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To find the increase or decrease in the value of a certain quantity over a period of time; rather than quoting numbers, you can express this comparison as a percent increase or decrease in quantity. Also, if there’s any discrepancy in a value, instead of saying how much a quantity differs from the actual, we can use percent to express the error....Read MoreRead Less

The percent change of a number refers to the quantity by which it has changed from its original value. To calculate the percent change, we must first calculate the value of change, which is the difference between the final and original amount. The percent change is calculated by dividing this change in value by the original amount and multiplying by 100.

Additionally, a ** percent error** is the percent by which the observed amount is different from the actual amount. To calculate the percent error, divide the error amount by the actual amount and multiply by 100.

There are a couple of formulas that will help in calculating the percent change and percent error and they are as follows:

1. Percent change \( =\frac{(\text{new amount}~-~ \text{original amount})}{(\text{original amount})} \times 100\)

(1) When the new amount is greater than the original amount, then it is a percent increase.

(2) When the original amount is greater than the new amount, then it is a percent decrease.

2. Percent error \( =\frac{(\text{amount of error})}{(\text{Actual amount})} \times 100\)

Here, the amount of error = actual amount – observed amount

(1) When the actual amount is greater than the observed amount, then we get a positive amount of error.

(2) When the observed amount is greater than the actual amount, then we get a negative amount of error.

**Example 1:**

**The number 35 is mistakenly read as the number 53. Calculate the percent change between the two numbers.**

**Solution:**

Given,

Original number

Misread number (new number)

The percentage change is computed as follows:

Percent change \( =\frac{(\text{Misread number (new number) – original number})}{(\text{original amount})} \times 100\)

\(=\frac{53~-~35}{35} \times 100\)

= 51%

**Example 2:**

**What would be the percent change in the depth of a pond if it increased from ****10**** ft to ****15**** ft because of heavy rains?**

**Solution:**

Given,

Original depth = 10 ft

New depth = 15 ft

15>10, so this represents a percent increase.

Percent change \( =\frac{(\text{new amount}~ -~ \text{original amount})}{(\text{original amount})} \times 100\)

\(=\frac{15~-~10}{10} \times 100\)

= 50%

Therefore, the depth of the pond has increased by 50%.

**Example 3:**

**A new year’s discount sale was held at the local clothes store, and the price of a dress was reduced from ****$25**** to ****$20****. Calculate the percentage decrease in the price of the dress.**

**Solution: **

Given,

Original cost of the dress =$25

New amount =$20

$20<$25, so it is a percent decrease.

Percent change \( =\frac{(\text{new amount}~ -~ \text{original amount})}{(\text{original amount})} \times 100\)

\(=\frac{20~-~25}{25} \times 100\)

\(=-20\)%

As a result, the price of the dress was reduced by 20%.

**Example 4:**

**I expected ****70**** people to attend a charity concert in a park near my house, but there were actually ****80**** people who attended. Determine the percent error.**

**Solution:**

Given,

Actual number of people who attended = 80

Observed number of people who attended = 70

Error amount = actual amount – observed amount

= 80 – 70

= 10

\(\text{Percent error} = \frac{\text{Error amount}}{\text{Actual amount}} \times 100\)

\(= \frac{10}{80} \times 100\)

\(= 0.125 \times 100\)

= 12.5%

Hence, the percent error is 12.5 %.

**Example 5:**

**John was ****4 ****feet tall when he measured his height. He later discovered his true height to be ****4.5**** feet after using another tape measure. Calculate the percent error in John’s height.**

**Solution:**

Given,

Actual height = 4.5 feet

Observed height = 4 feet

Error amount = actual height – observed height

= 4.5 – 4

= 0.5

\(\text{Percent error} = \frac{\text{Error amount}}{\text{Actual amount}} \times 100\)

\(= \frac{0.5}{4.5} \times 100\)

\(=0.1111 \times 100\)

= 11.11%

Hence, the percent error in John’s height is 11.11 %.

Frequently Asked Questions

The symbol percent “%” denotes that the denominator has a value of 100.

Percentage change and percentage difference are not the same things because percentage difference uses the average of the two numbers as a reference point. While the point of reference in percentage change is one of the numbers.

Percentage error is a measurement of a test’s or experiment’s accuracy. You can use this method to see if the data collection is going well or not. Mathematicians and business professionals alike use it. It is also crucial for students interested in studying economics.

The two kinds of percentage changes are ** incremental** , or increase, and

Divide the value by the total value, then multiply by 100 to get a percentage. The formula for calculating percentages is \(\(frac{value}{total value}) \times 100%\).

Yes, the percentage can be greater than 100 when a specific value is greater than the total value.