Letter-numbers and general rules
Meera is 4 years older than her brother Tarun. When Tarun is 9, Meera is 13; when he is 15, she is 19. For any age n of Tarun, Meera is n + 4.
A letter-number (variable) is a letter that stands for any number, and its value can change. Writing a rule with a letter gives a general rule that works for every number. Short forms: 5 × n = 5n, n × n = n², n ÷ 5 = n/5, 1 × n = n.
In the figure: Each row follows the same rule: Meera = Tarun + 4. The last row says it for every age n.
Rule 1: A letter stands for a number — Put a number in place of the letter to get an actual value.
- Easy: If n = 6, then n + 4 = 10.
- Medium: A square of side s has perimeter 4s.
- Exam-type: n boxes of 12 pencils hold 12n pencils; for n = 5 that is 60.
Rule 2: Writing products — Drop the × sign; write the number first, then the letters.
- Easy: 5 × n = 5n.
- Medium: n × n = n².
- Exam-type: a × b × 3 = 3ab.
Rule 3: Other short forms — Repeated addition becomes a multiple; division is written as a fraction.
- Easy: n + n + n = 3n.
- Medium: n ÷ 5 = n/5.
- Exam-type: 1 × n = n, and 0 × n = 0.
5n means 5 × n. If n = 3 then 5n = 15, not 53.
Reading 7a as "7 and a" and writing 73 when a = 3. Correct: 7 × 3 = 21.
Number first, letter next, no × sign. Memory clue: "Letters are hidden numbers; side by side means times."
Spot the difference
| Point | Constant | Letter-number |
|---|---|---|
| Value | fixed (4) | can change (n) |
| Example | 4 | n, 4n |
| Use | one situation | every situation |
Practice questions
- 1Write "3 more than n".
- 2Write 5 × n in the short form.
- 3If n = 6, find n + 4.
- 4Write the perimeter of a square of side s.
- 5Write n × n in the short form.
- 6Tarun is n years old and Meera is 4 years older. How old is Meera when Tarun is 20?
- 7What is 7a when a = 3?
- 8Write the number of pencils in n boxes of 12, and find it for n = 5.
Show answers (try them yourself first)
- 1n + 3
- 25n
- 310
- 44s
- 5n²
- 624 (n + 4 = 20 + 4)
- 721
- 812n; 60
Forming expressions with letters
A cab charges ₹50 booking plus ₹12 for every km. For k km the fare is 12k + 50.
An expression is made of terms joined by + or –. In 12k, the number 12 is the coefficient and k is the letter. A term with no letter, like 50, is a constant. The coefficient keeps the sign in front of it (–3y has coefficient –3).
In the figure: 12k + 50 has two terms: 12k (coefficient 12, letter k) and the constant 50.
Rule 1: Words to expression — "More" means +, "less" means –, "times" means ×.
- Easy: 5 more than x = x + 5.
- Medium: Twice a number, decreased by 3 = 2n – 3.
- Exam-type: 7 less than 3 times y = 3y – 7.
Rule 2: Terms and coefficients — Split at + and –; the number multiplying the letter, with its sign, is the coefficient.
- Easy: 6a + 2 has terms 6a and 2.
- Medium: 5x – 3y + 8: coefficients 5 and –3, constant 8.
- Exam-type: a – 7b: the coefficient of a is 1 and of b is –7.
Rule 3: Expressions from situations — Find what is fixed (constant) and what changes with the letter.
- Easy: n notebooks at ₹20 each cost 20n rupees.
- Medium: ₹30 fee plus ₹5 per page p: 5p + 30.
- Exam-type: Perimeter of a rectangle l by b: 2l + 2b.
The coefficient carries the sign in front of it: in 5x – 3y the coefficient of y is –3.
Writing "n less than 5" as n – 5. It is 5 – n; "5 less than n" is n – 5.
The number or letter after "than" stays in front: "3 less than x" = x – 3. Memory clue: "Less than flips": 3 less than x → x – 3.
Spot the difference
| Name | Meaning | In 5x – 3y + 8 |
|---|---|---|
| Term | part joined by + or – | 5x, –3y, 8 |
| Coefficient | number with the letter | 5, –3 |
| Constant | term with no letter | 8 |
Practice questions
- 1Write "5 more than x" as an expression.
- 2Write the cost of n notebooks at ₹20 each.
- 3Write the terms of 6a + 2.
- 4Write "twice a number, decreased by 3".
- 5Write "7 less than 3 times y".
- 6Name the coefficients and the constant in 5x – 3y + 8.
- 7A cab charges ₹50 booking and ₹12 per km. Write the fare for k km.
- 8Write the perimeter of a rectangle of length l and breadth b.
Show answers (try them yourself first)
- 1x + 5
- 220n
- 36a and 2
- 42n – 3
- 53y – 7
- 6coefficients 5, –3; constant 8
- 712k + 50
- 82l + 2b
Evaluating expressions by substitution
The cab fare is 12k + 50. For a 5 km ride the fare is 12 × 5 + 50 = ₹110. Putting a number in place of the letter is called substitution.
To evaluate an expression, replace every letter by its given number (use brackets for negative numbers) and then follow the order of operations. Different values of the letter give different values of the expression.
In the figure: For n = 4: 3n gives 12, then + 2 gives 14. So 3n + 2 = 14 when n = 4.
Rule 1: Substitute whole numbers — Write the number in place of the letter, with × between a number and a letter.
- Easy: x + 7 for x = 5: 5 + 7 = 12.
- Medium: 3n + 2 for n = 4: 3 × 4 + 2 = 14.
- Exam-type: 2a + 3b for a = 5, b = 4: 10 + 12 = 22.
Rule 2: Negative values — Put the negative number in a bracket first, then calculate.
- Easy: x – 3 for x = –2: (–2) – 3 = –5.
- Medium: 4n + 5 for n = –2: 4 × (–2) + 5 = –3.
- Exam-type: 10 – 3n for n = –4: 10 – 3 × (–4) = 10 + 12 = 22.
Rule 3: Table of values — Substitute n = 1, 2, 3, … one by one to see the pattern of the answers.
- Easy: 5n – 1: n = 1, 2, 3, 4 gives 4, 9, 14, 19 (adds 5 each time).
- Medium: 12k + 50 for k = 1, 2, 3 gives 62, 74, 86.
- Exam-type: n² + 1 for n = 1, 2, 3 gives 2, 5, 10.
Write 3n as 3 × n before substituting: with n = 4 it is 3 × 4.
Writing 3n as 34 when n = 4. It is 3 × 4 = 12.
Replace, bracket, calculate. Memory clue: "R-B-C": Replace the letter, Bracket negatives, Calculate in order.
Spot the difference
| Point | Letter form | After substitution |
|---|---|---|
| Example | 3n + 2 | 3 × 4 + 2 |
| Result | depends on n | 14 |
Practice questions
- 1Find x + 7 when x = 5.
- 2Find 3n + 2 when n = 4.
- 3Find 2a + 3b when a = 5, b = 4.
- 4Find x – 3 when x = –2.
- 5Find 4n + 5 when n = –2.
- 6Find the fare 12k + 50 for k = 5.
- 7Find 10 – 3n when n = –4.
- 8For which n is 3n + 2 equal to 17? Try values.
Show answers (try them yourself first)
- 112
- 214
- 322
- 4–5
- 5–3
- 6₹110
- 722
- 8n = 5 (3 × 5 + 2 = 17)
Simplifying expressions
You buy 3 pens at ₹n each and 5 more pens at ₹n each. The total is 3n + 5n = 8n: just 8 pens at ₹n.
Like terms have the same letter part (3n and 5n). Add or subtract their coefficients and keep the letter: pn + qn = (p + q)n. Unlike terms (3n and 5m) cannot be combined. Brackets: a(b + c) = ab + ac, and a minus before a bracket flips the signs.
In the figure: Three n-blocks and five n-blocks make eight n-blocks: 3n + 5n = 8n.
Simplifying expressions
In the app you can move this figure with your finger.
Open in the appRule 1: Combine like terms — Group the terms with the same letters and add their coefficients.
- Easy: 3n + 5n = 8n.
- Medium: 9x – 4x + 2 = 5x + 2.
- Exam-type: 7a + 3b – 2a + b = 5a + 4b.
Rule 2: Unlike terms stay apart — Terms with different letters (or a letter and a constant) cannot be merged.
- Easy: 3n + 5m cannot be simplified.
- Medium: 4x + 3 cannot be simplified.
- Exam-type: 2x + 3y + 4x = 6x + 3y (only the x terms merge).
Rule 3: Brackets — Multiply the outside number with every term inside; a minus sign outside flips the signs inside.
- Easy: 2(n + 3) = 2n + 6.
- Medium: 5(2a – 1) = 10a – 5.
- Exam-type: 3(x + 2) – (x – 4) = 3x + 6 – x + 4 = 2x + 10.
x alone means 1x, so x + 3x = 4x.
Writing 3a + 2b = 5ab. Different letters never merge; keep 3a + 2b.
Underline each kind of term in a different colour, then add the matching ones. Memory clue: "Same letters, same team: add the numbers, keep the letter."
Spot the difference
| Point | Like terms | Unlike terms |
|---|---|---|
| Letters | same | different |
| Example | 3n, 5n | 3n, 5m |
| Can add? | yes: 8n | no |
Practice questions
- 1Simplify 3n + 5n.
- 2Simplify 9x – 4x.
- 3Simplify 2(n + 3).
- 4Simplify 7a + 3b – 2a + b.
- 5Simplify 5(2a – 1).
- 6Simplify 2x + 3y + 4x.
- 7Simplify 3(x + 2) – (x – 4).
- 8Simplify 3(2x + 4) – 6x.
Show answers (try them yourself first)
- 18n
- 25x
- 32n + 6
- 45a + 4b
- 510a – 5
- 66x + 3y
- 72x + 10
- 812 (the x terms cancel)
Patterns and relationships
Matchstick triangles in a row: 1 triangle uses 3 sticks, 2 triangles use 5, 3 triangles use 7. For n triangles we need 2n + 1 sticks.
A pattern can be described by an expression in n, the position number (1, 2, 3, …). This general rule gives any term and also proves relationships. Even number = 2n, odd number = 2n + 1, multiple of 5 = 5n. In a pattern that grows by a fixed step, the step is the coefficient of n.
In the figure: 1 triangle → 3 sticks, 2 → 5, 3 → 7. Each new triangle adds 2 sticks, so the rule is 2n + 1.
Rule 1: Find the rule — Find the fixed step between terms; it is the coefficient of n. Then adjust with a constant.
- Easy: 3, 6, 9, 12 → step 3 → rule 3n.
- Medium: 4, 7, 10, 13 → step 3, first term 4 = 3 + 1 → rule 3n + 1.
- Exam-type: 5, 9, 13, 17 → step 4, first term 5 = 4 + 1 → rule 4n + 1.
Rule 2: Use the rule — Put the position number in place of n to get that term; or put the term equal to the rule to find the position.
- Easy: 10th term of 3n = 30.
- Medium: 20th term of 3n + 1 = 61.
- Exam-type: Which term of 4n + 1 is 41? 4n = 40, so n = 10.
Rule 3: Prove relationships — Write the numbers with letters; if the result is of a known form, the relationship is true for all numbers.
- Easy: n + (n + 1) = 2n + 1: the sum of two consecutive numbers is odd.
- Medium: n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1): a multiple of 3.
- Exam-type: (2a + 1) + (2b + 1) = 2(a + b + 1): the sum of two odd numbers is even.
n is the position number (1, 2, 3, …), not the value of the term.
Writing the rule of 3, 6, 9 as n + 3. Check: n = 1 gives 4, not 3. The rule is 3n.
Always test your rule on the first two terms. Memory clue: "Step = coefficient": the fixed jump is the number beside n.
Spot the difference
| Point | Even number | Odd number |
|---|---|---|
| General form | 2n | 2n + 1 |
| Examples (n = 1, 2, 3) | 2, 4, 6 | 3, 5, 7 |
| Divide by 2 | remainder 0 | remainder 1 |
Practice questions
- 1Write the rule for 3, 6, 9, 12, …
- 2Write the general form of an even number.
- 3Find the 10th term of the rule 3n.
- 4Write the rule for 4, 7, 10, 13, …
- 5How many sticks are needed for 5 triangles if n triangles need 2n + 1?
- 6Find the 20th term of 3n + 1.
- 7Which term of 4n + 1 is 41?
- 8Show that the sum of three consecutive numbers is a multiple of 3.
Show answers (try them yourself first)
- 13n
- 22n
- 330
- 43n + 1
- 511 (2 × 5 + 1)
- 661
- 710th (4n = 40)
- 8n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)