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State Board · Class 5 · Mathematics

Lesson 5: Fractions

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

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Parts of a whole: numerator and denominator

From daily life

A roti is folded into 4 equal parts and Meera eats 1 part. She has eaten 14 of the roti.

Simple definition

A fraction names a part of a whole that is cut into equal parts. The number below the line is the denominator: it tells how many equal parts the whole has. The number above the line is the numerator: it tells how many parts we take.

A pizza cut into 8 equal slices with 3 slices shaded: the fraction 3/8, with numerator and denominator labelled.38numerator: parts takendenominator: equal parts in all

In the figure : The pizza has 8 equal slices and 3 are shaded, so the shaded part is 38. Numerator 3, denominator 8.

Rule 1: Reading a fraction — Top number = parts taken. Bottom number = equal parts in all.

  • Easy : 25 means 2 parts taken out of 5 equal parts.
  • Medium : A ribbon is cut into 6 equal pieces and 5 are used. Used: 56. Left: 16.
  • Exam-type : A class has 10 children and 3 are absent. Absent: 310. Present: 710. The two fractions add up to 1 whole.
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Parts of a whole: numerator and denominator

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Rule 2: Parts must be equal — A fraction is true only when all the parts have the same size.

  • Easy : A bar cut into 4 equal parts with 1 shaded shows 14.
  • Medium : A bar cut into 4 unequal pieces with 1 shaded does not show 14.
  • Exam-type : A sheet of 12 squares is shared equally by 3 children. Each gets 4 squares, which is 13 of the sheet.
Remember

The denominator can never be 0, and the denominator tells the size of each part: a bigger denominator means smaller parts.

Common mistake

Writing 38 when 3 of 8 unequal pieces are shaded. Check first that all the pieces are equal.

Shortcut

Memory clue: the Denominator is Down, so the numerator is on top.

Spot the difference

PointNumeratorDenominator
Placeabove the linebelow the line
Tellsparts takenequal parts in the whole
In 3838

Practice questions

Easy
  1. 1What is the denominator of 59?
  2. 2What is the numerator of 712?
  3. 3A chocolate bar has 10 equal pieces and you eat 4. Write the fraction eaten.
Medium
  1. 4A pizza is cut into 8 equal slices and 5 are eaten. What fraction is left?
  2. 5Which is greater in size: one part of a whole cut into 4 equal parts, or one part of a whole cut into 8 equal parts?
  3. 6A garden has 15 plants and 6 have flowers. What fraction has flowers? What fraction has no flowers?
Challenging
  1. 7A rope is cut into 5 pieces: three are of equal length and two are shorter. Is each equal piece 15 of the rope? Why?
  2. 8Ravi has 12 marbles. He gives 5 to Sita and 4 to Amit. What fraction of the marbles does he still have?
Show answers (try them yourself first)
  1. 19
  2. 27
  3. 3410
  4. 48 − 5 = 3 slices are left, so 38.
  5. 5One part of the whole cut into 4 equal parts (14), because fewer parts means bigger parts.
  6. 6With flowers 615; without flowers 915.
  7. 7No, because the 5 pieces are not all equal.
  8. 812 − 5 − 4 = 3 marbles are left, so 312.

Proper, improper and mixed fractions

From daily life

Half a glass of milk is less than a full glass, but 3 glasses of milk are more than 2 full glasses plus half of one: 2 and a half glasses. Fractions can be less than 1, equal to 1, or more than 1.

Simple definition

A proper fraction has a numerator smaller than its denominator, like 34. An improper fraction has a numerator equal to or bigger than its denominator, like 74. A mixed fraction has a whole number and a proper fraction together, like 134.

Three bars: a proper fraction 3/4, an improper fraction 7/4 drawn with two bars, and the mixed fraction 1 and 3/4.properimpropermixed3474134top < bottomtop ≥ bottomwhole + fraction

In the figure : 34 is less than 1 bar. 74 needs two bars. The same amount written as a mixed fraction is 134.

Rule 1: Three kinds of fractions — Compare the numerator with the denominator to decide the kind.

  • Easy : 29 is proper because 2 < 9.
  • Medium : 99 and 116 are improper because the numerator is equal to or bigger than the denominator.
  • Exam-type : Sort 58, 85, 33, 212: proper 58; improper 85 and 33; mixed 212.

Rule 2: Improper to mixed — Divide the numerator by the denominator. The quotient is the whole number and the remainder is the new numerator.

  • Easy : 72: 7 ÷ 2 gives quotient 3 and remainder 1, so 72 = 312.
  • Medium : 175: 17 ÷ 5 gives quotient 3 and remainder 2, so 175 = 325.
  • Exam-type : Convert back: 423 = (4 × 3 + 2) over 3 = 143. Multiply the whole number by the denominator, add the numerator, keep the denominator.
Remember

A proper fraction is always less than 1. An improper fraction is 1 or more.

Common mistake

Writing 94 = 294 or forgetting the remainder. 9 ÷ 4 gives 2 and remainder 1, so 94 = 214.

Shortcut

Memory clue: Divide, Quotient, Remainder — the quotient is the whole number, the remainder goes on top of the same denominator.

Spot the difference

KindRuleExample
Propernumerator < denominator56
Impropernumerator ≥ denominator96
Mixedwhole + proper136

Practice questions

Easy
  1. 1Is 611 proper or improper?
  2. 2Write 134 as a mixed fraction.
  3. 3Write 235 as an improper fraction.
Medium
  1. 4Which of 49, 94, 99 are improper?
  2. 5Write 236 as a mixed fraction.
  3. 6Write 518 as an improper fraction.
Challenging
  1. 7Meena has 3 full rotis and half of another one. Write the amount as a mixed fraction and as an improper fraction.
  2. 8A bus ride takes 114 hours. Write this as a mixed fraction and say whether it is more than 2 hours.
Show answers (try them yourself first)
  1. 1Proper, because 6 < 11.
  2. 213 ÷ 4 = 3 remainder 1, so 314.
  3. 32 × 5 + 3 = 13, so 135.
  4. 494 and 99.
  5. 523 ÷ 6 = 3 remainder 5, so 356.
  6. 65 × 8 + 1 = 41, so 418.
  7. 7Mixed 312; improper (3 × 2 + 1) ÷ 2 = 72.
  8. 811 ÷ 4 = 2 remainder 3, so 234 hours. Yes, it is more than 2 hours.

Equivalent fractions and simplest form

From daily life

Two friends share a pizza. Asha cuts hers into 2 equal slices and eats 1. Ravi cuts his into 4 equal slices and eats 2. They have eaten the same amount, so 12 = 24.

Simple definition

Fractions that show the same part of a whole are equivalent fractions. A fraction is in its simplest form when its numerator and denominator have no common factor except 1.

Three equal-length bars with 1/2, 2/4 and 4/8 shaded; the shaded lengths are equal.122448Same shaded length → equivalent fractions

In the figure : Three bars of the same length with 12, 24 and 48 shaded. The shaded length is equal, so the fractions are equivalent.

Rule 1: Make an equivalent fraction — Multiply or divide the numerator and the denominator by the same number.

  • Easy : 13 = 1×23×2 = 26.
  • Medium : 35 = 3×45×4 = 1220.
  • Exam-type : Find the missing number: 27 = ?21. 7 × 3 = 21, so the numerator is 2 × 3 = 6: 27 = 621.
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Equivalent fractions and simplest form

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Rule 2: Simplest form — Divide the numerator and the denominator by their greatest common factor.

  • Easy : 48 = 4÷48÷4 = 12.
  • Medium : 1824: the greatest common factor is 6, so 1824 = 34.
  • Exam-type : Are 69 and 1015 equivalent? Simplify both: 69 = 23 and 1015 = 23. Yes, they are equivalent.
Remember

Multiply or divide the top and the bottom by the SAME number. The value of the fraction does not change.

Common mistake

Adding the same number: 12 becomes 34 by adding 2 to both. Wrong! Multiplying 1 and 2 by 2 gives 24.

Shortcut

Memory clue: Same on top, same on bottom — whatever you do to the top, do to the bottom.

Spot the difference

PointEquivalent fractionSimplest form
Howmultiply or divide both by the same numberdivide both by the greatest common factor
Example12 = 361216 = 34

Practice questions

Easy
  1. 1Write one fraction equivalent to 23.
  2. 2Fill in: 34 = ?12.
  3. 3Simplify 510.
Medium
  1. 4Simplify 1218.
  2. 5Is 46 in the simplest form? If not, simplify it.
  3. 6Are 37 and 1228 equivalent?
Challenging
  1. 7Write 2050 in the simplest form.
  2. 8Kiran ate 48 of a cake and Tara ate 36 of an equal cake. Who ate more?
Show answers (try them yourself first)
  1. 146 (multiply both by 2).
  2. 29, because 4 × 3 = 12 and 3 × 3 = 9.
  3. 312
  4. 423 (divide both by 6).
  5. 5No, 2 is a common factor; 46 = 23.
  6. 6Yes, 3 × 4 = 12 and 7 × 4 = 28.
  7. 7The greatest common factor is 10, so 25.
  8. 8Both ate the same amount, because 48 = 12 and 36 = 12.

Comparing and ordering fractions

From daily life

Two shopkeepers sell sugar. Anil gives 34 kg and Sunil gives 24 kg. With the same denominator it is easy to see that Anil gives more.

Simple definition

To compare fractions, look at the denominators first. Same denominators: the bigger numerator is the bigger fraction. Same numerators: the bigger denominator is the smaller fraction. Different denominators: first make the denominators equal.

A number line from 0 to 1 with 1/4, 1/2 and 3/4 marked; fractions further to the right are bigger.01412341Further right = bigger fraction

In the figure : On the number line 14 < 12 < 34. The further to the right, the bigger the fraction.

Rule 1: Same denominator — Compare the numerators only.

  • Easy : 59 > 29 because 5 > 2.
  • Medium : Arrange 710, 310, 910 from smallest to biggest: 310, 710, 910.
  • Exam-type : Which is more: 813 of a ribbon or 1113 of the same ribbon? 1113, as 11 > 8.

Rule 2: Same numerator — The bigger denominator gives the smaller fraction.

  • Easy : 18 < 14, because the whole cut into 8 parts has smaller parts.
  • Medium : 35 > 37 because 5 < 7.
  • Exam-type : Order 23, 29, 25 from biggest to smallest: 23, 25, 29.

Rule 3: Different denominators — Change both to equivalent fractions with the same denominator, then compare the numerators.

  • Easy : 23 and 34: 23 = 812 and 34 = 912, so 23 < 34.
  • Medium : 56 and 79: both become 18ths: 1518 and 1418, so 56 > 79.
  • Exam-type : Order 12, 23, 34 using 12ths: 612, 812, 912. So 12 < 23 < 34.
Remember

With the same denominator, the bigger numerator wins. With the same numerator, the smaller denominator wins.

Common mistake

Thinking 18 is bigger than 14 because 8 is bigger than 4. A bigger denominator means smaller parts.

Shortcut

Memory clue: same bottom, bigger top wins; same top, smaller bottom wins.

Spot the difference

SituationRuleExample
Same denominatorbigger numerator is bigger47 > 37
Same numeratorbigger denominator is smaller29 < 25
Different denominatorsmake denominators equal first12 = 36 > 26

Practice questions

Easy
  1. 1Which is bigger: 411 or 711?
  2. 2Which is smaller: 56 or 58?
  3. 3Put the symbol < or >: 29 __ 59.
Medium
  1. 4Compare 13 and 25 using 15ths.
  2. 5Arrange from smallest: 38, 18, 78, 58.
  3. 6Arrange from biggest: 12, 15, 13, 110.
Challenging
  1. 7Reena drank 34 of a bottle of water and Seema drank 58 of an equal bottle. Who drank more?
  2. 8Arrange 23, 56, 12 from smallest to biggest.
Show answers (try them yourself first)
  1. 1711
  2. 258, because 8 > 6.
  3. 3<
  4. 4515 and 615, so 13 < 25.
  5. 518, 38, 58, 78
  6. 612, 13, 15, 110
  7. 734 = 68 and 68 > 58, so Reena drank more.
  8. 8In 6ths: 46, 56, 36. Order: 12, 23, 56.

Adding and subtracting like fractions

From daily life

A shopkeeper has 27 kg of rice in one tin and 37 kg in another. Pouring them together gives 57 kg. Parts of the same size can be counted together.

Simple definition

Like fractions have the same denominator. To add like fractions, add the numerators and keep the denominator. To subtract, subtract the numerators and keep the denominator.

Two bars cut in 7 equal parts: 2 parts and 3 parts together make 5 parts, so 2/7 + 3/7 = 5/7.firstsecond27+37=57

In the figure : Two parts and three parts of a whole cut in 7 equal parts make 5 parts, so 27 + 37 = 57.

Rule 1: Adding like fractions — Add the numerators. The denominator stays the same.

  • Easy : 16 + 46 = 56.
  • Medium : 59 + 79 = 129 = 139 = 113.
  • Exam-type : Sita ran 310 km in the morning and 910 km in the evening. Total = 1210 = 1210 = 115 km.

Rule 2: Subtracting like fractions — Subtract the numerators. The denominator stays the same.

  • Easy : 78 − 28 = 58.
  • Medium : 1112 − 512 = 612 = 12.
  • Exam-type : A jug has 910 litre of juice and 410 litre is poured out. Left = 510 = 12 litre.
Remember

Add or subtract only the numerators. The denominator tells the size of the parts and does not change.

Common mistake

27 + 37 = 514. Wrong! The denominators are not added. The answer is 57.

Shortcut

Memory clue: the numerator moves, the denominator stays. Always simplify at the end.

Spot the difference

OperationRuleExample
Addadd numerators, same denominator38 + 28 = 58
Subtractsubtract numerators, same denominator68 − 18 = 58
Simplifydivide by the common factor48 = 12

Practice questions

Easy
  1. 1Add: 29 + 59.
  2. 2Subtract: 811 − 311.
  3. 3Add: 310 + 410. Write the answer in the simplest form.
Medium
  1. 4Subtract: 912 − 312. Simplify.
  2. 5Add: 58 + 78. Write the answer as a mixed fraction.
  3. 6A tank is 35 full. 15 more water is added. How full is it now?
Challenging
  1. 7Mohan reads 29 of a book on Monday and 49 on Tuesday. How much is left to read?
  2. 8A rope of 1112 m is cut: a piece of 412 m is used for a swing and a piece of 512 m is used for a bag. How much rope remains?
Show answers (try them yourself first)
  1. 179
  2. 2511
  3. 3710. It is already in the simplest form.
  4. 4612 = 12
  5. 5128 = 148 = 112
  6. 635 + 15 = 45 full.
  7. 7Read = 69; left = 99 − 69 = 39 = 13.
  8. 8Used = 912; left = 1112 − 912 = 212 = 16 m.