In this guide
Mixture and partnership questions look like two different chapters. One is about rice, tea, milk and water. The other is about people putting money into a business. Underneath, both are ratio questions. If you can split an amount in the ratio 3 : 2, you can solve nearly all of them.
Revise ratio and average first if they feel shaky. A mixture price is just a weighted average.
Part A: mixtures
The mixture price
When you mix two items, the price of the mixture per kg is:
Mixture price = total cost ÷ total quantity
It always lies between the two prices, and closer to the price of the item you used more of.
The alligation rule (mixing to hit a target price)
To find the ratio in which two items must be mixed to get a given mixture price:
Cheaper : dearer = (dearer price − mixture price) : (mixture price − cheaper price)
Draw it as a small cross on your rough sheet: the two prices at the top, the mixture price in the middle, and the two differences at the bottom. Notice that the difference is taken across. The cheaper item's share comes from the dearer price.
Milk and water
A ratio like 3 : 1 means 3 parts milk for every 1 part water. Find one part, then everything else.
When you add only water, the milk does not change. When you add only milk, the water does not change. Always start from the quantity that stays fixed.
When you take out some of the mixture, you take out milk and water in the same ratio as the mixture.
Part B: partnership
When people invest in a business, profit is shared in the ratio of their investment.
- Same time period: share in the ratio of the amounts invested.
- Different time periods: share in the ratio of investment × time (in months).
| Situation | Ratio to use |
|---|---|
| A ₹20,000, B ₹30,000, both for a year | 2 : 3 |
| A ₹10,000 for 12 months, B ₹15,000 for 6 months | 1,20,000 : 90,000 = 4 : 3 |
| B joins after 4 months | B's time = 12 − 4 = 8 months |
Solved examples
Example 1. 2 kg of rice at ₹40/kg is mixed with 3 kg at ₹50/kg. Find the price of the mixture per kg.
- Total cost = 2 × 40 + 3 × 50 = 80 + 150 = ₹230.
- Total weight = 5 kg.
- Price = 230 ÷ 5 = ₹46/kg.
Example 2. In what ratio should tea at ₹200/kg be mixed with tea at ₹300/kg to get tea worth ₹240/kg? How much of the cheaper tea is in 25 kg of the blend?
- Cheaper : dearer = (300 − 240) : (240 − 200) = 60 : 40 = 3 : 2.
- In 25 kg: 5 parts = 25 kg, so 1 part = 5 kg.
- Cheaper tea = 3 × 5 = 15 kg.
- Check: (15 × 200 + 10 × 300) ÷ 25 = 6,000 ÷ 25 = 240.
Example 3. A 20-litre mixture has milk and water in the ratio 3 : 1. How much water must be added to make the ratio 3 : 2?
- One part = 20 ÷ 4 = 5 litres. Milk = 15 litres; water = 5 litres.
- Milk stays 15. For 3 : 2, water must be 10 litres.
- Add 10 − 5 = 5 litres of water.
Example 4. A 40-litre mixture has milk and water in the ratio 3 : 1. 8 litres of mixture is taken out and replaced with 8 litres of water. Find the new ratio.
- At the start: milk 30, water 10.
- The 8 litres removed are in the ratio 3 : 1: milk 6, water 2.
- Left: milk 24, water 8. Add 8 water: water 16.
- New ratio = 24 : 16 = 3 : 2.
Example 5. A invests ₹10,000 for 12 months and B invests ₹15,000 for 6 months. The profit is ₹7,000. Find each share.
- A = 10,000 × 12 = 1,20,000. B = 15,000 × 6 = 90,000.
- Ratio = 1,20,000 : 90,000 = 4 : 3.
- A = 4/7 × 7,000 = ₹4,000; B = ₹3,000.
Example 6. A starts a business with ₹30,000. After 4 months, B joins with ₹40,000. At the end of the year the profit is ₹17,000. Find each share.
- A's months = 12; B's months = 12 − 4 = 8.
- A = 30,000 × 12 = 3,60,000. B = 40,000 × 8 = 3,20,000.
- Ratio = 36 : 32 = 9 : 8.
- A = 9/17 × 17,000 = ₹9,000; B = ₹8,000.
Common mistakes
| Mistake | Correct way |
|---|---|
| Averaging two prices directly (40 and 50 gives 45) | Weight by quantity: total cost ÷ total kg |
| Writing the alligation differences on the wrong side | Cheaper share = dearer − mean |
| Changing the milk when only water is added | Keep the unchanged liquid fixed |
| Removing only milk when mixture is taken out | Remove both in the mixture's ratio |
| Using 12 months for a partner who joined late | Months = 12 − months missed |
Practice set
- 4 kg of sugar at ₹40/kg is mixed with 6 kg at ₹45/kg. Find the price of the mixture per kg.
- In what ratio should wheat at ₹20/kg be mixed with wheat at ₹30/kg to get a mixture worth ₹24/kg?
- A 30-litre mixture has milk and water in the ratio 2 : 1. How much water is there?
- In the same 30-litre mixture, how much milk must be added to make the ratio 4 : 1?
- Rice at ₹30/kg and ₹45/kg is mixed in the ratio 2 : 1. Find the price of the mixture per kg.
- A and B invest ₹40,000 and ₹60,000 for a year. The profit is ₹15,000. Find A's share.
- X invests ₹5,000 for 8 months and Y invests ₹8,000 for 5 months. What is the ratio of their profits?
- A starts with ₹24,000. After 6 months, B joins with ₹36,000. The year's profit is ₹14,000. Find each share.
Answers:
- (160 + 270) ÷ 10 = ₹43/kg.
- (30 − 24) : (24 − 20) = 6 : 4 = 3 : 2.
- 1 part = 10 litres. Water = 10 litres.
- Milk 20, water 10. For 4 : 1 with water fixed at 10, milk must be 40. Add 20 litres of milk.
- (2 × 30 + 1 × 45) ÷ 3 = 105 ÷ 3 = ₹35/kg.
- Ratio 2 : 3. A = 2/5 × 15,000 = ₹6,000.
- 40,000 : 40,000 = 1 : 1.
- A = 24,000 × 12 = 2,88,000; B = 36,000 × 6 = 2,16,000. Ratio 4 : 3. A ₹8,000, B ₹6,000.
What to do next
- Solve ten alligation questions, drawing the cross every time.
- Practise five "add water" and five "remove and replace" questions.
- Solve five partnership questions where one partner joins late or leaves early.
- Check every answer by putting it back into the question, as in Example 2.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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