Ratio & Proportion

🧠

Quick recall

• Make the common term equal before combining ratios.
• For ages, add or subtract years from every person's age.

On this page

Formula Cards

1. Proportion and cross multiplication

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

2. Divide Q in ratio a:b

$$Q_1=\frac{aQ}{a+b},\qquad Q_2=\frac{bQ}{a+b}$$

3. Divide Q in ratio a:b:c

$$Q_i=\frac{\text{corresponding ratio term}}{a+b+c}\,Q$$

4. Duplicate and triplicate ratios

$$a^2:b^2,\qquad a^3:b^3$$

5. Sub-duplicate ratio

$$\sqrt{a}:\sqrt{b}$$

For positive ratio terms.

6. Compound ratio

$$(a:b)\times(c:d)=ac:bd$$

7. Mean proportional

$$a:b=b:c\quad\Rightarrow\quad b=\sqrt{ac}$$

For positive quantities.

8. Third proportional

$$c=\frac{b^2}{a}$$

9. Fourth proportional

$$d=\frac{bc}{a}$$

10. Componendo-dividendo

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

All displayed denominators must be nonzero.

Direct Proportion

1. Rule

$$x=ky,\qquad\frac{x_1}{y_1}=\frac{x_2}{y_2}$$

A constant ratio means both quantities change by the same factor.

Worked examples

Example 1

Question: Five apples cost Rs. 100. Find the cost of eight apples.

Step 1: Find cost per apple.

$$100\div5=20$$

Step 2: Multiply by eight.

$$8\times20=160$$

Answer: Rs. 160.

Example 2

Question: A car travels 150 km using 10 L of fuel. How far on 25 L at the same consumption rate?

Step 1: Find distance per liter.

$$150\div10=15\text{ km/L}$$

Step 2: Multiply by 25 L.

$$15\times25=375\text{ km}$$

Answer: 375 km.

Inverse Proportion

1. Rule

$$xy=k,\qquad x_1y_1=x_2y_2$$

One quantity increases while the other decreases by the reciprocal factor. For a fixed distance, doubling speed halves travel time.

Worked example

Example 1

Question: Five equally efficient workers finish in 12 days. How long for ten workers?

Step 1: Keep worker-days constant.

$$5\times12=10D$$

Step 2: Solve.

$$D=6$$

Answer: 6 days.

Compound Ratio & Multiple Ratios

1. Chain through a common term

$$A:B=2:3,\quad B:C=4:5\quad\Rightarrow\quad A:B:C=8:12:15$$

Scale both ratios until B is 12.

2. Inverse ratio

$$\frac{1}{a}:\frac{1}{b}=b:a$$

Example: inverse of 3:5 is 5:3.

3. Second chain example

$$A:B=3:4,\quad B:C=2:5\quad\Rightarrow\quad A:B:C=3:4:10$$

Equivalent unsimplified form: 6:8:20.

4. Compound ratio example

$$(2:3)\times(5:7)=10:21$$

Multiply corresponding terms.

Partnership Problems (profit sharing)

1. Same investment time

$$\text{Profit}_A:\text{Profit}_B=C_A:C_B$$

2. Different investment times

$$\text{Profit}_A:\text{Profit}_B=C_AT_A:C_BT_B$$

Worked examples

Example 1

Question: A invests Rs. 4,000 for 12 months; B invests Rs. 6,000 for 8 months. Find the profit ratio.

Step 1: Calculate capital × time.

$$4000(12)=48000,\qquad6000(8)=48000$$

Step 2: Simplify the ratio.

$$48000:48000=1:1$$

Answer: 1:1.

Example 2

Question: A, B, C invest Rs. 2,000, 3,000, 5,000 for the same year. Split profit of Rs. 10,000.

Step 1: Find the capital ratio and total units.

$$2:3:5,\qquad2+3+5=10$$

Step 2: Multiply total profit by each share fraction.

$$A=2000,\qquad B=3000,\qquad C=5000$$

Answer: A Rs. 2,000; B Rs. 3,000; C Rs. 5,000.

Age-based Ratio Problems

Set present ages as the given ratio terms multiplied by one unknown. Add or subtract the same number of years from both ages.

Worked example

Example 1

Question: Present ages are in ratio 3:4. After five years, the ratio is 4:5. Find present ages.

Step 1: Let the ages be 3k and 4k.

$$\frac{3k+5}{4k+5}=\frac{4}{5}$$

Step 2: Cross-multiply.

$$15k+25=16k+20$$

Step 3: Solve and recover both ages.

$$k=5,\qquad3k=15,\qquad4k=20$$

Answer: 15 and 20 years.

Mixed Direct + Inverse (wages-type)

1. Equal efficiency and work comparison

$$\frac{M_1D_1H_1}{W_1}=\frac{M_2D_2H_2}{W_2}$$

M = workers; D = days; H = hours per day; W = work. Wages are proportional to worker-hours only if the hourly pay rate is the same.

Worked example

Example 1

Question: Six workers earn Rs. 9,600 working 8 h/day for 10 days. What do eight workers earn working 6 h/day for 12 days, at the same pay per worker-hour?

Step 1: Find original and new worker-hours.

$$6(8)(10)=480,\qquad8(6)(12)=576$$

Step 2: Scale earnings by the worker-hour ratio.

$$9600\times\frac{576}{480}=11520$$

Answer: Rs. 11,520.

Ratio ↔ Percentage Conversion

1. Shares of the total

$$\text{First share}=\frac{a}{a+b}\times100\%,\qquad\text{Second share}=\frac{b}{a+b}\times100\%$$

2. Example: ratio 2:3

$$\frac{2}{5}\times100=40\%,\qquad\frac{3}{5}\times100=60\%$$

Worked Examples

Example 1

Question: Divide 300 in the ratio 2:3.

Step 1: Find total ratio units.

$$2+3=5$$

Step 2: Calculate each share.

$$300\times\frac{2}{5}=120,\qquad300\times\frac{3}{5}=180$$

Answer: 120 and 180.

Example 2

Question: Divide 900 in the ratio 2:3:4.

Step 1: Find the value of one ratio unit.

$$\frac{900}{2+3+4}=100$$

Step 2: Multiply by each ratio term.

$$2(100)=200,\quad3(100)=300,\quad4(100)=400$$

Answer: 200, 300, 400.

Example 3

Question: Five workers finish a job in 12 days. How long for 10 equally efficient workers?

Step 1: Keep worker-days constant.

$$5\times12=10D$$

Step 2: Solve.

$$D=6$$

Answer: 6 days.

Example 4

Question: a:b = 3:4 and b:c = 5:6. Find a:b:c.

Step 1: Match b at 20.

$$a:b=15:20,\qquad b:c=20:24$$

Step 2: Combine.

$$a:b:c=15:20:24$$

Answer: 15:20:24.

Example 5

Question: Find the mean proportional between 4 and 9.

Step 1: Use the continued-proportion relation.

$$b^2=4\times9$$

Step 2: Take the positive square root.

$$b=\sqrt{36}=6$$

Answer: 6.

Example 6

Question: Find the third proportional to 3 and 6.

Step 1: Set up continued proportion.

$$3:6=6:c$$

Step 2: Cross-multiply and solve.

$$c=\frac{6^2}{3}=12$$

Answer: 12.

Example 7

Question: Find the fourth proportional to 2, 3, 8.

Step 1: Set up the proportion.

$$2:3=8:d$$

Step 2: Cross-multiply.

$$d=\frac{3\times8}{2}=12$$

Answer: 12.

Example 8

Question: If a/b = 3/5, find (a+b)/(a−b).

Step 1: Apply componendo-dividendo.

$$\frac{a+b}{a-b}=\frac{3+5}{3-5}$$

Step 2: Simplify.

$$\frac{8}{-2}=-4$$

Answer: −4.

Drill

Try each question before opening its answer.

Question 1

Divide 500 in ratio 3:2.

• Show answer

  300, 200

Question 2

If a:b=2:5, and a=14, find b.

• Show answer

  35

Question 3

Divide 720 among A,B,C in ratio 2:3:4.

• Show answer

  160, 240, 320

Question 4

If 8 men build a wall in 10 days, how many men needed for 5 days (inverse)?

• Show answer

  16 men

Question 5

Find the ratio equivalent to 4:6 in simplest form.

• Show answer

  2:3

Question 6

If x:y=3:4 and y:z=2:5, find x:y:z.

• Show answer

  6:8:20 = 3:4:10

Question 7

Two numbers are in ratio 5:7. Their sum is 144. Find the numbers.

• Show answer

  60, 84

Question 8

A sum is divided among A and B in ratio 3:5. If A gets 300, find total sum.

• Show answer

  800

Question 9

Speed and time are inversely proportional for a fixed distance. If speed doubles, what happens to time?

• Show answer

  It halves.

Question 10

If 3 pens cost as much as 5 pencils, find ratio of cost of 1 pen to 1 pencil.

• Show answer

  5:3

Question 11

Find the duplicate ratio of 3:5.

• Show answer

  9:25

Question 12

Find the sub-duplicate ratio of 16:25.

• Show answer

  4:5

Question 13

Compound the ratios 3:4 and 8:9.

• Show answer

  24:36 = 2:3

Question 14

A and B invest Rs. 5000 and Rs. 7000 for a year. Profit is Rs. 6000. Find B's share.

• Show answer

  Rs. 3500

Question 15

Ages of P and Q are in ratio 5:6. Six years ago the ratio was 4:5. Find present ages.

• Show answer

  30, 36

Question 16

Find the inverse ratio of 7:9.

• Show answer

  9:7

Question 17

8 workers finish a job in 15 days. How many workers for 10 days (inverse)?

• Show answer

  12 workers

Question 18

If A:B=5:6 and B:C=2:3, find A:B:C.

• Show answer

  10:12:18 = 5:6:9

Question 19

6 men working 8 hrs/day finish a job in 10 days. In how many days will 8 men working 6 hrs/day finish it?

• Show answer

  10 days

  $

  6(10)(8)=8D(6)quadRightarrowquad D=10

  $

Question 20

Convert ratio 3:7 to percentages of the total.

• Show answer

  30%, 70%

Problem-solving template (copy-paste)

Problem

• Statement:
• Given:
• Asked:

Solution

• Steps:
• Final answer:

Mistake / learning

• What went wrong:
• Shortcut / rule:

Practice tracker

YYYY-MM-DD

• Attempted/Correct:
• Mistake pattern: