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Linear word problems

Algebra word problems — name the unknown before you translate

Algebra word problems are solved by translating one clause at a time. Name the unknown explicitly, convert each phrase into a symbol in the order it appears, then solve the single equation that results. For consecutive integers, letting the smallest be n and writing n, n + 1, n + 2 collapses the whole question into one linear equation.

The arithmetic in these is trivial; the translation is where the marks move. A phrase like 'multiplied by 2 and then increased by 6' has to become 2x + 6 in that order, and questions are written so that the wrong order produces a number that is also on the option list. Writing down what x stands for is the single highest-value habit here.

Try it against the clock. Linear Word Problems runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading.

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How to translate algebra word problems clause by clause

  1. Write down what the unknown represents

    Put 'let n be the smallest integer' on the page. Most wrong answers in this topic are correct values of a quantity the question did not ask for.

  2. Translate in the order the sentence runs

    'Multiplied by 2 and then increased by 6 gives 28' becomes 2x + 6 = 28. Reversing the operations gives 2(x + 6), a different equation with a different answer.

  3. Express consecutive integers from a single letter

    Three consecutive integers are n, n + 1 and n + 2. Summing them gives 3n + 3, so a total of 18 means 3n = 15 and n = 5.

  4. Re-read the question before choosing

    You may have solved for the smallest while the question wanted the largest. Check which quantity is being asked for against what your letter stands for.

Turning English phrases into symbols

The phraseBecomes
Increased by 6+ 6
Multiplied by 2 and then increased by 62x + 6
6 more than twice a number2x + 6
Twice the sum of a number and 62(x + 6)
Three consecutive integersn, n + 1, n + 2

3 sample algebra word problems, with worked solutions

Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.

Question 1

A number multiplied by 3 and then increased by 8 gives 44.

What is the number?

  1. 12
  2. 20
  3. 41
  4. 44
  5. 17
Show the worked solution
  1. 3x + 8 = 44.
  2. x = 12.
  3. Answer: 12.

Question 2

Three consecutive integers add to 30.

What is the smallest of the three integers?

  1. 9
  2. 10
  3. 5
  4. 8
  5. 2
Show the worked solution
  1. Let the smallest be n. Then n + (n+1) + (n+2) = 30.
  2. 3n + 3 = 30, so 3n = 27, n = 9.
  3. Answer: 9.

Question 3

5 times a number, increased by 3, equals 58.

What is the number?

  1. 12
  2. 58
  3. 3
  4. 11
  5. 55
Show the worked solution
  1. 5x + 3 = 58.
  2. 5x = 55.
  3. x = 11.
  4. Answer: 11.

Common mistakes this drill is built from

The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.

Answering for the wrong quantity

Solving correctly for the smallest integer and selecting it when the largest was asked for. The value you found is always on the option list.

Reversing the order of operations

2x + 6 and 2(x + 6) translate two different sentences. The word 'then' fixes the order and is easy to read past.

Using a separate letter for each consecutive integer

Three letters need three equations. Expressing all of them from one unknown leaves a single equation to solve.

Why your answer came out wrong

The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here.

Three consecutive integers add to 18 — is the answer 6?

6 is the middle integer, and the question usually asks for the smallest, which is 5. Because three consecutive integers average to the middle one, dividing the total by 3 lands you on the middle value every time. Check which one is wanted before selecting.

'A number multiplied by 2 and then increased by 6 gives 28' — is the number 8?

It is 11. The equation is 2x + 6 = 28, so 2x = 22 and x = 11. Getting 8 means solving 2(x + 6) = 28 instead, which translates a different sentence. The word 'then' places the addition after the multiplication.

FAQ

How should I set up a consecutive integer problem?

Use one letter. Let the smallest be n, then the next two are n + 1 and n + 2, and their sum is 3n + 3. If they total 18, then 3n = 15 and n = 5. Introducing three separate variables turns a one-line problem into a system you do not need.

What is the difference between '6 more than twice a number' and 'twice the sum of a number and 6'?

The first is 2x + 6 and the second is 2(x + 6), which expands to 2x + 12. They differ by 6 and both results are typically offered as options, so the phrase order has to be read carefully rather than skimmed.

Why do I keep getting the right value but the wrong mark?

Almost always because the question asked for a different quantity than the one you named. If n is the smallest and the question wants the largest, the answer is n + 2. Writing the definition of n on the page makes the final check take two seconds.

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