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State Board · Class 6 · Mathematics

Lesson 6: Fractions

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

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Types of fractions

From daily life

A pizza is cut into 8 equal slices and you eat 3. You ate 38 of the pizza. A fraction shows some equal parts out of a whole.

Simple definition

A fraction is written as numeratordenominator. The top number is the numerator (अंश) and the bottom number is the denominator (छेद).

Rule 1: Kinds of fractions — A unit fraction (एकक अपूर्णांक) has numerator 1: 15, 18, 110. The bigger the denominator, the smaller the part. Like fractions (सजातीय) have the same denominator: 59 and 29. Unlike fractions (विजातीय) have different ones: 14 and 16. If the numerator is less than the denominator it is a proper fraction (व्यवहारी); if it is not less, it is an improper fraction (अव्यवहारी) like 114. A mixed fraction (मिश्र अपूर्णांक) like 234 has a whole part and a fraction part.

Unit fractions: the bigger the denominator, the smaller the part.11/21/31/41/5More parts: each part is smaller

In the figure : In the figure unit fractions: more parts, smaller part.

  • Easy : Easy: 16 is larger than 19 because 6 is smaller than 9.
  • Medium : Medium: In 77, 38, 95: only 38 is proper; 77 and 95 are improper.
  • Exam-type : Exam: A pizza has 8 slices and 3 are eaten. Eaten 38; left 88 − 38 = 58.

Rule 2: Equivalent fractions and simplest form — Fractions with the same value are equivalent (समतुल्य): 25 = 410 = 615. Multiply or divide the numerator and denominator by the same number. To get the simplest form (सोपे रूप), divide both by their GCD: 1824 = 34 (GCD is 6).

  • Easy : Easy: 13 = 26 (multiply by 2).
  • Medium : Medium: 1520 = 34 (divide by 5).
  • Exam-type : Exam: Reduce 4256: GCD is 14, so 4256 = 34.
Live figure

Types of fractions

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Remember

Numerator on top, denominator below. Same value = equivalent. Divide by GCD for the simplest form.

Common mistake

Adding to top and bottom to make equivalent fractions. Multiply or divide instead.

Shortcut

Memory clue: "अंश वर, छेद खाली; दोघांना सारखे गुणा-भागा" (top = numerator, bottom = denominator; same × ÷ on both).

Spot the difference

TypeExample
Unit15
Like59, 29
Improper114
Mixed234

Practice questions

Easy
  1. 1Which is larger, 14 or 19?
  2. 2Write one like fraction of 311.
  3. 3Is 83 proper or improper?
Medium
  1. 4Write 12 with denominator 8.
  2. 5Reduce 1624 to the simplest form.
  3. 6Reduce 2736 to the simplest form.
Challenging
  1. 7A rope is cut into 12 equal pieces and 5 are used. What part is used and what part is left?
  2. 8Why is 13 = 26?
Show answers (try them yourself first)
  1. 114.
  2. 2711 (any numerator with denominator 11).
  3. 3Improper.
  4. 448.
  5. 5GCD 8, so 23.
  6. 6GCD 9, so 34.
  7. 7Used 512; left 712.
  8. 8Multiplying the top and bottom by 2 does not change the value.

Comparing fractions

From daily life

Meena drinks 12 glass of milk and Sita drinks 13 glass. Who drank more? To compare fractions we need parts of the same size.

Simple definition

To compare (तुलना करणे) fractions, make the denominators equal using their LCM, then compare the numerators.

Rule 1: Like fractions and unit fractions — For like fractions the larger numerator means the larger fraction: 59 > 49. For unit fractions the smaller denominator means the larger fraction: 15 > 18.

  • Easy : Easy: 710 > 310.
  • Medium : Medium: 16 and 14: 4 < 6, so 14 is larger.
  • Exam-type : Exam: The smallest of 211, 911, 511 is 211.

Rule 2: Unlike fractions — Take the LCM of the denominators and make equivalent fractions. 23 and 35: LCM is 15, so 1015 and 915. Since 10 > 9, 23 > 35.

1/2 = 3/6 is longer than 1/3 = 2/6.1/21/33/6>2/6Make denominators equal (LCM 6), then compare numerators

In the figure : In the figure 12 = 36 is longer than 13 = 26.

  • Easy : Easy: 12 and 25: LCM 10, so 510 > 410.
  • Medium : Medium: 34 and 56: LCM 12, so 912 < 1012.
  • Exam-type : Exam: Do not compare numerators alone. 56 > 79 although 5 < 7, because 1518 > 1418.
Live figure

Comparing fractions

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Remember

Like: compare numerators. Unit: smaller denominator is larger. Unlike: use LCM first.

Common mistake

Saying 29 is larger than 13 because 9 is big. Compare 29 and 39.

Shortcut

Memory clue: "छेद सारखे करा, मग अंश पाहा" (make denominators equal, then look at numerators).

Spot the difference

FractionsLarger
49 and 7979
110 and 1313
23 and 3523

Practice questions

Easy
  1. 1Which is larger, 49 or 79?
  2. 2Which is larger, 110 or 13?
  3. 3Compare 25 and 12.
Medium
  1. 4Compare 37 and 23.
  2. 5Put > or < : 58 __ 34.
  3. 6Ram studies 23 hour and Shyam 34 hour. Who studies more?
Challenging
  1. 7Is 78 greater than 89?
  2. 8Why can't we compare unlike fractions by numerators alone?
Show answers (try them yourself first)
  1. 179.
  2. 213.
  3. 3410 < 510, so 25 < 12.
  4. 4LCM 21: 921 < 1421, so 23 is larger.
  5. 558 < 68, so 58 < 34.
  6. 6812 < 912, so Shyam.
  7. 7No: 6372 < 6472, so 89 is larger.
  8. 8The parts are of different sizes. For example 56 > 79 although 5 < 7.

Ascending and descending order

From daily life

Three friends drank 12, 38 and 34 litre of juice. Arranging them from least to most needs the same method as comparing.

Simple definition

Ascending order (चढता क्रम) goes from smallest to largest and descending order (उतरता क्रम) from largest to smallest. On a number line, the fraction on the right is larger.

Rule 1: Arranging fractions — Like fractions: arrange by numerators. Unlike fractions: change all to the LCM of the denominators. 34, 16, 512 become 912, 212, 512, so ascending order is 16, 512, 34.

1/6, 5/9 and 2/3 on a number line: 1/6 < 5/9 < 2/3.011/63/185/910/182/312/18In eighteenths: 3 < 10 < 12

In the figure : In the figure 16, 59, 23 on a number line.

  • Easy : Easy: Ascending order of 27, 67, 47 is 27, 47, 67.
  • Medium : Medium: Descending order of 12, 38, 34: 48, 38, 68 gives 34, 12, 38.
  • Exam-type : Exam: Ascending order of 23, 56, 79: LCM 18 gives 1218, 1518, 1418, so 23, 79, 56.
Remember

Ascending = small to large. Descending = large to small. Unlike: LCM first.

Common mistake

Arranging the new fractions but writing the answer in the changed form. Write the original fractions in the answer.

Shortcut

Memory clue: "चढता = जिना चढणे, लहान पायरीपासून मोठीकडे" (ascending = climbing stairs, small step to big).

Spot the difference

OrderDirection
AscendingSmallest → largest
DescendingLargest → smallest

Practice questions

Easy
  1. 1Arrange in ascending order: 59, 29, 79.
  2. 2Arrange in descending order: 310, 910, 110.
  3. 3On a number line, is the fraction on the right smaller or larger?
Medium
  1. 4Ascending order of 12, 13, 14.
  2. 5Ascending order of 25, 310, 12.
  3. 6Descending order of 56, 23, 712.
Challenging
  1. 7Ropes are 34 m, 58 m and 78 m. Which is the shortest?
  2. 8Ascending order of 45, 34, 710, 920.
Show answers (try them yourself first)
  1. 129, 59, 79.
  2. 2910, 310, 110.
  3. 3Larger.
  4. 414, 13, 12.
  5. 5410, 310, 510, so 310, 25, 12.
  6. 61012, 812, 712, so 56, 23, 712.
  7. 768, 58, 78: the 58 m rope.
  8. 8LCM 20: 1620, 1520, 1420, 920, so 920, 710, 34, 45.

Adding and subtracting fractions

From daily life

Mother uses 14 kg sugar for tea and 24 kg for sweets. Altogether she used 34 kg. Adding fractions is like adding equal pieces.

Simple definition

Only parts of the same size can be added or subtracted, so the denominators must be equal first.

Rule 1: Like fractions — Keep the denominator and add or subtract the numerators. 29 + 49 = 69 = 23, 910 − 310 = 610 = 35 and 38 + 58 = 1.

  • Easy : Easy: 310 + 410 = 710.
  • Medium : Medium: 1112 − 512 = 612 = 12.
  • Exam-type : Exam: 37 of a plot has a house. The rest is garden: 1 − 37 = 77 − 37 = 47.

Rule 2: Unlike fractions (Brahmagupta's method) — Find the LCM of the denominators, write equivalent fractions with it, then add or subtract. 14 + 16 = 312 + 212 = 512. 56 − 14 = 1012 − 312 = 712.

1/2 + 1/3: both become sixths, then add the numerators: 5/6.1/2+1/3= 3/6= 2/63/6 + 2/6 = 5/6

In the figure : In the figure 12 + 13 = 56 with bars.

  • Easy : Easy: 12 + 14 = 24 + 14 = 34.
  • Medium : Medium: 35 − 12 = 610 − 510 = 110.
  • Exam-type : Exam: 78 − 13 = 2124 − 824 = 1324.
Remember

Same denominator: add or subtract only the numerators. Different: LCM first.

Common mistake

Adding the denominators too: 14 + 14 = 28. The answer is 24.

Shortcut

Memory clue: "छेद सारखा तर अंश मिळवा; छेद वेगळा तर लसावि काढा".

Spot the difference

CaseMethod
Like fractionsAdd / subtract numerators
Unlike fractionsLCM, then add / subtract
The rest1 − part

Practice questions

Easy
  1. 1Add 27 + 37.
  2. 2Subtract 89 − 59.
  3. 3Add 15 + 310.
Medium
  1. 4Subtract 34 − 13.
  2. 5Add 16 + 14.
  3. 6A tank is 25 full. What part is empty?
Challenging
  1. 7Rita reads 14 of a book on Monday and 13 on Tuesday. How much is read and how much is left?
  2. 8Find 56 − 12 + 13.
Show answers (try them yourself first)
  1. 157.
  2. 239 = 13.
  3. 3210 + 310 = 510 = 12.
  4. 4912 − 412 = 512.
  5. 5212 + 312 = 512.
  6. 61 − 25 = 35.
  7. 7Read 312 + 412 = 712; left 512.
  8. 856 − 36 + 26 = 46 = 23.

Mixed fractions and word problems

From daily life

A recipe needs 212 cups of flour. That is two whole cups and one half cup, a mixed fraction. To add or subtract such numbers we change them to improper fractions.

Simple definition

Mixed to improper: whole × denominator + numerator, over the same denominator. Improper to mixed: divide the numerator by the denominator.

Rule 1: Changing the form — 235 = (2 × 5 + 3)5 = 135. 236: 23 ÷ 6 gives quotient 3 and remainder 5, so 236 = 356.

The mixed fraction 3 3/4 equals the improper fraction 15/4.3 wholes3/43 3/4 = (3 × 4 + 3)/4 = 15/43 wholes = 12 quarters, plus 3 quarters = 15 quarters

In the figure : In the figure 334 = 154.

  • Easy : Easy: 112 = 32.
  • Medium : Medium: 423 = (4 × 3 + 2)3 = 143.
  • Exam-type : Exam: 298: 29 ÷ 8 gives 3 and remainder 5, so 298 = 358.

Rule 2: Adding and subtracting mixed fractions — Change to improper fractions, make the denominators equal, then add or subtract. 212 + 123 = 52 + 53 = 156 + 106 = 256 = 416. 514 − 212 = 214 − 104 = 114 = 234. Word problem: 4 friends each eat 13 of a pizza, so 43 = 113 pizzas are eaten.

  • Easy : Easy: 114 + 214 = 54 + 94 = 144 = 312.
  • Medium : Medium: 313 − 116 = 103 − 76 = 206 − 76 = 136 = 216.
  • Exam-type : Exam: 234 + 156 = 114 + 116 = 3312 + 2212 = 5512 = 4712.
Live figure

Mixed fractions and word problems

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Remember

Mixed → improper: whole × denominator + numerator. Improper → mixed: divide. Add/subtract as improper fractions.

Common mistake

Adding the whole parts and the fraction parts separately without checking if the fraction part crosses 1. Use improper fractions to be safe.

Shortcut

Memory clue: "पूर्ण गुणिले छेद, अधिक अंश" (whole × denominator + numerator).

Spot the difference

ChangeMethod
Mixed → improperwhole × denominator + numerator
Improper → mixeddivide numerator by denominator

Practice questions

Easy
  1. 1Write 213 as an improper fraction.
  2. 2Write 175 as a mixed fraction.
  3. 3Write 512 as an improper fraction.
Medium
  1. 4Add 112 + 212.
  2. 5Subtract 413 − 123.
  3. 6Add 214 + 112.
Challenging
  1. 7A ribbon is 312 m long and 134 m is used. How much is left?
  2. 8Six children each get 14 of a mango. How many mangoes are given?
Show answers (try them yourself first)
  1. 173.
  2. 2325.
  3. 3112.
  4. 432 + 52 = 82 = 4.
  5. 5133 − 53 = 83 = 223.
  6. 694 + 64 = 154 = 334.
  7. 7144 − 74 = 74 = 134 m.
  8. 864 = 112 mangoes.