Replacement Method: Formula, Rules and Solved Questions
Replacement Method is used when a part of a liquid mixture is removed and replaced with another liquid, usually water. The method finds the quantity of the original liquid left after one or more operations. Its main formula uses the fraction of the mixture remaining after each replacement and extends directly to repeated replacement.
What is the Replacement Method?
In every operation, the fraction of the original mixture that remains is (V − q)/V. Therefore, if the initial quantity of the original liquid is A, the quantity left after n identical replacements is A[(V − q)/V]^n. This method assumes that the mixture is uniform before each quantity is removed. For example, from a 20-litre vessel, removing and replacing 5 litres twice leaves 20 × (15/20)^2 = 11.25 litres of the original liquid.
Replacement Method Formula & Tricks
Important Formulas
V is the total volume of the vessel and q is the quantity removed and replaced in one operation.
A is the initial quantity of the original liquid, and n is the number of identical replacement operations.
Subtract the quantity of original liquid left after replacement from its initial quantity.
If different quantities q₁, q₂, ..., q_n are removed in successive operations, multiply the remaining fractions for all operations.
Quick Tricks
If q/V of the mixture is removed, the fraction left is 1 − q/V. Raise this remaining fraction to the number of identical operations.
When the original liquid is pure and the added liquid contains none of it, the final concentration of the original liquid is the remaining fraction multiplied by its initial concentration.
Do not add different removed quantities before calculating. Apply each operation successively using its own remaining fraction.
Replacement Method Concepts
Single Replacement in a Uniform Mixture
If a vessel contains A units of the original liquid, the amount of that liquid removed is Aq/V. Hence, the amount left is A − Aq/V = A(V − q)/V. The same fraction applies to each component of the mixture.
Repeated Replacement Formula
After the first operation, the original quantity becomes A(V − q)/V. Each later operation multiplies the current quantity by the same fraction. Therefore, repeated replacement forms a geometric progression.
Quantity of New Liquid Added
If the initial vessel volume is V and the original liquid remaining after n operations is A_n, then new liquid = V − A_n when A = V initially. If the initial mixture already contains other components, calculate the new liquid component separately from its initial amount and its remaining fraction.
Replacement with Different Quantities
The vessel volume remains V after every replacement, but the fraction removed can change. Each operation acts on the quantity left after the previous operation, so the factors must be applied successively.
Finding the Number of Replacements
For identical operations, the equation is (1 − q/V)^n = A_n/A. When the value is a simple power, n can be found by inspection. For other values, logarithms may be used: n = log(A_n/A) ÷ log(1 − q/V).
Replacement Method Video Lessons
Watch short topic-wise lessons for quick revision.
Repeated Dilution Formula in Replacement
Learn the repeated dilution formula used in replacement method problems, including how successive removals and refills change the concentration or quantity of a mixture.
Practice Replacement Method Questions
Practise published questions related to this topic.
Replacement Method Quick Quiz
Attempt 5 questions and check your score instantly.
Keep practising
Practice more Replacement Method questions in the PrepShots app and continue from your current topic.
Practice More Questions - Start ₹1 Trial →Quick Revision Notes
Replacement Method Revision Points
Use these rules to solve mixture replacement questions quickly and accurately.
- For vessel volume V and quantity q replaced each time, the remaining fraction is (V − q)/V.
- After n identical replacements, original quantity left = A[(V − q)/V]^n.
- The mixture must be uniform before each quantity is removed.
- Quantity of original liquid removed = initial quantity − final quantity.
- For unequal replacements, multiply all remaining fractions: A × ∏(1 − q_i/V).
- If the vessel initially contains only the original liquid, new liquid present = vessel volume − original liquid left.
- For concentration, multiply the initial concentration by the remaining fraction of the original component.
Replacement Method FAQs
What is the mixture replacement formula?
If V is the vessel volume, q is replaced each time, A is the initial quantity of the original liquid, and n is the number of operations, then original liquid left = A[(V − q)/V]^n.
How much milk remains when 5 litres are replaced twice from a 20-litre vessel initially full of milk?
Milk left = 20[(20 − 5)/20]^2 = 20(3/4)^2 = 11.25 litres.
What fraction of the original liquid remains after replacing one-fifth of a mixture three times?
The remaining fraction in each operation is 4/5. After three operations, the fraction left is (4/5)^3 = 64/125.
How is the amount of new liquid found after repeated replacement?
If the vessel initially contains only the original liquid, subtract the original liquid left from the vessel volume. For a 20-litre vessel with 11.25 litres of original liquid left, the new liquid is 8.75 litres.
Can the same replacement formula be used when different quantities are removed?
Use the product form: final original quantity = A(1 − q₁/V)(1 − q₂/V) ... (1 − q_n/V). The quantities must be applied in sequence.
What happens if half the mixture is removed and replaced twice?
The original fraction left is (1/2)^2 = 1/4. Thus, one-fourth of the original liquid remains and three-fourths has been replaced by the new liquid.