Compound Variation (Chain Rule) explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
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Notes
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Formulae
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Examples
Step 4
Practice
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Compound variation means one quantity depends on two or more other quantities.
It combines direct variation and inverse variation in the same problem.
This topic is useful in work, speed, cost, production, and comparison problems.
Direct variation means two quantities increase or decrease together.
Inverse variation means one quantity increases while the other decreases.
Compound variation uses more than one relationship at once.
In chaiule problems, compare each change one at a time and decide whether to multiply by the direct ratio or inverse ratio.
A reasonableness check is important. More workers should usually reduce time. More work should usually increase time.
Definition: One quantity varies directly with another if increasing one increases the other in the same ratio.
More items usually cost more money.
Meaning: Two quantities move in opposite directions.
Definition: One quantity varies inversely with another if increasing one decreases the other in the same ratio.
More workers usually take less time for the same job.
Meaning: One quantity depends on more than one other quantity.
Definition: Compound variation combines two or more direct or inverse relationships.
Time may depend directly on amount of work and inversely on number of workers.
Meaning: Chain rule solves multtep ratio comparisons.
Definition: Chain rule compares changed quantities using a product of ratios.
If work increases, time increases. If workers increase, time decreases.
Easy
Question: If 4 books cost Rs 200, what is the cost of 6 books?
Medium
Question: 6 workers finish a job in 10 days. How many days will 12 workers take for the same job?
Exam-level
Question: 8 workers make 120 toys in 6 days. How many days will 12 workers take to make 180 toys?
Treating every relationship as direct
Why it happens: Ratios are copied without meaning.
Correct approach: Decide direct or inverse for each quantity.
Skipping the reasonableness check
Why it happens: The calculation is trusted blindly.
Correct approach: Ask whether the answer should increase or decrease.
Mixing old and new quantities
Why it happens: The table is not labelled.
Correct approach: Write old and new values in separate rows.
How do I know if a relationship is inverse?
If one quantity increases and the target quantity should decrease, it is inverse.
Why use chain rule?
It handles more than one changing quantity in one problem.
Can a problem have both direct and inverse variation?
Yes. That is compound variation.
Practise the same concept from this article. Your student profile may guide entitlement and follop recommendations, but it does not replace this concept.