Learning Objectives
Chapter Overview
Key skills you will develop by the end of Chapter 1.
Explain direct proportion and use \(y = kx\) to find unknown values
Solve other forms of direct proportion: \(y = kx^{2}\), \(y = kx^{3}\), \(y = k\sqrt{x}\)
Explain inverse proportion and use \(y = \frac{k}{x}\) with the product rule
Solve other forms of inverse proportion: \(y = \frac{k}{x^{2}}\), inverse-square laws
Read proportion graphs: a straight line through the origin vs a hyperbola
Find the constant of proportionality \(k\) from a given pair of values
Mathematics: Direct and Inverse Proportions
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Part 1
Direct Proportion
When two quantities increase together in the same ratio.
1. What is direct proportion? State it as an equation.
• Define direct proportion• What does \(y \propto x\) mean?
Two quantities are in direct proportion when they increase or decrease in the same ratio — double one and the other doubles. We write \(y \propto x\), which becomes:
\(y = kx\) or \(\frac{y}{x} = k\)
Here \(k\) is the constant of proportionality and \(k \neq 0\). The ratio \(\frac{y}{x}\) stays the same for every pair of values.
📝 The graph of \(y\) against \(x\) is a straight line that passes through the origin \((0,0)\).
2. What is the method to solve a direct-proportion problem?
• How do you find k?• Steps for y = kx problems
Step 1: Write the model \(y = kx\).
Step 2: Substitute the given pair of values and solve for \(k\).
Step 3: Write the full equation, then substitute the new value.
Worked example: \(y \propto x\) and \(y = 12\) when \(x = 3\). Find \(y\) when \(x = 7\).
\(12 = k \times 3\) so \(k = 4\). Then \(y = 4x = 4 \times 7 = 28\).
Step 2: Substitute the given pair of values and solve for \(k\).
Step 3: Write the full equation, then substitute the new value.
Worked example: \(y \propto x\) and \(y = 12\) when \(x = 3\). Find \(y\) when \(x = 7\).
\(12 = k \times 3\) so \(k = 4\). Then \(y = 4x = 4 \times 7 = 28\).
⛔ Common Mistake: don't add or subtract the change. Proportion works by multiplying by the constant \(k\), not by adding a fixed amount.
3. How do you read the graph of a direct proportion?
• What does the direct-proportion graph look like?• Why does it pass through the origin?
For \(y = kx\), plotting \(y\) against \(x\) gives a straight line through the origin. The steeper the line, the larger the constant \(k\) (the gradient of the line is \(k\)).
When \(x = 0\), \(y = k \times 0 = 0\), so the line must pass through \((0, 0)\).
When \(x = 0\), \(y = k \times 0 = 0\), so the line must pass through \((0, 0)\).
📝 If a straight line does not pass through the origin, it is not a direct proportion.
Part 2
Other Forms of Direct Proportion
When y is proportional to a power of x.
1. What are the other forms of direct proportion?
• y proportional to x squared• Direct proportion with powers and roots
\(y\) can be directly proportional to a power of \(x\):
For example \(y = 3x^{2}\) means \(y\) is directly proportional to \(x^{2}\), because \(\frac{y}{x^{2}} = 3\) is constant.
Square
\(y = kx^{2}\)
Cube
\(y = kx^{3}\)
Square root
\(y = k\sqrt{x}\)
📝 Plotting \(y\) against \(x^{2}\) (not \(x\)) gives a straight line through the origin.
2. Worked example: y is directly proportional to x².
• Solve a y = kx² problem• Find y when y proportional to x squared
\(y \propto x^{2}\) and \(y = 18\) when \(x = 3\). Find \(y\) when \(x = 5\).
\(y = kx^{2}\), so \(18 = k \times 3^{2} = 9k\), giving \(k = 2\).
Then \(y = 2x^{2} = 2 \times 5^{2} = 2 \times 25 = 50\).
\(y = kx^{2}\), so \(18 = k \times 3^{2} = 9k\), giving \(k = 2\).
Then \(y = 2x^{2} = 2 \times 5^{2} = 2 \times 25 = 50\).
⛔ Common Mistake: square the value BEFORE multiplying by \(k\). \(3^{2} = 9\), not \(6\).
Part 3
Inverse Proportion
When one quantity increases as the other decreases.
1. What is inverse proportion? State it as an equation.
• Define inverse proportion• What is the product rule?
Two quantities are in inverse proportion when one increases as the other decreases, so their product stays constant. We write:
\(xy = k\) or \(y = \frac{k}{x}\)
The product rule \(x_{1}y_{1} = x_{2}y_{2}\) is very useful: the product of \(x\) and \(y\) is the same for every pair.
📝 The graph of \(y\) against \(x\) is a hyperbola (a smooth curve). The graph of \(y\) against \(\frac{1}{x}\) is a straight line through the origin.
2. What is the method to solve an inverse-proportion problem?
• How do you find k in inverse proportion?• Steps for y = k/x problems
Step 1: Write the model \(xy = k\) (i.e. \(y = \frac{k}{x}\)).
Step 2: Substitute the given pair to find \(k\).
Step 3: Use \(y = \frac{k}{x}\) with the new value.
Worked example: \(y \propto \frac{1}{x}\) and \(y = 8\) when \(x = 3\). Find \(y\) when \(x = 6\).
\(k = xy = 3 \times 8 = 24\). Then \(y = \frac{24}{6} = 4\).
Step 2: Substitute the given pair to find \(k\).
Step 3: Use \(y = \frac{k}{x}\) with the new value.
Worked example: \(y \propto \frac{1}{x}\) and \(y = 8\) when \(x = 3\). Find \(y\) when \(x = 6\).
\(k = xy = 3 \times 8 = 24\). Then \(y = \frac{24}{6} = 4\).
⛔ Common Mistake: in inverse proportion you multiply \(x\) and \(y\) to get \(k\) — you do not divide \(y\) by \(x\) (that is for direct proportion).
3. In inverse proportion, what happens to y when x is doubled?
• If x doubles, what happens to y?• Reasoning about inverse proportion
Because \(xy = k\) is constant, if \(x\) is doubled then \(y\) is halved; if \(x\) is tripled then \(y\) becomes one third.
Example: 4 workers take 6 days (\(k = 24\)). With 8 workers, \(y = \frac{24}{8} = 3\) days — half the time.
Example: 4 workers take 6 days (\(k = 24\)). With 8 workers, \(y = \frac{24}{8} = 3\) days — half the time.
📝 Direct: same ratio (both rise). Inverse: constant product (one rises, the other falls).
Part 4
Other Forms of Inverse Proportion
When y is inversely proportional to a power of x.
1. What are the other forms of inverse proportion?
• y inversely proportional to x squared• Inverse-square relationships
\(y\) can be inversely proportional to a power of \(x\):
The product \(x^{n} \times y = k\) stays constant. For \(y = \frac{k}{x^{2}}\), doubling \(x\) divides \(y\) by \(4\).
Inverse square
\(y = \frac{k}{x^{2}}\)
Inverse cube
\(y = \frac{k}{x^{3}}\)
Inverse root
\(y = \frac{k}{\sqrt{x}}\)
📝 Real inverse-square laws include the brightness of light and the force of gravity, which both fall as \(\frac{1}{x^{2}}\).
2. Worked example: y is inversely proportional to x².
• Solve a y = k/x² problem• Find y when y inversely proportional to x squared
\(y \propto \frac{1}{x^{2}}\) and \(y = 9\) when \(x = 2\). Find \(y\) when \(x = 3\).
\(k = x^{2}y = 2^{2} \times 9 = 4 \times 9 = 36\).
Then \(y = \frac{36}{x^{2}} = \frac{36}{3^{2}} = \frac{36}{9} = 4\).
\(k = x^{2}y = 2^{2} \times 9 = 4 \times 9 = 36\).
Then \(y = \frac{36}{x^{2}} = \frac{36}{3^{2}} = \frac{36}{9} = 4\).
⛔ Common Mistake: square \(x\) first, then divide. \(3^{2} = 9\), so \(y = \frac{36}{9} = 4\).
Vocabulary
Key Terms
Quick reference for the important words in this chapter.
| Term | Meaning |
|---|---|
| Direct proportion | A relationship where \(y = kx\); the ratio \(\frac{y}{x}\) is constant. |
| Inverse proportion | A relationship where \(xy = k\); the product \(xy\) is constant. |
| Constant of proportionality (k) | The fixed number linking the two quantities; found from one known pair. |
| Proportional \((\propto)\) | The symbol meaning "is proportional to". |
| Product rule | \(x_{1}y_{1} = x_{2}y_{2}\) for inverse proportion. |
| Hyperbola | The curved graph of \(y\) against \(x\) for inverse proportion. |
| Origin | The point \((0,0)\); a direct-proportion line always passes through it. |
Chapter Summary
Direct vs Inverse
Quick reference — the two relationships side by side.
| Direct Proportion | Inverse Proportion |
|---|---|
| \(y = kx\), ratio \(\frac{y}{x} = k\) | \(y = \frac{k}{x}\), product \(xy = k\) |
| As \(x\) increases, \(y\) increases | As \(x\) increases, \(y\) decreases |
| Double \(x\) ⇒ double \(y\) | Double \(x\) ⇒ halve \(y\) |
| Graph: straight line through origin | Graph: hyperbola (curve) |
| Powers: \(y = kx^{2}\), \(y = k\sqrt{x}\) | Powers: \(y = \frac{k}{x^{2}}\), \(y = \frac{k}{x^{3}}\) |
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