Direct Proportion

TL;DR

Direct proportion means two quantities change together at the same rate, so if one doubles, the other doubles too. You can describe this relationship with a simple multiplication, where one value is always a fixed multiple of the other. It's super useful for predicting outcomes when you know how things scale.

1. The Mental Model

Imagine you're buying apples: the more apples you buy, the more you pay. If one apple costs $1, two apples cost $2, and ten apples cost $10. The cost and the number of apples increase together, always by the same factor.

2. The Core Material

Direct proportion is when two quantities, let's call them 'A' and 'B', are linked such that if A increases, B increases by the same factor, and if A decreases, B decreases by the same factor. Think of it as a constant ratio or a consistent scaling.

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

This "fish" symbol ($\propto$) means "is directly proportional to". To turn this into an equation we can work with, we introduce a constant of proportionality, usually 'k':

$A = k \times B$

Here, 'k' is just a number that tells you how much A changes for every unit change in B. It never changes for a given relationship.

Finding the Constant of Proportionality (k)

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If you know one pair of A and B values that are directly proportional, you can always find 'k'. Just rearrange the formula:

$k = \frac{A}{B}$

Once you have 'k', you can find any other A or B value in that relationship.

graph TD
    A["Quantity A increases"] --> B["Quantity B increases (by same factor)"]
    B --> C{"Is the ratio A/B constant?"}
    C -- "Yes" --> D["Direct Proportion"]
    C -- "No" --> E["Not Direct Proportion"]
    D --> F["A = k * B"]
    F --> G["Find 'k' by A/B"]
    G --> H["Use 'k' to predict other values"]

Using 'k' to Solve Problems

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Once you've found 'k', you can use it to calculate unknown values.

Example: If the cost (C) of fabric is directly proportional to its length (L), and 5 meters cost $10, what would 12 meters cost?

1. Identify the relationship: $C \propto L$, so $C = k \times L$.
2. Find 'k' using known values:
   $k = \frac{C}{L} = \frac{$10}{5 \text{ meters}} = $2 per meter.
3. Use 'k' to find the unknown:
   $C = k \times L = $2/meter $\times$ 12 meters = $24.

So, 12 meters of fabric would cost $24.

3. Worked Example

Let's say the amount of paint (P) needed is directly proportional to the area (A) you want to paint. You know that 2 litres of paint can cover 20 square meters. How much paint do you need for a wall that is 35 square meters?

1. State the direct proportion: $P \propto A$, which means $P = k \times A$.
2. Use the given information to find 'k':
   We have $P = 2$ litres when $A = 20$ square meters.
   $k = \frac{P}{A} = \frac{2 \text{ litres}}{20 \text{ sq meters}} = 0.1 \text{ litres/sq meter}$.
   This 'k' means you need 0.1 litres of paint for every square meter.
3. Now, use 'k' to answer the question:
   We want to find $P$ when $A = 35$ square meters.
   $P = k \times A = 0.1 \text{ litres/sq meter} \times 35 \text{ sq meters}$.
   $P = 3.5 \text{ litres}$.

You would need 3.5 litres of paint for a 35 square meter wall.

4. Key Takeaways

• Direct proportion means two quantities increase or decrease at the same rate.
• The relationship can be written as $A \propto B$, or $A = k \times B$.
• 'k' is the constant of proportionality, representing the unchanging ratio between the quantities.
• You can find 'k' by dividing one quantity by the other: $k = A/B