Fractions

Welcome dear students! Today we are going to learn about Fractions from Class 6 Maths.

Recall that when some whole number of things are shared equally among some number of people, fractions tell us how much each share is. Let us imagine a conversation between Shabnam and Mukta. Shabnam asks if you remember how much roti each child gets when one roti is divided equally between two children. Mukta answers that each child gets half a roti. Shabnam explains that the fraction one half is written as 1/2, and we sometimes read it as one upon two. Mukta then asks how much one child gets if one roti is equally shared among four children. Shabnam replies that each child gets 1/4 roti. Mukta wonders which is more, 1/2 roti or 1/4 roti. Shabnam explains that when two children share one roti equally, each gets 1/2 roti. When four children share the same one roti equally, each gets 1/4 roti. Since more children share the same one roti in the second group, each child gets a smaller share. So, 1/2 roti is more than 1/4 roti, which we write as 1/2 > 1/4.

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Now let us move to section 7.1, Fractional Units and Equal Shares. Beni asks which fraction is greater, 1/5 or 1/9. Arvin guesses that 1/9 is greater because 9 is bigger than 5. Beni corrects him, saying it is a common mistake, and tells him to think of these fractions as shares. Arvin reasons that if one roti is shared among five children, each gets 1/5 roti, and if shared among nine children, each gets 1/9 roti. Beni confirms this and asks which share is higher. Arvin realizes that sharing with more people means getting less, so 1/9 < 1/5. Beni praises him, and Arvin concludes that 1/100 is bigger than 1/200. When one unit is divided into several equal parts, each part is called a fractional unit. These are all fractional units: 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/50, 1/100, and so on. We also sometimes refer to fractional units as unit fractions.

Let us solve the Figure it Out questions for this section. First, three guavas together weigh 1 kg. If they are roughly the same size, each guava will roughly weigh 1/3 kg. Second, a wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is 1/4 kg. Third, four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank 3/4 glass of sugarcane juice. This is because 3 glasses divided by 4 friends gives a share of 3/4 for each person. Fourth, the big fish weighs 1/2 kg and the small one weighs 1/4 kg. Together they weigh 1/2 + 1/4. Converting to a common denominator, this is 2/4 + 1/4, which equals 3/4 kg.

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Next, we have a Math Talk box about knowledge from the past. [Math Talk] Fractions have been used in India since ancient times. In the Rig Veda, the fraction 3/4 is referred to as tri-pada. This matches words like teen paav in Hindi and mukkaal in Tamil. I encourage you to discuss with your family and classmates what words are used in your local languages for fractions like one half, one quarter, three quarters, one and a half, and two and a half. Write them down and arrange these fraction words from smallest to biggest: quarter, half, one and a quarter, three quarters, one and a half, two and a half.

Now we move to section 7.2, Fractional Units as Parts of a Whole. Imagine a picture showing a whole chikki. Another picture shows the chikki broken into two pieces. The bigger piece contains three pieces of 1/4 chikki. So we measure the bigger piece using the fractional unit 1/4, and we see it is 3/4 chikki. Another picture shows a whole chikki cut into six equal pieces. A different picture shows the same chikki cut into six equal pieces in a different way. By dividing the whole chikki into six equal parts in different ways, we get 1/6 chikki pieces of different shapes. A Math Talk prompt asks if they are of the same size. Yes, they are all equal in area, just different in shape. Next, a diagram shows a whole chikki divided into three equal vertical parts. One of those pieces is shown separately. We get this piece by breaking the chikki into three equal pieces, so this is 1/3 chikki.

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For the Figure it Out activity showing different chikki pieces, look at each diagram in your textbook. To find the fraction for parts a through h, count how many identical pieces would fit into the whole chikki. For example, if a piece is one of eight equal parts, it is 1/8. If it is one of sixteen, it is 1/16. Use your textbook diagrams to determine and fill in the correct fractions for each shape in the boxes provided.

Next, we learn about section 7.3, Measuring Using Fractional Units. Take a strip of paper. We consider this paper strip to be one unit long. Fold the strip into two equal parts and open it. Taking the strip as one unit, the lengths of the two new parts are each 1/2 unit. If you fold the previously folded strip again into two equal parts, you get four equal parts. Two times 1/4 equals 2/4. Three times 1/4 equals 3/4. Four times 1/4 equals 4/4. Let us do it once more with a diagram showing a rectangle divided into eight equal parts. Two times 1/8 equals 2/8. Four times 1/8 equals 4/8. Six times 1/8 equals 6/8. Eight times 1/8 equals 8/8, which equals 1. Fractional quantities can be measured using fractional units. Let us look at another example with circles. The first shows half a circle, representing 1/2. The second shows a full circle, representing 1/2 + 1/2, which is 2 times half. The third shows one and a half circles, representing 1/2 + 1/2 + 1/2, which is 3 times half. The fourth shows two full circles, representing 1/2 + 1/2 + 1/2 + 1/2, which is 4 times half. The fifth shows two and a half circles, representing 1/2 + 1/2 + 1/2 + 1/2 + 1/2, which is 5 times half. We can describe how much the quantity is by collecting together the fractional units.

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Let us solve the Figure it Out questions for this section. First, continue the table of 1/2 for two more steps. Six times half is 6/2. Seven times half is 7/2. Second, create a similar table for 1/4. One times 1/4 is 1/4. Two times 1/4 is 2/4. Three times 1/4 is 3/4. Four times 1/4 is 4/4, which equals 1. Five times 1/4 is 5/4. Six times 1/4 is 6/4. Seven times 1/4 is 7/4. Eight times 1/4 is 8/4, which equals 2. Third, make 1/3 using a paper strip by folding it into three equal parts. To make 1/6, fold each of those thirds in half again, giving six equal parts. Fourth, draw a picture and write an addition statement for five times 1/4 of a roti. The addition statement is 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 5/4. For nine times 1/4 of a roti, the addition statement is 1/4 added nine times, which equals 9/4. Fifth, match each fractional unit with the correct picture. 1/3 matches the circle divided into three equal parts with one shaded. 1/5 matches the circle divided into five equal parts with one shaded. 1/8 matches the circle divided into eight equal parts with one shaded. 1/6 matches the circle divided into six equal parts with one shaded.

We usually read the fraction 3/4 as three quarters or three upon four, but reading it as 3 times 1/4 helps us understand the size because it clearly shows the fractional unit is 1/4 and there are 3 such units. Recall what we call the top and bottom numbers. In the fraction 5/6, 5 is the numerator and 6 is the denominator. As a Teacher's Note, I encourage you to take some time to explore the idea of fractional units using different shapes like circles, squares, rectangles, and triangles. Try drawing and dividing them yourself to see how the fractional unit changes with each shape.

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Now let us move to section 7.4, Marking Fraction Lengths on the Number Line. We have marked lengths 1, 2, 3 on the number line. Now we mark fractions. A diagram shows a number line from 0 to 2. The distance between 0 and 1 is one unit, divided into two equal parts. Each part is 1/2 unit. A blue line from 0 to the first mark is 1/2 unit long. Next, a number line from 0 to 2 is divided into three equal parts between 0 and 1. The fractional unit is 1/3. A blue line extends to the second mark, so its length is 2/3. Next, a unit is divided into five equal parts. Blue lines extend to 1/5 and 3/5. Next, a unit is divided into eight equal parts. I encourage you to write the appropriate fractions for each mark in your notebook.

Let us solve the Figure it Out questions for this section. First, on a number line, draw lines of lengths 1/10, 3/10, and 4/5. Divide the unit into ten equal parts. Mark the first part for 1/10, the third part for 3/10, and the eighth part for 4/5. Second, write five more fractions and mark them. You can choose 1/2, 2/3, 5/6, 7/8, and 9/10, and mark them by dividing the unit into the appropriate number of equal parts. Third, how many fractions lie between 0 and 1? Infinitely many fractions lie between 0 and 1, because we can always divide the unit into more and more equal parts. Fourth, a number line from 0 to 2 shows a blue line from 0 to 1/2. The black line extends from 0 to the mark halfway between 1 and 2. Since the unit is divided into halves, the mark between 1 and 2 is 1 1/2, or 3/2. So the black line is 3/2 units long. Fifth, a number line from 0 to 2 is divided into fifths. The empty boxes between 1 and 2 represent 6/5, 7/5, 8/5, and 9/5. The black lines end at these marks, so their lengths are 6/5, 7/5, 8/5, and 9/5 respectively.

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Now we learn about section 7.5, Mixed Fractions. You marked fractions on the number line earlier. The blue lines were less than one, and the black lines were more than one. Let us classify them. Lengths less than 1 unit include 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 2/6, 3/6, 4/6, 5/6, 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8. Lengths more than 1 unit include 3/2, 5/2, 7/2, 4/3, 7/3, 5/4, 7/4, 9/4, 6/5, 7/5, 8/5, 9/5, 7/6, 8/6, 9/6, 10/6, 11/6, 9/8, 10/8, 11/8, 12/8, 13/8, 14/8, 15/8. In fractions less than 1, the numerator is smaller than the denominator. In fractions more than 1, the numerator is larger than the denominator. We know 3/2, 5/2, and 7/2 are greater than 1. We can see how many whole units they contain. 3/2 = 1/2 + 1/2 + 1/2 = 1 + 1/2. 5/2 = 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 2 + 1/2. Adding 1/3 + 1/3 + 1/3 = 3/3 = 1. Adding one more 1/3 gives 4/3, which is greater than 1.

Let us solve the Figure it Out questions. First, how many whole units are in 7/2? 7/2 = 1/2 added seven times = 3 + 1/2. So there are 3 whole units. Second, how many whole units in 4/3 and 7/3? 4/3 = 1/3 added four times = 1 + 1/3, so 1 whole unit. 7/3 = 1/3 added seven times = 2 + 1/3, so 2 whole units. Writing fractions greater than one as mixed numbers: 3/2 = 1 + 1/2. 4/3 = 1/3 + 1/3 + 1/3 + 1/3 = 1 + 1/3. A mixed number or mixed fraction contains a whole number, called the whole part, and a fraction that is less than 1, called the fractional part. Let us solve the next Figure it Out. First, figure out whole units: a. 8/3 = 2 + 2/3. b. 11/5 = 2 + 1/5. c. 9/4 = 2 + 1/4. Second, can all fractions greater than 1 be written as mixed numbers? Yes, because any improper fraction can be divided to give a whole number quotient and a remainder, which becomes the numerator of the fractional part. Third, write as mixed fractions: a. 9/2 = 4 1/2. b. 9/5 = 1 4/5. c. 21/19 = 1 2/19. d. 47/9 = 5 2/9. e. 12/11 = 1 1/11. f. 19/6 = 3 1/6.

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Can we write a mixed number as a regular fraction? Yes. Jaya explains that 3 + 3/4 means 1 + 1 + 1 + 3/4. Since 1 = 1/4 + 1/4 + 1/4 + 1/4, we get 4/4 + 4/4 + 4/4 + 3/4 = 15/4. Let us solve the Figure it Out. Write as fractions: a. 3 1/4 = 13/4. b. 7 2/3 = 23/3. c. 9 4/9 = 85/9. d. 3 1/6 = 19/6. e. 2 3/11 = 25/11. f. 3 9/10 = 39/10.

Now we move to section 7.6, Equivalent Fractions. Using a fraction wall, we observe that 1/2, 2/4, and 4/8 represent the same length. We say 1/2 = 2/4 = 4/8. These are equivalent fractions that denote the same length but are expressed in different fractional units. Check if 1/3 and 2/6 are equivalent using paper strips. Yes, they are. Make your own fraction wall. Looking at it, 1/2 and 3/6 are equal. 2/3 and 4/6 are equivalent because they cover the same length. Three pieces of 1/6 make 1/2. Two pieces of 1/6 make 1/3.

Let us solve the Figure it Out. First, are 3/6, 4/8, 5/10 equivalent? Yes, because 3/6 simplifies to 1/2, 4/8 simplifies to 1/2, and 5/10 simplifies to 1/2. Second, write two equivalent fractions for 2/6. They are 4/12 and 6/18. Third, 4/6 equals what? The textbook asks you to fill in the blanks and write as many equivalent fractions as you can. Try finding them by multiplying the numerator and denominator by the same number. Understanding equivalent fractions using equal shares: One roti shared by four children gives each 1/4. You can also express this event through division facts, addition facts, and multiplication facts. The division fact is 1 divided by 4 equals 1/4. The addition fact is 1 equals 1/4 plus 1/4 plus 1/4 plus 1/4. The multiplication fact is 1 equals 4 times 1/4.

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Let us solve the next Figure it Out. First, three rotis shared by four children. Each gets 3/4 roti. The division fact is 3 divided by 4 equals 3/4. The addition fact is 3 equals 3/4 plus 3/4 plus 3/4 plus 3/4. The multiplication fact is 3 equals 4 times 3/4. Second, two rotis shared by four children. Each gets 2/4 or 1/2 roti. The division fact is 2 divided by 4 equals 2/4. The addition fact is 2 equals 2/4 plus 2/4 plus 2/4 plus 2/4. The multiplication fact is 2 equals 4 times 2/4. Third, two cakes divided among five children gives each 2/5 cake. To give the same amount to ten children, we need 4 cakes, because 4/10 equals 2/5. Fractions where shares are equal are called equivalent fractions. So 1/2 = 2/4 = 3/6. Find more equivalent to 1/2: 4/8, 5/10, 6/12, 7/14. Equally divide rotis: 2 rotis among 3 children gives 2/3 each. 4 rotis among 6 children gives 4/6 each. 6 rotis among 9 children gives 6/9 each. The shares are the same because 2/3 = 4/6 = 6/9. 2/3 is the simplest form of 4/6 and 6/9.

Solve the next Figure it Out. a. 5 glasses among 4 friends equals 10 glasses among 8 friends. So 5/4 = 10/8. b. 4 kg in 3 bags equals 12 kg in 9 bags. So 4/3 = 12/9. c. 7 rotis among 5 children equals 14 rotis among 10 children. So 7/5 = 14/10. Comparing shares: 1 chikki among 2 children gives 1/2. 5 chikkis among 8 children gives 5/8. Since 1/2 = 4/8 and 4/8 < 5/8, the second group gets more. Next, 1 chikki among 2 children gives 1/2. 4 chikkis among 7 children gives 4/7. 1/2 = 4/8. Since 4/7 > 4/8, the second group gets more. Comparing groups: Group 1: 3 glasses among 4 children is 3/4. Group 2: 7 glasses among 10 children is 7/10. 3/4 = 30/40 and 7/10 = 28/40. Since 30/40 > 28/40, Group 1 gets more. Group 1: 4 glasses among 7 children is 4/7. Group 2: 5 glasses among 7 children is 5/7. Since denominators are equal, 5/7 > 4/7, so Group 2 gets more. [Math Talk] When the number of children is same, it is easier to compare, isn't it?

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Now, find equivalent fractions for the given pairs of fractions such that the fractional units are the same. The pairs are: a. 7/2 and 3/5, b. 8/3 and 5/6, c. 3/4 and 3/5, d. 6/7 and 8/5, e. 9/4 and 5/2, f. 1/10 and 2/9, g. 8/3 and 11/4, h. 13/6 and 1/9. Take your time to work these out in your notebook by finding a common denominator for each pair.

Expressing in lowest terms: In any fraction, if its numerator and denominator have no common factor except 1, then the fraction is said to be in lowest terms or in its simplest form. Worked example: Is 16/20 in lowest terms? No, 4 is a common factor of 16 and 20. Let us reduce 16/20 to lowest terms. Both 16 and 20 are divisible by 4. So 16 divided by 4 over 20 divided by 4 equals 4/5. There is no common factor between 4 and 5. Hence, 16/20 in lowest terms is 4/5. You can also do it in steps, like reducing 36/60 by dividing by 2 twice, then by 3, to get 3/5.

Let us solve the Figure it Out for lowest terms. a. 17/51: divide by 17 to get 1/3. b. 64/144: divide by 16 to get 4/9. e. 126/147: divide by 21 to get 6/7. d. 525/112: divide by 7 to get 75/16. Note that the textbook labels these items as a, b, e, and d.

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Now section 7.7, Comparing Fractions. Which is greater, 4/5 or 7/9? Find equivalent fractions with same denominator. 4/5 = 36/45. 7/9 = 35/45. Clearly 36/45 > 35/45, so 4/5 > 7/9. Another pair: 7/9 and 17/21. Common multiple is 63. 7/9 = 49/63. 17/21 = 51/63. Clearly 49/63 < 51/63, so 7/9 < 17/21. Steps to compare: Step 1: Change given fractions to equivalent fractions with same denominator. Step 2: Compare numerators.

Solve Figure it Out. 1. Compare: a. 8/3 and 5/2 become 16/6 and 15/6, so 8/3 > 5/2. b. 4/9 and 3/7 become 28/63 and 27/63, so 4/9 > 3/7. c. 7/10 and 9/14 become 49/70 and 45/70, so 7/10 > 9/14. d. 12/5 and 8/5, same denominator, so 12/5 > 8/5. e. 9/4 and 5/2 become 9/4 and 10/4, so 5/2 > 9/4. 2. Ascending order: a. 7/10, 11/15, 2/5 become 21/30, 22/30, 12/30. Order: 2/5, 7/10, 11/15. b. 19/24, 5/6, 7/12 become 19/24, 20/24, 14/24. Order: 7/12, 19/24, 5/6. 3. Descending order: a. 25/16, 7/8, 13/4, 17/32 become 50/32, 28/32, 104/32, 17/32. Order: 13/4, 25/16, 7/8, 17/32. b. 3/4, 12/5, 7/12, 5/4 become 45/60, 144/60, 35/60, 75/60. Order: 12/5, 5/4, 3/4, 7/12.

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Now section 7.8, Addition and Subtraction of Fractions. Meena ate 1/2 chikki and her brother ate 1/4. Total eaten is 1/2 + 1/4. Converting to common denominator, 2/4 + 1/4 = 3/4. Remaining is 1/4. Adding fractions with same denominator: 2/5 + 1/5 = 3/5. 4/7 + 6/7 = 10/7 = 1 3/7. Adding different denominators: 1/4 + 1/3. Common denominator is 12. 1/4 = 3/12. 1/3 = 4/12. Sum is 7/12. Brahmagupta's method for adding fractions: 1. Find equivalent fractions with common denominator. 2. Add numerators, keep denominator. 3. Express in lowest terms if needed. Example: 2/3 + 1/5. LCM of 3 and 5 is 15. 2/3 = 10/15. 1/5 = 3/15. Sum is 13/15. Example: 1/6 + 1/3. LCM is 6. 1/6 stays 1/6. 1/3 = 2/6. Sum is 3/6 = 1/2.

Solve Figure it Out. 1. a. 2/7 + 5/7 + 6/7 = 13/7. b. 3/4 + 1/3 = 9/12 + 4/12 = 13/12. c. 2/3 + 5/6 = 4/6 + 5/6 = 9/6 = 3/2. d. 2/3 + 2/7 = 14/21 + 6/21 = 20/21. e. 3/4 + 1/3 + 1/5 = 45/60 + 20/60 + 12/60 = 77/60. f. 2/3 + 4/5 = 10/15 + 12/15 = 22/15. g. 4/5 + 2/3 = 12/15 + 10/15 = 22/15. h. 3/5 + 5/8 = 24/40 + 25/40 = 49/40. i. 9/2 + 5/4 = 18/4 + 5/4 = 23/4. j. 8/3 + 2/7 = 56/21 + 6/21 = 62/21. k. 3/4 + 1/3 + 1/5 = 77/60. l. 2/3 + 4/5 + 3/7 = 70/105 + 84/105 + 45/105 = 199/105. m. 9/2 + 5/4 + 7/6 = 54/12 + 15/12 + 14/12 = 83/12. 2. 2/3 + 3/4 = 8/12 + 9/12 = 17/12 litres. 3. 2/5 + 3/4 = 8/20 + 15/20 = 23/20 meters. Since 23/20 > 1, the lace is sufficient.

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Subtraction with same denominator: 6/7 - 4/7 = 2/7. Subtraction with different denominators: 3/4 - 2/3 = 9/12 - 8/12 = 1/12. Brahmagupta's method for subtracting: 1. Convert to same denominator. 2. Subtract numerators. 3. Simplify. Solve Figure it Out. 1. a. 8/15 - 3/15 = 5/15 = 1/3. b. 2/5 - 4/15 = 6/15 - 4/15 = 2/15. c. 5/6 - 4/9 = 15/18 - 8/18 = 7/18. d. 2/3 - 1/2 = 4/6 - 3/6 = 1/6. 2. a. 10/3 - 13/4 = 40/12 - 39/12 = 1/12. b. 23/3 - 18/5 = 115/15 - 54/15 = 61/15. c. 45/7 - 29/7 = 16/7. 3. a. 7/10 - 1/2 = 7/10 - 5/10 = 2/10 = 1/5 km. b. 10/3 - 13/4 = 40/12 - 39/12 = 1/12 minutes. Jeevika takes less time by 1/12 minutes.

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Now section 7.9, A Pinch of History. Do you know what a fraction was called in ancient India? It was called bhinna in Sanskrit, which means broken. It was also called bhaga or ansha meaning part or piece. The way we write fractions today originated in India. In the Bakshali manuscript around 300 CE, 1/2 was written as 1/2. This continued with Aryabhata, Brahmagupta, Sridharacharya, and Mahaviracharya. The fraction bar was introduced by Al-Hassar in the 12th century. Ancient Egyptians and Babylonians used only fractional units, called Egyptian fractions. General fractions were first introduced in India with arithmetic rules. Brahmagupta codified them in 628 CE. His methods are still used today.

The puzzle asks to find different fractional units that add to 1. Two different units cannot add to 1. Three different units: 1/2 + 1/3 + 1/6 = 1. What if we look for four different fractional units that add up to 1? Can you find four different fractional units that add up to 1? It turns out that this problem has six solutions! Can you find at least one of them? Can you find them all? You can try using similar reasoning as in the cases of two and three fractional units, or find your own method. Once you find one solution, try to divide a circle into parts to visualise it! This is your Try This activity for this chapter.

Let us review the Summary. A fraction results when a whole number of units is divided into equal parts and shared equally. Fractional units are parts when one whole unit is divided into equal parts. In 5/6, 5 is the numerator and 6 is the denominator. Mixed fractions contain a whole number part and a fractional part. Fractions can be shown on a number line. Equivalent fractions represent the same share or number. Lowest terms mean no common factor other than 1. Brahmagupta's method for adding and subtracting requires converting to the same denominator, then adding or subtracting numerators.

Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]