Variation · 1.0.4
Solving direct variation problems
This standard asks you to apply direct variation (and simple joint variation) to solve real-life problems, not just find the relation. Using y = kx or y = kxz, you calculate an unknown value, compare two different situations, or work out how one quantity changes when another is scaled by a given factor.
The official learning standard (1.0.4)
“Solve problems involving direct variation. Joint variation is a direct variation in which one variable varies as a product of two or more variables.”
What it means
This standard asks you to apply direct variation (and simple joint variation) to solve real-life problems, not just find the relation. Using y = kx or y = kxz, you calculate an unknown value, compare two different situations, or work out how one quantity changes when another is scaled by a given factor.
How it is examined
This is a common Paper 2 structured question. It often has several parts: the first sets up the relation using given values, and later parts ask you to find a new value, compare two scenarios, or interpret what happens when a quantity is scaled, for example, doubled, tripled or increased by a percentage.
Worked example
A worker's wage, W (RM), varies directly as the number of hours he works, h. He earns RM90 for 6 hours of work.
Find his wage for 10 hours of work, and the number of hours he must work to earn RM180.
- Since W varies directly as h, write W = kh.
- Substitute W = 90, h = 6: 90 = k(6), so k = 15.
- The relation is W = 15h.
- For h = 10: W = 15(10) = RM150.
- For W = 180: 180 = 15h, so h = 12 hours.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I recognise that a word problem involves direct variation?
Look for phrases such as 'varies directly as' or 'directly proportional to', or a situation with a fixed rate that keeps the ratio between two quantities constant, such as a fixed price per item or a constant speed. If both quantities increase or decrease together in the same ratio, it is direct variation.
What if the problem gives a percentage change instead of a new value?
Convert the percentage into a multiplying factor first, for example, 'increased by 20%' means multiply by 1.2, and 'decreased by 10%' means multiply by 0.9. Since the relation is direct variation, you can apply this factor straight to the known quantity to find the new value.
Do I always need to write out the full equation y = kx?
Yes, writing the equation clearly and showing how you found k is the standard, mark-earning method in SPM. Jumping straight to an answer using an unexplained shortcut ratio can lose you method marks in Paper 2, even if the final number is correct.