Problem 1 - Algebra 1: Concepts and Skills

Question

What operation does decreased by indicate?

Step-by-Step Solution

Answer

'Decreased by' indicates the mathematical operation of subtraction.

Step 1: Identify the Term

Understand that the term 'decreased by' is used in word problems, mathematical sentences or phrases, typically to indicate a mathematical operation.

Step 2: Relate to Mathematical Operation

Relate the term 'decreased by' to the relevant mathematical operation. In mathematical terms, 'decreased by' is associated with subtraction.

Step 3: Formulate a Mathematical Sentence

Formulate a mathematical sentence using an example for clear understanding. If 'a' is decreased by 'b', this can be translated into a mathematical sentence as 'a - b'.

Key Concepts

Subtraction

Subtraction is one of the four basic operations in arithmetic, the others being addition, multiplication, and division. It represents the process of taking one number or quantity away from another. If we consider the numerical expression
\( a - b = c \),
it reads as 'a minus b equals c.' Subtraction is symbolized by the minus sign \( - \). It's important to recognize that subtraction is not commutative, meaning that \( a - b \) is not the same as \( b - a \), unless a and b are the same number.

In practical situations, we use subtraction to determine how much is left after we take some amount away from a total. For example, if you have 10 apples and you give away 3, you are left with 7 apples, which is calculated by the subtraction 10 - 3 = 7. Understanding subtraction is fundamental to solving various mathematical problems, including those that involve algebraic expressions and word problems.

Word Problems

Word problems are a way of expressing mathematical operations through a narrative or a real-life scenario. Solving word problems involves several steps: comprehending the text, identifying the relevant information, translating the words into a mathematical expression or equation, and then solving that equation.

For instance, a word problem might state, 'Sarah had 15 candies, and she gave 8 away. How many candies does she have now?' To solve this, you identify the operation (subtraction) and the numbers involved (15 and 8). You would then write the mathematical sentence 15 - 8 to find the answer. Word problems are designed to evaluate not only your computational skills but also your ability to apply math to real-world scenarios. They encourage critical thinking, and interpreting terms such as 'increased by,' 'decreased by,' or 'more than' is crucial in forming the correct algebraic expressions for the solution.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. For example,
\( 3x + 4 \)
is an algebraic expression where \( x \) is a variable that can represent any number. These expressions are the cornerstone of algebra and are used to create equations that model real-world problems.

Subtraction in Algebraic Expressions

In the context of the term 'decreased by,' we apply subtraction to algebraic expressions. The phrase 'decreased by' would translate to a minus sign in an expression. If you have an algebraic expression that reads 'the quantity of x decreased by 5,' it would be written as
\( x - 5 \).
Algebraic expressions can be manipulated according to the laws of algebra to solve for unknown variables, and understanding how to translate phrases into these expressions is pivotal in solving algebraic equations arising from word problems.