Problem 6 - Applied Calculus

Question

Find the relative, or percent, change. \(\underline{B}\) changes from 12,000 to 15,000

Step-by-Step Solution

Answer

The percent change is 25%.

Step 1: Identify the Original and New Values

First, identify the original value and the new value. Here, the original value of \( \underline{B} \) is 12,000, and the new value is 15,000.

Step 2: Calculate the Amount of Change

Subtract the original value from the new value to find the change in \( \underline{B} \). This gives us: \[\text{Change} = 15,000 - 12,000 = 3,000\]

Step 3: Divide the Change by the Original Value

To find the relative change, divide the change by the original value of \( \underline{B} \):\[\text{Relative Change} = \frac{3,000}{12,000}\]

Step 4: Convert to a Percentage

Multiply the relative change by 100 to express it as a percentage:\[\text{Percent Change} = \left(\frac{3,000}{12,000}\right) \times 100 = 25\%\]

Step 5: Conclude the Calculation

The relative or percent change in \( \underline{B} \) is 25\%. This means \,B\ increased by 25\% from its original value.

Key Concepts

Relative Change

When we talk about relative change, we are referring to how much something has increased or decreased in relation to its original value. It's like measuring the growth or decline in terms of percentages, making it easier to understand the extent of change. To calculate relative change, you first determine the difference between the new value and the original value. In our exercise, this difference, or change, was calculated as follows:

• Original value = 12,000
• New value = 15,000
• Change = 15,000 - 12,000 = 3,000

Relative change is essentially expressed as a fraction or a proportion of the original value. This provides a relative measurement rather than an absolute one. Once you have calculated the change, you simply divide it by the original value:

• Relative Change = \( \frac{3,000}{12,000} \)

This fraction tells you how much the original value has been altered in terms of its own size. So, the relative change isn't just about the numeric difference but understanding this difference relative to where you started.

Percentage Calculation

Percentage calculation is a powerful tool for expressing how large or small one quantity is relative to another. Using percentages makes it easy to compare changes at a glance. The percentage change, therefore, gives us a clearer picture of the amount of increase or decrease in terms of 100 parts. Once you have the relative change, converting it into a percentage is straightforward:

• Percent Change = \( \frac{3,000}{12,000} \times 100 \)

This equation turns the relative change into a percentage. Here, multiplying by 100 shifts the decimal two places to the right, converting a proportional value into more intuitive percentage terms.

Hence, the percentage calculation shows that \( \underline{B} \) has increased by 25%. This percentage is what we communicate when we say that \( \underline{B} \) has experienced a 25% increase, making it much easier for anyone to understand the degree of change.

Change in Value

The concept of change in value denotes the actual numerical difference between the final quantity and the initial one. It’s about finding out how much something has grown or shrunk directly in numbers, without yet considering this in the context of its initial size. In this exercise, to find the change in value:

• Subtract the original amount from the new amount: 15,000 - 12,000.
• Resulting in a change of 3,000.

Understanding this change helps to visualize the raw data of our transformation. While it gives us the quantitative shift, it's crucial not to confuse this with the relative or percentage change. The change in value is the first step, laying the groundwork to better interpret and quantify what this change means in relative terms.

This insight into change in value offers a foundation on which percentages are built, aligning the numeric transition with its relativity to the starting point.