Letters for unknown numbers
A packet of biscuits has some biscuits inside, but you do not know how many. If you call the number p, then 4 packets hold 4p biscuits, whatever p may be. A letter is standing in for a number.
In algebra a letter such as n or x stands for a number we do not know yet. A letter whose value can change is a variable (चल). A plain number like 5 is a constant (स्थिरांक). In 5y, the number 5 is the coefficient (सहगुणक) of y.
In the figure : A letter n hides a number; n + 3 is an expression.
Rule 1: Writing with letters — 3 × n is written 3n and n ÷ 4 is written n/4.
- Easy : A notebook costs ₹ x. 5 notebooks cost 5x.
- Medium : My age is n years. After 3 years it is n + 3.
- Exam-type : n sweets shared by 4 children: each gets n/4.
Rule 2: Variable, constant, coefficient — In 7y + 9: y is the variable, 7 is the coefficient, 9 is the constant.
- Easy : In 5y, the coefficient is 5.
- Medium : In 2x + 6: coefficient 2, constant 6.
- Exam-type : 4m means 4 × m; it does not mean 4 and m side by side as the number 4m.
A letter stands for a number. We can use any letter: n, x, p, y.
Thinking n + 3 is a single number. It is an expression that changes when n changes.
Memory clue: Letter = Lock; the number inside is the key.
Spot the difference
| Word | Meaning |
|---|---|
| Variable | letter whose value can change |
| Constant | fixed number |
| Coefficient | number multiplying the letter |
Practice questions
- 1A notebook costs ₹ x. Write the cost of 5 notebooks.
- 2My age is n years. Write my age after 3 years.
- 3n sweets are shared by 4 children. What does each get?
- 4What is the coefficient in 5y?
- 5Name the variable, coefficient and constant in 7y + 9.
- 6A packet has p biscuits. How many biscuits in 4 packets and in 10 packets?
- 7Write "a number x divided by 6" in algebra.
- 8Raju says 3n is a number between 30 and 39. What is his mistake?
Show answers (try them yourself first)
- 15x
- 2n + 3
- 3n/4
- 45
- 5y; 7; 9
- 64p and 10p
- 7x/6
- 83n means 3 × n. It is not "3 and n written together".
Writing expressions from words
"Ravi has 7 more marbles than Sita" becomes s + 7 when Sita has s marbles. Algebra is a short way to write sentences with numbers.
To write an expression from words, find the key word: plus, more than, increased by → +; less than, decreased by → −; twice, thrice, times → ×; half, one-fourth, shared → ÷.
In the figure : Words and the matching algebra expression.
Rule 1: One-step phrases — Find the key word and write the matching sign.
- Easy : "y increased by 9" is y + 9.
- Medium : "6 less than m" is m − 6 (not 6 − m).
- Exam-type : "one-fourth of p" is p/4 and "thrice k" is 3k.
Rule 2: Two-step phrases and formulae — Write the first operation, then the second, in the order of the words.
- Easy : "3 more than twice a number" is 2n + 3.
- Medium : "5 less than 3 times a number" is 3n − 5.
- Exam-type : Equilateral triangle of side a: perimeter 3a. Rectangle l by b: area lb.
Order matters for subtraction: m − 6 and 6 − m are different.
Writing "6 less than m" as 6 − m. Start with m, then take away 6.
Memory clue: "Less than" flips: 6 less than m = m − 6.
Spot the difference
| Words | Expression |
|---|---|
| y increased by 9 | y + 9 |
| 6 less than m | m − 6 |
| thrice k | 3k |
Practice questions
- 1Write "y increased by 9".
- 2Write "6 less than m".
- 3Write "thrice k".
- 4Write "one-fourth of p".
- 5Write "3 more than twice a number n".
- 6Write "5 less than 3 times a number n".
- 7Write the perimeter of an equilateral triangle of side a.
- 8Meena writes "6 less than m" as 6 − m. Correct her.
Show answers (try them yourself first)
- 1y + 9
- 2m − 6
- 33k
- 4p/4
- 52n + 3
- 63n − 5
- 73a
- 8Start with m and subtract 6: m − 6.
Finding the value of an expression
A shop gives a ₹ 10 coupon on every notebook costing ₹ 3n + 1. To find the cost for a given n, you put a number in place of n. This is finding the value of an expression.
To find the value of an expression, put the number in place of the letter and calculate. Multiply and divide before you add or subtract.
In the figure : Put n = 1, 2, 3, 4 in 2n + 3 to get 5, 7, 9, 11.
Rule 1: Substitute and calculate — Replace the letter with the number; remember 3n means 3 × n.
- Easy : For m = 3: 4m − 7 = 4 × 3 − 7 = 12 − 7 = 5.
- Medium : For n = 5: 3n + 2 = 15 + 2 = 17.
- Exam-type : For x = 12: x/2 + 5 = 6 + 5 = 11.
Rule 2: Table of values — Put n = 1, 2, 3, 4 one by one.
- Easy : n + 6 for n = 1, 2, 3, 4 gives 7, 8, 9, 10.
- Medium : 3n + 1 for n = 1, 2, 3, 4 gives 4, 7, 10, 13.
- Exam-type : 10 − 2x for x = 3 is 10 − 6 = 4.
Multiply first, then add or subtract. 4 × 3 − 7 = 5, not 4 × (3 − 7).
Reading 3n with n = 5 as 35. It is 3 × 5 = 15.
Memory clue: Replace, then multiply, then add.
Spot the difference
| Expression | Value at n = 4 |
|---|---|
| n + 6 | 10 |
| 3n + 1 | 13 |
| 2n + 3 | 11 |
Practice questions
- 1Find 4m − 7 for m = 3.
- 2Find 3n + 2 for n = 5.
- 3Find x/2 + 5 for x = 12.
- 4Find 10 − 2x for x = 3.
- 5Make a table for n + 6 with n = 1, 2, 3, 4.
- 6Make a table for 3n + 1 with n = 1, 2, 3, 4.
- 7A rectangle has length l = 8 cm and breadth b = 5 cm. Find the area lb and the perimeter 2l + 2b.
- 8Anil finds 3n for n = 5 as 35. What is his mistake?
Show answers (try them yourself first)
- 15
- 217
- 311
- 44
- 57, 8, 9, 10
- 64, 7, 10, 13
- 740 cm²; 16 + 10 = 26 cm
- 83n means 3 × 5 = 15.
Simple equations
A balance is level with a bag and 5 marbles on one side and 12 marbles on the other. To find the marbles in the bag, take away 5 from both sides.
An equation (समीकरण) says that two sides are equal, like x + 5 = 12. What you do to one side you must do to the other. To solve it, use the opposite operation.
In the figure : A balance: x + 5 = 12, so x = 7.
Rule 1: One-step equations — To undo adding, subtract. To undo subtracting, add. To undo multiplying, divide. To undo dividing, multiply.
- Easy : y + 8 = 15, so y = 15 − 8 = 7.
- Medium : m − 6 = 9, so m = 9 + 6 = 15.
- Exam-type : 4n = 36, so n = 36 ÷ 4 = 9. And k/5 = 6, so k = 6 × 5 = 30.
Rule 2: Two-step equations — First undo the adding or subtracting, then undo the multiplying. Check by putting the answer back.
- Easy : 2x + 3 = 11 → 2x = 8 → x = 4.
- Medium : 5x − 4 = 21 → 5x = 25 → x = 5.
- Exam-type : Check: 5 × 5 − 4 = 21. Left side = right side.
Simple equations
In the app you can move this figure with your finger.
Open in the appKeep the balance: do the same on both sides.
Doing the step on one side only. Subtract on both sides.
Memory clue: Opposite operation, both sides, then check.
Spot the difference
| Equation | Solution |
|---|---|
| y + 8 = 15 | y = 7 |
| 4n = 36 | n = 9 |
| 2x + 3 = 11 | x = 4 |
Practice questions
- 1Solve y + 8 = 15.
- 2Solve m − 6 = 9.
- 3Solve 4n = 36.
- 4Solve k/5 = 6.
- 5Solve 2x + 3 = 11.
- 6Solve 5x − 4 = 21 and check.
- 7Solve 3x + 2 = 20 and check.
- 8Sita solves x + 5 = 12 as x = 17. What did she do wrong?
Show answers (try them yourself first)
- 1y = 7
- 2m = 15
- 3n = 9
- 4k = 30
- 5x = 4
- 6x = 5; 5 × 5 − 4 = 21
- 73x = 18; x = 6; 3 × 6 + 2 = 20
- 8She added 5. She should subtract: x = 12 − 5 = 7.
Number puzzles and word problems
A magician says: "Think of a number, triple it and subtract 2. I get 19." You can find the number by writing an equation and solving it, or by working backwards.
To solve a word problem: let a letter stand for the unknown, write an equation from the story, solve it and check with the story. A function machine follows a rule; to find the input, work backwards.
In the figure : A function machine: put in 6, get 19.
Rule 1: Story to equation — Translate each phrase into algebra, then solve.
- Easy : "Triple a number, subtract 2, get 19": 3n − 2 = 19 → 3n = 21 → n = 7.
- Medium : "A box with b pencils and 5 loose ones makes 29": b + 5 = 29 → b = 24.
- Exam-type : "Mother is 28 years older than Rita, mother is 40": x + 28 = 40 → x = 12.
Rule 2: Consecutive numbers and machines — Consecutive numbers are n and n + 1. For a machine, work backwards with opposite operations.
- Easy : n + (n + 1) = 41 → 2n + 1 = 41 → n = 20. The numbers are 20 and 21.
- Medium : Machine 4n + 2 gives 22: 22 − 2 = 20; 20 ÷ 4 = 5. Input = 5.
- Exam-type : Check: 4 × 5 + 2 = 22.
Always check the answer in the story, not only in the equation.
Writing 2n + 1 = 19 and getting n = 20 by adding. Subtract 1, then divide by 2: n = 9.
Memory clue: Think – Write – Solve – Check.
Spot the difference
| Step | What to do |
|---|---|
| 1 | Let a letter be the unknown |
| 2 | Write the equation |
| 3 | Solve and check |
Practice questions
- 1I think of a number, triple it and subtract 2 to get 19. Find the number.
- 2A box with b pencils and 5 loose ones make 29 pencils. Find b.
- 3Mother is 28 years older than Rita and mother is 40. Find Rita's age.
- 4Two consecutive numbers add to 41. Find them.
- 5A machine 4n + 2 gives 22. Find the input.
- 6Write and solve: "double a number and add 1 gives 19".
- 7A machine 3n + 1 gives 19. Find the input and check.
- 8Raju solves 2n + 1 = 19 and gets n = 20. Find his error and the right answer.
Show answers (try them yourself first)
- 17
- 224
- 312 years
- 420 and 21
- 55
- 62n + 1 = 19; n = 9
- 719 − 1 = 18; 18 ÷ 3 = 6. Check: 3 × 6 + 1 = 19.
- 8He added 1. Subtract 1 to get 2n = 18, then divide: n = 9.