1. Chapter at a Glance
Suppose a pen costs ₹12. Then 2 pens cost 2 × 12, 5 pens cost 5 × 12, and n pens cost 12 × n. The letter n stands for "any number of pens". Such a letter is a letter-number (a variable). Using letters, one short expression can describe infinitely many situations at once. This is the beginning of algebra. In this chapter you will form expressions using letters, find their values, simplify them by combining like terms, and use them to describe patterns.
- Using letters to stand for numbers.
- Writing expressions for situations involving unknown or changing quantities.
- Finding the value of an expression for given values of the letters.
- Terms, coefficients, like and unlike terms.
- Simplifying expressions by combining like terms.
- Describing patterns and formulas with letter-numbers.
2. Letters as Numbers
A letter-number is a letter that stands for a number. The number may be unknown or may vary. Common letters are a, b, c, n, x, y.
- If Shabnam is 3 years older than Aftab and Aftab's age is a years, Shabnam's age is a + 3. This expression works for every year of their lives.
- If a box holds 6 mangoes, then b boxes hold 6 × b mangoes. We usually write this as 6b (the × sign is dropped between a number and a letter).
- If x sweets are shared equally among 4 children, each gets x ÷ 4, written x/4.
- 6 × b is written 6b (number first). b × 6 is also written 6b.
- 1 × a is written simply a.
- a × b is written ab; a × a is written a².
- Division is written as a fraction: y ÷ 3 = y/3.
3. Forming Expressions from Situations
| Situation | Expression |
|---|---|
| 5 more than a number n | n + 5 |
| 7 less than a number p | p − 7 |
| Three times a number y | 3y |
| Half of a number m | m/2 |
| 4 more than twice a number k | 2k + 4 |
| Cost of x kg rice at ₹55/kg and 2 kg dal at ₹d/kg | 55x + 2d |
| Perimeter of a square of side s | 4s |
| Perimeter of a rectangle (length l, breadth b) | 2l + 2b = 2(l + b) |
"7 less than p" is p − 7, not 7 − p. "p subtracted from 20" is 20 − p.
Tickets for a show cost ₹80 for adults and ₹50 for children. Write the total cost for a adults and c children.
Solution: Cost = 80a + 50c. For 3 adults and 4 children: 80 × 3 + 50 × 4 = 240 + 200 = ₹440.
4. Finding the Value of an Expression
To evaluate an expression, replace each letter by its given value (substitution) and calculate, following the order of operations.
Find the value of 3x + 2y − 5 when x = 4 and y = 7.
Solution: 3 × 4 + 2 × 7 − 5 = 12 + 14 − 5 = 21.
The formula for the perimeter of a rectangle is 2(l + b). Find it for l = 12 cm, b = 7.5 cm.
Solution: 2 × (12 + 7.5) = 2 × 19.5 = 39 cm.
5. Terms, Coefficients, Like and Unlike Terms
Just as with arithmetic expressions, algebraic expressions are made of terms joined by + or −. In 5x − 3y + 8:
- The terms are 5x, −3y and 8.
- The number multiplying a letter is its coefficient: the coefficient of x is 5, of y is −3.
- A term with no letter (8) is a constant term.
Like terms have the same letter part (same letters with the same powers): 4a and −9a are like; 3xy and 7yx are like. Unlike terms have different letter parts: 2x and 2y; 5a and 5a².
6. Simplifying Expressions
Like terms can be added or subtracted by adding or subtracting their coefficients — this follows from the distributive property: 4a + 9a = (4 + 9)a = 13a. Unlike terms cannot be combined into one term.
- Remove brackets (change signs if there is a minus before the bracket).
- Group like terms together, each with its sign.
- Add or subtract the coefficients of like terms.
- Write the result with the letter terms first and the constant last.
Simplify 7x + 3y − 2x + 5 − y + 4.
Solution: (7x − 2x) + (3y − y) + (5 + 4) = 5x + 2y + 9.
Simplify 4(2a + 3) − (a − 6).
Solution: 8a + 12 − a + 6 = 7a + 18.
7. Describing Patterns with Letters
Letters are powerful for describing patterns. Look at a row of squares made with matchsticks:
The first square needs 4 sticks; each new square needs only 3 more (it shares one side). So n squares need 1 + 3n sticks. Checking: n = 10 gives 31 sticks.
Other patterns
- Odd numbers: the nth odd number is 2n − 1.
- Multiples of 7: the nth multiple is 7n.
- A calendar: in any 2 × 2 block of dates with top-left a, the dates are a, a + 1, a + 7, a + 8. Diagonal sums are equal: a + (a + 8) = (a + 1) + (a + 7) = 2a + 8.
- Sum of three consecutive numbers n, n + 1, n + 2 is 3n + 3 = 3(n + 1) — always three times the middle number.
Show that the sum of any two consecutive odd numbers is a multiple of 4.
Solution: Two consecutive odd numbers are 2n − 1 and 2n + 1. Their sum = 4n, which is always a multiple of 4.
8. Solved Problems (NCERT Style)
Write expressions for: (a) the cost of p pencils at ₹6 each and q erasers at ₹4 each; (b) the number of days in w weeks and 3 days; (c) the change from ₹100 after buying 3 items costing ₹m each.
Solution: (a) 6p + 4q; (b) 7w + 3; (c) 100 − 3m.
Are 2(x + 3) and 2x + 3 equal for all values of x?
Solution: Try x = 1: 2(4) = 8 but 2 + 3 = 5. They differ, so they are not equal. In fact 2(x + 3) = 2x + 6, which is always 3 more than 2x + 3.
Simplify 6a + 4 − 2(a − 3) and find its value when a = 5.
Solution: 6a + 4 − 2a + 6 = 4a + 10. For a = 5: 20 + 10 = 30. (Check with the original: 30 + 4 − 2 × 2 = 30 ✔.)
A triangle has sides x cm, (x + 2) cm and (2x − 1) cm. Write its perimeter and find it for x = 6.
Solution: Perimeter = x + x + 2 + 2x − 1 = 4x + 1. For x = 6: 25 cm.
Think of a number, add 5, double it, subtract 10 and divide by 2. Show that you always get back your number.
Solution: Let the number be n. Steps: n + 5 → 2n + 10 → 2n → n. The final answer is always n, which explains the trick.
A tiled path has 2 grey tiles in the first design, 5 in the second and 8 in the third. How many tiles are in the 20th design?
Solution: The number goes up by 3 each time, so the rule is 3n − 1 (check: n = 1 → 2, n = 2 → 5). For n = 20: 60 − 1 = 59 tiles.
9. Master Formula Table
| Situation / rule | Expression |
|---|---|
| n more than x | x + n |
| n less than x | x − n |
| n times x | nx |
| Square perimeter / area | 4s / s² |
| Rectangle perimeter / area | 2(l + b) / lb |
| nth odd number / nth even number | 2n − 1 / 2n |
| Matchstick squares in a row | 3n + 1 |
| Combining like terms | ax + bx = (a + b)x |
| Removing bracket after minus | −(a − b) = −a + b |
| Three consecutive numbers' sum | 3(n + 1) |
10. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Letter-numbers | Stand for unknown or varying numbers | Writing conventions |
| Forming expressions | Translate words into symbols | Word problems |
| Evaluation | Substitute and calculate | Formulas |
| Like terms | Same letter part | Identify, combine |
| Simplification | Group like terms | Brackets with signs |
| Patterns | General rule with n | Matchsticks, calendars |
11. Mind Map
- Unknowns, variables
- 6b, ab, a², x/4
- Ages, costs
- Perimeter, area
- Substitute values
- Like terms
- Simplify
- 3n + 1 sticks
- Calendar blocks
- Consecutive numbers
12. Tips and Tricks
- Test your expression with a small number: if 1 box has 6 mangoes, does your expression give 6 when b = 1?
- To find a pattern rule, look at how much is added each step — that number multiplies n.
- Put brackets around a substituted negative or decimal value to avoid sign errors.
- Combine like terms one letter at a time.
- Adding unlike terms: 3x + 2y is not 5xy.
- Writing "5 less than x" as 5 − x.
- Thinking 2a means a number with digits 2 and a (it means 2 × a).
- Forgetting to multiply every term inside a bracket: 3(x + 4) is 3x + 12, not 3x + 4.
13. Important One-liners
- A letter-number stands for a number that may be unknown or may change.
- 5 × n is written 5n.
- An expression with letters takes different values for different values of the letters.
- Terms are joined by + or − signs.
- The coefficient is the number multiplying the letter.
- Like terms have the same letter part.
- Only like terms can be combined into a single term.
- The perimeter of a square of side s is 4s.
- n squares in a row need 3n + 1 matchsticks.
- The sum of three consecutive numbers is three times the middle one.
14. Exam-pattern Questions with Detailed Answers
Which expression means "8 less than three times y"?
- 8 − 3y
- 3y − 8
- 3(y − 8)
- 3 − 8y
Show answer
Three times y is 3y; 8 less than that is 3y − 8.
Answer: (B) 3y − 8
Find the value of 5a − 2b when a = 6 and b = 4.
Show answer
5 × 6 − 2 × 4 = 30 − 8 = 22.
Answer: 22
Pick out the like terms: 4xy, 3x, −7xy, 9y, 2yx, x².
Show answer
4xy, −7xy and 2yx all have letter part xy (order does not matter).
Answer: 4xy, −7xy, 2yx
Simplify: 9m − 4n + 3 − 5m + 7n − 8.
Show answer
(9m − 5m) + (−4n + 7n) + (3 − 8) = 4m + 3n − 5.
Answer: 4m + 3n − 5
Rina is r years old. Her father is 4 years more than 3 times her age. Write his age and find it when Rina is 11.
Show answer
Father's age = 3r + 4. For r = 11: 33 + 4 = 37 years.
Answer: 3r + 4; 37 years
Simplify 3(2x − 5) − 2(x − 4).
- 4x − 7
- 4x − 23
- 8x − 7
- 4x + 7
Show answer
6x − 15 − 2x + 8 = 4x − 7.
Answer: (A) 4x − 7
Matchsticks form triangles in a row, each new triangle sharing a side with the previous one. The first needs 3 sticks, two need 5. Write a rule for n triangles.
Show answer
Each new triangle adds 2 sticks. Starting with 1 stick and adding 2 per triangle: 2n + 1. Check: n = 1 → 3, n = 2 → 5 ✔.
Answer: 2n + 1 matchsticks
The length of a rectangle is 3 cm more than its breadth b. Write expressions for its perimeter and area.
Show answer
Length = b + 3. Perimeter = 2(b + 3 + b) = 2(2b + 3) = 4b + 6. Area = b(b + 3).
Answer: Perimeter 4b + 6; area b(b + 3)
In a calendar, a 3 × 3 block of dates has centre date c. Write all nine dates in terms of c and show that their sum is 9c.
Show answer
Rows: c − 8, c − 7, c − 6; c − 1, c, c + 1; c + 6, c + 7, c + 8.
The positive and negative parts cancel in pairs: sum = 9c.
Answer: Sum = 9c
A taxi charges ₹50 as a fixed fare plus ₹14 per km. Write an expression for a journey of d km. How far can you travel for ₹414?
Show answer
Fare = 50 + 14d. For ₹414: 14d = 414 − 50 = 364, so d = 26 km.
Answer: 50 + 14d; 26 km
Show that the difference between a two-digit number and the number formed by reversing its digits is always a multiple of 9.
Show answer
A two-digit number with tens digit a and ones digit b is 10a + b. Reversed: 10b + a.
Difference = (10a + b) − (10b + a) = 9a − 9b = 9(a − b), a multiple of 9.
Answer: Difference = 9(a − b)
Find the value of 2p² − 3pq + q when p = 2 and q = 3.
Show answer
2p² = 2 × 2 × 2 = 8. 3pq = 3 × 2 × 3 = 18. q = 3.
Value = 8 − 18 + 3 = −7.
Answer: −7
15. Quick Revision Summary
- Letters stand for numbers; write 5 × n as 5n, a × a as a².
- Translate words carefully: "7 less than p" = p − 7.
- Evaluate by substituting values and following the order of operations.
- Like terms (same letter part) combine by adding coefficients; unlike terms stay separate.
- Letters describe patterns: n squares → 3n + 1 sticks; nth odd number 2n − 1.