Problem 75 - Elementary Algebra Within Reach

Question

Determine which verbal phrase(s) is (are) equivalent to the expression \(n+4\). (a) 4 more than \(n\) (b) the sum of \(n\) and 4 (c) \(n\) less than 4 (d) the ratio of \(n\) to 4 (e) the total of 4 and \(n\)

Step-by-Step Solution

Answer

The verbal phrases equivalent to the expression \(n+4\) are (a) '4 more than \(n\)', (b) 'the sum of \(n\) and 4', and (e) 'the total of 4 and \(n\)'.

Step 1: Identifying phrases equivalent to the expression \(n+4\)

Analyze each option individually: (a) '4 more than \(n\)' indicates \(n+4\). (b) 'The sum of \(n\) and 4' is the straightforward description of \(n+4\). (c) '\(n\) less than 4' gives \(4-n\), which is different from the given expression. (d) 'The ratio of \(n\) to 4' suggests division, \(n/4\), not addition. (e) 'The total of 4 and \(n\)' stands for \(4+n\) which is equivalent to \(n+4\) due to the commutative property of addition.

Step 2: Determining the correct verbal phrases

The equivalent verbal phrases for \(n+4\) from the options given are (a) '4 more than \(n\)', (b) 'the sum of \(n\) and 4', and (e) 'the total of 4 and \(n\)'.

Key Concepts

Verbal Phrases in Math

Understanding verbal phrases in mathematics is crucial for translating written language into mathematical expressions. These phrases are often used in word problems and require careful interpretation to accurately transform them into algebraic expressions. Let's explore some common verbal phrases that you might encounter:

• **"More than"**: This phrase usually indicates addition. For example, "4 more than \( n \)" translates to \( n + 4 \).
• **"Total of" or "Sum of"**: Both phrases suggest adding two or more quantities. For example, "the sum of \( n \) and 4" means \( n + 4 \).
• **"Less than"**: This phrase often implies subtraction, but the order is important. For instance, "\( n \) less than 4" actually means \( 4 - n \), which is different from \( n + 4 \).
• **"Ratio of"**: Indicates division, such as in "the ratio of \( n \) to 4" which becomes \( n/4 \).

Interpreting these phrases correctly is key to solving mathematical problems, ensuring that they are converted into equivalent algebraic expressions. Keeping a list of common translations can be very helpful.

Equivalent Expressions

In algebra, two expressions are equivalent if they have the same value for all possible values of the variables involved. Identifying equivalent expressions allows you to recognize different forms of the same mathematical quantity.For the expression \( n + 4 \):

• "4 more than \( n \)" directly translates to \( n + 4 \).
• "The sum of \( n \) and 4" is a straightforward way to describe \( n + 4 \).
• "The total of 4 and \( n \)" emphasizes that order does not matter, confirming \( 4 + n \) is equivalent to \( n + 4 \).

These equivalent expressions reflect the same mathematical operation but in different forms, showing the diversity in which one can express the same quantity. Recognizing equivalent expressions aids in simplifying complex equations and solving problems efficiently.

Commutative Property of Addition

The commutative property of addition is a fundamental principle that states that numbers can be added in any order without changing the sum. Mathematically, it is represented as:\[a + b = b + a\]This property is helpful when working with expressions such as \( n + 4 \). It shows that this expression is equivalent to \( 4 + n \) due to the commutative property:

• The expression ''4 more than \( n \)'' is \( n + 4 \).
• ''The total of 4 and \( n \)'' can be written as \( 4 + n \); both are equivalent due to the commutative property.

Understanding this property allows you to rearrange terms to make calculations simpler or to combine like terms more easily. It's important in algebra as it provides flexibility in formulating and solving mathematical problems.