Variations
Direct, inverse, joint and partial variation, and how to find the constants from given data. The unit is all Foundation and is often combined with other topics in DSE questions.
What you will learn
- 6.1 Direct and inverse variation; real-life problems.
- 6.2 Graphs of direct and inverse variation.
- 6.3 Joint and partial variation; real-life problems.
You need first: Number and algebra basics, Percentages
Key terms
| English | 中文 |
|---|---|
| varies directly as | 隨……正變 |
| varies inversely as | 隨……反變 |
| joint variation | 聯變 |
| partial variation | 部分變 |
| variation constant | 變分常數 |
Standard steps
Words to equation
| Statement | Equation |
|---|---|
| y varies directly as x | y = kx |
| y varies directly as x² | y = kx² |
| y varies inversely as x | y = k/x |
| y varies jointly as x and z | y = kxz |
| y varies directly as x and inversely as z | y = kx/z |
| y is partly constant and partly varies as x | y = a + bx (two constants) |
| y partly varies as x and partly as x² | y = ax + bx² |
Solve a variation problem
- Write the equation with k (or a and b for partial variation).
- Put in the given values to find k. Partial variation needs two sets of values and simultaneous equations.
- Write the full equation with k's value.
- Use it to answer the question.
Percentage change
Write each new value as a factor (+20% → 1.2), put the factors into the equation, and compare with 1. No need to find k.
Worked examples
Example 1: y varies directly as x² and inversely as z. y = 12 when x = 2 and z = 3. Find y when x = 3 and z = 27.
y = kx²/z
12 = k(4)/3, so k = 9
y = 9(3²)/27 = 3Example 2: In example 1, x increases by 20% and z decreases by 10%. Find the percentage change in y.
new y = k(1.2x)² / (0.9z) = (1.44 / 0.9) × kx²/z = 1.6y
y increases by 60%Example 3: The cost $C of printing n books is partly constant and partly varies as n. 100 books cost $3500 and 250 books cost $5000. Find C in terms of n.
C = a + bn
3500 = a + 100b
5000 = a + 250b
subtract: 1500 = 150b, so b = 10, a = 2500
C = 2500 + 10nCommon mistakes
- Using y = x + k for "varies directly" (it is y = kx).
- Partial variation with only one constant.
- "Varies inversely as the square root of x" written as k/x² instead of k/√x.
- Percentage change: adding the percentages (+20% and −10%) instead of multiplying factors.
How the DSE tests it
- A(2): partial variation with two data points, then a follow-up such as the number of items for a given cost, or a range of values found from a discriminant.
- Paper 2: percentage change in y; choose the graph of a direct or inverse variation (a line through the origin, or a curve like y = k/x).
Questions you might not ask in class
What is k?
A fixed number that links x and y in this particular situation. Find it once, then use it for every other question part.
How do I know it's partial variation?
The words "partly … and partly …", or a cost with a fixed part plus a part that grows.
Why don't I need k for percentage change?
k is the same before and after, so it cancels when you compare.