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 students new to algebra and word problems, is "decreased by." This article will delve deep into the meaning of "decreased by" in a mathematical context, exploring its applications, providing clear examples, and offering strategies for solving problems involving this phrase. This guide aims to demystify this seemingly simple yet important concept, equipping you with the skills to confidently tackle any mathematical problem involving decrease.

Meta Description: Learn the precise mathematical meaning of "decreased by," explore its applications in various problem types, and master strategies for solving problems involving this phrase. This comprehensive guide provides clear examples and ensures you confidently tackle any decrease-related math problem.

Understanding the Meaning of "Decreased By"

In mathematics, "decreased by" signifies a subtraction operation. When a quantity is "decreased by" a certain amount, it means that the second amount is subtracted from the first. This is a fundamental concept in arithmetic and algebra, forming the basis for solving numerous word problems and equations.

It's important to differentiate "decreased by" from other phrases like "decreased to" or "reduced to." "Decreased to" implies a final result, while "decreased by" focuses on the amount of reduction. We'll explore these distinctions with examples later in the article.

Examples of "Decreased By" in Simple Arithmetic

Let's begin with some straightforward examples to solidify the understanding of "decreased by."

• Example 1: A number is decreased by 5. If the original number is 12, what is the result?

  This translates to 12 - 5 = 7. The number 12 is decreased by 5, resulting in 7.

• Example 2: John had 20 apples. He decreased the number of apples by 8. How many apples does John have left?

  This problem translates to 20 - 8 = 12. John now has 12 apples.

• Example 3: The temperature was 30°C. It decreased by 10°C. What is the new temperature?

  The calculation is 30°C - 10°C = 20°C. The new temperature is 20°C.

These examples showcase the straightforward nature of "decreased by" in simple arithmetic scenarios. The operation always involves subtracting the second number from the first.

"Decreased By" in Algebra and Variable Expressions

The concept extends seamlessly into algebra, where variables represent unknown quantities.

• Example 4: A variable 'x' is decreased by 7. Express this mathematically.

  The expression would be x - 7. This represents the result after decreasing x by 7.

• Example 5: The expression 2y is decreased by 5. Write this as a mathematical expression.

  The expression becomes 2y - 5. This shows the result of decreasing 2y by 5.

These examples demonstrate how "decreased by" translates directly into algebraic expressions, maintaining the core concept of subtraction.

Distinguishing "Decreased By" from "Decreased To"

It's crucial to distinguish between "decreased by" and "decreased to." These phrases have fundamentally different meanings and lead to distinct mathematical operations.

• "Decreased by" indicates the amount of decrease. The operation is subtraction from the original value.

• "Decreased to" indicates the final value after the decrease. To find the amount of decrease, you need to subtract the final value from the initial value.

Let's illustrate this difference:

• Example 6: The price of a shirt was decreased by $10. If the original price was $50, what is the new price?

  This uses "decreased by," so we subtract: $50 - $10 = $40. The new price is $40.

• Example 7: The price of a shirt was decreased to $40. If the original price was $50, by how much was it decreased?

  This uses "decreased to." We find the difference between the initial and final prices: $50 - $40 = $10. The price was decreased by $10.

These examples clearly highlight the difference in meaning and the resulting mathematical operations. Paying attention to these subtle differences is vital for accurate problem-solving.

Solving Word Problems Involving "Decreased By"

Word problems frequently incorporate the phrase "decreased by." Successfully solving these problems requires carefully identifying the key information and translating the words into mathematical expressions.

Example 8: Sarah had $150 in her bank account. She spent $35 on groceries and then decreased her remaining balance by $20 on a new book. How much money does Sarah have left?

1. Identify the initial value: Sarah starts with $150.
2. First decrease: She spends $35 on groceries: $150 - $35 = $115.
3. Second decrease: She decreases her balance by $20: $115 - $20 = $95.
4. Final answer: Sarah has $95 left.

Example 9: A rectangular garden has a length of 25 meters. Its length is decreased by 5 meters during a renovation. What is the new length of the garden?

1. Identify the initial value: The initial length is 25 meters.
2. Decrease: The length is decreased by 5 meters.
3. Calculation: 25 meters - 5 meters = 20 meters.
4. Final answer: The new length of the garden is 20 meters.

These examples demonstrate the step-by-step process of solving word problems involving "decreased by." Always break down the problem into manageable steps, clearly identifying the initial value and the amount of decrease.

"Decreased By" in Percentages

The phrase "decreased by" also appears in percentage problems. This adds another layer of complexity but follows the same underlying principle of subtraction.

Example 10: The price of a laptop was decreased by 20%. If the original price was $800, what is the new price?

1. Calculate the amount of decrease: 20% of $800 is (20/100) * $800 = $160.
2. Subtract the decrease from the original price: $800 - $160 = $640.
3. Final answer: The new price of the laptop is $640.

Example 11: A shop's sales decreased by 15% compared to last month. If last month's sales were $5000, what were this month's sales?

1. Calculate the amount of decrease: 15% of $5000 is (15/100) * $5000 = $750.
2. Subtract the decrease from last month's sales: $5000 - $750 = $4250.
3. Final answer: This month's sales were $4250.

These percentage problems show that "decreased by" still means subtraction, but the amount of decrease needs to be calculated first using the percentage.

Advanced Applications and Further Exploration

The concept of "decreased by" extends to more complex mathematical scenarios, including:

• Sequences and series: Analyzing decreasing arithmetic or geometric sequences involves repeatedly subtracting a constant value or a constant ratio.
• Calculus: The concept of a derivative in calculus deals with the instantaneous rate of change, which often involves analyzing how a function's value "decreases by" a certain amount over a very small interval.
• Financial mathematics: Calculating depreciation, compound interest, or analyzing declining balances all involve applying the concept of "decreased by" in various contexts.

Mastering the understanding of "decreased by" lays a solid foundation for tackling increasingly complex mathematical problems. Its consistent application in different areas emphasizes its importance as a fundamental mathematical operation.

Conclusion

The phrase "decreased by" signifies a straightforward subtraction operation in mathematics. Understanding its meaning is vital for solving various problems, from simple arithmetic to complex algebraic expressions and percentage calculations. By carefully analyzing the problem statement, correctly identifying the initial value and the amount of decrease, and applying the subtraction operation, one can confidently solve any mathematical problem involving this phrase. Remember to distinguish "decreased by" from "decreased to," ensuring accuracy in your calculations and problem-solving approach. With practice and attention to detail, you can master the concept of "decreased by" and confidently tackle a wide range of mathematical challenges.