Understanding Percentages: Decrease, Increase, and Calculation

Understanding Percentages of Quantities

Perhaps you've seen a sale sign that boasts a 50% discount or a company's stock price skyrocketing by 100%. These numbers aren't just random digits; they're percentages, a way of expressing changes or ratios as a fraction of 100%. In this article, we'll dive into the fundamental concepts of percentage decrease, increase, and calculating percentages, providing a solid foundation for navigating this useful mathematical tool.

Percentage Decrease

A percentage decrease, also known as a percentage reduction or a percentage decline, indicates the amount by which a quantity has been reduced. This is expressed as a fraction of the original quantity. For instance, if the price of a textbook falls from $50 to $40, a 20% decrease has occurred.

To calculate a percentage decrease, find the quotient of the original quantity minus the new quantity divided by the original quantity. Multiply this value by 100 to obtain the percentage.

[ \text{Percentage decrease} = \left(\frac{\text{original quantity} - \text{new quantity}}{\text{original quantity}}\right) \times 100 ]

In the textbook example,

[ \text{Percentage decrease} = \left(\frac{50 - 40}{50}\right) \times 100 = 0.2 \times 100 = 20% ]

Percentage Increase

A percentage increase, also known as a percentage gain or a percentage rise, indicates the amount by which a quantity has been increased. This is expressed as a fraction of the initial quantity. For example, if your monthly salary climbs from $2000 to $2400, a 20% increase has occurred.

To calculate a percentage increase, find the quotient of the new quantity divided by the original quantity minus 1, then multiply this value by 100 to obtain the percentage.

[ \text{Percentage increase} = \left(\frac{\text{new quantity}}{\text{original quantity} - 1}\right) \times 100 ]

In the monthly salary example,

[ \text{Percentage increase} = \left(\frac{2400}{2000 - 1}\right) \times 100 = \frac{2400}{1999} \times 100 \approx 120.45% ]

Since a percentage increase cannot exceed 100%, round to 120% and note that this simplifies to 120/100 = 1.2, or 120% as a decimal or fraction.

Calculating Percentages

Once you understand these two concepts, calculating a percentage is straightforward.

1. To calculate a percentage of a quantity, multiply the quantity by the percentage expressed as a decimal or fraction, then round as needed.

2. To find a percentage of a quantity, multiply the percentage expressed as a decimal or fraction by the quantity, then divide by 100.

Here are examples for each:

• Find 15% of 500: ( 0.15 \times 500 = 75 )
• Calculate 7% of a $100: ( \frac{7}{100} \times 100 = 7 )

Incorporating percentages into everyday life and decision making can be a valuable skill. Remember that a percentage is a fraction of 100%, and use the techniques discussed here to work with percentages effectively and accurately!

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