Direct Proportion: Fundamentals and Applications

TL;DR

Direct proportion describes a relationship where two quantities increase or decrease together at a constant rate. This means their ratio always stays the same. You'll learn to recognize, write, and solve problems involving these types of relationships.

1. The Mental Model

Imagine you're buying identical candies. If you buy more candies, you'll pay more money, and if you buy fewer, you'll pay less. The cost and the number of candies change in the same direction, and the price per candy stays constant. That's direct proportion!

2. The Core Material

When two quantities are directly proportional, it means that as one quantity increases, the other increases by the same factor, and vice-versa. Think of it as a constant multiplier.

Understanding the Relationship

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If quantity $y$ is directly proportional to quantity $x$, we write it as:
$y \propto x$

This symbol $\propto$ means "is proportional to." To turn this into an equation, we introduce a constant of proportionality, usually denoted by $k$.
So, $y = kx$

Here, $k$ is a fixed number. It tells you how much $y$ changes for every unit change in $x$. To find $k$, you can rearrange the equation:
$k = \frac{y}{x}$

This means the ratio of $y$ to $x$ is always constant.

Identifying Direct Proportion

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You'll know if a situation is direct proportion if:
1. Both quantities increase or decrease together.
2. The ratio $\frac{y}{x}$ (or $\frac{\text{quantity 1}}{\text{quantity 2}}$) is always the same.
3. Its graph is a straight line passing through the origin (0,0).

Let's look at an example:

Here, the constant of proportionality $k = 0.50$. So, the equation is $y = 0.50x$.

Solving Direct Proportion Problems

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There are usually two main ways to solve these:

Method 1: Finding the Constant of Proportionality ($k$)

1. Set up the relationship: $y = kx$.
2. Use the given pair of values (x and y) to find $k$.
3. Use the found $k$ with the new value of $x$ (or $y$) to find the unknown.

Method 2: Using Ratios (Unitary Method / Proportion Method)

1. Set up a proportion: $\frac{y_1}{x_1} = \frac{y_2}{x_2}$.
2. Substitute the known values.
3. Solve for the unknown. This works because the ratio $y/x$ is constant.

Here's how the problem-solving process generally flows:

graph TD
    A["Identify quantities (x and y)"] --> B{"Is it Direct Proportion?"}
    B -- Yes --> C["Write the relationship: y = kx"]
    C --> D["Find 'k' using known values (k = y/x)"]
    D --> E["Use 'k' to find unknown y or x"]
    B -- No --> F["Not Direct Proportion (use other methods)"]

3. Worked Example

Problem: If 3 kg of apples cost $7.50, how much would 5 kg of apples cost?

Solution using Method 1 (Constant of Proportionality):

1. Let $m$ be the mass of apples (in kg) and $C$ be the cost (in $).
   Since cost is directly proportional to mass, $C = km$.

2. We're given that 3 kg of apples cost $7.50.
   So, $7.50 = k \times 3$
   To find $k$: $k = \frac{7.50}{3} = 2.50$
   This means the cost per kg is $2.50.

3. Now, we want to find the cost of 5 kg of apples. Use the equation $C = km$ with our found $k$:
   $C = 2.50 \times 5$
   $C = 12.50$

So, 5 kg of apples would cost $12.50.

Solution using Method 2 (Ratios):

1. Set up the proportion: $\frac{\text{Cost}_1}{\text{Mass}_1} = \frac{\text{Cost}_2}{\text{Mass}_2}$

2. Substitute the known values:
   $\frac{7.50}{3} = \frac{C_2}{5}$

3. Solve for $C_2$:
   $C_2 = \frac{7.50}{3} \times 5$
   $C_2 = 2.50 \times 5$
   $C_2 = 12.50$

Both methods give the same answer.

4. Key Takeaways

• Direct proportion means two quantities change together at a constant rate.
• You can write direct proportion as $y \propto x$ or $y = kx$.
• The constant of proportionality, $k$, is found by dividing $y$ by $x$ ($k = y/x$).
• The graph of a direct proportion is always a straight line passing through the origin (0,0).
• You can solve direct proportion problems by finding $k$ or by setting up a ratio.

Common Mistakes to Avoid:
- Don't assume everything that increases together is directly proportional; check the ratio.
- Forgetting that the graph must pass through the origin to be direct proportion.
- Mixing up which variable is $x$ and which is $y$ when calculating $k$.
- Not labeling units in your final answer when dealing with real-world problems.

5. Now Try It

A machine prints 150 pages in 5 minutes. Assuming the printing speed is constant (a direct proportion!), calculate how many pages the machine can print in 12 minutes. Then, figure out how long it would take to print 450 pages.

What success looks like: You'll have two answers, one for pages printed in 12 minutes and one for the time taken to print 450 pages, both calculated correctly using either the constant of proportionality or ratio method.