Problems on Ages and Partnership Notes

Problems on Ages and Partnership are two short, scoring chapters of Quantitative Aptitude that are almost entirely built on ratio and simple equations. Age questions appear regularly in SSC CGL, CHSL, MTS, bank exams and state exams, and they also show up in UPSC CSAT as word puzzles. Partnership is a favourite of bank exams and SSC because it tests ratio in a business setting: how should profit be divided among partners who invested different amounts for different periods? Both chapters can be mastered in a few days with the right method, and both reward students who form equations carefully rather than guessing.

Quick Facts at a Glance

• Age difference never changes: If A is 5 years older than B today, A will always be 5 years older.
• Everyone ages equally: After t years, every person's age increases by t.
• Ratio method: Take ages as multiples of a common factor (for example 3x and 5x).
• Partnership profit ratio = Capital × Time ratio.
• Working partner gets a salary or share for managing the business before the rest is divided.

Part A: Problems on Ages

1. The Basic Framework

Every age question talks about three points of time: past (t years ago), present and future (t years hence). Set up a small table with present ages as unknowns, then write past and future ages by subtracting or adding.

2. Important Rules

• If the present age is x, then n times the age = nx, and age n years ago = x − n.
• The difference between two people's ages is constant at all times.
• The sum of ages of a group of n people increases by n × t after t years.
• If the ratio of ages is given, use a common multiplier; do not assume the ratio values are the ages themselves.
• Age questions must give whole, positive and sensible answers. If you get a negative or fractional age, check your equation.

3. Solved Examples on Ages

Example 1: The ratio of present ages of A and B is 4 : 5. After 6 years, the ratio will be 5 : 6. Find their present ages.
Let ages be 4x and 5x. Then (4x + 6) ÷ (5x + 6) = 5/6 → 24x + 36 = 25x + 30 → x = 6. Ages are 24 and 30 years.

Shortcut: The difference of ratio terms is 1 in both ratios (5 − 4 and 6 − 5), so each "part" is constant. The ratio moved from 4 to 5 for A, a change of 1 part = 6 years. So A = 4 × 6 = 24 and B = 30.

Example 2: A father is three times as old as his son. Ten years later, he will be twice as old. Find their present ages.
Let the son be x. Father = 3x. Then 3x + 10 = 2(x + 10) → x = 10. Son = 10, father = 30.

Example 3: The average age of a family of 5 members is 24 years. If the youngest is 8 years old, what was the average age of the family when the youngest was born?
Present total = 120. Eight years ago, the other 4 members' total = 120 − 8 − (4 × 8) = 80. Average = 80 ÷ 4 = 20 years.

Example 4: Five years ago, the sum of the ages of a mother and her daughter was 40 years. Today, the mother is four times as old as the daughter. Find their present ages.
Present sum = 40 + 5 + 5 = 50. Let the daughter be d; mother = 4d. Then 5d = 50, so d = 10. Daughter = 10 years, mother = 40 years. Tip: always translate each sentence into one equation, and if you get a fraction or negative age, re-read the question because exam questions are designed to give sensible whole numbers.

4. Common Age Question Types

• Ratio of ages now and after or before some years.
• Sum and difference of ages.
• Average age of a family or group, with births, deaths or new members.
• Product of ages (solve by factor pairs).
• Age-based puzzles in CSAT, such as "When I was as old as you are now..." statements.

"When I was as old as you are now" trick: If A says to B, "When I was as old as you are now, you were x," then the age difference d = A − B = B − x. Writing it as equal differences solves most such puzzles in one line.

Part B: Partnership

1. What Is Partnership?

When two or more people invest money together in a business, they are partners. The profit (or loss) is shared according to how much each person invested and for how long.

• Simple partnership: All partners invest for the same period. Profit ratio = ratio of capitals.
• Compound partnership: Partners invest for different periods. Profit ratio = ratio of capital × time.
• Working (active) partner: Manages the business and may get a salary or a fixed share before profit sharing.
• Sleeping partner: Only invests money and does not take part in running the business.

2. Key Formulas

3. Solved Examples on Partnership

Example 1: A and B invest ₹40,000 and ₹60,000 for a year. Profit is ₹25,000. Ratio = 2 : 3. A gets ₹10,000 and B gets ₹15,000.

Example 2: A invests ₹30,000 for 12 months and B invests ₹45,000 for 8 months. Ratio = 30,000 × 12 : 45,000 × 8 = 3,60,000 : 3,60,000 = 1 : 1. They share profit equally.

Example 3 (changing capital): A starts with ₹50,000 and after 4 months withdraws ₹10,000. B invests ₹45,000 for the whole year. A's investment = 50,000 × 4 + 40,000 × 8 = 5,20,000. B's = 45,000 × 12 = 5,40,000. Ratio = 26 : 27.

Example 4 (working partner): A and B invest equally. A is the working partner and gets 10% of the profit as salary. Out of a profit of ₹20,000, A first gets ₹2,000. The remaining ₹18,000 is shared equally. A's total = 2,000 + 9,000 = ₹11,000; B gets ₹9,000.

Example 5 (finding time): A and B invest in the ratio 3 : 4 and their profits are in the ratio 9 : 16. If A invested for 6 months, for how long did B invest? 3 × 6 : 4 × T = 9 : 16 → 18/4T = 9/16 → T = 8 months.

Common Mistakes to Avoid

1. Treating ratio numbers as actual ages.
2. Adding the time only to one person's age in "after t years" questions.
3. Forgetting that average age of a group rises by t after t years, unless members change.
4. In partnership, using capitals alone when time periods differ.
5. Forgetting to deduct a working partner's salary before dividing the profit.

Preparation Strategy

• Revise ratio and proportion first, since both chapters depend on it.
• Solve 25 age questions using both the equation method and the ratio-part shortcut.
• Solve 20 partnership questions with different capital and time combinations.
• In the exam, check your answer by putting it back into the question; it takes 10 seconds and removes silly errors.

Practice Questions with Answers

1. The sum of ages of A and B is 50 and the difference is 10. Find their ages. Answer: 30 and 20.
2. Four years ago, the ratio of ages of P and Q was 3 : 4. Their present ages total 64. Find P's present age. Answer: 3x + 4x + 8 = 64 → x = 8; P = 28.
3. A, B and C invest ₹20,000, ₹30,000 and ₹50,000. Profit is ₹40,000. Find C's share. Answer: 5/10 × 40,000 = ₹20,000.
4. X invests ₹12,000 for a year and Y joins after 4 months with ₹18,000. Find their profit ratio. Answer: 12,000 × 12 : 18,000 × 8 = 1 : 1.

Related Notes

• Ratio and Proportion Notes
• Average, Mixture and Alligation Notes
• Profit, Loss and Interest Notes
• All Quantitative Aptitude Notes

Frequently Asked Questions

What is the most important rule in age problems?

The difference between two people's ages always remains the same, no matter how many years pass.

How do you solve age problems using ratio?

Take present ages as multiples of a common variable, such as 3x and 5x, form an equation from the given condition and solve for x.

How does the average age of a group change over time?

If no member joins or leaves, the average age increases by exactly the number of years that pass.

Are age problems asked in UPSC CSAT?

Yes, mostly as simple word problems or logical puzzles that need forming one or two equations.

What is partnership in mathematics?

It is a business arrangement where two or more people invest money and share profit or loss according to their investment and time.

What is the formula for profit sharing in partnership?

Profit is shared in the ratio of capital multiplied by time for each partner.

What is the difference between simple and compound partnership?

In simple partnership, all partners invest for the same time. In compound partnership, the time periods differ.

Who is a sleeping partner?

A sleeping partner invests money but does not take part in running the business.

Who is a working partner?

A working partner manages the business and may receive a salary or extra share before the remaining profit is divided.

How do you handle a change in capital during the year?

Multiply each capital amount by the number of months it remained invested and add them to get the effective investment.

Why should I check my answer in age problems?

Putting the answer back into the question quickly confirms whether all conditions are satisfied and catches equation errors.

Which chapter should I study before ages and partnership?

Ratio and proportion, because both chapters are direct applications of ratios.