What Does Decreased By Mean In Math

What Does "Decreased By" Mean in Math? A Comprehensive Guide

Understanding mathematical terminology is crucial for solving problems accurately. One phrase that often causes confusion, especially for beginners, is "decreased by." This comprehensive guide will delve into the meaning of "decreased by" in various mathematical contexts, providing clear explanations, examples, and practical applications. We'll also explore related concepts and address common misconceptions.

Understanding the Meaning of "Decreased By"

In mathematics, "decreased by" signifies subtraction. It indicates that a certain value is being reduced or lessened by another value. The phrase implies a process of taking away or diminishing a quantity. It's a fundamental operation, essential for solving numerous problems across various mathematical fields.

Key takeaway: When you see "decreased by," you should immediately think of subtraction.

Examples of "Decreased By" in Different Contexts

Let's examine how "decreased by" is used in different mathematical scenarios:

1. Simple Subtraction Problems:

• Problem: A store had 25 apples. They sold 10 apples. How many apples are left?

• Solution: This problem can be expressed as "25 decreased by 10." The calculation is 25 - 10 = 15 apples.

• Another Example: A bank account had $500. A withdrawal of $150 was made. How much money remains?

• Solution: This translates to "$500 decreased by $150." The calculation is $500 - $150 = $350.

2. Word Problems Involving Percentages:

• Problem: The price of a shirt was $50. It was decreased by 20%. What is the new price?

• Solution: First, calculate the amount of the decrease: 20% of $50 = (20/100) * $50 = $10. Then, subtract this amount from the original price: $50 - $10 = $40. The new price is $40. This demonstrates that "decreased by 20%" means reducing the original value by 20% of itself.

3. Algebraic Expressions:

• Problem: Express the phrase "x decreased by 5" algebraically.

• Solution: This is simply written as x - 5. The variable 'x' represents an unknown value, and "decreased by 5" translates directly to subtracting 5 from x.

• Another Example: Express "The quantity (3y + 2) decreased by 7" algebraically.

• Solution: This becomes (3y + 2) - 7, which can be simplified to 3y - 5.

4. Real-world Applications:

The phrase "decreased by" finds frequent use in various real-world scenarios, including:

• Finance: Calculating interest, discounts, profit margins, and changes in stock prices.
• Science: Measuring changes in temperature, pressure, or other physical quantities.
• Engineering: Determining reductions in material strength or efficiency.
• Economics: Analyzing changes in GDP, inflation, or unemployment rates.

Distinguishing "Decreased By" from Other Mathematical Phrases

It's crucial to differentiate "decreased by" from similar-sounding phrases to avoid misinterpretations.

1. "Decreased To":

"Decreased to" implies a final result, not the amount of decrease. For example: "The temperature decreased to 10°C" doesn't tell us how much the temperature fell. We only know the final temperature. This is different from "The temperature decreased by 5°C," which tells us the amount of change.

2. "Reduced By":

"Reduced by" is essentially synonymous with "decreased by." Both indicate a subtraction operation. They are interchangeable in most contexts.

3. "Less Than":

"Less than" indicates a comparison, not a subtraction operation. For instance, "5 less than 10" translates to 10 - 5, but it's not the same as "10 decreased by 5," although they yield the same result.

Common Mistakes and Misconceptions

A frequent error involves misinterpreting percentage decreases. Remember, a percentage decrease is calculated based on the original value, not the final value.

Advanced Applications and Extensions

The concept of "decreased by" extends into more advanced mathematical concepts:

1. Calculus:

In calculus, the concept of decrease is central to understanding derivatives and rates of change. The derivative of a function at a point represents the instantaneous rate of decrease (or increase) of the function at that point.

2. Statistics:

In statistics, "decreased by" is used to describe changes in variables over time or across different groups. For example, one might analyze how a company's sales figures decreased by a certain percentage year over year.

3. Linear Algebra:

In linear algebra, decreasing a vector by another vector involves subtracting the components of one vector from the corresponding components of the other.

Conclusion: Mastering "Decreased By"

Understanding the phrase "decreased by" is fundamental to mathematical proficiency. It's a simple concept, but grasping its nuances is critical for accurate problem-solving in various contexts. By carefully examining the different applications and avoiding common mistakes, you can confidently tackle mathematical problems involving decreases and confidently interpret word problems accurately. Remember the key takeaway: "decreased by" always means subtraction. Apply this knowledge consistently, and your mathematical skills will significantly improve. Practice with diverse examples, and you'll soon master this crucial concept. This detailed explanation, enriched with examples and real-world applications, should provide a solid foundation for your understanding. Continue exploring different mathematical concepts, and your abilities will only grow stronger.