/subtraction.

Compounding Two Ratios — LCM Trick

LCM links two ratios in seconds.

Takeaway. LCM of the common term is the fastest bridge between two ratios.

Fourth, Third and Mean Proportional

Three named proportions, one pattern.

Takeaway. All three come from $a:b = c:d$ — just plug in the knowns.

Partnership: Profit Splits by Capital × Time

Business partners split profit by ratio.

In a partnership, profit is shared in the ratio of **capital × time invested** — not just capital.

Takeaway. Partnership profit ratio = (capital × time) ratio.

Mixtures & Alligations Use Ratios

Mixing liquids is a ratio game.

If milk:water = $3:2$ in 40 L, milk = $\frac{3}{5} \times 40 = 24$ L, water = 16 L.

Takeaway. Every mixture question is a ratio split in disguise.

SSC CGL: Use the Multiplier, Skip the Algebra

Let $k$ do the heavy lifting.

Takeaway. One multiplier $k$ turns ratio problems into simple linear equations.

Remember: Extremes × Extremes = Means × Means

Extremes hug the outside.

In $a:b :: c:d$, the **extremes** ($a, d$) are on the outside; the **means** ($b, c$) are in the middle. Their products are equal.

Takeaway. Outside × Outside = Inside × Inside.

Inverse Proportion: Don't Use the Direct Formula

Inverse means multiply, not divide.

Takeaway. Direct → ratio equal. Inverse → product equal.

Convert Ratios to Fractions for Quick Comparison

Fractions beat ratios in comparisons.

Takeaway. Cross-multiplication compares ratios without decimals.