Translate Words Into Math Expressions

What does translating words into math mean?

Translating words into math means converting written or spoken language into a mathematical format that can be calculated or solved.

We change the math language to expressions by systematically replacing vocabulary words with addition, subtraction, multiplication, and division symbols. We also replace unknown quantities with letters called variables.

This translation is the foundation of algebra. It allows us to take a real-world scenario, represent its facts symbolically, and then apply mathematical rules to find a solution.

Identify quantities and relationships

The first step in choosing operations in word problems is recognizing the specific vocabulary words that indicate how numbers interact.

We must also identify any unknown value. We assign a letter, such as $n$, $x$, or $y$, to represent this unknown quantity.

Once the variables are set, we look for math operation words and symbols that tell us whether to add, subtract, multiply, or divide.

When you translate words into math expressions, read the problem carefully. Note every given number and determine exactly how it connects to the unknown variable.

Expressions versus equations

An expression represents a single mathematical value, whereas an equation contains an equals sign and states that two values are exactly the same.

When you translate numerical expressions or algebraic expressions, you combine numbers, variables, and operations without an equals sign. The phrase "four times a number" translates to the expression $4x$.

An equation occurs when the sentence includes verbs like "is", "equals", "yields", or "results in". The sentence "Four times a number is twenty" translates to the equation $4x=20$.

Identify whether the prompt asks you to write an expression or an equation before you begin translating.

Phrases that reverse order

Certain subtraction phrases reverse the order of the numbers as they appear in the English sentence. This is a crucial rule when you write expressions from words.

The phrases "less than" and "subtracted from" mean the first value is taken away from the second value.

For example, the phrase "seven less than a number" does not translate to $7-n$. Because the seven is being taken away from the number, the correct translation is $n-7$.

The operation of subtraction is not commutative, meaning the order fundamentally changes the result. Always swap the order for "less than" and "subtracted from". However, the phrase "decreased by" maintains the original order. "A number decreased by seven" is $n-7$.

Use grouping symbols

Sentences that contain phrases like "the sum of" or "the difference of" alongside multiplication or division often require parentheses. Grouping symbols dictate the order of operations when evaluating numerical expressions.

If a sentence says "five times the sum of a number and two", the multiplication applies to the entire sum, not just the number.

Without parentheses, writing $5x+2$ means "five times a number, plus two". To correctly show that the sum happens first, we use parentheses: $5(x+2)$.

Use parentheses whenever you multiply or divide by a quantity that is described as a "sum" or a "difference".

Worked examples

Translating word phrases into algebra becomes easier with practice. Following a step-by-step method helps you avoid missing hidden operations when solving multi-step word problems.

Example 1: Translating a basic relationship

Question: Translate the following phrase into an algebraic expression: "Twelve more than the product of a number and three."

Method:

1. Assign a variable to the unknown. Let $y$ be "a number".
2. Identify the first operation: "the product of a number and three" translates to $3y$.
3. Identify the second operation: "Twelve more than" means we add $12$ to the previous term.

Answer: $3y+12$.

Check: The expression preserves the order: the product happens first, and then $12$ is added to it.

Example 2: Using reversal phrases in an equation

Question: Translate the following sentence into an algebraic equation: "Nine less than half of a number is fifteen."

Method:

1. Assign a variable. Let $m$ be the number.
2. Translate the first portion: "half of a number" is $\frac{1}{2}m$ or $\frac{m}{2}$.
3. Apply the reversal phrase: "Nine less than" means we subtract $9$ from the first portion. This gives $\frac{1}{2}m-9$.
4. Translate the equals sign: The word "is" translates to $=$.
5. Set the expression equal to the result: fifteen.

Answer: $\frac{1}{2}m-9=15$.

Check: Since "less than" reverses order, the subtraction of $9$ correctly appears after the half term.

Example 3: Translating with grouping symbols

Question: Translate the following phrase into an algebraic expression: "One quarter of the difference between a number and eight."

Method:

1. Let $w$ be the unknown number.
2. Identify the grouped operation: "the difference between a number and eight". Because this happens together, place it in parentheses: $(w-8)$.
3. Identify the outside operation: "One quarter of" means multiplying by $\frac{1}{4}$ or dividing by $4$.
4. Combine the steps, multiplying the fraction by the grouped difference.

Answer: $\frac{1}{4}(w-8)$.

Check: If the parentheses were omitted, $\frac{1}{4}w-8$ would wrongly mean one quarter of a number, minus eight. The parentheses correctly group the difference.

Common mistakes

When you translate words into expressions, watch out for these frequent errors:

• Ignoring order in subtraction: Translating "six less than a number" as $6-x$ instead of $x-6$.
• Missing parentheses: Translating "double the sum of a number and four" as $2n+4$ instead of $2(n+4)$.
• Confusing expressions and equations: Adding an equals sign when the phrase does not contain verbs like "is" or "equals".
• Mixing up quotient order: Translating "the quotient of five and a number" as $\frac{n}{5}$ instead of $\frac{5}{n}$. The first value mentioned is always the numerator.

Frequently asked questions

What are the key words for subtraction?

Common words for subtraction include difference, minus, decreased by, less than, subtracted from, and deduct. Remember that "less than" and "subtracted from" swap the order of the numbers.

How do I know what letter to use for a variable?

Unless the problem specifies a letter, you can choose any letter. The letters $x$, $y$, and $n$ are the most common, but choosing a letter that matches the context (like $c$ for cost or $d$ for distance) can be helpful.

Does "of" always mean multiplication?

When used with fractions or percentages, "of" usually indicates multiplication. For example, "half of a number" translates to $\frac{1}{2}\times n$ or $\frac{1}{2}n$.

What is the difference between "less" and "less than"?

"Less" maintains the order, while "less than" reverses it. "Ten less a number" translates to $10-n$. "Ten less than a number" translates to $n-10$.