Ordered comparison
The ratio of \(a\) to \(b\) is
\[a:b=\frac ab\quad(b\ne0)\]
First term: antecedent. Second: consequent.
4 SSC CGL · Mathematics
Ratio, Proportion & Variation
Handwritten-style notes for SSC CGL Tier I and Tier II.
01MAP IT!
The Big Picture — Graphic Mind Map
Ratio compares, proportion equates two ratios, and variation explains how quantities change together.
RATIO
⇄
PROPORTION
⇄
PROPORTION
1. Build Ratiosorder, units and simplification
2. Compareequivalent and cross products
3. Proportionmeans, extremes and missing term
4. Combinecompound and continued ratios
5. Divideshares, difference and totals
6. Varydirect, inverse, joint and change
\[a:b=\frac ab\qquad a:b::c:d\iff ad=bc\]
02FOUNDATION
The Foundation (Basics) — Core Concepts
Learn ratio as an ordered comparison, then build proportion and variation one rule at a time.
AWhat is a ratio?
Same kind and same units
Convert comparable quantities to one unit before forming a ratio.
\[2\text{ m}:50\text{ cm}=200:50=4:1\]
Order cannot be reversed
\[a:b\ne b:a\ \text{in general}\]
“Boys to girls” and “girls to boys” are reciprocal comparisons.
A ratio is a labelled comparison bridge
Equal-sized objects make the parts visible. If \(6\) blue blocks and \(9\) green blocks are compared, the simplified ratio is \(2:3\), but the total has \(5\) ratio-parts.
\[6:9=2:3\qquad\text{total parts}=2+3=5\]
BSimplifying, scaling and equivalent ratios
Simplest form
Divide all terms by their HCF.
\[24:36=\frac{24}{12}:\frac{36}{12}=2:3\]
Equivalent ratios
\[a:b=ka:kb\quad(k\ne0)\]
Multiplying or dividing every term by the same non-zero number preserves the ratio.
Fractions or decimals in a ratio
Multiply every term by the LCM of denominators or a suitable power of \(10\).
\[\frac12:\frac34:1.25=2:3:5\]
CComparing ratios
Cross-product comparison
For positive denominators:
\[a:b\ ?\ c:d\Longleftrightarrow ad\ ?\ bc\]
Common consequent
Scale both ratios until their second terms match; then compare first terms.
\[2:3=8:12,\quad3:4=9:12\]
As fractions
Ratio order equals fraction order when consequents are positive.
\[\frac23\lt\frac34\Rightarrow2:3\lt3:4\]
Cross-products sit on a comparison balance
Cross multiplication avoids decimals. Compare \(ad\) with \(bc\), but keep the denominator signs positive before using the direction of the inequality.
DProportion — equality of two ratios
Basic proportion
\[a:b::c:d\iff\frac ab=\frac cd\iff ad=bc\]
\(a,d\) are extremes; \(b,c\) are means.
Fourth proportional
If \(a:b=c:x\), then
\[x=\frac{bc}{a}\]
Third and mean proportional
\[a:b=b:c\Rightarrow c=\frac{b^2}{a}\]
\[a:x=x:b\Rightarrow x=\sqrt{ab}\]
| Transformation | From \(a/b=c/d\) | Condition / use |
|---|---|---|
| Invertendo | \(b/a=d/c\) | All denominators non-zero. |
| Alternendo | \(a/c=b/d\) | Useful for regrouping corresponding terms. |
| Componendo | \((a+b)/b=(c+d)/d\) | Add denominator to numerator on both sides. |
| Dividendo | \((a-b)/b=(c-d)/d\) | Subtract denominator from numerator. |
| Componendo–dividendo | \((a+b)/(a-b)=(c+d)/(c-d)\) | Requires non-zero new denominators. |
Proportion is a four-seat table
The outer seats are extremes and the inner seats are means. A valid proportion balances when the product of extremes equals the product of means.
\[\text{extremes product}=\text{means product}\]
ECompound, duplicate and continued ratios
Compound ratio
Multiply corresponding terms.
\[(a:b)\circ(c:d)=ac:bd\]
For many ratios, multiply all antecedents and all consequents.
Duplicate families
\[\text{duplicate}=a^2:b^2,\quad\text{triplicate}=a^3:b^3\]
\[\text{subduplicate}=\sqrt a:\sqrt b\]
Continued ratio
To combine \(A:B=a:b\) and \(B:C=c:d\), equalise \(B\).
\[A:B:C=ac:bc:bd\]
Continued ratio needs one common connector
Scale both ratios until the shared quantity has the same number of parts. Then read all terms in one chain.
FDividing a quantity in a ratio
Total is known
If \(Q\) is divided in \(a:b:c\), one part is
\[k=\frac{Q}{a+b+c}\]
Shares are \(ak,bk,ck\).
Difference is known
If shares in \(a:b\) differ by \(D\):
\[k=\frac{D}{|a-b|}\]
One share is known
If \(a\) parts equal \(S\), then
\[1\text{ part}=\frac Sa\]
The parts method slices one whole
Add ratio terms to count total equal parts. Find the value of one part, then multiply by each term. Never divide the quantity by each ratio term separately.
\[Q\text{ in }2:3:5\Rightarrow\text{one part}=Q/10\]
GDirect, inverse, joint and combined variation
Direct variation
\[y\propto x\iff y=kx\iff\frac{y_1}{x_1}=\frac{y_2}{x_2}\]
Both move in the same direction by the same factor.
Inverse variation
\[y\propto\frac1x\iff xy=k\iff x_1y_1=x_2y_2\]
One rises while the other falls.
Joint variation
\[y\propto xz\iff y=kxz\]
Combined variation
\[y\propto\frac{x^a}{z^b}\iff y=k\frac{x^a}{z^b}\]
| Signal language | Likely model | Quick test |
|---|---|---|
| More–more or less–less | Direct proportion | \(y/x\) remains constant. |
| More–less with fixed product | Inverse proportion | \(xy\) remains constant. |
| Varies as product | Joint variation | \(y/(xz)\) remains constant. |
| Varies directly as one and inversely as another | Combined variation | Build a single constant equation first. |
Watch the arrows, then test the constant
Direction is a clue, not proof. Translate the relation into a constant expression and compare two states.
HRatio after addition, removal or transfer
Original amounts
If \(A:B=a:b\), write
\[A=ak,\qquad B=bk\]
Then apply every stated change to the actual expressions.
Same addition to both
\[\frac{ak+x}{bk+x}=\frac cd\]
Cross multiply and solve for \(k\) or \(x\).
Transfer from \(A\) to \(B\)
\[\frac{ak-x}{bk+x}=\frac cd\]
Total remains unchanged; one decreases as the other increases.
Same subtraction
\[\frac{ak-x}{bk-x}=\frac cd\]
Both quantities must remain meaningful and denominators non-zero.
Different changes
\[\frac{ak+p}{bk+q}=\frac cd\]
Keep signs attached to their specific quantities; do not alter the ratio terms directly.
IUseful ratio identities and applications
Ratio to fraction of total
\[A:B=a:b\Rightarrow\frac{A}{A+B}=\frac{a}{a+b}\]
Ratio to percentage share
\[A\text{ share}=\frac{a}{a+b}\times100\%\]
Products and squares
\[a:b=c:d\Rightarrow ac:bd=a^2:b^2\ \text{when }c:d=a:b\]
| Problem phrase | Fast representation | Core move |
|---|---|---|
| Income : expenditure | \(I:E=a:b\) | Saving is \((a-b)k\), not a ratio term unless converted. |
| Present ages | \(A=ak,B=bk\) | Add/subtract the same number of years, not the same ratio. |
| Two numbers and their sum | \(ak,bk\) | Use \((a+b)k=S\). |
| Two numbers and their difference | \(ak,bk\) | Use \(|a-b|k=D\). |
| Speed and time for fixed distance | \(S\propto1/T\) | Use inverse ratio. |
| Workers and days for fixed work | \(W\propto1/D\) | Use inverse ratio when efficiency is equal. |
03SAVE TIME
Short Tricks & Magic Formulas
Represent unknown amounts as ratio-parts and postpone arithmetic until the final step.
1The \(ak,bk\) master substitution
Sum given
\[A:B=a:b,\ A+B=S\Rightarrow k=\frac S{a+b}\]
Difference given
\[|A-B|=D\Rightarrow k=\frac D{|a-b|}\]
Product given
\[AB=P\Rightarrow k=\sqrt{\frac P{ab}}\]
2Merge two ratios instantly
Shared middle term
\[A:B=a:b,\quad B:C=c:d\]
\[A:B:C=ac:bc:bd\]
LCM method
Find the LCM of the two values assigned to the shared quantity, scale both ratios to that value, then combine.
3Change-in-ratio equation
Addition
\[d(ak+x)=c(bk+x)\]
Transfer
\[d(ak-x)=c(bk+x)\]
Quick invariant
A transfer preserves the total. Equal addition changes the total by twice the addition.
4Option-first checks
The ratio speed dashboard
Before solving: match units, preserve order, count total parts, and decide whether the relation is direct or inverse.
04DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
Wrong options usually reverse the order, misuse total parts, or confuse direct and inverse change.
Trap 1: ratio has no total by itself
From \(A:B=2:3\), you know relative parts, not the actual values. A sum, difference, product or one share is needed to find \(k\).
Trap 2: inverse ratio reverses
For a fixed distance, speeds \(2:3\) correspond to times \(3:2\). Copying the same ratio is the standard distractor.
Trap 3: add years, not ratio parts
After \(t\) years, ages \(3k,5k\) become \(3k+t,5k+t\), not \(3(k+t),5(k+t)\).
05LOCK IT IN
Memory Hooks & Mnemonics
Attach each structural rule to a visual sentence for rapid recall.
“Ratio remembers order”
Antecedent comes first; reversing the words reciprocates the ratio.
“Extremes hug the edge; means meet inside”
In a proportion, the cross-products of extremes and means are equal.
“Direct dances together”
Both quantities rise or fall by the same factor; their quotient stays fixed.
“Inverse moves opposite”
One rises while the other falls; their product stays fixed.
“Slice by parts, not by terms”
Add ratio terms, find one part, then multiply to obtain each share.
“Match the middle to build the chain”
Equalise the shared term before forming a continued ratio.
RatioOrdered comparison
SimplifyDivide by HCF
ProportionCross-products equal
DirectQuotient constant
InverseProduct constant
Change problemsWrite \(ak,bk\) first