Linear equations in one unknown · Ages 12–13
Word problems with equations
Choose, form, solve, then check it makes sense.
How to do it
Method
Choose, form, solve, then check it makes sense
5x + 2(10 − x) = 35, 3x + 20 = 35, x = 5
Another way
Suppose them all alike
All $2: $20; (35 − 20) ÷ 3 = 5
In full
Four steps: choose the unknown, write the other quantities with it, form the equation, solve it; then check that the answer makes sense (a number of people can't be a fraction). Ages: everyone gains 1 a year, so a gap in ages never changes. Giving: when A gives B k things, A loses k and B gains k; the total stays, and the gap between them falls by 2k. Coins: total value = number × face value. Say $5 and $2 coins, 10 in all, worth $35. With x $5 coins: 5x + 2(10 − x) = 35, x = 5. Or suppose they were all $2: that is $20, $15 short; each swap to a $5 adds $3, so 15 ÷ 3 = 5 swaps. The same answer.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Dad is 48 and his child is 6. In how many years will Dad be 4 times the child's age?
Show the answer and working
Answer: 8 years
48 + x = 4(6 + x) gives x = 8. Check: in 8 years, 48 + 8 and 6 + 8.
Example 2
$5 and $2 coins, 9 in all, can be worth exactly $25.
True or false?
Show the answer and working
Answer: False
5x + 2(9 − x) = 25 must give a whole x: 25 − 2 × 9 must divide by 3. A number of coins can't be a fraction.
Example 3
Tsz-hin has 36 books, Tsz-ching 12. How many to give for 2 times Tsz-ching's?
Show the answer and working
Answer: 4 books
36 − x = 2(12 + x) gives x = 4. Both change: Tsz-hin loses x, Tsz-ching gains x.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Ming has 2 times Chi-keung's, 45 in all. Chi-keung's is x: x = 45, x = .
Fill in each blank
Answer:
2.Ming buys 5 identical notebooks for $55 in all. With $x for one, the equation is 5x = 55, so x = .
Answer:
3.Ka-lok has some shells, is given 15 more, and then has 27. Say he had x at first: x + = 27, x = .
Fill in each blank
Answer:
4.May has 14 marbles, Ka-yan has 8. How many must May give Ka-yan to match?
Answer:
5.$5 and $2 coins, 18 in all, can be worth exactly $45.
True or false?
Answer:
All the answers
- 1. 3, 15
- 2. 11
- 3. 15, 12
- 4. 3 marbles
- 5. True
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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