Ratio and Proportion Concepts

1. Income, Expenditure & Savings

• Formula:

$$\text{Savings}=\text{Income}-\text{Expenditure}$$

If income and expenditure are given in ratios, represent them as multiples of unknowns and form equations using the savings information.

• Example: Incomes of A and B are in the ratio (3:4), expenditures are in the ratio (2:3), and both save ₹2,000. Find their incomes. Solution:

Let incomes be:

$$3x,\ 4x$$

Let expenditures be:

$$2y,\ 3y$$

Since both save ₹2,000:

$$3x-2y=2000 \tag{1}$$ $$4x-3y=2000 \tag{2}$$

Subtract (1) from (2):

$$x-y=0$$ $$x=y$$

Substitute in (1):

$$3x-2x=2000$$ $$x=2000$$

Therefore incomes are:

$$\boxed{₹6000,\ ₹8000}$$

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2. Income, Expenditure & Savings Percentage

• Formula:

$$\text{Savings}=\text{Income}-\text{Expenditure}$$ \text{Savings%}=\frac{\text{Savings}}{\text{Income}}\times100 \text{Expenditure%}=100-\text{Savings%}

• Example: A earns ₹40,000 and saves 25% of his income. Find his monthly expenditure. Solution:

Savings:

$$=25$$

Expenditure:

$$=40000-10000$$ $$\boxed{₹30000}$$

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3. Change in Income and Expenditure

• Formula:

$$S=I-E$$

After changes:

$$S' = I\left(1+\frac{x}{100}\right) -E\left(1+\frac{y}{100}\right)$$

Use when income and expenditure change by different percentages.

• Example: A person’s income is ₹30,000 and expenditure is ₹24,000. Income increases by 20% and expenditure by 10%. Find the percentage change in savings. Solution:

Original savings:

$$30000-24000=₹6000$$

New income:

$$30000(1.2)=₹36000$$

New expenditure:

$$24000(1.1)=₹26400$$

New savings:

$$36000-26400=₹9600$$

Increase in savings:

$$9600-6000=₹3600$$

Percentage increase:

\frac{3600}{6000}\times100 =\boxed{60%}

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4. Proportional Scaling

• Formula:

$$a:b=c:d\Rightarrow ad=bc$$

Fourth proportional:

$$a:b=c:x \Rightarrow x=\frac{bc}{a}$$

Use for direct proportion and missing terms in ratios.

• Example: Find the fourth proportional to 8, 12 and 18. Solution:

$$8:12=18:x$$ $$8x=12\times18$$ $$x=\frac{216}{8}$$ $$\boxed{27}$$

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5. Third Proportional

• Formula: If:

$$a:b=b:x$$

then:

$$\boxed{x=\frac{b^2}{a}}$$

• Example: Find the third proportional to 6 and 12. Solution:

$$6:12=12:x$$ $$6x=144$$ $$\boxed{x=24}$$

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6. Mean Proportional

• Formula: If (x) is the mean proportional between (a) and (b):

$$a:x=x:b$$

Therefore:

$$\boxed{x=\sqrt{ab}}$$

• Example: Find the mean proportional between 4 and 25. Solution:

$$x=\sqrt{4\times25}$$ $$x=\sqrt{100}$$ $$\boxed{10}$$

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7. Direct Proportion

• Formula:

$$y\propto x \Rightarrow \frac{y_1}{x_1}=\frac{y_2}{x_2}$$

Hence:

$$y_2=y_1\frac{x_2}{x_1}$$

Use when both quantities increase or decrease together in the same ratio.

• Example: 5 pens cost ₹75. What is the cost of 8 pens? Solution:

$$\frac{75}{5}=\frac{x}{8}$$ $$x=75\times\frac85$$ $$\boxed{₹120}$$

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8. Inverse Proportion

• Formula:

$$y\propto\frac1x \Rightarrow x_1y_1=x_2y_2$$

Use when one quantity increases while the other decreases proportionally, such as workers and days.

• Example: 8 workers complete a job in 15 days. How many days will 12 workers take, assuming equal efficiency? Solution:

$$8\times15=12\times d$$ $$d=\frac{120}{12}$$ $$\boxed{10\text{ days}}$$

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9. Chain Ratios

• Formula: To combine:

$$A:B=x:y,\qquad B:C=p:q$$

make the values of (B) equal by multiplying the ratios appropriately.

• Example: If (A:B=3:4) and (B:C=5:6), find (A:B:C). Solution:

$$A:B=3:4$$ $$B:C=5:6$$

LCM of 4 and 5 is 20.

$$A:B=15:20$$ $$B:C=20:24$$

Therefore:

$$\boxed{A:B:C=15:20:24}$$

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10. Multiple Chain Ratios

• Formula: Combine consecutive ratios by making every common term equal.
• Example: If (A:B=2:3), (B:C=4:5), and (C:D=10:7), find (A:B:C:D). Solution:

First:

$$A:B=2:3,\quad B:C=4:5$$

Make (B=12):

$$A:B=8:12$$ $$B:C=12:15$$

Thus:

$$A:B:C=8:12:15$$

Now (C:D=10:7). Make (C=30):

$$A:B:C=16:24:30$$ $$C:D=30:21$$

Therefore:

$$\boxed{A:B:C:D=16:24:30:21}$$

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11. Ratio Division of a Quantity

• Formula: If a quantity (Q) is divided in ratio:

$$a:b:c$$

then:

$$\text{Shares}= \frac{a}{a+b+c}Q,\ \frac{b}{a+b+c}Q,\ \frac{c}{a+b+c}Q$$

• Example: Divide ₹4,500 among A, B and C in the ratio (2:3:4). Solution:

Total parts:

$$2+3+4=9$$

A:

$$4500\times\frac29=\boxed{₹1000}$$

B:

$$4500\times\frac39=\boxed{₹1500}$$

C:

$$4500\times\frac49=\boxed{₹2000}$$

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12. Ratio Change After Addition/Subtraction

• Formula: If two quantities are in ratio (a:b), write them as:

$$ax,\ bx$$

Then apply the given addition/subtraction and form the new ratio.

• Example: Two numbers are in the ratio (3:5). If 8 is added to each, their ratio becomes (5:7). Find the numbers. Solution:

Let numbers be:

$$3x,\ 5x$$

After adding 8:

$$\frac{3x+8}{5x+8}=\frac57$$ $$7(3x+8)=5(5x+8)$$ $$21x+56=25x+40$$ $$4x=16$$ $$x=4$$

Numbers:

$$\boxed{12,\ 20}$$

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13. Compound Proportion

• Formula: If a quantity depends directly on some variables and inversely on others:

$$Q\propto \frac{A\times B}{C\times D}$$

Compare two situations using the corresponding ratios.

• Example: 8 workers working 6 hours per day complete a job in 10 days. How many days will 12 workers working 8 hours per day take? Solution:

Work is constant:

$$8\times6\times10 ================ 12\times8\times d$$ $$d=\frac{8\times6\times10}{12\times8}$$ $$\boxed{d=5\text{ days}}$$

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14. Partnership Ratios

• Formula:

$$\text{Profit share}\propto\text{Capital}\times\text{Time}$$

Therefore:

$$P_1:P_2=I_1T_1:I_2T_2$$

• Example: A invests ₹5,000 for 12 months and B invests ₹8,000 for 9 months. Find their profit-sharing ratio. Solution:

$$A:B=(5000\times12):(8000\times9)$$ $$=60000:72000$$ $$\boxed{5:6}$$

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15. Partnership with Change in Capital

• Formula:

$$\text{Profit ratio}=I_1T_1:I_2T_2$$

If capital changes during the year, divide the investment period into separate intervals and calculate:

$$\text{Effective capital} =\sum(\text{Capital}\times\text{Time})$$

• Example: A invests ₹10,000 for 12 months. B invests ₹8,000 for 6 months and then increases it to ₹12,000 for the remaining 6 months. Find the profit ratio. Solution:

A’s effective investment:

$$10000\times12=120000$$

B’s:

$$8000\times6+12000\times6$$ $$=48000+72000=120000$$

Therefore:

$$\boxed{A:B=1:1}$$

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16. Dilution & Mixture Ratio

• Formula: If a mixture contains two components in ratio (a:b):

$$\text{Quantity of first}: \text{Quantity of second}=a:b$$

For replacement/dilution, track the amount of the original component remaining after each operation.

• Example: A vessel contains milk and water in the ratio (3:1). If 20 L of the mixture is taken out and replaced with 20 L water, find the amount of milk removed when the vessel initially contains 80 L. Solution:

Initial milk:

$$80\times\frac34=60\text{ L}$$

Milk fraction:

$$\frac34$$

Milk removed:

$$20\times\frac34=15\text{ L}$$

Milk remaining:

$$60-15=\boxed{45\text{ L}}$$

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Advanced Variants

17. Ratio-Based Income, Expenditure & Savings

• Formula: If:

$$I_A:I_B=a:b,\qquad E_A:E_B=c:d$$

and savings are known, represent:

$$I_A=ax,\ I_B=bx,\quad E_A=cy,\ E_B=dy$$

Then use:

$$I_A-E_A=S_A,\qquad I_B-E_B=S_B$$

• Example: Incomes of A and B are in ratio (4:5), expenditures in ratio (3:4), and their savings are ₹3,000 and ₹2,000 respectively. Find their incomes. Solution:

Let:

$$I_A=4x,\quad I_B=5x$$ $$E_A=3y,\quad E_B=4y$$

Then:

$$4x-3y=3000$$ $$5x-4y=2000$$

Multiply first by 4:

$$16x-12y=12000$$

Second by 3:

$$15x-12y=6000$$

Subtract:

$$x=6000$$

Therefore:

$$I_A=4(6000)=\boxed{₹24000}$$ $$I_B=5(6000)=\boxed{₹30000}$$

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18. Continued Proportion

• Formula: If:

$$a:b=b:c$$

then:

$$b^2=ac$$

This is useful for identifying geometric-progressive relationships.

• Example: Find (x) if (4:x=x:25). Solution:

$$x^2=4\times25$$ $$x^2=100$$

For a positive quantity:

$$\boxed{x=10}$$

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19. Componendo and Dividendo

• Formula: If:

$$\frac ab=\frac cd$$

then:

$$\frac{a+b}{a-b} =============== \frac{c+d}{c-d}$$

Use to simplify ratio equations involving sums and differences.

• Example: If (a:b=3:2), find ((a+b):(a-b)). Solution:

Let:

$$a=3x,\quad b=2x$$

Then:

$$a+b=5x$$ $$a-b=x$$

Therefore:

$$\boxed{(a+b):(a-b)=5:1}$$

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20. Direct and Inverse Proportion Combined

• Formula:

$$Q\propto\frac{A\times B}{C}$$

Therefore:

$$\frac{Q_1}{Q_2} =============== \frac{A_1}{A_2} \times \frac{B_1}{B_2} \times \frac{C_2}{C_1}$$

• Example: The time required for a job is directly proportional to the amount of work and inversely proportional to the number of workers. If work doubles and workers increase by 50%, how does the time change? Solution:

$$T\propto\frac{W}{M}$$

Thus:

$$\frac{T_2}{T_1} =============== \frac{2}{1}\times\frac{1}{1.5}$$ $$=\frac43$$

Therefore time becomes:

$$\boxed{\frac43\text{ times the original}}$$

or a:

\boxed{33\frac13%\text{ increase}}