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 scientific notation in algebra.

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 (−) 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 dot (·) or parentheses, especially when variables are involved. Keywords such as "product," "times," and "of" signal 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 operations, the phrase "the quotient of a number and three increased by seven" can be translated into 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 expression formation, which is fundamental in applying algebra to various practical scenarios.

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 3x. Adding the fixed cost of one pencil, \$1.50, to this product results in the expression 3x + 1.5. This algebraic expression effectively models the total cost of buying x notebooks and one pencil.

Writing algebraic expressions from word problems is a fundamental skill in algebra, enabling the translation of real-world situations into mathematical language. In this case, the coefficient 3 represents the price per notebook, and placing it before the variable x follows standard algebraic conventions. The constant term 1.5 accounts for the pencil's cost, which is independent of the number of notebooks. Understanding how to construct and interpret such expressions is essential for solving a wide range of problems involving linear relationships and total costs.

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 for the total volume:

\[10 + r t\]

This expression effectively models the linear relationship between time and the volume of water in the tank, where the constant 10 represents the starting amount and the term r t accounts for the increase over time. Understanding how to construct such expressions is fundamental in algebra, as it allows for the translation of real-world situations into mathematical language, enabling further analysis and problem-solving.

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 word phrases into subtraction expressions.

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 variable is $y$, the expression is $2\cdot y$ or $2\left(y\right)$. Similarly, "the product of a number and one half" translates to $y\cdot \frac{1}{2}$. Understanding these keywords helps you form correct multiplication expressions.

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

The phrase "11 divided by a number" involves division. The keyword "divided by" indicates the division operation. We represent the unknown number with a variable, commonly $x$. The expression is written as 11 divided by $x$, which can be represented as a fraction: $\frac{11}{x}$. Alternatively, it can be written using a division slash as $11/x$. This expression captures the idea of dividing 11 by an unknown number.

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 accurately represents the phrase.