Ratio and Proportion: Rules and Worked Examples

Ratio and proportion is one of the most useful topics in the Lok Sewa maths and mental ability paper. The same ideas appear again in sharing money, partnership, time and work, speed and mixtures. This note explains each rule in simple steps, with worked examples in rupees, so you can solve these questions quickly and without mistakes.

1. Meaning of ratio

A ratio compares two quantities of the same kind by division. The ratio of a to b is written a : b and means a/b.

• The first term (a) is called the antecedent and the second term (b) is called the consequent.
• Both quantities must be in the same unit before you compare them.
• A ratio has no unit of its own.

Worked examples:

• Ram has Rs 600 and Shyam has Rs 900. Ratio of Ram’s money to Shyam’s money = 600 : 900 = 2 : 3.
• Ratio of 50 paisa to Rs 2. Change both to paisa: 50 : 200 = 1 : 4.
• Ratio of 45 minutes to 2 hours = 45 : 120 = 3 : 8.

2. Simplest form

A ratio does not change when both terms are multiplied or divided by the same non-zero number. To write a ratio in its simplest form, divide both terms by their HCF.

Tip: with fractions, multiply every term by the LCM of the denominators. With decimals, multiply by 10, 100 and so on until all terms are whole numbers.

3. Comparing ratios

To find the bigger of two ratios, write them as fractions and cross-multiply.

• Compare 3 : 4 and 5 : 7. Cross products: 3 × 7 = 21 and 4 × 5 = 20. Since 21 is more than 20, 3 : 4 is bigger.
• To arrange several ratios, change each to a decimal. For 2 : 3, 3 : 5 and 5 : 8 the values are 0.667, 0.6 and 0.625. Ascending order: 3 : 5, 5 : 8, 2 : 3.

Adding the same number to both terms: what number added to each term of 3 : 5 makes it 5 : 6?

(3 + x)/(5 + x) = 5/6, so 18 + 6x = 25 + 5x, which gives x = 7. Check: 10 : 12 = 5 : 6.

4. Dividing a sum in a given ratio

To divide an amount in the ratio a : b, add the parts, find the value of one part, then multiply.

Share of first = total × a/(a + b) and share of second = total × b/(a + b)

Worked examples:

• Divide Rs 12,000 between Gita and Rita in the ratio 3 : 5. Total parts = 8, one part = 12,000 ÷ 8 = Rs 1,500. Gita gets Rs 4,500 and Rita gets Rs 7,500.
• Divide Rs 18,000 among A, B and C in the ratio 2 : 3 : 4. Total parts = 9, one part = Rs 2,000. Shares: Rs 4,000, Rs 6,000 and Rs 8,000.
• Two shares are in the ratio 4 : 7 and differ by Rs 1,800. The difference is 3 parts, so one part = Rs 600. Shares: Rs 2,400 and Rs 4,200.

Joining two ratios: if A : B = 2 : 3 and B : C = 4 : 5, make the common term B equal. Multiply the first ratio by 4 and the second by 3: A : B = 8 : 12 and B : C = 12 : 15. So A : B : C = 8 : 12 : 15.

5. Compound, duplicate and inverse ratio

• Compound ratio: multiply all the antecedents together and all the consequents together. Compound ratio of 2 : 3, 4 : 5 and 5 : 8 = (2 × 4 × 5) : (3 × 5 × 8) = 40 : 120 = 1 : 3.
• Duplicate ratio of a : b is a² : b². Duplicate ratio of 3 : 4 = 9 : 16.
• Triplicate ratio of a : b is a³ : b³. Triplicate ratio of 2 : 3 = 8 : 27.
• Sub-duplicate ratio of a : b is √a : √b. Sub-duplicate ratio of 16 : 25 = 4 : 5.
• Inverse (reciprocal) ratio of a : b is b : a. Inverse ratio of 3 : 4 = 4 : 3.

6. Proportion: means and extremes

Four numbers a, b, c, d are in proportion when a : b = c : d. This is also written a : b :: c : d.

• a and d (the outer terms) are the extremes.
• b and c (the inner terms) are the means.
• Rule: product of extremes = product of means, that is a × d = b × c.

Worked examples:

• Are 3, 5, 12, 20 in proportion? 3 × 20 = 60 and 5 × 12 = 60. Yes.
• Find x if 4 : x = 10 : 25. Then 10 × x = 4 × 25 = 100, so x = 10.

7. Fourth, third and mean proportional

Another example: the mean proportional between 9 and 16 is √144 = 12.

8. Direct and inverse proportion

Direct proportion: when one quantity increases, the other increases in the same ratio. The ratio of the two stays fixed.

• 12 pens cost Rs 300. One pen costs Rs 25, so 20 pens cost 20 × 25 = Rs 500.

Inverse proportion: when one quantity increases, the other decreases in the same ratio. The product of the two stays fixed.

• 15 workers build a wall in 8 days. Total work = 15 × 8 = 120 worker-days. 20 workers need 120 ÷ 20 = 6 days.
• A car at 60 km per hour takes 4 hours for a journey of 240 km. At 80 km per hour it takes 240 ÷ 80 = 3 hours.

Tip: ask yourself, “If one goes up, does the other go up or down?” Up and up means direct; up and down means inverse.

9. Ages and mixtures

Ages:

• The ages of a father and son are in the ratio 7 : 2 and their sum is 54 years. Total parts = 9, one part = 6. Ages: 42 years and 12 years.
• Present ages of two sisters are in the ratio 5 : 3. After 4 years the ratio will be 3 : 2. Let the ages be 5x and 3x. Then (5x + 4)/(3x + 4) = 3/2, so 10x + 8 = 9x + 12 and x = 4. Present ages: 20 and 12 years. Check: after 4 years, 24 : 16 = 3 : 2.

Remember that both people grow older by the same number of years, so add the same number to both terms.

Mixtures:

• 40 litres of a mixture has milk and water in the ratio 3 : 1. Milk = 30 litres, water = 10 litres.
• How much water must be added to make the ratio 3 : 2? Milk stays 30 litres. 30/(10 + w) = 3/2, so 60 = 30 + 3w and w = 10 litres. Check: 30 : 20 = 3 : 2.
• Mixing two prices: rice at Rs 60 per kg is mixed with rice at Rs 80 per kg to get a mixture worth Rs 65 per kg. Ratio of cheaper to dearer = (80 − 65) : (65 − 60) = 15 : 5 = 3 : 1.