Translating Phrases to Expressions: Videos & Practice Problems

Translating phrases into algebraic expressions involves identifying variables and operations like addition, subtraction, multiplication, and division. Keywords such as "sum," "difference," "product," and "quotient" guide the formation of expressions using variables (e.g., $x$, $y$) and operations like $+$, $-$, $\cdot$, and fractions for division. Understanding terms like quotient, dividend, and divisor is essential for constructing expressions accurately, which is foundational for working with polynomials, exponents, and more complex algebraic concepts.

Translating Word Phrases to Expressions

Algebra plays a crucial role in everyday life, from calculating exam grades to budgeting vacations and managing home renovations. To effectively apply algebra in real-world situations, it is essential to translate verbal phrases into algebraic expressions. This process involves identifying key words that correspond to variables and mathematical operations.

A variable represents an unknown number or quantity and is typically denoted by letters such as x or y. When translating phrases, words like "a number," "a quantity," or "an unknown value" often indicate the use of a variable. However, algebraic expressions are not just variables alone; they include operations such as addition, subtraction, multiplication, and division.

Addition is represented by the plus sign (+). Keywords indicating addition include "some," "increased by," "more than," and "plus." For example, the phrase "five more than a number" translates to the algebraic expression \$5 + x\(, where \)x\( is the variable representing the unknown number.

Subtraction uses the minus sign (−). Words like "difference," "decreased by," "less than," and "minus" signal subtraction. For instance, "a number decreased by seven" becomes \)a - 7\(, with \)a\( as the variable.

Multiplication can be shown using a multiplication sign (×), a dot (·), or parentheses. Keywords such as "product," "times," and "of" indicate multiplication. Terms like "twice," "double," or "triple" specify the multiplier, meaning multiply by 2 or 3 respectively. For example, "the product of an unknown value and one half" is expressed as \)y \cdot \frac{1}{2}\(, where \)y\( is the variable.

Division is often represented as a fraction or with a slash (/). Keywords include "quotient," "divided by," "per," and "out of." For example, "11 divided by a number" translates to \)\frac{11}{x}\(, where \)x\( is the variable.

Combining these concepts, the phrase "the quotient of a number and three increased by seven" translates to the algebraic expression \)\frac{x}{3} + 7\(. Here, the quotient indicates division of the variable \)x$ by 3, and "increased by seven" adds 7 to the result.

Mastering the translation of verbal phrases into algebraic expressions enhances problem-solving skills and deepens understanding of algebraic concepts. Recognizing keywords and their corresponding operations allows for accurate and efficient formulation of expressions, which is fundamental in both academic and practical applications.

Let $A$ be the number. Translate the following phrase into an algebraic expression.
the sum of $17$ and a number

• A. $17-A$
• B. $17A$
• C. $17+A$
• D. $\frac{17}{A}$

Answer: (C)

Let $A$ be the number. Translate the following phrase into an algebraic expression.
negative nineteen times a number

• A. $-19A$
• B. $-19$
• C. $19A$
• D. $-\frac{A}{19}$

Answer: (A)

Let $A$ be the number. Translate the following phrase into an algebraic expression.
a number divided by $50$.

• A. $\frac{50}{A}$
• B. $\frac{A}{50}$
• C. $50$
• D. $50A$

Answer: (B)

Write the following as an algebraic expression.
negative three times the value of $x$ increased by $5$

• A. $-15x$
• B. $3x+5$
• C. $15x$
• D. $-3x+5$

Answer: (D)

Write the following as an algebraic expression.
one half of the difference between $y$ and $4$

• A. $\frac{1}{2}y-4$
• B. $y-2$
• C. $\frac{1}{2}(y-4)$
• D. $\frac{1}{2}(y+4)$

Answer: (C)

Write the following as an algebraic expression.
negative four divided by the sum of $a$ and $\frac{5}{2}$

• A. $\frac{-4}{(a+\frac{5}{2})}$
• B. $\frac{a+\frac{5}{2}}{-4}$
• C. $\frac{-4}{a}+\frac{5}{2}$
• D. $-4(a+\frac{5}{2})$

Answer: (A)

Translating Word Phrases to Expressions Example 1

Noah buys x notebooks, each costing \$3, and one pencil priced at \$1.50. To represent the total amount of money Noah spends, we create an algebraic expression that combines these costs. Since the number of notebooks is unknown, x stands for the quantity of notebooks purchased. Multiplying the number of notebooks by the cost per notebook gives the total cost for notebooks as \$3 times x. Adding the fixed cost of one pencil, \$1.50, completes the total cost expression.

Mathematically, this is expressed as \$3x + 1.5$, where the coefficient 3 precedes the variable x to follow standard algebraic notation. This expression effectively models the total expenditure based on the number of notebooks bought and the single pencil's price. Understanding how to translate real-world situations into algebraic expressions is a fundamental skill in solving word problems and forms the basis for more complex mathematical reasoning.

Translating Word Phrases to Expressions Example 2

To represent the amount of water in a tank after a certain time, we start with the initial volume and add the water added over time. If the tank initially contains 10 liters of water and is being filled at a rate of r liters per minute, then after t minutes, the additional water added is the product of the rate and time, expressed as r × t. Combining this with the initial 10 liters, the total amount of water in the tank after t minutes is given by the algebraic expression:

\[10 + r t\]

This expression effectively models the situation by accounting for the starting quantity and the continuous increase over time, allowing you to calculate the water volume at any given minute t.

Here’s what students ask on this topic:

How do you translate the phrase "five more than a number" into an algebraic expression?

To translate the phrase "five more than a number" into an algebraic expression, first identify the key components. "Five" is a constant number, so it is represented as 5. The phrase "more than" indicates addition. "A number" is an unknown value, which we represent with a variable, commonly $x$. Putting this together, the expression becomes 5 plus $x$, written as $5+x$. This expression captures the idea of adding five to an unknown number.

What keywords indicate subtraction when translating phrases into algebraic expressions?

When translating phrases into algebraic expressions, keywords that indicate subtraction include "difference," "decreased by," "less than," and "minus." For example, the phrase "a number decreased by seven" uses "decreased by" to signal subtraction. If we let the variable $a$ represent the unknown number, the expression becomes $a-7$. Recognizing these keywords helps you correctly translate verbal phrases into algebraic expressions involving subtraction.

How do you represent multiplication in algebraic expressions when translating phrases?

Multiplication in algebraic expressions is often represented by a dot ($\cdot$) or parentheses rather than the letter "x" to avoid confusion with variables. Keywords that indicate multiplication include "product," "times," "of," as well as terms like "twice," "double," and "triple." For example, "the product of an unknown value and one half" translates to multiplying a variable, say $y$, by $\frac{1}{2}$. The expression is $y\cdot \frac{1}{2}$. Understanding these keywords helps you form correct multiplication expressions.

What is the algebraic expression for "11 divided by a number"?

The phrase "11 divided by a number" involves division. The number 11 is the dividend, and "a number" is an unknown value represented by a variable, commonly $x$. Division is often shown as a fraction or with a division symbol. The algebraic expression is $\frac{11}{x}$, which means 11 divided by $x$. This expression accurately represents the phrase in algebraic form.

How do you translate the phrase "the quotient of a number and three increased by seven" into an algebraic expression?

To translate "the quotient of a number and three increased by seven," break it into parts. "Quotient" indicates division, so divide the variable representing the number, say $x$, by 3, written as $\frac{x}{3}$. The phrase "increased by seven" means add 7 to this quotient. Combining these, the algebraic expression is $\frac{x}{3}+7$. This expression captures the entire phrase accurately.