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Word Problems: from Story to Equations

Read the story, choose letters for the unknowns, and write the equations it describes. The page tells you whether your equations say what the story says and, if not, which part of the story is missing, then solves them step by step. New numbers give a new story of the same kind.

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Lesson

The theory — Word Problems: from Story to Equations

A word problem describes quantities in sentences. Translating it means choosing unknowns, letters for the quantities the story does not give, and writing each fact the story states as an equation. When every equation is linear, each unknown appearing only multiplied by a number, the system is solved by elimination: add multiples of one equation to another until one unknown is left. A system has one solution, none, or infinitely many. Which of the three the system has, the elimination finds; whether the story allows the answer, a negative volume or a month before a package starts, is checked against the story itself.

What each symbol means

x, y, z
the unknowns. What each stands for is written down first, and a system is only right for the letters it was written in
a₁x + b₁y = c₁
one linear equation. Each fact in the story becomes one
0 = c
what elimination leaves when c ≠ 0: the facts contradict each other and there is no answer
0 = 0
what elimination leaves when one fact repeats another. It adds nothing; if fewer equations than unknowns remain, there are infinitely many answers

A jar of coins and a mixture are the same problem 🖖

Press A jar of coins and write its equations: x + y = 21 for the count, 0.10x + 0.20y = 2.90 for the value. Now press Mixing two strengths: x + y = 45 for the volume, 0.25x + 0.60y = 20.7 for the amount of the substance. Two stories with nothing in common, and one shape of equations: a total, and the same total with each part weighted by what one unit of it is worth. Tickets at two prices and money invested at two rates are that shape again. The algebra is identical each time; what each story tests is whether you see which total is being weighted by what.

A correct solution can answer the wrong question 🖖

Press Two workers. The job is done by 4 h of one worker and 10 h of the other, or by 6 h and 5 h. With the hours alone as the unknowns the equations are not linear: 4/A + 10/B = 1. Written in rates, the share of the job each does in an hour, it is: 4x + 10y = 1 and 6x + 5y = 1, which give x = 1/8 and y = 1/20. The question asks how long each would take alone, and the answer is the reciprocal: 8 h and 20 h, not 1/8 and 1/20. Press Catching up and give Jonas's time the letter meant for Lena's: the system still solves, cleanly, and the page says your equations fit the story only with x and y swapped, because the answer gives each driver the other one's time.

Some systems have no answer, and the elimination says why 🖖

Press Plans that never meet. Gym A costs €25 to join and Gym B €55, and both charge €19 a month. Eliminate and everything with an unknown cancels, leaving 0 = −30 (or 0 = 30, with Gym B written first): the €30 gap is there on day one and no number of months closes it. Press One plan, two descriptions. Gym B's package is €166 for 6 months, which is exactly Gym A's €10 plus six months at €26, and from then on both charge €26. Eliminate and you get 0 = 0: the two equations are one line. Gym B's equation only describes its bill from month 6, so it is the story, not the elimination, that says every month from the sixth is an answer. A contradiction, and too few independent facts, are the two ways a linear system can fail to have one solution, and each has a plain reason in the story.

Problem solved in full

  1. The jar of coins, written and solved 5 steps

    A jar holds 21 coins, all 10-cent or 20-cent coins, worth €2.90 together. Write the equations and find how many coins of each kind there are. Press A jar of coins and type them in to check.

    Open this problem in the tool

    1. Choose the unknowns and say what each stands for:

    2. One fact counts the coins, the other counts their value in euros:

    3. Multiply the second equation by 10 to clear the decimals, then take away the first:

    4. Put y back into the first equation:

    5. Check against the story, not only against the equations:

    Answer

    There are 13 ten-cent coins and 8 twenty-cent coins. The check in the last step is the one that counts: 21 coins, worth €2.90, which is what the story said, rather than a pair of numbers that merely satisfies the equations someone wrote.

· Reviewed against published sources. Corrections welcome.