1. Chapter at a Glance
"Think of a number, double it, add 10, halve it, subtract the number you thought of — your answer is 5!" How does the magician know? The secret is algebra. By using a letter for the unknown number, we can see exactly why such tricks always work. In this chapter we play with number tricks, puzzles and patterns, explain them algebraically, and learn to solve linear equations — the most useful algebra skill of all.
- Using letters to explain "think of a number" tricks.
- Proving patterns with algebra (two-digit and three-digit number puzzles).
- Writing expressions and equations from word statements.
- Solving linear equations in one variable, including variables on both sides.
- Word problems on ages, numbers, money and geometry.
2. Explaining Number Tricks
Let the number be x and follow each step:
| Step | Instruction | Expression |
|---|---|---|
| 1 | Think of a number | x |
| 2 | Double it | 2x |
| 3 | Add 10 | 2x + 10 |
| 4 | Halve it | x + 5 |
| 5 | Subtract your number | 5 |
The x disappears, so the answer is 5 whatever number you start with. Every such trick is designed so that the unknown cancels out.
"Think of a number, add 3, multiply by 4, subtract 12, divide by your number." What is the answer?
Solution: x → x + 3 → 4x + 12 → 4x → 4x ÷ x = 4 (for any non-zero number).
3. Digit Puzzles
A two-digit number with tens digit a and units digit b is 10a + b (not "ab", which would mean a × b).
Take any two-digit number, reverse its digits and add the two numbers. Why is the result always divisible by 11?
Solution: (10a + b) + (10b + a) = 11a + 11b = 11(a + b). The result is 11 times the sum of the digits.
Show that (10a + b) − (10b + a) is always divisible by 9.
Solution: = 9a − 9b = 9(a − b).
For a three-digit number abc: value = 100a + 10b + c. Reversing gives 100c + 10b + a. Difference = 99(a − c), always divisible by 99.
Choose a 2 × 2 square of dates on a calendar. Show that the products of the diagonal pairs always differ by 7.
Solution: Dates: n, n + 1 (top row), n + 7, n + 8 (bottom row). n(n + 8) = n² + 8n; (n + 1)(n + 7) = n² + 8n + 7. Difference = 7.
4. Linear Equations
An equation is a statement that two expressions are equal. A linear equation in one variable has the variable only to the power 1, e.g. 3x + 5 = 20. The value that makes it true is the solution.
Whatever you do to one side of an equation, do the same to the other side. Equivalently, a term can be transposed (moved) to the other side by changing its sign: + becomes −, × becomes ÷.
Solve 3x + 5 = 20.
Solution: 3x = 15 → x = 5. Check: 3(5) + 5 = 20 ✓.
Solve 5x − 7 = 2x + 8.
Solution: 5x − 2x = 8 + 7 → 3x = 15 → x = 5.
Solve (x + 4)/3 − (x − 2)/4 = 2.
Solution: Multiply by 12: 4(x + 4) − 3(x − 2) = 24 → 4x + 16 − 3x + 6 = 24 → x + 22 = 24 → x = 2.
- Remove brackets.
- Clear fractions by multiplying by the LCM of denominators.
- Collect variable terms on one side, numbers on the other.
- Divide by the coefficient.
- Check by substituting back.
5. Word Problems
The sum of three consecutive integers is 72. Find them.
Solution: x + (x + 1) + (x + 2) = 72 → 3x + 3 = 72 → x = 23. Numbers: 23, 24, 25.
A mother is 3 times as old as her son. In 12 years she will be twice as old as him. Find their present ages.
Solution: Son = x, mother = 3x. 3x + 12 = 2(x + 12) → 3x + 12 = 2x + 24 → x = 12. Son 12, mother 36.
The length of a rectangle is 4 cm more than its breadth. Its perimeter is 48 cm. Find its dimensions.
Solution: 2(b + 4 + b) = 48 → 4b + 8 = 48 → b = 10. Breadth 10 cm, length 14 cm.
6. Solved Problems (NCERT Style)
A purse has ₹5 and ₹2 coins, 25 coins in all, worth ₹86. How many of each?
Solution: Let ₹5 coins = x; ₹2 coins = 25 − x. 5x + 2(25 − x) = 86 → 3x + 50 = 86 → x = 12. So 12 coins of ₹5 and 13 coins of ₹2.
The digits of a two-digit number add up to 9. Reversing the digits increases the number by 27. Find the number.
Solution: 9(b − a) = 27 → b − a = 3, and a + b = 9 → b = 6, a = 3. Number = 36.
7. Master Formula Table
| Idea | Expression |
|---|---|
| Two-digit number | 10a + b |
| Three-digit number | 100a + 10b + c |
| Number + reverse (2-digit) | 11(a + b) |
| Number − reverse (2-digit) | 9(a − b) |
| Number − reverse (3-digit) | 99(a − c) |
| Consecutive integers | x, x + 1, x + 2 |
| Consecutive even/odd | x, x + 2, x + 4 |
| Transposing | a + b = c ⇒ a = c − b; ax = c ⇒ x = c/a |
8. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Number tricks | The variable cancels out | Explain why the answer is fixed |
| Digit puzzles | Place value: 10a + b | Divisibility proofs |
| Linear equations | Balance / transposing | Solve and check |
| Fractions in equations | Multiply by LCM | Avoid sign errors |
| Word problems | Translate words → equation | Ages, coins, digits, perimeter |
9. Mind Map
- Let the number be x
- x cancels
- 10a + b
- 11(a+b), 9(a−b)
- Balance rule
- Transpose, check
- Ages, coins
- Perimeter, consecutive
10. Tips and Tricks
- Always write "Let … = x" clearly before forming an equation.
- For three consecutive numbers, using x − 1, x, x + 1 makes the sum simply 3x.
- Substitute your answer back into the original word problem, not just the equation.
- Writing a two-digit number as ab instead of 10a + b.
- Forgetting to change the sign when transposing.
- Multiplying only the first term by the LCM when clearing fractions.
- Mistakes with −(x − 2): it becomes −x + 2.
11. Important One-liners
- A linear equation has its variable only to the first power.
- The solution of an equation makes both sides equal.
- A two-digit number plus its reverse is always a multiple of 11.
- A two-digit number minus its reverse is always a multiple of 9.
- Number tricks work because the starting number cancels out.
- Transposing a term changes its sign.
12. Exam-pattern Questions with Detailed Answers
Solve 7x − 9 = 26.
Show answer
7x = 35 → x = 5.
Answer: x = 5
Write as an equation: "Four more than twice a number is 18."
Show answer
2x + 4 = 18 (so x = 7).
Answer: 2x + 4 = 18
Which of these is a linear equation?
- x² + 1 = 5
- 3x − 2 = 10
- 1/x = 4
- x³ = 8
Show answer
Only (B) has x to the power 1.
Answer: (B)
Solve 4(x − 3) = 2x + 6.
Show answer
4x − 12 = 2x + 6 → 2x = 18 → x = 9.
Answer: x = 9
Solve x/2 + x/3 = 10.
Show answer
Multiply by 6: 3x + 2x = 60 → x = 12.
Answer: x = 12
The sum of three consecutive odd numbers is 81. Find them.
Show answer
x + (x + 2) + (x + 4) = 81 → 3x = 75 → x = 25. Numbers: 25, 27, 29.
Answer: 25, 27, 29
Explain the trick: "Think of a number, multiply by 3, add 6, divide by 3, subtract your number." What is the answer and why?
Show answer
x → 3x → 3x + 6 → x + 2 → 2. The x cancels, so the answer is always 2.
Answer: Always 2
Riya is 6 years older than Sam. The sum of their ages is 38. Find their ages.
Show answer
x + (x + 6) = 38 → x = 16. Sam 16, Riya 22.
Answer: Sam 16, Riya 22
Solve (2x − 1)/3 − (x + 2)/5 = 1.
Show answer
Multiply by 15: 5(2x − 1) − 3(x + 2) = 15 → 10x − 5 − 3x − 6 = 15 → 7x = 26 → x = 26/7.
Answer: x = 26/7
Prove that the sum of any three consecutive integers is divisible by 3.
Show answer
(n − 1) + n + (n + 1) = 3n, which is a multiple of 3 for any integer n.
Answer: Sum = 3n, always divisible by 3
The denominator of a fraction is 3 more than its numerator. If 2 is added to both, the fraction becomes 2/3. Find the fraction.
Show answer
(x + 2)/(x + 5) = 2/3 → 3x + 6 = 2x + 10 → x = 4. Fraction = 4/7.
Answer: 4/7
A three-digit number has digits a, b, c. Show that the number minus the number with its digits reversed is a multiple of 99, and find the difference for 742.
Show answer
(100a + 10b + c) − (100c + 10b + a) = 99a − 99c = 99(a − c). For 742: 99 × (7 − 2) = 495. Check: 742 − 247 = 495 ✓.
Answer: 99(a − c); for 742 the difference is 495
13. Quick Revision Summary
- Use a letter for the unknown; tricks work because it cancels.
- Two-digit number = 10a + b; three-digit = 100a + 10b + c.
- Number ± reverse gives multiples of 11, 9 or 99.
- Solve equations by the balance rule; clear fractions using the LCM.
- Translate words carefully and always check the answer.