Direct and Inverse Proportionality

TL;DR

Direct proportionality means two things increase or decrease together at a constant rate. Inverse proportionality means one thing increases as the other decreases. We use these concepts to predict how changes in one quantity affect another.

1. The Mental Model

Imagine a seesaw. If it's perfectly balanced, and you add weight to one side, that side goes down. If you want it to balance again, you have to add weight to the other side (direct). Now imagine a teeter-totter where one person gets heavier, the other person needs to get lighter to stay balanced (inverse).

2. The Core Material

When two quantities are related, they can either change in the same direction or in opposite directions. This relationship is called proportionality.

Direct Proportionality

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Two quantities, let's say 'A' and 'B', are directly proportional if an increase in 'A' causes a proportional increase in 'B', and a decrease in 'A' causes a proportional decrease in 'B'. Think of it as a constant multiplier.

Mathematically, we write this as:
$A \propto B$

This means that $A = kB$, where 'k' is a non-zero constant of proportionality. You can find 'k' by dividing A by B ($k = A/B$). If you double B, A doubles. If you halve B, A halves.

Example: The more hours you work (A), the more money you earn (B), assuming a fixed hourly wage (k).

Inverse Proportionality

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Two quantities, 'A' and 'B', are inversely proportional if an increase in 'A' causes a proportional decrease in 'B', and a decrease in 'A' causes a proportional increase in 'B'. Here, their product is constant.

Mathematically, we write this as:
$A \propto \frac{1}{B}$

This means that $A = \frac{k}{B}$, or equivalently, $AB = k$, where 'k' is a non-zero constant of proportionality. You can find 'k' by multiplying A and B ($k = AB$). If you double B, A halves. If you halve B, A doubles.

Example: The more people sharing a pizza (A), the smaller the slice each person gets (B).

Here's how these relationships work:

graph LR
    A["Quantity 1 (A)"]

    subgraph "Direct Proportionality"
        A_Direct_Increase["A increases"] --> B_Direct_Increase["B increases proportionally"]
        A_Direct_Decrease["A decreases"] --> B_Direct_Decrease["B decreases proportionally"]
    end

    subgraph "Inverse Proportionality"
        A_Inverse_Increase["A increases"] --> B_Inverse_Decrease["B decreases proportionally"]
        A_Inverse_Decrease["A decreases"] --> B_Inverse_Increase["B increases proportionally"]
    end

3. Worked Example

Let's say the time it takes to paint a fence (T) is inversely proportional to the number of painters (P). If 2 painters can paint a fence in 6 hours, how long would it take 3 painters?

1. Identify the relationship: Inverse proportionality. This means $T \propto \frac{1}{P}$, or $T = \frac{k}{P}$, which can be rewritten as $TP = k$.

2. Find the constant 'k': We know that when $T = 6$ hours, $P = 2$ painters.
   $k = T \times P = 6 \text{ hours} \times 2 \text{ painters} = 12$.
   So, the constant of proportionality is 12.

3. Use 'k' to solve the new problem: We want to find T when $P = 3$ painters.
   Since $TP = k$, we have $T \times 3 = 12$.
   $T = \frac{12}{3} = 4$ hours.

So, it would take 3 painters 4 hours to paint the fence.

4. Key Takeaways

• Direct proportionality means $y = kx$; as one quantity increases, the other increases.
• Inverse proportionality means $y = \frac{k}{x}$ (or $xy=k$); as one quantity increases, the other decreases.
• The constant 'k' (constant of proportionality) links the two quantities.
• You find 'k' by using a known pair of values for the quantities.
• Once 'k' is found, you can predict unknown values for either quantity.
• Direct relationships often involve multiplication or division to find the constant.
• Inverse relationships often involve the product of the two variables to find the constant.

5. Now Try It

A baker finds that the number of loaves of bread (L) she bakes is directly proportional to the amount of flour (F) she uses. If she uses 5 kg of flour to bake 10 loaves, how many loaves can she bake with 8 kg of flour? Calculate 'k' first, then use it to find the answer. What does 'k' represent in this context? You should arrive at an answer in loaves.