Applying equations

1Summary 重點摘要

Applications in four steps: define x, set up the equation, solve, answer. Answers must be sensible, with units, matching the question.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is a word problem?A word problem describes an everyday situation in words, such as 'Mary has 5 apples, 2 more than John — how many does John have?' No equation is given; you must translate the words into an equation and solve it. Word problems test reading and translation: first work out who has more, who has less, and what the total is, then decide whether addition, subtraction or multiplication fits. After a few problems the patterns repeat, and mastering the patterns is the whole game.
2. 2. Step 1: Define unknownStep one is defining the unknown: set the quantity the question asks for as x, with its unit stated. Since the question asks 'how many does John have?', write 'let John have x apples'. The principle is to set x to exactly what is asked; choosing some other quantity forces an extra conversion step and invites mistakes. Common errors are omitting the unit, or setting x to the wrong person and then answering about the wrong one. Begin the line with 'Let' and end it with the unit.
3. 3. Step 2: Write equationStep two is translating the words into an equation, turning words like 'more than', 'less than', 'times' and 'in total' into operations. Mary has 5 apples and 2 more than John means John's amount plus 2 equals 5, so the equation is x + 2 = 5. A good check is to read the equation back into the words: John has 3, and 3 plus 2 equals 5, which matches 'Mary has 2 more than John'. If the sentence read back contradicts the problem, the equation is mistranslated and must be fixed before solving.
4. 4. Step 3: SolveStep three is solving the equation by moving terms to isolate the unknown. From x + 2 = 5, move +2 across to become −2, giving x = 5 − 2 = 3. The rule is that a term changes sign when it crosses the equals sign. Two mistakes recur: forgetting to flip the sign and writing x = 5 + 2 = 7, and simple arithmetic slips. Do one thing per line, write the steps out, and both checking and method marks take care of themselves.
5. 5. Step 4: Write answerStep four is writing the answer, and a bare x = 3 is not an answer. The complete answer is 'John has 3 apples', containing the number 3, the unit, and a full sentence responding to the question. Applied problems always require the unit back in the answer; answers without units lose marks, so 'how many dollars' means dollars and 'how many centimetres' means centimetres. Writing the full sentence also exposes any mistake in what x was set to.
6. 6. CheckChecking means substituting the answer back into the original situation: John has 3, Mary has 2 more, and 3 + 2 = 5 matches the stated 5, so the answer is correct. Use the words and numbers of the original problem, not the equation, because that is the only way to check the translation as well as the arithmetic. Re-solving the equation only rechecks the algebra, while word problems go wrong in translation far more often than in calculation.
7. 7. Two unknownsFor two unknowns, set one as x and express the other through their relationship. If A and B have $100 in total and A has $20 more than B, let B be x so A is x + 20; the total gives x + (x + 20) = 100, simplifying to 2x + 20 = 100, so x = 40, and B has $40 while A has $60. Check: 40 + 60 = 100 and 60 − 40 = 20, so both conditions hold. The common error is assigning letters to both people in a one-unknown equation — the second quantity must be written in terms of x.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Word problem: math in words
• Step 1: Define unknown (let x = what?)
• Step 2: Write equation (from problem)
• Step 3: Solve
• Step 4: Write answer (with units)