Problem 76 - Elementary Algebra Within Reach

Question

Determine whether order is important when translating each verbal phrase into an algebraic expression. Explain. (a) \(x\) increased by 10 (b) 10 decreased by \(x\) (c) The product of \(x\) and 10 (d) The quotient of \(x\) and 10

Step-by-Step Solution

Answer

In case of (a) \(x + 10\) and (c) \(10x\), order is not important because addition and multiplication are commutative operations. However, in case of (b) \(10 - x\) and (d) \(\frac{x}{10}\), order is important because subtraction and division are not commutative operations.

Step 1: Translating '(a) \(x\) increased by 10'

The phrase 'increased by' suggests addition. Thus, '\(x\) increased by 10' can be written as \(x + 10\). Here, the order is not important as the operation of addition is commutative, i.e., the order of numbers does not change the result.

Step 2: Translating '(b) 10 decreased by \(x\)'

The phrase 'decreased by' suggests subtraction. Thus, '10 decreased by \(x\)' can be written as \(10 - x\). Here, the order is important as the operation of subtraction is not commutative. Swapping the order would change the expression to \(x - 10\) which would not represent the same quantity.

Step 3: Translating '(c) The product of \(x\) and 10'

The phrase 'the product of' suggests multiplication. Thus, 'the product of \(x\) and 10' can be written as \(10x\). Here, the order is not important as the operation of multiplication is commutative.

Step 4: Translating '(d) The quotient of \(x\) and 10'

The phrase 'the quotient of' suggests division. Thus, 'the quotient of \(x\) and 10' can be written as \(\frac{x}{10}\). Here, the order is important as the operation of division is not commutative. Swapping the order would change the expression to \(\frac{10}{x}\) which would not represent the same quantity.

Key Concepts

Addition and Subtraction

Understanding addition and subtraction is vital when dealing with algebraic expressions.

• Addition: When we "increase" a number, we're using addition. The phrase "x increased by 10" becomes "x + 10." Here, the order isn't important due to the commutative property of addition, which means that regardless of the order, the result remains the same. So, "x + 10" is the same as "10 + x".
• Subtraction: Subtraction, however, is not commutative. "10 decreased by x" becomes "10 - x" and here the order is crucial. If you swap the numbers, you'll get a different expression: "x - 10" which changes the meaning of the calculation completely. Think of subtraction as a specific process of "taking away," where the starting point matters.

Remember, identifying whether the concept involves "adding to" or "taking away from" something will dictate whether order matters in your expression.

Multiplication and Division

Multiplication and division also have unique properties that affect the translation of verbal expressions into algebraic ones.

• Multiplication: The process of "finding the product" means multiplying numbers. "The product of x and 10" translates to "10x" or "x \times 10." Thanks to the commutative property, you can write "10x" as "x10" if needed, because multiplication allows for order flexibility.
• Division: For division, expressed as the "quotient," the order becomes significant. "The quotient of x and 10" turns into "\(\frac{x}{10}\)" - here, position matters. If reversed, as "\(\frac{10}{x}\)", the numerical relationship shifts entirely. Division, like subtraction, is not commutative, meaning that the numbers must stay in their given order to maintain the correct relationship.

Keep in mind whether you are dealing with "multiplying" or "dividing" and remember which operations allow for flexibility.

Commutative Property

The commutative property is an especially handy concept in math that applies to certain operations, allowing numbers to switch places without changing their result.

• This property holds for both addition and multiplication. For example, the expression "a + b" is equivalent to "b + a," and similarly, "a \times b" means the same as "b \times a." This interchangeability simplifies calculations and expression translations.
• However, subtraction and division do not enjoy this property, meaning "a - b" is not the same as "b - a," and "\(\frac{a}{b}\)" is not equal to "\(\frac{b}{a}\)." The order in these cases is essential to maintain the integrity of what each operation signifies.

Understanding which operations the commutative property affects ensures accurate and effective problem-solving.

Expression Translation

Translating expressions involves converting verbal phrases into mathematical statements, a skill essential for mastering algebra.

• This process requires recognizing key terms and their associated operations: "increased by" implies addition, "decreased by" implies subtraction, "product of" indicates multiplication, and "quotient of" signals division.
• The order and choice of terms are paramount depending on whether an operation is commutative or not. For expressions like "x increased by 10," understanding this allows flexibility, but for "10 decreased by x," order rules must be strictly observed.
• Being able to effectively translate between verbal and algebraic descriptions enhances one’s ability to interpret and solve mathematical situations accurately.

In realizing the language of algebra, pay attention to detail and the rules governing operations, and translating becomes a simpler task.