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Ratio and Proportion
Understanding Ratios
A ratio compares two or more quantities by division. Written as $a : b$, it tells you how many times one quantity contains another. Both quantities must be in the same unit.
$$a : b = \frac{a}{b}$$
A ratio expresses the relative size of two quantities using a colon or fraction.
$a$=First quantity (antecedent)(same as $b$)
$b$=Second quantity (consequent)(same as $a$)
$a = b$
→The ratio is $1 : 1$ — both quantities are equal.
$b = 0$
→The ratio is undefined — you cannot divide by zero.
Same Units Required: Rs. 50 and Rs. 30 give $50 : 30$. But Rs. 50 and 30 cents must first be converted to the same unit.
No Units in the Answer: A ratio is unitless — the units cancel out during division.
Order Matters: The ratio of boys to girls ($3 : 2$) is NOT the same as girls to boys ($2 : 3$).
Simplifying a ratio means dividing both terms by their greatest common divisor (GCD) to express the ratio in its smallest whole-number form. A simplified ratio is easier to work with and interpret.
$$\frac{a}{b} = \frac{a \div \text{GCD}}{b \div \text{GCD}}$$
Divide both terms of the ratio by their GCD to get the simplest form.
$\text{GCD}$=Greatest common divisor of $a$ and $b$(unitless)
Fraction Method: Write the ratio as a fraction $\frac{a}{b}$, then simplify the fraction — the simplified numerator and denominator form the ratio.
HCF Shortcut: Find the HCF of both numbers and divide each by it. For $24 : 36$, HCF $= 12$, so the simplified ratio is $2 : 3$.
Three-Term Ratios: For $a : b : c$, find the GCD of all three and divide each term. E.g., $15 : 25 : 35 \to 3 : 5 : 7$.
Comparing ratios helps you determine which of two ratios is larger. Convert both ratios to fractions and cross-multiply to compare without dealing with decimals.
$$a : b \text{ vs } c : d \implies \text{compare } a \times d \text{ and } b \times c$$
Cross-multiply: if $ad > bc$, then $a : b$ is the larger ratio.
$a \times d$=Product of the extremes (first term of each ratio × second term of the other)(unitless)
$b \times c$=Product of the means (second term of first ratio × first term of second)(unitless)
Convert to Common Base: Alternatively, make both ratios have the same second term. If comparing $3 : 4$ and $5 : 7$, convert to $21 : 28$ vs $20 : 28$ — now the first terms are directly comparable.
Decimal Check: For quick estimates, convert to decimals: $3 : 4 = 0.75$ vs $5 : 7 \approx 0.714$.
Proportion: The Equality of Ratios
A proportion states that two ratios are equal. When four quantities form a proportion, the relationship between them is governed by a simple but powerful cross-multiplication rule.
$$\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c$$
In a proportion, the product of the extremes equals the product of the means.
$a, d$=Extremes — the first and last terms of the proportion $a : b = c : d$(unitless)
$b, c$=Means — the middle two terms(unitless)
Finding the Missing Term: If three of four values are known, the fourth is found by cross-multiplication. Given $3 : 5 = x : 20$, solve $3 \times 20 = 5x \implies x = 12$.
Verification Check: To verify a proportion, simply check if $ad = bc$. If $2 : 3 = 8 : 12$, then $2 \times 12 = 24 = 3 \times 8$ ✓
Proportionality Symbol: Written as $a : b :: c : d$, read as "$a$ is to $b$ as $c$ is to $d$".
In continued proportion, the middle term of three quantities is the mean proportional (geometric mean) of the other two. This creates a chain where $a : b = b : c$.
$$a : b = b : c \implies b^2 = a \times c$$
The mean proportional is the square root of the product of the extremes.
$b$=Mean proportional between $a$ and $c$(same as $a$ and $c$)
$a, c$=The extremes of the continued proportion(same as $b$)
Finding the Mean Proportional: $b = \sqrt{a \times c}$. For example, the mean proportional between $4$ and $9$ is $\sqrt{36} = 6$, giving $4 : 6 = 6 : 9$.
Extended Chains: In $a : b = b : c = c : d$, every adjacent pair is in proportion. This means $b^2 = ac$, $c^2 = bd$, and the common ratio is $\frac{b}{a}$.
Business Application: If revenue grew proportionally each year from Rs. 100K to Rs. 400K in 2 years, the middle year's revenue is $\sqrt{100 \times 400} = 200$K.
Types of Proportion
In direct proportion, when one quantity increases, the other increases by the same factor. The ratio between corresponding values is always constant.
$$\frac{x_1}{y_1} = \frac{x_2}{y_2} = k$$
Both quantities maintain a constant ratio $k$, so $y = kx$.
$k$=Constant of proportionality(depends on context)
$x_1, x_2$=Values of the first quantity in two scenarios(any)
$y_1, y_2$=Corresponding values of the second quantity(any)
$k = 1$
→Both quantities are always equal: $x = y$.
Recognition Pattern: "More of A means more of B" at the same rate — e.g., more workers produce more output (assuming equal efficiency).
Unitary Method: Find the value for 1 unit first, then scale up. If 5 notebooks cost Rs. 200, then 1 notebook costs Rs. 40, so 8 notebooks cost $8 \times 40 = 320$.
Graph: A direct proportion plots as a straight line through the origin with slope $k$.
In inverse proportion, when one quantity increases, the other decreases so that their product remains constant. The two quantities move in opposite directions.
$$x_1 \times y_1 = x_2 \times y_2 = k$$
The product of the two quantities is constant, so $y = \frac{k}{x}$.
$k$=Constant product(depends on context)
$x_1, x_2$=Values of the first quantity(any)
$y_1, y_2$=Corresponding values of the second quantity(any)
Recognition Pattern: "More of A means less of B" — e.g., more workers finish a job in fewer days.
The Flip Trick: Convert inverse proportion to direct by flipping one quantity. If $6$ pipes fill a tank in $8$ hours, then $9$ pipes fill it in $\frac{6 \times 8}{9} = \frac{16}{3}$ hours.
Graph: An inverse proportion plots as a hyperbola — as $x$ increases, $y$ decreases but never reaches zero.
Distinguishing from Direct: Ask yourself: does doubling one quantity double the other (direct) or halve it (inverse)?
Proportion in Business Problems
In partnership problems, profits are distributed in proportion to each partner's investment and the duration for which it was invested. The key quantity is capital × time.
$$\text{Share of profit} \propto \text{Investment} \times \text{Time}$$
Each partner's profit share equals their investment-weighted time divided by the total.
$\text{Investment}$=Amount of money a partner puts into the business(currency)
$\text{Time}$=Duration the money is invested(months or years)
Profit Ratio: The ratio of profits equals the ratio of (investment × time). Partner A invests Rs. 50,000 for 6 months, Partner B invests Rs. 30,000 for 10 months — profit ratio is $(50 \times 6) : (30 \times 10) = 300 : 300 = 1 : 1$.
When Times Are Equal: If all partners invest for the same duration, profit is shared in the ratio of investments alone.
When Investments Are Equal: If all partners invest the same amount, profit is shared in the ratio of time periods.
Mid-Year Changes: If a partner joins or leaves mid-year, calculate their investment × time for the actual duration only.
Mixture problems involve combining two or more ingredients in a given ratio to form a new mixture. The key is tracking the quantity of each component individually.
$$\text{Fraction of ingredient} = \frac{\text{its ratio part}}{\text{sum of all parts}}$$
In a mixture with ratio $a : b$, ingredient A makes up $\frac{a}{a+b}$ of the total.
$a$=Parts of the first ingredient(parts)
$b$=Parts of the second ingredient(parts)
$a + b$=Total parts in the mixture(parts)
Finding Component Amounts: If 60 litres of mixture has water:milk in ratio $2 : 3$, water $= 60 \times \frac{2}{5} = 24$ litres and milk $= 60 \times \frac{3}{5} = 36$ litres.
Adding or Removing: When you add more of one component, update the amounts and find the new ratio. Adding 4 litres of water gives water $= 28$, milk $= 36$, ratio $= 28 : 36 = 7 : 9$.
Replacement Method: Removing a fraction of the mixture removes that same fraction of each component. Remove $\frac{1}{4}$ of a $2 : 3$ mixture — you lose $\frac{1}{4}$ of both water and milk.
Rule of Alligation: Used to find the ratio of two ingredients when mixing them to get a target value (like a target price or concentration).
Advanced Ratio Techniques
To divide a quantity $Q$ in the ratio $a : b$, each share is calculated by multiplying $Q$ by the respective fraction of the total parts.
$$\text{First share} = Q \times \frac{a}{a + b}, \quad \text{Second share} = Q \times \frac{b}{a + b}$$
Each party receives a fraction of the total proportional to their ratio share.
$Q$=Total quantity to be divided(any)
$a + b$=Total number of parts in the ratio(parts)
One Unit Method: One part $= \frac{Q}{a + b}$. First person gets $a$ units, second gets $b$ units. For Rs. 5000 in ratio $3 : 7$, one part $= 500$, so shares are Rs. 1500 and Rs. 3500.
Three-Way Split: For ratio $a : b : c$, shares are $Q \times \frac{a}{a+b+c}$, $Q \times \frac{b}{a+b+c}$, and $Q \times \frac{c}{a+b+c}$.
Verification: The shares must always add up to $Q$. If they don't, check your arithmetic.
The compound ratio of two or more ratios is obtained by multiplying the corresponding terms. It combines successive proportional changes into a single ratio.
$$(a : b) \text{ compounded with } (c : d) = ac : bd$$
Multiply corresponding terms of all ratios to get the compound ratio.
$ac$=Product of all first terms(unitless)
$bd$=Product of all second terms(unitless)
Duplicate Ratio: The compound ratio of $a : b$ with itself is $a^2 : b^2$, called the duplicate ratio.
Sub-Duplicate Ratio: The ratio of square roots $\sqrt{a} : \sqrt{b}$ is the sub-duplicate of $a : b$.
Triplicate Ratio: Compounding $a : b$ three times gives $a^3 : b^3$.
Business Use: If price increases by ratio $6 : 5$ and then again by $3 : 2$, the overall change is the compound ratio $(6 \times 3) : (5 \times 2) = 18 : 10 = 9 : 5$, meaning an $80\%$ increase.