Questions: Writing and Interpreting Algebraic Expressions

Question 1

A student translates 'five less than a number' as 5 − n. What is the correct expression, and what went wrong?

• A. Correct expression: n − 5; the student wrote the numbers in reading order, but 'less than' reverses the subtraction
• B. Correct expression: 5 − n; the student is actually right
• C. Correct expression: n + 5; 'less than' should be treated as addition
• D. Correct expression: −5n; 'less than' signals a negative coefficient

Answer: (A)

'Five less than a number' means: start with the number and remove 5 from it — that's n − 5. The phrase 'less than' describes what the result is relative to n, so n comes first in the subtraction. The student read left-to-right and wrote 5 − n, which would mean something different: a number that is n less than 5. Whenever you see 'less than' or 'subtracted from,' the reading order reverses the operation order.

Question 2

Which expression correctly translates 'twice the sum of a number and 4'?

• A. 2n + 4
• B. 2(n + 4)
• C. 2n + 8
• D. n + 8

Answer: (B)

'The sum of a number and 4' signals a grouping — n + 4 must be computed first. 'Twice the sum' means that whole group is multiplied by 2, giving 2(n + 4). Compare this to 'twice a number, plus 4' which gives 2n + 4 — here you double first, then add. The phrase 'the sum of…' acts like parentheses in language, indicating that everything inside belongs together before other operations are applied.

Question 3

The phrase 'the quotient of n and 5, decreased by 4' translates to n/5 − 4, not 4 − n/5.

• T. True
• F. False

Answer: True

'The quotient of n and 5' means n divided by 5, giving n/5. 'Decreased by 4' means subtract 4 from what came before: n/5 − 4. If the expression were '4 decreased by the quotient of n and 5,' that would be 4 − n/5. The phrase 'decreased by' follows the same reading-order logic as 'less than' — you subtract from what was stated before, not the other way around.

Question 4

In the expression 4(n − 7) + 2, the parentheses are optional because multiplication distributes, so you could write 4n − 7 + 2 instead.

• T. True
• F. False

Answer: False

The parentheses change the meaning. 4(n − 7) + 2 distributes to 4n − 28 + 2 = 4n − 26. Without parentheses, 4n − 7 + 2 = 4n − 5, which is a different expression. Removing the parentheses would only multiply the n by 4, leaving the −7 unchanged. The parentheses indicate that the entire quantity (n − 7) is multiplied by 4 — this is exactly the distinction between 'twice the sum of a number and 7' (2(n+7)) and 'twice a number, plus 7' (2n+7).

Question 5

Explain why 'five less than a number' translates to n − 5 and not 5 − n. What is it about the phrase 'less than' that reverses the order?

Model answer: 'Five less than a number' means the result is 5 fewer than the number — so you start with n and subtract 5, giving n − 5. The phrase 'less than' describes the result in relation to the number: the number n is the reference point, and we remove 5 from it. In English, 'less than' reads with the amount first ('five less than') but the math puts the reference quantity first (n − 5). This reversal occurs because 'less than' is a comparison phrase — it identifies the base value that comes second in the sentence but first in the subtraction.

This is one of the most consistently confused translations in algebra. A useful check: substitute a number. 'Five less than 10' should be 5 (ten minus five), not −5 (five minus ten). So n − 5 is correct. The same reversal applies to 'subtracted from': 'n subtracted from 10' is 10 − n, not n − 10.