Ratio, proportion and partnership for IBPS PO

A ratio tells you how a quantity is split, not how big it is. That single idea runs through the topic. Give a ratio one real number, and every part becomes known. Miss that real number, and no amount of calculation will find the answer, which is exactly what data sufficiency questions test.

In IBPS PO, ratio shows up everywhere: as standalone word problems on incomes, ages and mixtures, in partnership profit-sharing, and throughout DI, where "what is the ratio of…" is one of the most common questions. The methods below turn each story into one equation in one unknown.

Ratio basics

• A ratio a : b means a parts and b parts. Write the quantities as ax and bx, where x is the size of one part.
• Dividing an amount in a ratio. To split ₹7,000 in 2 : 3 : 2, there are 7 parts, so one part is ₹1,000.
• Simplifying. Divide both terms by their HCF. 126 : 153 = 14 : 17 (both divided by 9).
• Proportion. a : b = c : d means a × d = b × c. The fourth proportional to a, b, c is b × c ÷ a.

Why the multiplier works: a ratio only fixes the shares. Writing the parts as 3x and 5x keeps the shares right while leaving the actual size, x, to be found from one real fact.

Combining ratios

To join A : B and B : C, make B the same in both.

• A : B = 2 : 3 and B : C = 4 : 5.
• The LCM of 3 and 4 is 12. Scale: A : B = 8 : 12 and B : C = 12 : 15.
• A : B : C = 8 : 12 : 15.

Changing ratios

When the same amount is added to or taken from both quantities, the ratio changes but the difference does not.

• Write the original quantities as ax and bx.
• Set up (ax + k) ÷ (bx + k) = new ratio, and cross-multiply.

Why a shortcut exists: the difference (b − a)x stays fixed, so in simple cases you can match the differences between the old and new ratios instead of solving.

Partnership

Profit share is in proportion to investment × time. Money invested for longer earns a bigger share, and a partner who joins late or withdraws midway is paid only for what was actually in the business.

Worked examples

Example 1: A and B invest ₹30,000 and ₹45,000 for a year. The profit is ₹25,000. Find A's share.

• Ratio 30 : 45 = 2 : 3. Five parts.
• A's share = 2/5 × 25,000 = ₹10,000.

Example 2: A invests ₹40,000 for 12 months and B invests ₹60,000 for 8 months. In what ratio should they share the profit?

• 40,000 × 12 = 4,80,000. 60,000 × 8 = 4,80,000.
• Ratio 1 : 1. More money for less time can equal less money for more time.

Example 3: The incomes of P and Q are in the ratio 5 : 4 and their expenses in the ratio 3 : 2. Each saves ₹2,000. Find their incomes.

• Incomes 5x and 4x; expenses 3y and 2y.
• 5x − 3y = 2,000 and 4x − 2y = 2,000. From the second, y = 2x − 1,000.
• 5x − 3(2x − 1,000) = 2,000, so −x + 3,000 = 2,000 and x = 1,000.
• Incomes: ₹5,000 and ₹4,000. Expenses ₹3,000 and ₹2,000; both save ₹2,000.

Example 4: Two salaries are in the ratio 3 : 5. After a rise of ₹4,000 each, the ratio becomes 5 : 7. Find the salaries.

• (3x + 4,000) ÷ (5x + 4,000) = 5 ÷ 7.
• 21x + 28,000 = 25x + 20,000, so x = 2,000.
• Salaries: ₹6,000 and ₹10,000. Check: 10,000 : 14,000 = 5 : 7.

Example 5: A starts a business with ₹60,000. B joins after 4 months with ₹80,000. Six months after the start, A withdraws ₹20,000. The profit at the end of the year is ₹31,000. Find each share.

• A: 60,000 × 6 + 40,000 × 6 = 3,60,000 + 2,40,000 = 6,00,000.
• B: 80,000 × 8 = 6,40,000.
• Ratio 600 : 640 = 15 : 16. Thirty-one parts, each ₹1,000.
• A gets ₹15,000 and B gets ₹16,000.

Example 6: A and B invest ₹50,000 and ₹30,000 for a year. A manages the business and takes 10% of the profit as a salary; the rest is shared by investment. The profit is ₹40,000. How much does A receive in all?

• Salary = 10% of 40,000 = 4,000. Remaining 36,000.
• Split 5 : 3: A gets 36,000 × 5/8 = 22,500; B gets 13,500.
• A's total = 4,000 + 22,500 = ₹26,500.

Common mistakes

• Ignoring time in partnership questions.
• Joining ratios without equalising the common term.
• Reporting x instead of the quantities. In Example 4, x = 2,000, but the answer is ₹6,000 and ₹10,000.
• Splitting the whole profit by investment when a working partner's salary must come out first.
• Missing the month a partner joined. "Joins after 4 months" means 8 months in a 12-month year.

Practice set

1. A and B invest ₹20,000 and ₹30,000 for the same period. The profit is ₹15,000. Find A's share.
2. A invests ₹50,000 for 6 months and B invests ₹30,000 for 12 months. Find the profit-sharing ratio.
3. Two numbers are in the ratio 3 : 5. If 10 is added to each, the ratio becomes 5 : 7. Find the numbers.
4. A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C.
5. ₹7,000 is divided among A, B and C in the ratio 2 : 3 : 2. Find each share.
6. Find the fourth proportional to 4, 6 and 10.
7. Two numbers are in the ratio 4 : 7. If 6 is subtracted from each, the ratio becomes 1 : 2. Find the numbers.
8. P and Q start a business with ₹24,000 and ₹36,000. After 6 months, Q withdraws ₹12,000. The year's profit is ₹27,000. Find each share.

Answers:

1. ₹6,000. Ratio 2 : 3; 2/5 × 15,000.
2. 5 : 6. 3,00,000 : 3,60,000.
3. 15 and 25. 21x + 70 = 25x + 50, so x = 5.
4. 8 : 12 : 15. Make B equal to 12 in both.
5. ₹2,000, ₹3,000 and ₹2,000. Seven parts of ₹1,000.
6. 15. 6 × 10 ÷ 4.
7. 24 and 42. 2(4x − 6) = 7x − 6, so x = 6.
8. P ₹12,000; Q ₹15,000. P: 24,000 × 12 = 2,88,000. Q: 36,000 × 6 + 24,000 × 6 = 3,60,000. Ratio 4 : 5.

What to do next

• Solve eight ratio and eight partnership questions a day for a week, writing "how much × how long" for every partner.
• Practise ratio questions inside table DI, where simplifying fast saves real time.
• Move on to averages and ages, where age ratios use the same multiplier method.