Question

1. give examples of a 100% increase, 100% decrease, and 100% error. explain each.

Explanation:

100% Increase Example:

Step 1: Define the concept

A 100% increase means the new value is the original value plus 100% of the original value (which is the original value itself). So the formula for a 100% increase of a value \( x \) is \( \text{New Value} = x + 100\%\text{ of }x=x + x=2x \).

Step 2: Provide an example

Let's take the original number of apples as \( x = 5 \). A 100% increase in the number of apples means we add 100% of 5 (which is 5) to the original number. So the new number of apples \(=5 + 5=10\). We can also calculate it using the formula: \( 5\times(1 + 100\%)=5\times(1 + 1)=5\times2 = 10 \). So starting with 5 apples and having a 100% increase results in 10 apples.

100% Decrease Example:

Step 1: Define the concept

A 100% decrease means the new value is the original value minus 100% of the original value. The formula for a 100% decrease of a value \( x \) is \( \text{New Value}=x- 100\%\text{ of }x=x - x = 0 \).

Step 2: Provide an example

Suppose we have \( x = 8 \) dollars in a piggy bank. A 100% decrease in the amount of money means we subtract 100% of 8 (which is 8) from the original amount. So the new amount of money \(=8 - 8 = 0\) dollars. Using the formula: \( 8\times(1- 100\%)=8\times(1 - 1)=8\times0=0 \). So a 100% decrease of 8 dollars leaves us with 0 dollars.

100% Error Example:

Step 1: Define the concept

The percentage error formula is \( \text{Percentage Error}=\frac{\vert\text{Measured Value}-\text{True Value}\vert}{\text{True Value}}\times100\% \). A 100% error means \( \frac{\vert\text{Measured Value}-\text{True Value}\vert}{\text{True Value}}\times100\%=100\% \), which implies \( \vert\text{Measured Value}-\text{True Value}\vert=\text{True Value} \).

Step 2: Provide an example

Let the true value of the length of a stick be \( \text{True Value}=10 \) cm. Suppose a student measures the length as \( \text{Measured Value} = 20 \) cm (or \( \text{Measured Value}=0 \) cm, both cases work). Let's take \( \text{Measured Value} = 20 \) cm. Then the absolute difference \(=\vert20 - 10\vert = 10 \) cm, which is equal to the true value (10 cm). Using the percentage error formula: \( \frac{\vert20 - 10\vert}{10}\times100\%=\frac{10}{10}\times100\% = 100\% \). So if the true length is 10 cm and the measured length is 20 cm, the percentage error is 100%.

Summary of Explanations:

• 100% Increase: Occurs when a quantity doubles (since we add the entire original quantity to itself). For example, 5 apples become 10 apples.
• 100% Decrease: Occurs when a quantity is reduced to zero (since we subtract the entire original quantity from itself). For example, 8 dollars become 0 dollars.
• 100% Error: Occurs when the absolute difference between the measured and true value equals the true value (so the measured value is either double the true value or zero, in simple cases). For example, a true length of 10 cm measured as 20 cm has a 100% error.

Answer:

100% Increase Example:

Step 1: Define the concept

A 100% increase means the new value is the original value plus 100% of the original value (which is the original value itself). So the formula for a 100% increase of a value \( x \) is \( \text{New Value} = x + 100\%\text{ of }x=x + x=2x \).

Step 2: Provide an example

Let's take the original number of apples as \( x = 5 \). A 100% increase in the number of apples means we add 100% of 5 (which is 5) to the original number. So the new number of apples \(=5 + 5=10\). We can also calculate it using the formula: \( 5\times(1 + 100\%)=5\times(1 + 1)=5\times2 = 10 \). So starting with 5 apples and having a 100% increase results in 10 apples.

100% Decrease Example:

Step 1: Define the concept

A 100% decrease means the new value is the original value minus 100% of the original value. The formula for a 100% decrease of a value \( x \) is \( \text{New Value}=x- 100\%\text{ of }x=x - x = 0 \).

Step 2: Provide an example

Suppose we have \( x = 8 \) dollars in a piggy bank. A 100% decrease in the amount of money means we subtract 100% of 8 (which is 8) from the original amount. So the new amount of money \(=8 - 8 = 0\) dollars. Using the formula: \( 8\times(1- 100\%)=8\times(1 - 1)=8\times0=0 \). So a 100% decrease of 8 dollars leaves us with 0 dollars.

100% Error Example:

Step 1: Define the concept

The percentage error formula is \( \text{Percentage Error}=\frac{\vert\text{Measured Value}-\text{True Value}\vert}{\text{True Value}}\times100\% \). A 100% error means \( \frac{\vert\text{Measured Value}-\text{True Value}\vert}{\text{True Value}}\times100\%=100\% \), which implies \( \vert\text{Measured Value}-\text{True Value}\vert=\text{True Value} \).

Step 2: Provide an example

Let the true value of the length of a stick be \( \text{True Value}=10 \) cm. Suppose a student measures the length as \( \text{Measured Value} = 20 \) cm (or \( \text{Measured Value}=0 \) cm, both cases work). Let's take \( \text{Measured Value} = 20 \) cm. Then the absolute difference \(=\vert20 - 10\vert = 10 \) cm, which is equal to the true value (10 cm). Using the percentage error formula: \( \frac{\vert20 - 10\vert}{10}\times100\%=\frac{10}{10}\times100\% = 100\% \). So if the true length is 10 cm and the measured length is 20 cm, the percentage error is 100%.

Summary of Explanations:

• 100% Increase: Occurs when a quantity doubles (since we add the entire original quantity to itself). For example, 5 apples become 10 apples.
• 100% Decrease: Occurs when a quantity is reduced to zero (since we subtract the entire original quantity from itself). For example, 8 dollars become 0 dollars.
• 100% Error: Occurs when the absolute difference between the measured and true value equals the true value (so the measured value is either double the true value or zero, in simple cases). For example, a true length of 10 cm measured as 20 cm has a 100% error.