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State Board · Class 6 · Mathematics

Lesson 9: Equations

5 concepts · 40 practice questions with answers · Free to read

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Order of operations

From daily life

A shopkeeper sells a notebook for ₹ 20 and 2 pens for ₹ 5 each. The bill is 20 + 2 × 5 = ₹ 30, not 110. The order in which we do operations decides the answer.

Simple definition

An expression (राशी) is an arrangement of numbers and operations. To get one answer for everybody, we follow one fixed order: brackets first, then × and ÷ from left to right, then + and − from left to right.

Rule 1: Order of operations — Brackets → × and ÷ (left to right) → + and − (left to right). Example: 5 + 3 × 4 = 5 + 12 = 17, but (5 + 3) × 4 = 8 × 4 = 32.

Order of operations: brackets, then ÷ and ×, then + and −.1Brackets2÷ and ×3+ and −8 + 4 × 2 − 6= 8 + 8 − 6= 16 − 6 = 10Multiply first, then add and subtract

In the figure : In the figure the order of operations with an example.

  • Easy : Easy: 9 − 2 × 3 = 9 − 6 = 3.
  • Medium : Medium: 48 ÷ 6 × 2 = 8 × 2 = 16 (÷ and × go left to right).
  • Exam-type : Exam: 72 ÷ (3 + 6) − 2 × 3 = 72 ÷ 9 − 6 = 8 − 6 = 2.

Rule 2: Many brackets — Start from the innermost bracket and move outwards: 30 − [6 + (28 ÷ 4 − 5)] = 30 − [6 + (7 − 5)] = 30 − [6 + 2] = 30 − 8 = 22.

  • Easy : Easy: 10 − (2 + 3) = 10 − 5 = 5.
  • Medium : Medium: 25 − [3 + (12 ÷ 3)] = 25 − [3 + 4] = 25 − 7 = 18.
  • Exam-type : Exam: 50 − [8 + (45 ÷ 5 − 4)] = 50 − [8 + 5] = 50 − 13 = 37.
Remember

Brackets first. Then × and ÷. Then + and −. Same level: left to right.

Common mistake

Adding first in 8 + 4 × 2 and getting 24. Multiply first: 8 + 8 = 16.

Shortcut

Memory clue: "कंस, गुणा-भागा, मग बेरीज-वजाबाकी" (Brackets, then × ÷, then + −).

Spot the difference

StepWhat to do
1Solve the brackets
2Multiply / divide, left to right
3Add / subtract, left to right

Practice questions

Easy
  1. 1Find 6 + 2 × 5.
  2. 2Find (6 + 2) × 5.
  3. 3Find 20 − 12 ÷ 4.
Medium
  1. 4Find 36 ÷ 6 × 3.
  2. 5Find 100 − (20 + 5 × 4).
  3. 6Find 45 ÷ (2 + 3) + 7 × 2.
Challenging
  1. 7Find 40 − [5 + (18 ÷ 3 − 2)].
  2. 8Why is 8 + 4 × 2 equal to 16 and not 24?
Show answers (try them yourself first)
  1. 16 + 10 = 16.
  2. 28 × 5 = 40.
  3. 320 − 3 = 17.
  4. 46 × 3 = 18.
  5. 5100 − (20 + 20) = 100 − 40 = 60.
  6. 69 + 14 = 23.
  7. 740 − [5 + 4] = 40 − 9 = 31.
  8. 8Multiplication is done before addition: 4 × 2 = 8, then 8 + 8 = 16.

Equality and its properties

From daily life

A balance scale has a 5 kg weight on each pan, so it stays level. Two things that are equal are like a balanced scale.

Simple definition

Two expressions are equal (समान) when their values are the same. Doing the same thing to both sides keeps them equal.

Rule 1: Checking equality — Find the value of each side. 4 × 6 = 12 + 12 (both are 24), but 18 ÷ 3 ≠ 10 − 3 (6 and 7).

  • Easy : Easy: 9 + 6 = 3 × 5 (both 15), so they are equal.
  • Medium : Medium: 40 ÷ 8 and 2 + 4 are 5 and 6, so they are not equal.
  • Exam-type : Exam: Find the missing number: 25 + 5 = 18 + ☐. Left side is 30, so ☐ = 12.

Rule 2: Same operation on both sides — Add, subtract, multiply by or divide by (a non-zero number) the same number on both sides and the equality stays. From 8 + 2 = 5 × 2 (10 = 10), adding 4 gives 14 = 14 and multiplying by 3 gives 30 = 30.

Doing the same thing to both sides keeps the balance.6 + 45 × 210 = 1010 + 510 + 5Add 5 to both sides: still equal (15 = 15)

In the figure : In the figure the same operation on both sides keeps the balance.

  • Easy : Easy: 7 + 3 = 10; subtract 2 from both sides: 8 = 8.
  • Medium : Medium: 12 = 3 × 4; divide both sides by 2: 6 = 6.
  • Exam-type : Exam: 9 + 6 = 5 × 3 (15 = 15). Multiply both sides by 2: 30 = 30, still equal.
Remember

Same change on both sides = balance stays. Never divide by 0.

Common mistake

Adding 3 only on one side. The scale then tilts.

Shortcut

Memory clue: "जे डावीला, तेच उजवीला" (What you do to the left, do to the right).

Spot the difference

OperationEquality stays?
Add same number to both sidesYes
Multiply both sides by same numberYes
Add only on one sideNo

Practice questions

Easy
  1. 1Is 7 + 8 = 3 × 5?
  2. 2Is 24 ÷ 4 = 9 − 2?
  3. 3Find ☐: 14 + 6 = 11 + ☐.
Medium
  1. 410 = 4 + 6. Add 5 to both sides. What do you get?
  2. 56 × 4 = 24. Divide both sides by 4.
  3. 6Find ☐: 36 ÷ 6 = 2 × ☐.
Challenging
  1. 75 + 4 = 9. Multiply the left side by 2 and the right side by 3. Is it still equal?
  2. 8Why can we not divide both sides by 0?
Show answers (try them yourself first)
  1. 1Yes, both are 15.
  2. 2No, 6 ≠ 7.
  3. 320 = 11 + ☐, so ☐ = 9.
  4. 415 = 15.
  5. 56 = 6.
  6. 66 = 2 × ☐, so ☐ = 3.
  7. 7No: 18 ≠ 27. The same number must be used on both sides.
  8. 8Division by 0 has no meaning.

Letters for numbers and formulas

From daily life

One notebook costs ₹ 12. Then n notebooks cost 12 × n rupees. The letter n stands for any number of notebooks.

Simple definition

A letter that stands for an unknown or changing number is a variable (चल). Rules written with letters are formulas (सूत्र).

Rule 1: Words to expressions — "5 more than x" is x + 5, "3 less than x" is x − 3, "4 times x" is 4x, "20 minus x" is 20 − x. The number goes before the letter and the × sign is dropped: 3 × a = 3a.

  • Easy : Easy: "9 more than m" is m + 9.
  • Medium : Medium: "20 minus n" is 20 − n; if n = 8 it is 12.
  • Exam-type : Exam: "7 times a number, then 2 more" is 7x + 2; for x = 3 it is 23.

Rule 2: Formulas — The diameter of a circle is twice the radius: d = 2r. The perimeter of a square is P = 4a. Perimeter of a rectangle is P = 2(l + b). Profit = S − C (selling price minus cost price).

A letter rule: the diameter is two radii, d = 2r.Orrdd = r + rd = 2 × rd = 2r

In the figure : In the figure the rule d = 2r.

  • Easy : Easy: r = 7 cm gives d = 2 × 7 = 14 cm; a = 5 cm gives P = 4 × 5 = 20 cm.
  • Medium : Medium: Rectangle l = 8 cm, b = 5 cm: P = 2 × (8 + 5) = 26 cm.
  • Exam-type : Exam: S = ₹ 540 and C = ₹ 450 give profit = 540 − 450 = ₹ 90.
Remember

Letter = unknown number. Number before letter. 3 × a = 3a.

Common mistake

Writing "5 less than x" as 5 − x. It is x − 5.

Shortcut

Memory clue: "पेक्षा जास्त → +; पेक्षा कमी → −; पट → गुणाकार" ("more than" → +, "less than" → −, "times" → ×).

Spot the difference

WordsExpression
5 more than xx + 5
3 less than xx − 3
4 times x4x

Practice questions

Easy
  1. 1Write "6 more than y".
  2. 2Write "5 less than a".
  3. 3Write "8 times p".
Medium
  1. 4Write "15 minus x".
  2. 5Find the value of 3a + 2 when a = 4.
  3. 6A square has side 6 cm. Find its perimeter using P = 4a.
Challenging
  1. 7One pen costs ₹ p. Write the cost of 7 pens. If p = 9, how much is it?
  2. 8Write "5 times a number, then 3 subtracted".
Show answers (try them yourself first)
  1. 1y + 6.
  2. 2a − 5.
  3. 38p.
  4. 415 − x.
  5. 53 × 4 + 2 = 14.
  6. 64 × 6 = 24 cm.
  7. 77p; 7 × 9 = ₹ 63.
  8. 85x − 3.

Equations in one variable

From daily life

A bag of unknown weight x and 5 kg are on one pan; 12 kg is on the other, and the scale is level. So x + 5 = 12. The weight of the bag is the number we must find.

Simple definition

An equation (समीकरण) says that two sides are equal and has a variable. The value of the variable that makes both sides equal is the solution (उकल).

Rule 1: Solution by trial — Put values in place of the variable. For x + 3 = 8: x = 4 gives 7 (not 8), x = 5 gives 8, so the solution is x = 5.

The equation y + 6 = 7 balances only when y = 1.y + 67y = 0: 6 ≠ 7 y = 1: 7 = 7 ✓The solution is y = 1

In the figure : In the figure y + 6 = 7 balances for y = 1.

Live figure

Equations in one variable

In the app you can move this figure with your finger.

Open in the app
  • Easy : Easy: x + 2 = 9 is true for x = 7.
  • Medium : Medium: 2y = 14 is true for y = 7 (2 × 7 = 14).
  • Exam-type : Exam: Is z = 8 a solution of z − 4 = 6? 8 − 4 = 4 ≠ 6, so no; z = 10 gives 6 = 6, so yes.

Rule 2: Words to equations — "3 times a number is 21" gives 3x = 21. "A number subtracted from 40 leaves 25" gives 40 − x = 25.

  • Easy : Easy: "A number plus 8 is 15" gives x + 8 = 15.
  • Medium : Medium: "6 less than a number is 9" gives n − 6 = 9.
  • Exam-type : Exam: "Twice a number, increased by 3, is 17" gives 2x + 3 = 17.
Remember

Equation = equal sides + a variable. Solution makes both sides equal.

Common mistake

Calling x + 5 an equation. It has no "=" sign, so it is only an expression.

Shortcut

Memory clue: "समीकरण = तराजू; उकल = समतोल करणारी किंमत" (equation = scale, solution = the value that balances it).

Spot the difference

ItemExample
Expressionx + 5
Equationx + 5 = 12
Solutionx = 7

Practice questions

Easy
  1. 1Solve x + 7 = 12.
  2. 2Solve y − 3 = 9.
  3. 3Solve 4a = 20.
Medium
  1. 4Is x = 3 a solution of 2x + 1 = 7?
  2. 5Write an equation: 5 times a number is 35.
  3. 6Is m = 6 a solution of m + 9 = 14?
Challenging
  1. 7A number subtracted from 30 leaves 18. Write the equation and find the number.
  2. 8A pan has 2 equal bags and 1 kg; the other has 9 kg; the scale is level. Write the equation and find a bag's weight.
Show answers (try them yourself first)
  1. 1x = 5.
  2. 2y = 12.
  3. 3a = 5.
  4. 4Yes, 2 × 3 + 1 = 7.
  5. 55x = 35.
  6. 6No, 6 + 9 = 15 ≠ 14.
  7. 730 − x = 18, so x = 12.
  8. 82x + 1 = 9, so 2x = 8 and x = 4 kg.

Solving equations and word problems

From daily life

A shopkeeper says a notebook and a bag together cost ₹ 700 and the bag costs ₹ 100 more. To find the price of each, we write an equation and solve it.

Simple definition

To solve an equation, do the same operation on both sides until the variable is alone. Always check by putting the answer back.

Rule 1: Solving equations — x + 6 = 20 gives x = 20 − 6 = 14. 5x = 45 gives x = 45 ÷ 5 = 9. k ÷ 4 = 7 gives k = 7 × 4 = 28.

  • Easy : Easy: x + 5 = 13 gives x = 13 − 5 = 8.
  • Medium : Medium: x − 9 = 11 gives x = 11 + 9 = 20.
  • Exam-type : Exam: 3x + 4 = 25. Subtract 4: 3x = 21. Divide by 3: x = 7.

Rule 2: Word problems — Notebook costs x and bag costs x + 100; together ₹ 700. So 2x + 100 = 700, 2x = 600, x = 300. Notebook ₹ 300, bag ₹ 400, and 300 + 400 = 700 ✓.

Shirt x, pant x + 250, together 1250: the equation gives x = 500.ShirtxPantx250x + (x + 250) = 12502x = 1000, x = 500Shirt ₹ 500, pant ₹ 750

In the figure : In the figure the shirt costs x and the pant costs x + 250.

  • Easy : Easy: Ravi is x years old; in 5 years he will be 17. x + 5 = 17, so x = 12.
  • Medium : Medium: 3 equal pens cost ₹ 36. 3x = 36, so x = 12.
  • Exam-type : Exam: A number plus its double is 45. x + 2x = 45, 3x = 45, x = 15; check: 15 + 30 = 45 ✓.
Remember

Same operation on both sides. Variable alone. Check the answer.

Common mistake

Forgetting to check. Put the answer back: 3 × 7 + 4 = 25 ✓.

Shortcut

Memory clue: "उलटी क्रिया करा — + चा −, × चा ÷" (use the opposite operation: − for +, ÷ for ×).

Spot the difference

Equation hasDo
x + aSubtract a from both sides
x − aAdd a to both sides
axDivide both sides by a

Practice questions

Easy
  1. 1Solve x + 9 = 20.
  2. 2Solve 6x = 48.
  3. 3Solve x − 7 = 15.
Medium
  1. 4Solve 2x + 5 = 17.
  2. 5Solve 4x − 3 = 21.
  3. 6A pencil costs x and a pen costs x + 4. Together ₹ 16. Find both prices.
Challenging
  1. 7A number plus its double is 36. Find the number.
  2. 8Check whether x = 7 solves 3x + 4 = 25.
Show answers (try them yourself first)
  1. 1x = 11.
  2. 2x = 8.
  3. 3x = 22.
  4. 42x = 12, x = 6.
  5. 54x = 24, x = 6.
  6. 62x + 4 = 16, x = 6. Pencil ₹ 6, pen ₹ 10.
  7. 73x = 36, x = 12.
  8. 83 × 7 + 4 = 21 + 4 = 25, yes.