What Does Decreased By Mean In Math

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

The phrase "decreased by" in mathematics signifies a subtraction operation. It indicates that a certain value is being reduced or lessened by another value. Understanding this seemingly simple concept is crucial for mastering various mathematical problems, from basic arithmetic to more complex algebraic equations and word problems. This comprehensive guide will delve into the meaning of "decreased by," exploring its application in different contexts and providing numerous examples to solidify your understanding.

Understanding the Subtraction Operation

Before we dive into the specifics of "decreased by," let's revisit the fundamental concept of subtraction. Subtraction is one of the four basic arithmetic operations, alongside addition, multiplication, and division. It represents the process of finding the difference between two numbers. The number being subtracted is called the subtrahend, and the number from which it's subtracted is the minuend. The result is the difference.

For instance, in the expression 10 - 5 = 5, 10 is the minuend, 5 is the subtrahend, and 5 is the difference. This simple example embodies the core principle of "decreased by": the minuend (10) is decreased by the subtrahend (5), resulting in the difference (5).

"Decreased By" in Word Problems

The phrase "decreased by" frequently appears in word problems, requiring careful interpretation and translation into mathematical expressions. Let's examine several examples to illustrate how to effectively handle these problems:

Example 1: Simple Subtraction

• Problem: John had 20 apples. He gave away 7 apples. How many apples does John have left?

• Translation: This problem directly translates to a subtraction problem. John's initial number of apples (20) is decreased by the number of apples he gave away (7).

• Equation: 20 - 7 = 13

• Solution: John has 13 apples left.

Example 2: Incorporating Units

• Problem: The temperature was 30°C. It decreased by 8°C. What is the new temperature?

• Translation: The initial temperature (30°C) is decreased by 8°C. Note the importance of including units in the problem and the solution.

• Equation: 30°C - 8°C = 22°C

• Solution: The new temperature is 22°C.

Example 3: Multi-Step Problems

• Problem: Sarah had $50. She bought a book for $15 and a pen for $5. How much money does she have left?

• Translation: This problem involves multiple subtraction steps. First, we find the total amount spent ($15 + $5 = $20). Then, we subtract this total from her initial amount. Sarah's initial amount ($50) is decreased by the total amount she spent ($20).

• Equation: $50 - ($15 + $5) = $50 - $20 = $30

• Solution: Sarah has $30 left.

Example 4: Percentage Decrease

• Problem: A shop is having a 20% off sale. A shirt originally costs $40. What is the sale price?

• Translation: This problem involves calculating a percentage decrease. First, we find the amount of the discount (20% of $40 = $8). Then, we subtract this discount from the original price. The original price ($40) is decreased by the discount ($8).

• Equation: $40 - (0.20 * $40) = $40 - $8 = $32

• Solution: The sale price of the shirt is $32.

"Decreased By" in Algebraic Expressions

The phrase "decreased by" also finds its place in algebraic expressions, where variables represent unknown quantities. Let's look at some examples:

Example 5: Simple Algebraic Expression

• Problem: A number, x, is decreased by 5. Write an algebraic expression for the result.

• Translation: The unknown number (x) is decreased by 5.

• Equation: x - 5

Example 6: More Complex Algebraic Expression

• Problem: The sum of two numbers, a and b, is decreased by 10. Write an algebraic expression for the result.

• Translation: The sum of a and b (a + b) is decreased by 10.

• Equation: (a + b) - 10

Common Mistakes to Avoid

While the concept of "decreased by" seems straightforward, some common mistakes can lead to incorrect solutions. These include:

• Confusing Subtraction with Addition: The most common mistake is confusing "decreased by" with "increased by," which indicates addition. Always carefully read the problem statement to identify the correct operation.

• Incorrect Order of Operations: In multi-step problems, make sure to follow the order of operations (PEMDAS/BODMAS) to avoid errors. Parentheses, exponents, multiplication and division (from left to right), then addition and subtraction (from left to right).

• Misinterpreting Percentage Decrease: When dealing with percentage decreases, remember to calculate the amount of the decrease before subtracting it from the original value.

Practical Applications of "Decreased By"

The concept of "decreased by" is widely applicable in various real-world scenarios, including:

• Finance: Calculating discounts, interest deductions, tax reductions, and profit margins.

• Science: Measuring changes in temperature, pressure, volume, and other physical quantities.

• Engineering: Determining the reduction in strength or capacity of materials or structures.

• Everyday Life: Calculating remaining quantities after spending money, consuming resources, or completing tasks.

Conclusion

The phrase "decreased by" in mathematics simply translates to subtraction. While seemingly basic, understanding its application within various mathematical contexts, from simple arithmetic to complex algebraic expressions and word problems, is essential. Mastering this concept strengthens foundational math skills and provides the building blocks for tackling more advanced mathematical concepts. By diligently practicing and paying attention to detail, you can confidently interpret and solve problems involving "decreased by" in any situation. Remember to carefully read the problem, translate the words into a mathematical equation, and solve it systematically, following the order of operations to arrive at the correct answer. Consistent practice and attention to detail are key to mastering this important mathematical concept.