What are some examples of direct variation in real life?

Direct variation occurs when two quantities are directly proportional to each other, meaning as one quantity increases or decreases, the other does so at the same rate. This relationship can be expressed mathematically as \( y = kx \), where \( k \) is the constant of proportionality.

Here are some real-life examples of direct variation:

1. **Distance and Time at Constant Speed**:
- Example: If a car is traveling at a constant speed, the distance traveled is directly proportional to the time driven. If the speed is 60 km/h, the distance (\( d \)) covered after \( t \) hours is \( d = 60t \).
- Explanation: Doubling the time doubles the distance, halving the time halves the distance.

2. **Work Done and Number of Workers**:
- Example: If the number of workers increases while the work efficiency and hours remain constant, the amount of work done is directly proportional to the number of workers.
- Explanation: If one worker completes a task in 5 hours, two workers will complete double the amount of work in the same time frame.

3. **Cost and Quantity**:
- Example: The cost of items is directly proportional to the quantity purchased if the price per item remains constant. If one apple costs $2, buying \( x \) apples will cost \( 2x \).
- Explanation: Buying more apples increases the cost proportionally.

4. **Weight and Volume of a Substance**:
- Example: The weight of a substance is directly proportional to its volume if the density is constant. For example, if 1 liter of water weighs 1 kg, then 2 liters will weigh 2 kg.
- Explanation: Doubling the volume doubles the weight.

5. **Wages and Hours Worked**:
- Example: A worker earning an hourly wage of $15 will have total wages \( W \) directly proportional to the hours worked (\( h \)): \( W = 15h \).
- Explanation: Working more hours increases the wages proportionally.

6. **Electricity Usage and Time**:
- Example: If an appliance consumes a fixed amount of electricity per hour, the total electricity usage is directly proportional to the time the appliance is used.
- Explanation: Running the appliance for twice as long doubles the electricity consumed.

In all these examples, the key feature of direct variation is the constant rate of change between the two variables, maintaining proportionality.