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State Board · Class 7 · Mathematics

Lesson 5: Operations on Rational Numbers

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

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Rational numbers and their sets

From daily life

If one roti is shared between two children, each gets half, that is 12. If you owe 8 rupees we write −8, and −8 = −81. All such numbers are rational.

Simple definition

A number that can be written in the form of a fraction is a rational number.

  • 11 = 111
  • −6 = −61
  • 0 = 01
  • 0.75 = 34
Remember

the denominator is never zero. 50 is not a number.

Types of numbers

Type of numberExamples
Natural1, 2, 3, 4, ...
Whole0, 1, 2, 3, ...
Integers..., −2, −1, 0, 1, 2, ...
Rational12, −34, 5, 0
Order

natural → whole → integers → rational. Each set is part of the next one, so 5 is natural, whole, an integer and rational — all four.

Closure: if an operation on two numbers of a set always gives an answer in the same set, the set is closed under that operation.

  • Example: 3 − 5 = −2. The answer is not a whole number, so whole numbers are not closed under subtraction.
  • Example: 8 ÷ 3 = 83. The answer is not an integer, so integers are not closed under division.
SetAdditionSubtractionMultiplicationDivision
Natural✓✗✓✗
Whole✓✗✓✗
Integers✓✓✓✗
Rational✓✓✓✓
Remember

rational numbers are closed under division, but we cannot divide by 0.

Common mistake

thinking "only fractions are rational". Integers and 0 are rational too.

Spot the difference

PointIntegersRational numbers
Examples−2, 0, 7−2, 0, 7, 12, −34
Can be a fraction?NoYes
Closed under division?NoYes (except dividing by 0)

Practice questions

Easy
  1. 1Is 9 a rational number?
  2. 2Is 0 rational?
  3. 3Is 30 a rational number?
Medium
  1. 4Which sets does −7 belong to?
  2. 5Is the answer of 4 − 9 a whole number?
  3. 6Is the answer of 10 ÷ 4 an integer?
Challenging
  1. 7Which two operations on whole numbers always give a whole number?
  2. 8Name a set that is closed under division and state the condition.
Show answers (try them yourself first)
  1. 1Yes, 9 = 91.
  2. 2Yes, 0 = 01.
  3. 3No, the denominator is zero.
  4. 4Integers and rational numbers.
  5. 5No, 4 − 9 = −5.
  6. 6No, 10 ÷ 4 = 52.
  7. 7Addition and multiplication (e.g. 6 + 15 = 21, 6 × 15 = 90; but 6 − 15 = −9 is not a whole number).
  8. 8Rational numbers; the divisor must not be 0.

Operations on rational numbers

From daily life

You ate 14 of a cake and your brother ate 24. Together you ate 34. All operations on fractions work the same way for rational numbers.

Rule 1: Addition — make the denominators equal, then add the numerators.

  • Easy: 14 + 24 = 34.
  • Medium: 25 + 34 → denominator 20 → 820 + 1520 = 2320.
  • Exam type: −34 + 56 → denominator 12 → −912 + 1012 = 112.

Rule 2: Subtraction — make the denominators equal, then subtract the numerators.

  • Easy: 79 − 29 = 59.
  • Medium: 56 − 14 → denominator 12 → 1012 − 312 = 712.
  • Exam type: 13 − 56 → 26 − 56 = −36 = −12.

Rule 3: Multiplication — numerator × numerator, denominator × denominator. Cancel first to keep numbers small.

  • Easy: 38 × 49 = 1272 = 16.
  • Medium: −23 × 37 = −621 = −27.
  • Exam type: −56 × −910: cancel 5 and 10 → 1 and 2; cancel 9 and 6 → 3 and 2. Answer 34 (two negatives make a positive).

Rule 4: Division — take the reciprocal of the second number and multiply.

  • Easy: 35 ÷ 910 = 35 × 109 = 23.
  • Medium: −47 ÷ 23 = −47 × 32 = −1214 = −67.
  • Exam type: −34 ÷ −98 = −34 × −89 = 2436 = 23.

Mixed numbers: change to improper fractions first.

  • 1 12 + 2 13 = 32 + 73 = 96 + 146 = 236 = 3 56.
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Operations on rational numbers

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Remember

add/subtract = make denominators equal. Multiply = straight across. Divide = flip the second number and multiply.

Common mistake

adding numerators and denominators separately when the denominators differ. 12 + 13 ≠ 25; the correct answer is 56.

Sign shortcut

multiplying or dividing two negatives gives a positive; one negative and one positive gives a negative.

Spot the difference

OperationNeed equal denominators?What about the second number?
Addition / subtractionYesUsed as it is
MultiplicationNoUsed as it is
DivisionNoFlip it (reciprocal) and multiply

Practice questions

Easy
  1. 127 + 37
  2. 2911 − 411
  3. 323 × 35
Medium
  1. 412 + 23
  2. 534 − 25
  3. 645 ÷ 215
Challenging
  1. 7−23 + 14 − 56
  2. 8−35 × 109 ÷ −23
Show answers (try them yourself first)
  1. 157
  2. 2511
  3. 3615 = 25
  4. 436 + 46 = 76
  5. 51520 − 820 = 720
  6. 645 × 152 = 6010 = 6
  7. 7denominator 12: −812 + 312 − 1012 = −1512 = −54
  8. 8−35 × 109 = −23; then −23 ÷ −23 = 1

Multiplicative inverse and numbers between two rationals

From daily life

If you fill 12-litre bags from a 1-litre pack of milk you get 2 bags. So the "flip" of 12 is 2. That is the reciprocal.

Simple definition

swap the numerator and denominator of a fraction to get its reciprocal. Number × its reciprocal = 1.

  • Easy: the reciprocal of 35 is 53, because 35 × 53 = 1.
  • Medium: 4 = 41, so the reciprocal of 4 is 14.
  • Exam type: the reciprocal of −310 is −103 (the sign stays the same).
Remember

0 has no reciprocal.

Rational numbers between two numbers: however close two rational numbers are, there are always endlessly many more rational numbers between them. To find one, add the two and divide by 2 (the average).

  • Easy: 14 and 12: (14 + 24) ÷ 2 = 34 ÷ 2 = 38. So 14 < 38 < 12.
  • Medium: 23 and 34: (812 + 912) ÷ 2 = 1712 ÷ 2 = 1724.
  • Exam type: three numbers between 15 and 25: make the denominator 50 → 1050 and 2050. Numbers in between: 1150, 1250, 1350.
Shortcut

to get many numbers, make the denominators bigger (multiply by 10 or 100). That creates more room in between.

Common mistake

thinking there are infinitely many integers between two integers too. Between −3 and 4 there are only 6: −2, −1, 0, 1, 2, 3.

Spot the difference

PointReciprocalOpposite number
For 3553−35
How to get itswap numerator and denominatorchange the sign
Relationproduct = 1sum = 0

Practice questions

Easy
  1. 1Reciprocal of 59
  2. 2Reciprocal of 8
  3. 3Reciprocal of −13
Medium
  1. 4What is the reciprocal of 0?
  2. 5One number between 13 and 12
  3. 6One number between 12 and 23
Challenging
  1. 7Three rational numbers between 14 and 13
  2. 8Which two numbers are their own reciprocals?
Show answers (try them yourself first)
  1. 195
  2. 218
  3. 3−3
  4. 4None; 0 has no reciprocal.
  5. 5(26 + 36) ÷ 2 = 512
  6. 6(36 + 46) ÷ 2 = 712
  7. 7denominator 48: 1248 and 1648; in between 1348, 1448, 1548.
  8. 81 and −1.

Decimal form of rational numbers

From daily life

Half a rupee is 0.50 rupee and a quarter rupee is 0.25 rupee. So 12 = 0.5 and 14 = 0.25.

Rule: to get the decimal form of a fraction, divide the numerator by the denominator.

  • Easy: 94 = 9 ÷ 4 = 2.25.
  • Medium: 98 = 9 ÷ 8 = 1.125; with a negative sign, −98 = −1.125.
  • Exam type: 83 = 2.666... (6 repeats), 511 = 0.4545... (45 repeats).

Two kinds

KindMeaningExamples
Terminatingthe division ends0.5, 0.25, 0.375
Recurringsome digits repeat again and again0.333..., 0.1666...

Shortcut to tell without dividing: simplify the fraction and look at the denominator. If its only prime factors are 2 and 5, the decimal terminates.

  • 720: 20 = 2 × 2 × 5 → terminates (0.35).
  • 712: 12 = 2 × 2 × 3 → a 3 appears, so recurring (0.58333...).
Remember

even if the decimal never ends, the number is still rational, because it can be written as a fraction.

Common mistake

calling 615 recurring because of 15. Simplify first: 615 = 25 = 0.4, which terminates.

Spot the difference: see the table above — terminating = the division ends; recurring = digits repeat.

Practice questions

Easy
  1. 1Decimal form of 14
  2. 2of 35
  3. 3of 18
Medium
  1. 4of 118
  2. 5of 23
  3. 6of 920
Challenging
  1. 7Write the decimal form of 1112.
  2. 8Which of 715, 925, 1118 has a terminating decimal? (without dividing)
Show answers (try them yourself first)
  1. 10.25
  2. 20.6
  3. 30.125
  4. 41.375
  5. 50.666...
  6. 60.45
  7. 70.91666...
  8. 8925 (25 = 5 × 5, only 5); the others have a 3 in the denominator. 925 = 0.36.

Simplifying expressions in the correct order

From daily life

You buy 5 pens at 4 rupees each and a 10-rupee notebook. Total = 5 × 4 + 10. Multiply first (20), then add = 30 rupees. Change the order and the answer is wrong.

Rule (BODMAS)

  • Brackets first: ( ) then [ ] then { } — innermost first.
  • Then "of" and powers.
  • Then division and multiplication — left to right.
  • Lastly addition and subtraction — left to right.

Solved examples

  • Easy: 8 + 3 × 4 = 8 + 12 = 20 (multiply first).
  • Easy: 36 ÷ 4 × 3 = 9 × 3 = 27 (left to right).
  • Medium: 2 × (5 + 4) − 6 = 2 × 9 − 6 = 18 − 6 = 12.
  • Medium: 50 − [20 − (8 − 3)] = 50 − [20 − 5] = 50 − 15 = 35.
  • Exam type: 100 ÷ 5 × 2 + 3 × (6 − 2) = 100 ÷ 5 × 2 + 3 × 4 = 20 × 2 + 12 = 40 + 12 = 52.
  • With fractions: 12 + 13 × 34 = 12 + 14 = 34.
  • With "of": 12 of 40 + 6 = 20 + 6 = 26.
Remember

division and multiplication are on the same level; so are addition and subtraction. Solve them left to right.

Common mistake

in 36 ÷ 4 × 3, multiplying first (4 × 3 = 12; 36 ÷ 12 = 3). That is wrong; the correct answer is 27.

Spot the difference

ExpressionWrong orderRight orderAnswer
8 + 3 × 4(8 + 3) × 4 = 443 × 4 first20
36 ÷ 4 × 336 ÷ 12 = 3left to right27

Practice questions

Easy
  1. 17 + 2 × 5
  2. 224 ÷ 6 + 4
  3. 315 − 3 × 2
Medium
  1. 4(12 − 4) × 3 + 5
  2. 548 ÷ 6 × 2
  3. 630 − [10 + (8 − 3)]
Challenging
  1. 723 + 12 × 45
  2. 85 + [20 − {12 ÷ (8 − 4) + 2}] × 2
Show answers (try them yourself first)
  1. 117
  2. 28
  3. 39
  4. 48 × 3 + 5 = 29
  5. 58 × 2 = 16
  6. 630 − [10 + 5] = 15
  7. 723 + 25 = 1015 + 615 = 1615
  8. 88 − 4 = 4; 12 ÷ 4 = 3; 3 + 2 = 5; 20 − 5 = 15; 15 × 2 = 30; 5 + 30 = 35.