IB MYP 4-5 Maths- Direct and inverse proportion - Study Notes - New Syllabus
IB MYP 4-5 Maths- Direct and inverse proportion – Study Notes
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- Direct and inverse proportion
IB MYP 4-5 Maths- Direct and inverse proportion – Study Notes – All topics
Direct and inverse proportion
Direct Proportion
Two quantities are said to be in direct proportion when they increase or decrease together in the same ratio.
If \( x \) is directly proportional to \( y \), we write:
\( x \propto y \) or \( x = ky \)
where \( k \) is the constant of proportionality.
- The ratio \( \frac{x}{y} \) is constant.
- If one doubles, the other doubles.
- The graph is a straight line through the origin.
Formula: If \( x \propto y \), then:
\( \frac{x_1}{y_1} = \frac{x_2}{y_2} \)
Example:
Given that \( x \propto y \), and \( x = 12 \) when \( y = 4 \), find:
a) The constant of proportionality
b) The value of \( x \) when \( y = 9 \)
▶️ Answer/Explanation
Use the formula \( x = ky \)
\( 12 = k \cdot 4 \Rightarrow k = \frac{12}{4} = 3 \)
Use the constant \( k = 3 \) to find \( x \) when \( y = 9 \)
\( x = 3 \cdot 9 = 27 \)
a) \( \boxed{k = 3} \)
b) \( \boxed{x = 27} \)
Example:
If \( a \propto b \), and \( a = 7 \) when \( b = 2 \), find the value of \( b \) when \( a = 21 \).
▶️ Answer/Explanation
Set up a proportion:
\( \frac{a_1}{b_1} = \frac{a_2}{b_2} \Rightarrow \frac{7}{2} = \frac{21}{b} \)
Cross multiply
\( 7b = 42 \Rightarrow b = \frac{42}{7} = 6 \)
\(\boxed{b = 6}\)
Inverse Proportion
Two quantities are said to be in inverse proportion when one increases as the other decreases such that their product is constant.
If \( x \) is inversely proportional to \( y \), we write:
\( x \propto \frac{1}{y} \) or \( x = \frac{k}{y} \)
where \( k \) is the constant of proportionality.
- The product \( x \cdot y \) is constant.
- If one doubles, the other is halved.
- The graph is a curved hyperbola (not a straight line).
Formula: If \( x \propto \frac{1}{y} \), then:
\( x_1 \cdot y_1 = x_2 \cdot y_2 = k \)
Example:
If \( x \propto \frac{1}{y} \), and \( x = 12 \) when \( y = 3 \), find:
a) The constant of proportionality
b) The value of \( x \) when \( y = 8 \)
▶️ Answer/Explanation
Use the formula \( x \cdot y = k \)
\( 12 \cdot 3 = 36 \Rightarrow k = 36 \)
Use \( x = \frac{36}{y} \) to find \( x \) when \( y = 8 \)
\( x = \frac{36}{8} = 4.5 \)
a) \( \boxed{k = 36} \)
b) \( \boxed{x = 4.5} \)
Example:
If 6 workers can complete a task in 10 days, how many days will it take 15 workers to complete the same task, assuming they work at the same rate?
▶️ Answer/Explanation
Use inverse proportion: workers × days = constant
\( 6 \cdot 10 = 60 \Rightarrow \text{constant} = 60 \)
Use \( \text{workers} \cdot \text{days} = 60 \)
\( 15 \cdot x = 60 \Rightarrow x = \frac{60}{15} = 4 \)
\(\boxed{4 \text{ days}}\)