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From Words to Algebra: Translating Statements into Expressions and Equations

Capítulo 5

Estimated reading time: 6 minutes

A Reliable 4-Step Method for Translating Words into Algebra

Translating a sentence into algebra is less about memorizing keywords and more about matching the relationship described in the sentence. Use this repeatable process each time:

Identify the unknown and define a variable.
Choose operations based on how quantities are related.
Write the expression or equation.
Quick reasonableness check by plugging in a simple value.

This chapter focuses on turning common phrases into correct expressions and equations, especially when word order can trick you.

Step 1: Identify the Unknown and Define a Variable

Start by asking: “What number or quantity am I trying to find?” Then name it with a variable.

“A number” → let n be the number.
“The length of a rectangle” → let L be the length.
“The total cost” → let C be the total cost.

If there are two unknowns, define both clearly.

“The number of adults and children” → let a be adults and c be children.

Step 2: Choose Operations Based on Relationships (Meaning First)

Many words suggest operations, but the safest approach is to read the relationship in plain language and then choose the operation that matches it.

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Phrase	Meaning	Typical algebra form
sum of	add	`a + b`
difference of	subtract (order matters)	`a - b`
product of	multiply	`ab` or `a(b)`
quotient of	divide (order matters)	`a / b`
more than	add to something	`x + k`
less than	subtract from something (order matters)	`x - k` or `k - x` depending on wording
twice	multiply by 2	`2x`
half	divide by 2	`x/2`
at least	greater than or equal to	`x ≥ k`
at most	less than or equal to	`x ≤ k`

Important comparison language:

“is”, “equals”, “is the same as” → write an equals sign =.
“at least” → ≥ (could be equal or bigger).
“at most” → ≤ (could be equal or smaller).

Step 3: Write the Expression or Equation

An expression names a quantity (no equals sign). An equation states two quantities are equal (has =). An inequality compares using <, >, ≤, or ≥.

Step 4: Quick Reasonableness Check (Plug In an Easy Value)

After writing your algebra, test it by choosing a simple number for the variable (like 10). Then compare the result to the sentence’s meaning.

Example check: If the sentence says “5 less than a number,” and you let n = 10, the result should be 5 less than 10, which is 5. That helps you confirm the correct order.

Common Translation Patterns (with Step-by-Step Examples)

“Sum,” “Difference,” “Product,” “Quotient”

Example A (sum): “The sum of a number and 7.”

Unknown: the number → let n be the number.
Relationship: sum means add.
Expression: n + 7
Check: if n = 10, then n + 7 = 17, which matches “10 plus 7.”

Example B (difference): “The difference of a number and 7.”

Let n be the number.
Difference means subtract, but order depends on wording; here it’s written in the order given.
Expression: n - 7
Check: if n = 10, result is 3, which is “10 minus 7.”

Example C (product): “The product of 4 and a number.”

Let n be the number.
Product means multiply.
Expression: 4n
Check: if n = 10, 4n = 40.

Example D (quotient): “The quotient of a number and 5.”

Let n be the number.
Quotient means divide; keep the order given.
Expression: n/5
Check: if n = 10, n/5 = 2.

“More than” and “Less than” (Order Can Flip)

These phrases are common sources of mistakes because English word order doesn’t always match algebra order.

Example E: “8 more than a number.”

Let n be the number.
“More than” means add 8 to the number.
Expression: n + 8
Check: if n = 10, you get 18, which is 8 more than 10.

Example F (tricky): “5 less than a number.”

Let n be the number.
“5 less than” means subtract 5 from the number (the number comes first in subtraction).
Expression: n - 5 (not 5 - n)
Check: if n = 10, then n - 5 = 5. That matches “5 less than 10.” If you had written 5 - n, you’d get -5, which doesn’t match the phrase.

Example G (another tricky one): “A number is 12 less than 50.”

Let n be the number.
“12 less than 50” means 50 - 12.
Equation: n = 50 - 12
Check: right side is 38, so the sentence says the number is 38.

“Twice” and “Half”

Example H: “Twice a number increased by 3.”

Let n be the number.
Twice a number is 2n, then increased by 3 means add 3.
Expression: 2n + 3
Check: if n = 10, then 2n + 3 = 23. That matches “twice 10 is 20, plus 3 is 23.”

Example I: “Half of a number minus 4.”

Let n be the number.
Half of a number is n/2, then subtract 4.
Expression: n/2 - 4
Check: if n = 10, then 10/2 - 4 = 1.

Turning Statements into Equations: “Is” Means Equals

When a sentence says one quantity is another quantity, you are being told they are equal.

Example J: “The sum of a number and 9 is 20.”

Let n be the number.
Sum of a number and 9 → n + 9.
“is 20” means equals 20 → n + 9 = 20.
Check: try n = 11. Then 11 + 9 = 20, true.

Example K: “Three times a number is 27.”

Let n be the number.
Three times a number → 3n.
Equation: 3n = 27.
Check: n = 9 makes it true.

At Least / At Most: Intro to Inequalities

Some statements don’t give one exact value; they give a range.

At least means the value is not smaller than the number: ≥.
At most means the value is not larger than the number: ≤.

Example L: “A number is at least 12.”

Let n be the number.
At least 12 → n ≥ 12.
Check: n = 12 works; n = 20 works; n = 5 does not.

Example M: “The total cost is at most 50 dollars.”

Let C be the total cost.
At most 50 → C ≤ 50.
Check: C = 50 works; C = 40 works; C = 60 does not.

Targeted Practice: Tricky Phrasing and Comparisons

Practice Set 1: “Less than” and “More than” (Write an Expression)

For each phrase: (1) define a variable, (2) write the expression, (3) check using n = 10 (or another easy value).

1) “7 less than a number”
2) “7 more than a number”
3) “A number less 7” (common shorthand)
4) “12 less than twice a number”
5) “Half a number more than 3”

Answer key (expressions)

1) n - 7
2) n + 7
3) n - 7
4) 2n - 12
5) n/2 + 3

Practice Set 2: “Is” Means Equals (Write an Equation)

Translate each into an equation. Then do a quick check by plugging in a value that makes the equation true if you can guess one.

6) “The product of a number and 6 is 42.”
7) “A number decreased by 9 is 15.”
8) “The quotient of a number and 4 is 3.”
9) “Twice a number plus 1 is 17.”

Answer key (equations)

6) 6n = 42
7) n - 9 = 15
8) n/4 = 3
9) 2n + 1 = 17

Practice Set 3: At Least / At Most (Write an Inequality)

10) “A number is at least 0.”
11) “The temperature is at most 30 degrees.”
12) “You need at least 8 points.”

Answer key (inequalities)

10) n ≥ 0
11) T ≤ 30
12) p ≥ 8

A Mini Checklist for Avoiding Common Translation Errors

Circle the unknown in the sentence and name it with a variable before writing anything else.
Watch subtraction and division order: “less than” and “quotient of” often require careful ordering.
Look for “is”: it usually signals an equation with =.
Do a plug-in check with an easy value (like 10) to see if the expression matches the sentence’s meaning.

Now answer the exercise about the content:

Which algebraic expression correctly translates the phrase “5 less than a number” (let n be the number)?

You are right! Congratulations, now go to the next page

You missed! Try again.

“5 less than a number” means subtract 5 from the number, so the number comes first: n - 5. A quick check: if n = 10, n - 5 = 5, which is 5 less than 10.