9. Mixture and alligation
Free study material · concepts, shortcuts & solved questions
The mean value of a mixture is total value ÷ total quantity. Alligation compares cheaper, dearer and mean: quantity of cheaper : quantity of dearer = (dearer−mean):(mean−cheaper). Use the cross-differences opposite their prices.
Method before formula Alligation gives a quantity ratio, not the amount of money. If there are more than two components or wastage, use total-cost equations instead. |
Formula and decision table
Example | Setup |
₹20 and ₹50 to mean ₹30 | Cheaper:dearer=(50−30):(30−20)=2:1 |
Replacement | Remove x of mixture and add water; track solute remaining |
Weighted mean | Total cost/total quantity |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
Mix ₹20 and ₹50 items to get ₹30 mean. Quantity ratio: | (50−30):(30−20)=20:10=2:1. | B. 2:1 |
A mixture has 30% milk. If 10 L mixture, milk is: | 30% of 10=3 L. | B. 3 L |
Alligation cross-differences represent: | They give the quantity ratio. | A. Quantities ratio |
What an examiner is testing
The options around Mixture and alligation usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: Mix ₹20 and ₹50 items to get ₹30 mean. Quantity ratio: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 2:1, because (50−30):(30−20)=20:10=2:1.
Example 2: A mixture has 30% milk. If 10 L mixture, milk is: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 3 L, because 30% of 10=3 L.
Example 3: Alligation cross-differences represent: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is Quantities ratio, because They give the quantity ratio.
Chapter practice
1. Mix ₹20 and ₹50 items to get ₹30 mean. Quantity ratio:
(A) 1:2 (B) 2:1 (C) 3:1 (D) 1:3
Answer: B. 2:1 | Explanation: (50−30):(30−20)=20:10=2:1.
2. A mixture has 30% milk. If 10 L mixture, milk is:
(A) 2 L (B) 3 L (C) 4 L (D) 7 L
Answer: B. 3 L | Explanation: 30% of 10=3 L.
3. Alligation cross-differences represent:
(A) Quantities ratio (B) Final prices (C) Discounts (D) Time ratio
Answer: A. Quantities ratio | Explanation: They give the quantity ratio.
SSC CGL speed and trap clinic
For Mixture and alligation, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Mixture and alligation? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Mixture and alligation question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |