Understanding Direct Proportion with Powers
Direct proportion describes a relationship where one quantity increases as another increases. This can be expressed algebraically as y ∝ x^n, meaning y is proportional to x raised to the power n.
The power n can be: 1 (linear), 2 (square), 3 (cube), or fractional such as 1/2 (square root) or 1/3 (cube root).
To convert the proportionality into an equation, introduce a constant of proportionality k: y = kx^n. This constant k remains fixed for the relationship and must be found using known values of x and y.
- Direct proportion means y increases as x increases.
- Expressed as y ∝ x^n, where n can be 1, 2, 3, 1/2, or 1/3.
- Equation form: y = kx^n, with constant k.
- Example: If y ∝ x^2 and y = 18 when x = 3, then k = 18 / 3^2 = 18 / 9 = 2.
- Use k to find y for other x values.
Understanding Inverse Proportion with Powers
Inverse proportion describes a relationship where one quantity decreases as another increases. Algebraically, this is written as y ∝ 1/x^n, meaning y is proportional to the reciprocal of x raised to the power n.
Again, n can be 1, 2, 3, or fractional powers like 1/2 or 1/3.
The equation form is y = k / x^n, where k is the constant of proportionality found from known values.
- Inverse proportion means y decreases as x increases.
- Expressed as y ∝ 1/x^n, with n as 1, 2, 3, 1/2, or 1/3.
- Equation form: y = k / x^n.
- Example: If y ∝ 1/x and y = 4 when x = 3, then k = 4 × 3 = 12.
- Use k to find y for other x values.
Using the Proportional Symbol (∝) and Finding Constants
The symbol ∝ indicates proportionality but is not an equation. To solve problems, convert it into an equation by introducing a constant k.
To find k, substitute known values of x and y into the equation and solve for k.
Once k is known, the equation can be used to find unknown values of y or x by substitution and algebraic manipulation.
- ∝ means 'is proportional to' but is not an equation.
- Convert y ∝ x^n to y = kx^n or y ∝ 1/x^n to y = k / x^n.
- Find k by substituting known x and y values.
- Use the equation to calculate unknown quantities.
- Check units and reasonableness of answers.
Worked Example: Direct Proportion with a Square Term
Suppose y is directly proportional to the square of x, and y = 50 when x = 5. Find y when x = 8.
Step 1: Write the equation y = kx^2 (since n = 2).
Step 2: Substitute known values to find k: 50 = k × 5^2 → 50 = 25k → k = 2.
- Step 3: Use k to find y when x = 8: y = 2 × 8^2 = 2 × 64 = 128.
- Answer: y = 128 when x = 8.
- Check: Since y increases with the square of x, increasing x from 5 to 8 increases y from 50 to 128, which is consistent.
Worked Example: Inverse Proportion with Cube Root
Suppose y is inversely proportional to the cube root of x, and y = 6 when x = 27. Find y when x = 8.
Step 1: Write the equation y = k / x^(1/3). Here n = 1/3.
Step 2: Substitute known values to find k: 6 = k / 27^(1/3) → 6 = k / 3 → k = 18.
- Step 3: Use k to find y when x = 8: y = 18 / 8^(1/3) = 18 / 2 = 9.
- Answer: y = 9 when x = 8.
- Check: Since y is inversely proportional to the cube root of x, decreasing x from 27 to 8 increases y from 6 to 9, which fits the inverse relationship.
Identifying and Solving Proportion Problems
To decide whether a problem involves direct or inverse proportion, observe how the quantities change:
If one quantity increases as the other increases, it is direct proportion.
If one quantity decreases as the other increases, it is inverse proportion.
- Express the relationship using the proportional symbol ∝ and identify the power n.
- Convert the proportionality into an equation with constant k.
- Use known values to find k.
- Substitute to find unknown quantities.
- Use algebraic manipulation to solve for variables if needed.
- Always check answers for consistency and units.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify relationship | Determine if quantities increase or decrease together | If y doubles when x doubles, direct proportion | State type of proportion clearly in answer |
| Write proportionality | Use y ∝ x^n or y ∝ 1/x^n with correct power n | y ∝ x^2 for square relationship | Use correct notation and explain power |
| Convert to equation | Introduce constant k: y = kx^n or y = k/x^n | y = kx^2 | Show formula before substitution |
| Find constant k | Substitute known x and y values and solve for k | If y=18 when x=3 and y ∝ x^2, then k=2 | Show clear substitution and calculation |
| Calculate unknowns | Use equation with k to find missing values | Find y when x=5: y=2×5^2=50 | Write full working and units |
| Check answer | Verify units and reasonableness | Check if y increases as expected | Explain why answer makes sense |
Active recall
Quick check
1. If y is directly proportional to the square of x and y = 18 when x = 3, what is y when x = 5?
y = kx^2; find k: 18 = k × 3^2 → k = 18/9 = 2; then y = 2 × 5^2 = 2 × 25 = 50.
2. If y is inversely proportional to the cube of x and y = 4 when x = 2, find y when x = 4.
y = k / x^3; find k: 4 = k / 2^3 → k = 4 × 8 = 32; then y = 32 / 4^3 = 32 / 64 = 0.5.
3. Explain the meaning of y ∝ 1/x^(1/2).
y is inversely proportional to the square root of x; as x increases, y decreases proportionally to 1 divided by the square root of x.