How do I translate phrases such as “less than” into algebra?
Quick answer
For a phrase with “less than,” first identify the quantity that is being reduced. “Five less than x” means take 5 away from x, so write x − 5. In “x less than 5,” take x away from 5, so write 5 − x. Don’t translate by swapping in a subtraction sign where the words appear: ask what amount starts the comparison and what is taken from it.
The question, the details, the next step.
I keep reversing subtraction when a problem says “less than.” Is there a reliable way to translate it?
Yes: find the quantity that is being reduced, then subtract the other quantity from it. In “8 less than a number n,” the number n is reduced by 8, so the expression is n − 8. The phrase “8 less than n” does not mean 8 − n.
A useful paraphrase is “take 8 away from n.” That makes the order clear before you write anything. This follows the same idea as “twenty less than a number n”: take 20 from n. [S1]
Also check whether you have a phrase or a complete statement. A phrase such as “8 less than n” translates to an expression, n − 8. A sentence such as “8 less than n is 13” says two quantities are equal, so it translates to the equation n − 8 = 13. [S1]
What changes when the variable comes first, as in “x less than 12”?
The phrase still tells you what is taken away from what. In “x less than 12,” the 12 is reduced by x, so write 12 − x. Compare these two:
- “4 less than x” → x − 4.
- “x less than 4” → 4 − x.
The order of the quantities in the phrase matters. To check your translation, choose a value for x. If x = 10, then “4 less than x” gives 10 − 4 = 6, while “x less than 4” gives 4 − 10 = −6. The different results make it clear why swapping the terms changes the expression. This is a check on the example, not a rule that every “less than” phrase must be interpreted without context.
Can you show how to handle a longer comparison, and how I can tell if the answer makes sense?
Try “7 less than twice a number p.” First identify the quantity being reduced: “twice a number p” is 2p. Then take 7 from that quantity: 2p − 7.
For a quick check, let p = 6. Twice the number is 12; 7 less than 12 is 5. The expression gives 2(6) − 7 = 5, so it matches the paraphrase. By contrast, 7 − 2p gives −5 for this value and describes taking 12 away from 7, not taking 7 away from 12.
Here is a copyable method:
- Mark the two quantities in the phrase.
- Ask: “What amount do I start with, and what is taken away?”
- Write start amount − amount taken away.
- Substitute a simple value if you want to check the order.
What about “the difference of 12 and x”? Is that always the same as “x less than 12”?
In the introductory example “difference of 47 and 19,” subtraction follows the stated order: 47 − 19. [S1] Using that same wording pattern, “the difference of 12 and x” is 12 − x. That also matches “x less than 12.”
However, don’t rely on a familiar phrase alone if the surrounding sentence gives a different comparison or defines the quantities differently. Rewrite it in plain language, identify what is reduced, and check that your expression represents that relationship. The cited introductory material supports the examples above; it does not cover every possible use of “less than,” so context still matters.
A practical checklist
Identify both quantities in the phrase, including any quantity such as “twice a number.”
Ask which quantity starts the comparison and is being reduced.
Write that starting quantity first, then subtract the amount taken away.
Paraphrase your expression in words to see whether it matches the original phrase.
Substitute a simple value for the variable when you need to check subtraction order.
Use an equals sign only when the wording states that two quantities are equal.
A few more useful answers
Does “less than” always mean I should reverse the order of the words?
No. Don’t apply a word-order shortcut by itself. Identify what is being reduced: “5 less than x” is x − 5, while “x less than 5” is 5 − x.
How do I translate “the difference of A and B”?
In the example “difference of 47 and 19,” the subtraction follows the stated order: 47 − 19. Using that wording pattern, “the difference of 12 and x” is 12 − x. [S1]
When should I use an equation instead of an expression?
Use an expression for a quantity, such as “6 less than n” → n − 6. Use an equation when the wording says that quantity equals another value, such as “6 less than n is 10” → n − 6 = 10. [S1]
Sources
References used for this guide.
- [S1] OpenStax Contemporary Mathematics: Algebraic Expressions and Translating Words openstax.org · Checked