Lesson 31: Equation Word Problems

§Text Passage

1Summary 重點摘要

Use equations to solve word problems: translate word problem into equation, solve for answer. Learn to turn life problems into maths.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Shopping ProblemExample: buy 2 books at $x each, pay $100 and receive $32 change — how much is each book? The relation 'cost of two books + change = money paid' gives 2x + 32 = 100. Subtract 32 from both sides for 2x = 68, then divide by 2 for x = 34, so each book is $34. Check: 34 × 2 = 68 and 68 + 32 = 100, matching the payment; the unknown could instead be the change — a different set-up should give the same answer, a good test of whether the equation is sound.
2. Block 2: Sharing MoneyExample: 5 classmates buy a gift together, each contributing $x, and the gift costs $180 — how much each? The relation '5 shares = total price' gives 5x = 180. Dividing both sides by 5 gives x = 36, so each contributes $36. This type needs only one multiplication step. Check: 36 × 5 = 180, exactly the price of the gift; in any pooled-purchase question of 'x dollars each, n people in all', the equation is always n lots of x equal to the total — a fixed form.
3. Block 3: Age ProblemExample: Dad is x years old this year, his son is 12, and Dad is 28 years older — how old is Dad? The relation 'Dad's age - son's age = the difference' gives x - 12 = 28. Adding 12 to both sides gives x = 40, so Dad is 40 this year. Check: 40 - 12 = 28, matching the given difference; age problems have one special feature — the difference between two ages never changes over the years, and this is often the key to the equation.
4. Block 4: Number ProblemExample: a number multiplied by 3 and then increased by 5 equals 20 — what is the number? Let the number be x; 'times 3 plus 5' is written 3x + 5, so the equation is 3x + 5 = 20. Subtract 5 for 3x = 15, then divide by 3 for x = 5. The number is 5; checking, 3 × 5 + 5 = 20, correct. Name the unknown by translating the wording phrase by phrase: 'times 3' is written 3x, 'plus 5' is written + 5, and 'equals 20' is written = 20; matching phrase to phrase keeps the equation faithful.
5. Block 5: How to Write?Turning a word problem into an equation takes three steps: first decide what the unknown is and call it x, such as the price of one book; second find the equal relation in the problem, such as two books plus change equaling the payment; third write the equation from that relation. A right relation is half the battle. Name the unknown after exactly what the question asks: if it asks the price of one book, let one book be x dollars, which makes both the equation and the check more direct. When hunting for the equal relation, watch words like 'in total', 'difference' and 'times as many' — they hint at adding, subtracting and multiplying.
6. Block 6: Check ImportantAlways check after solving by putting the answer back into the story. Example: if each book is $34, two books are $68, and paying $100 returns $32 of change, matching the question exactly, so the solution is right. Checking catches a wrongly named unknown or a wrongly written equation — the final gate for word problems. Put the answer back into the story and recompute, rather than only re-checking the algebra: at $34 each, two books are $68, and $68 plus $32 change is exactly $100. With this habit, most slips are caught before you hand in.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Word problem: set x → write equation → solve → check
• Unknown = what the question asks
• Shopping: total + change = paid
• Sharing: total = each × people
• Always check