Pattern 20·Advanced·L3

Replacement & Composition

Medium FrequencyImportant 90–120 sec

USE WHEN

Part of a mixture/group is removed and replaced, changing percentage composition.

BEST METHOD

Component Tracking

TARGET

90–120 sec

MASTERY

not started

Recognition Cues

remove and replace
mixture composition changes
repeated replacement

How to Recognise

Part of a mixture/group is removed and replaced, changing percentage composition.

Core Concept

When a random sample is removed from a uniform mixture, each component is removed in the same proportion.

After one replacement: original component remaining = Original component \times (1−x/V). Repeated n times: \times (1−x/V)^n.

Athena Shortcut

For one replacement, component tracking is often clearer than formula use.

Alternative Method · Component Tracking

Track exact amount of the key component after every replacement.

Timed drill

Identify the base or invariant before you calculate.

2:00

Attempt first · Standard Example

40 L mixture has 75% milk. 8 L is removed and replaced with water. Milk left?

Attempt first · CAT-Level Example

A 50 L mixture is 60% alcohol. 10 L is replaced by water twice. Alcohol left?

Common Trap

Removed liquid has the same composition as the mixture at that moment.

3-level practice ladder

Attempt all three before checking solutions

Foundation

An 80 L mixture contains 50% milk. 20 L is removed and replaced with water once. How much milk remains?

CAT Standard

A 50 L mixture contains 60% alcohol. 10 L is removed and replaced with water twice. How much alcohol remains?

Advanced / Hybrid

A 64 L mixture contains 75% milk. 16 L is removed and replaced with water three times. How much milk remains?

PYQ Mapping

Related archetypeCAT

This pattern is a recurring CAT arithmetic archetype.

Pattern Connections

#15 Percentage Distribution #27 Percentage + Mixture #32 Multi-Constraint CAT Percentage Problems

Core Concept

When a random sample is removed from a uniform mixture, each component is removed in the same proportion.

After one replacement: original component remaining = Original component \times (1−x/V). Repeated n times: \times (1−x/V)^n.

Attempt all three before checking solutions

Foundation

An 80 L mixture contains 50% milk. 20 L is removed and replaced with water once. How much milk remains?

CAT Standard

A 50 L mixture contains 60% alcohol. 10 L is removed and replaced with water twice. How much alcohol remains?

Advanced / Hybrid