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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.
Application Strategy
Worked Examples
Problem Statement
Ali invests Rs. 60,000 in a business for 8 months and Sara invests Rs. 45,000 for 10 months. If the total profit is Rs. 78,000, how much does each person receive?
partnershipratio
Strategic Toolkit
$$\text{Profit ratio} = (I_1 \times T_1) : (I_2 \times T_2)$$
Step 1
Calculate investment × time for each partner
Ali's share: $60{,}000 \times 8 = 480{,}000$. Sara's share: $45{,}000 \times 10 = 450{,}000$.
$$\text{Ali} = 480{,}000, \quad \text{Sara} = 450{,}000$$
Step 2
Find the profit-sharing ratio
The ratio is $480{,}000 : 450{,}000$. Divide both by their GCD (30,000) to simplify.
$$480 : 450 = 16 : 15$$
Step 3
Divide the profit in this ratio
Total parts $= 16 + 15 = 31$. Ali gets $\frac{16}{31}$ of Rs. 78,000 and Sara gets $\frac{15}{31}$.
$$\text{Ali} = \frac{16}{31} \times 78{,}000 = 40{,}258 \approx \text{Rs. } 40{,}258.06$$
Problem Statement
A container holds 80 litres of a mixture of milk and water in the ratio $5 : 3$. If 16 litres of the mixture is removed and replaced with water, what is the new ratio of milk to water?
mixture_problemsratio
Strategic Toolkit
$$\text{Component amount} = \text{Total} \times \frac{\text{its parts}}{\text{total parts}}$$
Step 1
Find the initial amounts of milk and water
Total parts $= 5 + 3 = 8$. Milk $= 80 \times \frac{5}{8} = 50$ litres. Water $= 80 \times \frac{3}{8} = 30$ litres.
$$\text{Milk} = 50 \text{ L}, \quad \text{Water} = 30 \text{ L}$$
Step 2
Calculate how much of each is removed
Removing 16 litres removes $\frac{16}{80} = \frac{1}{5}$ of each component. Milk removed $= 50 \times \frac{1}{5} = 10$ L. Water removed $= 30 \times \frac{1}{5} = 6$ L.
$$\text{Remaining: Milk} = 50 - 10 = 40 \text{ L}, \quad \text{Water} = 30 - 6 = 24 \text{ L}$$
Step 3
Add replacement water and find the new ratio
After adding 16 litres of water: Milk $= 40$ L, Water $= 24 + 16 = 40$ L.
$$\text{New ratio} = 40 : 40 = 1 : 1$$
Problem Statement
A team of 12 workers can complete a project in 18 days. How many days will it take 8 workers to complete the same project, assuming all workers work at the same rate?
inverse_proportion
Strategic Toolkit
$$W_1 \times D_1 = W_2 \times D_2$$
Step 1
Identify this as an inverse proportion problem
More workers means fewer days — this is inverse proportion. The total work (worker-days) is constant.
Step 2
Set up the equation using the constant product
The total work is $12 \times 18 = 216$ worker-days. This must equal the new product.
$$12 \times 18 = 8 \times D_2$$
Step 3
Solve for the unknown
$D_2 = \frac{12 \times 18}{8} = \frac{216}{8} = 27$ days.
$$D_2 = 27 \text{ days}$$
Problem Statement
Three numbers are in continued proportion. The first number is 9 and the third is 25. Find the middle number and verify the proportion.
continued_proportionmean_proportional
Strategic Toolkit
$$b = \sqrt{a \times c}$$
Step 1
Apply the mean proportional formula
For continued proportion $a : b = b : c$, the middle number $b = \sqrt{a \times c}$.
$$b = \sqrt{9 \times 25} = \sqrt{225} = 15$$
Step 2
Verify the proportion
Check: $\frac{9}{15} = \frac{3}{5}$ and $\frac{15}{25} = \frac{3}{5}$. Both ratios are equal, confirming continued proportion.
$$\frac{9}{15} = \frac{15}{25} = \frac{3}{5} \checkmark$$
Step 3
Verify using the cross-multiplication property
Check $b^2 = ac$: $15^2 = 225$ and $9 \times 25 = 225$. Both are equal.
$$b^2 = 225 = a \times c \checkmark$$
Problem Statement
The price of a product first increases in the ratio $5 : 4$ and then decreases in the ratio $6 : 7$. What is the overall ratio of the original price to the final price?
compound_ratiodirect_proportion
Strategic Toolkit
$$\text{Compound ratio} = (5 \times 6) : (4 \times 7)$$
Step 1
Write both changes as ratios from original to new
First change: price goes from $4$ to $5$ (ratio $5 : 4$ means new : old). Second change: price goes from $7$ to $6$ (ratio $6 : 7$ means new : old).
Step 2
Compute the compound ratio
Multiply corresponding terms: first terms (new prices) and second terms (old prices) separately.
$$(5 \times 6) : (4 \times 7) = 30 : 28 = 15 : 14$$
Step 3
Interpret the result
The overall ratio of final price to original price is $15 : 14$. This means the final price is $\frac{15}{14}$ times the original — a net increase of $\frac{1}{14} \approx 7.14\%$.
$$\text{Net change} = \frac{15 - 14}{14} = \frac{1}{14} \approx 7.14\% \text{ increase}$$
Memorize formulas
Cross-Multiplication
Solve any proportion or compare two ratios
Formula
$\frac{a}{b} = \frac{c}{d} \implies ad = bc$
Mean Proportional
Find the middle term in continued proportion $a : b : c$
Formula
$b = \sqrt{a \times c}$
Direct Proportion
Same rate: more of one means more of the other
Formula
$\frac{x_1}{y_1} = \frac{x_2}{y_2}$
Inverse Proportion
Fixed total: more of one means less of the other
Formula
$x_1 \times y_1 = x_2 \times y_2$
Partnership Profit Ratio
Distribute profits based on investment and time
Formula
$\text{Profit ratio} = I_1 T_1 : I_2 T_2 : \cdots$
Component in a Mixture
Find the amount of one ingredient in a ratio mixture
Formula
$\text{Amount} = \text{Total} \times \frac{\text{its parts}}{\text{total parts}}$
Compound Ratio
Combine successive proportional changes into one ratio
Formula
$(a : b) \circ (c : d) = ac : bd$
Review
1
A recipe requires flour, sugar, and butter in the ratio 5 : 2 : 1. If a baker uses 300 grams of flour, how many grams of butter are needed?
Ans
60 g
Review
Tap to reveal
Set up the proportion
The ratio of flour to butter is 5 : 1, so $\frac{\text{butter}}{\text{flour}} = \frac{1}{5}$.
$$\frac{b}{300} = \frac{1}{5}$$
Solve for butter
Cross-multiply to find the butter quantity.
$$b = 300 \times \frac{1}{5} = 60 \text{ g}$$
2
Three partners — Ali, Sara, and Tahir — invest Rs. 50,000, Rs. 30,000, and Rs. 20,000 respectively in a business for 12, 8, and 6 months. If the total profit is Rs. 15,840, how much does Sara receive?
Ans
Rs. 3,960
Review
Tap to reveal
Calculate each partner's investment-time product
Multiply each investment by the number of months to get the effective capital contribution.
$$\text{Ali} = 50000 \times 12 = 600000, \quad \text{Sara} = 30000 \times 8 = 240000, \quad \text{Tahir} = 20000 \times 6 = 120000$$
Determine the profit-sharing ratio
Simplify the ratio of investment-time products to get the profit distribution ratio.
$$5 : 2 : 1 \quad (\text{total } 8 \text{ parts})$$
Compute Sara's share
Apply the component formula: Sara's share = Total profit $\times$ (Sara's parts / total parts).
$$\frac{2}{8} \times 15840 = \text{Rs. } 3960$$
3
If 6 machines can produce 240 units in 5 hours, how many hours would it take 10 machines to produce 400 units?
Ans
5 hours
Review
Tap to reveal
Find the production rate per machine
Calculate how many units one machine produces in one hour.
$$\text{Rate} = \frac{240}{6 \times 5} = 8 \text{ units/machine/hour}$$
Set up the new condition
With 10 machines producing 80 units per hour, find the time needed for 400 units.
$$t = \frac{400}{10 \times 8} = 5 \text{ hours}$$
4
The ratio of the ages of a father and son is 7 : 3. In 12 years, the ratio will be 5 : 3. What is the son's current age?
Ans
12 years
Review
Tap to reveal
Express current ages using the ratio
Let the current ages be $7x$ (father) and $3x$ (son).
$$\text{Father} = 7x, \quad \text{Son} = 3x$$
Set up the proportion for future ages
In 12 years, the ratio becomes $5 : 3$, so cross-multiply to form an equation in $x$.
$$\frac{7x + 12}{3x + 12} = \frac{5}{3}$$
Solve for the son's age
Cross-multiply and solve for $x$, then substitute to find the son's current age.
$$21x + 36 = 15x + 60 \implies 6x = 24 \implies x = 4 \implies \text{Son} = 12 \text{ years}$$
5
If $\frac{a}{b} = \frac{3}{7}$ and $\frac{b}{c} = \frac{14}{5}$, what is the compound ratio $a : c$?
Ans
6 : 5
Review
Tap to reveal
Multiply the given ratios
Chain the ratios: $\frac{a}{c} = \frac{a}{b} \times \frac{b}{c}$.
$$\frac{a}{c} = \frac{3}{7} \times \frac{14}{5} = \frac{42}{35}$$
Simplify the result
Reduce the fraction to its simplest form to get the compound ratio.
$$a : c = 6 : 5$$
6
A container has a mixture of milk and water in the ratio 3 : 2. If 10 litres of the mixture is removed and replaced with 10 litres of milk, the new ratio of milk to water becomes 2 : 1. How many litres were in the container originally?
Ans
60 litres
Review
Tap to reveal
Express quantities before and after replacement
Let total = $5x$ litres. Removing 10 litres removes 6 litres of milk and 4 litres of water. Adding 10 litres of milk gives new quantities.
$$\text{Milk} = 3x - 6 + 10 = 3x + 4, \quad \text{Water} = 2x - 4$$
Set up the new ratio equation
The new ratio of milk to water is 2 : 1, so cross-multiply.
$$\frac{3x + 4}{2x - 4} = 2 \implies 3x + 4 = 4x - 8 \implies x = 12$$
Find the original total
Substitute $x = 12$ to get the original volume.
$$5x = 60 \text{ litres}$$
7
If 8 workers complete a project in 15 days, how many days would 12 workers take to complete the same project?
Ans
10 days
Review
Tap to reveal
Identify the proportion type
More workers means fewer days — this is an inverse proportion.
$$x_1 \times y_1 = x_2 \times y_2$$
Solve for the unknown
Substitute the known values and solve for the number of days.
$$8 \times 15 = 12 \times d \implies d = 10 \text{ days}$$
8
If $2, x, 8$ are in continued proportion, what is the value of $x$?
Ans
4
Review
Tap to reveal
Apply the continued proportion property
For three numbers $a, b, c$ in continued proportion: $\frac{a}{b} = \frac{b}{c}$, giving $b^2 = ac$.
$$x^2 = 2 \times 8 = 16$$
Solve for the mean proportional
Using $b = \sqrt{ac}$, find the positive value of $x$.
$$x = \sqrt{16} = 4$$
9
A shopkeeper mixes two types of rice costing Rs. 40/kg and Rs. 60/kg in the ratio 3 : 2. He sells the mixture at Rs. 54/kg. What is his profit percentage on the mixture?
Ans
12.5%
Review
Tap to reveal
Find the cost price of the mixture per kg
Use the component formula: cost per kg = (sum of each component's cost $\times$ its parts) / total parts.
$$\text{CP} = \frac{3(40) + 2(60)}{3 + 2} = \frac{240}{5} = 48 \text{ Rs/kg}$$
Calculate the profit percentage
Profit = SP − CP. Profit percentage = (Profit / CP) $\times$ 100.
$$\text{Profit\%} = \frac{54 - 48}{48} \times 100 = 12.5\%$$
10
If $\frac{x + 3}{x - 2} = \frac{7}{3}$, what is the value of $x$?
Ans
$\frac{23}{4}$
Review
Tap to reveal
Cross-multiply to eliminate fractions
Apply $\frac{a}{b} = \frac{c}{d} \implies ad = bc$ to form a linear equation in $x$.
$$3(x + 3) = 7(x - 2)$$
Solve for $x$
Expand and collect like terms.
$$3x + 9 = 7x - 14 \implies 4x = 23 \implies x = \frac{23}{4}$$
11
A car travels 240 km in 4 hours. At the same speed, how far will it travel in 7 hours?
Ans
420 km
Review
Tap to reveal
Use direct proportion
Distance is directly proportional to time at constant speed.
$$\frac{d}{7} = \frac{240}{4} = 60$$
Calculate the distance
Multiply the constant speed by the new time.
$$d = 60 \times 7 = 420 \text{ km}$$
12
Two numbers are in the ratio 4 : 7. If each number is increased by 8, the ratio becomes 2 : 3. What is the smaller number?
Ans
16
Review
Tap to reveal
Express the numbers in terms of a common variable
Let the two numbers be $4x$ and $7x$, and set up the proportion after adding 8 to each.
$$\frac{4x + 8}{7x + 8} = \frac{2}{3}$$
Cross-multiply and solve
Apply cross-multiplication to solve for $x$.
$$3(4x + 8) = 2(7x + 8) \implies 12x + 24 = 14x + 16 \implies x = 4$$
Find the smaller number
Substitute $x = 4$ to get the smaller original number.
$$4x = 4(4) = 16$$
Question 1 of 12
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1
A recipe requires flour, sugar, and butter in the ratio 5 : 2 : 1. If a baker uses 300 grams of flour, how many grams of butter are needed?