Translating Phrases to Expressions: Videos & Practice Problems

Translating phrases into algebraic expressions involves identifying variables and operations such as addition, subtraction, multiplication, and division. Keywords like "sum," "difference," "product," and "quotient" guide the formation of expressions using variables (e.g., $x$, $y$) and operations (e.g., $+$, $-$, $\cdot$, $\frac{a}{b}$). Understanding these terms is essential for constructing polynomials and expressions in standard form, facilitating problem-solving in algebra and real-world contexts.

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" indicate the use of variables. However, algebraic expressions always combine variables with operations such as addition, subtraction, multiplication, or division.

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

Subtraction uses the minus sign (−) and is indicated by words like "difference," "decreased by," "less than," or "minus." 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" suggest 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 (/). Words like "quotient," "divided by," "per," or "out of" indicate division. 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. Multiplying the number of notebooks by the price per notebook gives the total cost for notebooks as \$3x\(. Adding the cost of one pencil, \)1.50, to this product results in the expression for the total cost.

Thus, the algebraic expression representing the total cost is \$3x + 1.5$. This expression clearly shows the relationship between the number of notebooks purchased and the overall amount spent, integrating both variable and constant terms. Writing expressions in this form, with the coefficient preceding the variable, is a standard convention in algebra and helps simplify further calculations or problem-solving steps.

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, the total amount of water after t minutes can be expressed algebraically. Since the rate r indicates how many liters are added each minute, multiplying the rate by the time t gives the total liters added: r × t. Adding this to the initial 10 liters results in the expression 10 + r t. This formula effectively models the linear growth of water volume in the tank over time, where the constant 10 represents the starting amount and the term r t accounts for the increase based on the filling rate and elapsed time.

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 meaning of the phrase by adding 5 to an unknown number.

What keywords indicate addition when translating phrases into algebraic expressions?

When translating phrases into algebraic expressions, keywords that indicate addition include "some," "increased by," "more than," and "plus." These words signal that you should use the addition operation, represented by the plus sign $+$. For example, in the phrase "seven more than a number," the words "more than" tell us to add 7 to the variable representing the number, resulting in the expression $7+x$. Recognizing these keywords helps you correctly form algebraic expressions from verbal phrases.

How do you represent multiplication in algebraic expressions when given phrases like "twice a number" or "the product of a number and one half"?

Multiplication in algebraic expressions can be represented using a multiplication dot $\cdot$ or parentheses. Keywords such as "product," "times," "of," "twice," "double," and "triple" indicate multiplication. For example, "twice a number" means multiplying the number by 2, so if the number is $x$, the expression is $2\cdot x$ or $2\left(x\right)$. Similarly, "the product of a number and one half" translates to multiplying the variable by one half, written as $y\cdot \frac{1}{2}$. Understanding these keywords helps you correctly form multiplication expressions.

What are the common keywords for division when translating phrases into algebraic expressions?

Common keywords that indicate division in phrases include "quotient," "divided by," "per," and "out of." These words suggest using division, which can be represented as a fraction or with a division symbol. For example, the phrase "11 divided by a number" translates to the expression $\frac{11}{x}$, where $x$ is the variable representing the unknown number. Recognizing these keywords allows you to correctly write division expressions from verbal phrases.

How do you translate complex phrases like "the quotient of a number and three increased by seven" into algebraic expressions?

To translate "the quotient of a number and three increased by seven," break the phrase 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 step-by-step approach helps in accurately translating complex phrases into expressions.