Ratio and Proportion • Topic 3 of 3

Inverse Variation

What is inverse variation? Two quantities are said to vary inversely if one increases while the other decreases proportionally. If x and y vary inversely, then their product remains constant:

xy = k or y = k/x

where k is the constant of variation.

Key properties:

As x increases, y decreases proportionally
As x decreases, y increases proportionally
The graph is a rectangular hyperbola (never touches the axes)
If x doubles, y halves; if x is halved, y doubles

Real-life examples of inverse variation:

Speed and time for a fixed distance — faster speed = less time
Number of workers and time to complete a job — more workers = less time
Number of people sharing a bill and each person's share — more people = smaller share
Pressure and volume of a gas (Boyle's Law) — $P_{1}V_{1}$ = $P_{2}V_{2}$

Inverse variation equation: If y ∝ 1/x (reads "y is inversely proportional to x"), then y = k/x or xy = k.

┌─────────────────────────────────────────────────────────────┐
│              INVERSE VARIATION - VISUAL REPRESENTATION       │
└─────────────────────────────────────────────────────────────┘

GRAPH OF INVERSE VARIATION (y = k/x, k > 0):

    y
    │
   10┤          ●
     │        ●
    8┤      ●
     │    ●
    6┤  ●
     │●
    4┤
     │
    2┤
     │
    0└─────┬─────┬─────┬─────┬─────► x
          2     4     6     8     10
    
    As x increases, y decreases (curved shape)


REAL-LIFE TABLE (Speed vs Time for 120 km journey):

    Distance fixed = 120 km
    
    ┌────────────┬──────────────┬─────────────┐
    │ Speed(km/h)│ Time (hours) │  Speed×Time │
    ├────────────┼──────────────┼─────────────┤
    │     30     │      4       │     120     │
    │     40     │      3       │     120     │
    │     60     │      2       │     120     │
    │    120     │      1       │     120     │
    └────────────┴──────────────┴─────────────┘
    
    Speed × Time = Constant (120)


HALVING/DOUBLING EFFECT:

    Original: x = 4, y = 6  (k = 24)
    
    Double x: x = 8, y = 3 (y becomes half!)
    
    Halve x: x = 2, y = 12 (y becomes double!)


COMPARISON: DIRECT vs INVERSE VARIATION

┌──────────────────┬─────────────────────┬─────────────────────┐
│                  │   Direct Variation   │   Inverse Variation │
├──────────────────┼─────────────────────┼─────────────────────┤
│ Equation         │     y = kx           │     y = k/x         │
│ Product/ Ratio   │     y/x = k          │     xy = k          │
│ Graph shape      │     Straight line    │     Hyperbola       │
│ Passes through   │     Origin (0,0)     │     Never touches   │
│                  │                      │     axes            │
│ When x doubles   │     y doubles        │     y halves        │
└──────────────────┴─────────────────────┴─────────────────────┘


PROBLEM-SOLVING STEPS:

    Step 1: Write as y ∝ 1/x
    Step 2: Replace with y = k/x or xy = k
    Step 3: Find k using given values
    Step 4: Use k to find unknown value

Solved Examples

Worked Example

If y varies inversely as x, and y = 12 when x = 5, find y when x = 15.

Solution

Step 1: y ∝ 1/x → xy = k (constant)
Step 2: Find k: 5 × 12 = 60 → k = 60
Step 3: Equation: xy = 60
Step 4: When x = 15: 15 × y = 60 → y = 60/15 = 4

Answer: y = 4

Worked Example

8 workers can complete a job in 15 days. How many days will 12 workers take to complete the same job?

Solution

Step 1: Workers and days vary inversely (more workers = fewer days)
Step 2: $W_{1} \times D_{1}$ = $W_{2} \times D_{2}$ (constant = total work)
Step 3: 8 × 15 = 12 × \(D_{2}
Step\) 4: 120 = 12 × $D_{2}$ → $D_{2}$ = 120/12 = 10

Answer: 10 days

Worked Example

y varies inversely as the square of x. If y = 8 when x = 3, find y when x = 6.

Solution

Step 1: y ∝ 1/$x^{2}$ → y = k/\(x^{2}
Step\) 2: Find k: 8 = k/9 → k = 8 × 9 = 72
Step 3: Equation: y = 72/\(x^{2}
Step\) 4: When x = 6: y = 72/36 = 2

Answer: y = 2

Key Points

Inverse variation: y ∝ 1/x → xy = k (constant)
Graph is a hyperbola (curved, never touches axes)
Product xy remains constant for all pairs
If x doubles, y halves; if x halves, y doubles
To solve: find k using given pair, then use equation
Can extend to squares: y ∝ 1/$x^{2}$, cubes, etc.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.In inverse variation, $xy=$

Explanation: Product is constant.

Q2.If $x\propto\tfrac1y$ and $x$ doubles, then $y$:

Explanation: Inverse relationship.

Q3.If $xy=12$ and $x=3$, then $y=$

Explanation: $y=12/3=4$.

Q4."More workers, less time" is an example of:

Explanation: Inverse variation.

Practice more MCQs → 📝 Assignment · 35 marks