Fractions

Hello, my dear students! Welcome to today's mathematics lesson. I am so happy to see you all here, ready to learn something new and exciting. Today, we are going to study a very important chapter - Chapter 7 on Fractions. Now, I know some of you might have heard about fractions before, maybe from your parents or in earlier classes. But in this chapter, we are going to explore fractions in much greater depth. We will understand what fractions really mean, how to work with them, and how they appear in our everyday life. Are you ready? Let us begin!

Let me start by asking you a question. Imagine you have one whole roti, and you want to share it equally between two children. How much roti will each child get? Think about it for a moment. Yes, you are right! Each child will get half a roti. We write this as "one half" and mathematically, we write it as 1/2. We also sometimes read this as "one upon two." Now, what if one roti is shared equally among four children? Then each child gets 1/4 of the roti, which we call "one quarter" or "one fourth." Now, which is more - 1/2 roti or 1/4 roti? Let us think about this carefully. When 2 children share 1 roti equally, each child gets 1/2 roti. When 4 children share 1 roti equally, each child gets 1/4 roti. Since in the second group, more children share the same one roti, each child gets a smaller share. So, 1/2 is greater than 1/4. We write this as 1/2 > 1/4. This is the basic idea behind fractions - they tell us how much of something we have when we share it equally among some people.

Now, let us think about a common mistake that many students make. Suppose I ask you - which fraction is greater, 1/5 or 1/9? At first glance, some of you might think that 1/9 is greater because 9 is bigger than 5. But let us think about this using our sharing idea. If one roti is shared among 5 children, each child gets 1/5 of the roti. If one roti is shared among 9 children, each child gets 1/9 of the roti. Now, if I share with more people, do I get more or less? Obviously less! So, 1/9 is actually smaller than 1/5. We write this as 1/9 < 1/5. This is a very important point to remember - when we are dealing with unit fractions (fractions where the numerator is 1), the larger the denominator, the smaller the fraction.

Now, let us understand what we mean by "fractional units." When one unit is divided into several equal parts, each part is called a fractional unit. For example, when we divide something into 2 equal parts, each part is 1/2. When we divide it into 3 equal parts, each part is 1/3. When we divide it into 4 equal parts, each part is 1/4, and so on. 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 call these "unit fractions." So, 1/100 is bigger than 1/200 because when we divide something into fewer parts, each part is larger.

Now, let me give you some examples to practice. First question: Three guavas together weigh 1 kg. If they are roughly of the same size, each guava will roughly weigh how much? Since the three guavas are equal and together they weigh 1 kg, each guava weighs 1/3 kg. Second question: A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is 1/4 kg. Third question: Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank how much? They shared 3 glasses among 4 friends, so each friend gets 3/4 of a glass. Fourth question: The big fish weighs 1/2 kg and the small one weighs 1/4 kg. Together they weigh 1/2 + 1/4 kg. Let us calculate this: 1/2 = 2/4, so 2/4 + 1/4 = 3/4 kg. So together they weigh 3/4 kg.

I want to share something interesting with you. Fractions have been used and named in India since ancient times! In the Rig Veda, which is one of the oldest texts in the world, the fraction 3/4 is referred to as "tri-pada." This has the same meaning as the words for 3/4 in many Indian languages today. For example, in colloquial Hindi, we say "teen paav" for three quarters, and in Tamil, we say "mukkaal." Isn't that wonderful? The words for fractions that we use today in many Indian languages go back to ancient times. I encourage you to ask your grandparents, parents, teachers, and classmates what words they use for different fractions, such as one and a half, three quarters, one and a quarter, half, quarter, and two and a half. You will find that our rich linguistic heritage has different words for these fractions in different languages!

Now, let us move on to the next section. We are going to understand fractions as parts of a whole. Let me show you a picture in your mind. Imagine a whole chikki. Now, if we break this chikki into 2 equal pieces, how much of the original chikki is each piece? Each piece would be 1/2 of the whole chikki. If we break it into 4 equal pieces, each piece is 1/4 of the whole chikki. Now, look at the picture in your textbook that shows a chikki broken into pieces. You can see that the bigger piece has 3 pieces of 1/4 chikki in it. So we can measure the bigger piece using the fractional unit 1/4, and we see that the bigger piece is 3/4 chikki. This is very important - when we say 3/4, it means 3 times 1/4, or three quarters.

Now, here is an interesting question. By dividing the whole chikki into 6 equal parts in different ways, we get 1/6 chikki pieces of different shapes. Are they of the same size? Even though the shapes are different, the pieces are of the same size because we divided the whole into 6 equal parts. The shape does not matter - what matters is that each part is equal to 1/6 of the whole.

Now, let us learn about 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 then open up the strip again. Taking the strip to be one unit in length, what are the lengths of the two new parts of the strip created by the crease? Each part is 1/2 unit long. So we have two parts, each of length 1/2. What will you get if you fold the previously-folded strip again into two equal parts? You will now get four equal parts, each of length 1/4. So 2 times 1/4 = 2/4, 3 times 1/4 = 3/4, and 4 times 1/4 = 4/4, which equals 1. Let us do it once more! If we fold to get 8 equal parts, then 2 times 1/8 = 2/8, 4 times 1/8 = 4/8, 6 times 1/8 = 6/8, and 8 times 1/8 = 8/8 = 1. This shows us that fractional quantities can be measured using fractional units.

Let us look at another example with rotis. Imagine a full roti, which is our whole. If we take 1/2 of a roti, that is 1 times half. If we take 2 times 1/2, that is 2/2, which equals 1 whole roti. If we take 3 times 1/2, that is 3/2, which is 1 and 1/2 rotis. If we take 4 times 1/2, that is 4/2, which equals 2 whole rotis. And if we take 5 times 1/2, that is 5/2, which equals 2 and 1/2 rotis. We can describe how much the quantity is by collecting together the fractional units. This is a very powerful way to understand fractions!

Now, let me ask you to continue this table of 1/2 for 2 more steps. We already have 5 times 1/2 = 5/2 = 2 and 1/2. So 6 times 1/2 = 6/2 = 3, and 7 times 1/2 = 7/2 = 3 and 1/2. Can you create a similar table for 1/4? Let us try: 1/4 = 1 times quarter, 2 times 1/4 = 2/4 = 1/2, 3 times 1/4 = 3/4, and 4 times 1/4 = 4/4 = 1. Make 1/3 using a paper strip. Can you use this to also make 1/6? Yes! If we fold a strip into thirds, and then fold each third in half, we get sixths. So 1/6 is half of 1/3.

Now, let us learn about reading fractions. We usually read the fraction 3/4 as "three quarters" or "three upon four," but reading it as "3 times 1/4" helps us to understand the size of the fraction because it clearly shows what the fractional unit is (1/4) and how many such fractional units (3) there are. This way of reading fractions helps us understand what the fraction really means.

Now, let me ask you to recall what we call the top number and the bottom number of fractions. In the fraction 5/6, 5 is called the numerator and 6 is called the denominator. The numerator tells us how many fractional units we have, and the denominator tells us the size of each fractional unit, or into how many equal parts the whole has been divided.

Now, we are going to learn about marking fraction lengths on the number line. We have marked lengths equal to 1, 2, 3, ... units on the number line. Now, let us try to mark lengths equal to fractions on the number line. What is the length of the blue line shown in your textbook? The distance between 0 and 1 is one unit long. It is divided into two equal parts. So, the length of each part is 1/2 unit. So, the blue line is 1/2 unit long.

Now, can you find the lengths of the various blue lines shown below? Let us look at the first one. Here, the fractional unit is dividing a length of 1 unit into three equal parts. So each part is 1/3. If the blue line covers 2 of these parts, then its length is 2/3. In the second one, a unit is divided into 5 equal parts. So if the blue line covers 2 parts, it is 2/5, and if it covers 4 parts, it is 4/5. In the third one, a unit is divided into 8 equal parts, so we can have fractions like 1/8, 2/8, 3/8, and so on.

Now, let us think about fractions that are greater than one. You marked some fractions on the number line earlier. Did you notice that the lengths of all the blue lines were less than one and the lengths of all the black lines were more than 1? Let us classify these fractions in two groups: lengths less than 1 unit, and lengths more than 1 unit. Did you notice something common between the fractions that are greater than 1? In all the fractions that are less than 1 unit, the numerator is smaller than the denominator, while in the fractions that are more than 1 unit, the numerator is larger than the denominator.

We know that 3/2, 5/2, and 7/2 are all greater than 1 unit. But can we see how many whole units they contain? Let us break them down: 3/2 = 1/2 + 1/2 + 1/2 = 1 + 1/2. So 3/2 contains 1 whole unit and 1/2 more. Similarly, 5/2 = 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 2 + 1/2. So 5/2 contains 2 whole units and 1/2 more. Also, remember that 1/3 + 1/3 + 1/3 = 3/3 = 1. If I add one more 1/3, I will get more than 1 unit! So, 4/3 is greater than 1.

Now, let me ask you some questions. How many whole units are there in 7/2? We can think of 7/2 as 2 + 1/2 + 1/2 + 1/2, but actually it is easier to divide 7 by 2. 7 divided by 2 is 3 with a remainder of 1. So 7/2 = 3 and 1/2. So there are 3 whole units in 7/2. How many whole units are there in 4/3? 4 divided by 3 is 1 with a remainder of 1, so 4/3 = 1 and 1/3. So there is 1 whole unit in 4/3. And in 7/3? 7 divided by 3 is 2 with a remainder of 1, so 7/3 = 2 and 1/3. So there are 2 whole units in 7/3.

Now, let us learn about writing fractions greater than one as mixed numbers. We saw that 3/2 = 1 + 1/2. We can write other fractions in a similar way. For example, 4/3 = 1/3 + 1/3 + 1/3 + 1/3 = 1 + 1/3. Since 3 times 1/3 = 1, we have 4/3 = 1 + 1/3.

Let me give you more examples. How many whole units are there in 8/3? 8 divided by 3 is 2 with a remainder of 2, so 8/3 = 2 and 2/3. How about 11/5? 11 divided by 5 is 2 with a remainder of 1, so 11/5 = 2 and 1/5. And 9/4? 9 divided by 4 is 2 with a remainder of 1, so 9/4 = 2 and 1/4.

This number is thus also called "two and two thirds." We also write it as 2 2/3. 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).

Now, can all fractions greater than 1 be written as such mixed numbers? Most of them can, but not all. For example, 8/4 = 2, which is just a whole number and does not need to be written as a mixed number. But generally, fractions where the numerator is greater than the denominator can be written as mixed numbers.

Let me give you some practice questions. Write the following fractions as mixed fractions: 9/2, 9/5, 21/19, 47/9, 12/11, and 19/6. Let me work through each one with you.

For 9/2: 9 divided by 2 is 4 with a remainder of 1, so 9/2 = 4 1/2. For 9/5: 9 divided by 5 is 1 with a remainder of 4, so 9/5 = 1 4/5. For 21/19: 21 divided by 19 is 1 with a remainder of 2, so 21/19 = 1 2/19. For 47/9: 47 divided by 9 is 5 with a remainder of 2, so 47/9 = 5 2/9. For 12/11: 12 divided by 11 is 1 with a remainder of 1, so 12/11 = 1 1/11. For 19/6: 19 divided by 6 is 3 with a remainder of 1, so 19/6 = 3 1/6.

Now, can we write a mixed number as a regular fraction? Yes, we can! Let me show you how. When I have 3 1/4, this means 3 + 1/4. I know that 1 = 1/4 + 1/4 + 1/4 + 1/4. So 3 = 3 times 1 = 3 times (1/4 + 1/4 + 1/4 + 1/4) = (4 × 1/4) + (4 × 1/4) + (4 × 1/4) = 12/4. Then adding the 1/4, we get 12/4 + 1/4 = 13/4. So 3 1/4 = 13/4.

Let me give you more examples. Write the following mixed numbers as fractions: 3 1/4, 7 2/3, 9 4/9, 3 1/6, 2 3/11, and 3 9/10.

For 3 1/4: We have 3 + 1/4 = 12/4 + 1/4 = 13/4. For 7 2/3: We have 7 + 2/3 = 21/3 + 2/3 = 23/3. For 9 4/9: We have 9 + 4/9 = 81/9 + 4/9 = 85/9. For 3 1/6: We have 3 + 1/6 = 18/6 + 1/6 = 19/6. For 2 3/11: We have 2 + 3/11 = 22/11 + 3/11 = 25/11. For 3 9/10: We have 3 + 9/10 = 30/10 + 9/10 = 39/10.

Now, let us learn about equivalent fractions. In the previous section, we used paper folding to represent various fractions using fractional units. Let us do some more activities with the same paper strips.

What do we observe? Let us check whether 1/2 and 2/4 are equal. If we take a strip and fold it to show 1/2, and another strip and fold it to show 2/4, are the lengths equal? Yes, they are! Are the lengths 2/4 and 4/8 equal? Yes, they are also equal! So we can say that 1/2 = 2/4 = 4/8. These are "equivalent fractions" that denote the same length, but they are expressed in terms of different fractional units.

Now, check whether 1/3 and 2/6 are equivalent fractions or not, using paper strips. Yes, they are equivalent because 1/3 = 2/6.

You can also make your own fraction wall using paper strips as given in the picture in your textbook!

Now, let me ask you some questions based on the fraction wall. Are the lengths 1/2 and 3/6 equal? Yes! Are 2/3 and 4/6 equivalent fractions? Why? Yes, they are equivalent because 2/3 = 4/6. How many pieces of length 1/6 will make a length of 1/2? We need 3 pieces of 1/6 to make 1/2, because 3/6 = 1/2. How many pieces of length 1/6 will make a length of 1/3? We need 2 pieces of 1/6 to make 1/3, because 2/6 = 1/3.

Now, let me ask you: Are 3/6, 4/8, 5/10 equivalent fractions? Why? Yes, they are all equivalent because they all equal 1/2. Write two equivalent fractions for 2/6. Since 2/6 = 1/3, some equivalent fractions are 1/3, 3/9, 4/12, and so on.

Now, let us understand equivalent fractions using the idea of equal shares. One roti was shared equally by four children. What fraction of the whole did each child get? Each child gets 1/4 of the roti. The four shares must be equal to each other!

You can also express this event through division facts, addition facts, and multiplication facts. The division fact is 1 ÷ 4 = 1/4. The addition fact is 1 = 1/4 + 1/4 + 1/4 + 1/4. The multiplication fact is 1 = 4 × 1/4.

Now, let me ask you: Three rotis are shared equally by four children. Show the division in the picture and write a fraction for how much each child gets. Also, write the corresponding division facts, addition facts, and multiplication facts. Each child gets 3/4 roti. The division fact is 3 ÷ 4 = 3/4. The addition fact is 3 = 3/4 + 3/4 + 3/4 + 3/4. The multiplication fact is 3 = 4 × 3/4.

Now, draw a picture to show how much each child gets when 2 rotis are shared equally by 4 children. Each child gets 1/2 roti. The division fact is 2 ÷ 4 = 1/2. The addition fact is 2 = 1/2 + 1/2 + 1/2 + 1/2. The multiplication fact is 2 = 4 × 1/2.

Now, Anil was in a group where 2 cakes were divided equally among 5 children. How much cake would Anil get? He would get 2/5 of a cake. Now, if there are 10 children in my group, how many cakes will I need so that they get the same amount of cake as Anil? If we want each of 10 children to get the same amount as Anil got (which was 2/5), then we need 2 groups of 5 children, so we need 2 × 2 = 4 cakes. So 2/5 = 4/10! The share of each child is the same in both these situations!

Let us examine the shares of each child in the following situations: 1 roti divided equally between 2 children, 2 rotis divided equally among 4 children, and 3 rotis divided equally among 6 children. In each situation, the share of every child is the same, which is 1/2. So we can say that 1/2 = 2/4 = 3/6. Fractions where the shares are equal are called "equivalent fractions." So 1/2, 2/4, and 3/6 are all equivalent fractions.

Find some more fractions equivalent to 1/2. Some examples are 4/8, 5/10, 6/12, 7/14, and so on.

Now, let me ask you to equally divide the rotis in the situations shown below and write down the share of each child. Are the shares in each of these cases the same? Why? The first situation is 2 rotis divided equally among 3 children. Each child gets 2/3 roti. The second situation is 4 rotis divided equally among 6 children. Each child gets 4/6 = 2/3 roti. The third situation is 6 rotis divided equally among 9 children. Each child gets 6/9 = 2/3 roti. Yes, the shares are the same in all cases because 2/3 = 4/6 = 6/9. 2/3 is also called the simplest form of 4/6. It is also the simplest form of 6/9 as well. Do you notice anything about the relationship between the numerator and denominator in each of these fractions? In the simplest form, the numerator and denominator have no common factor other than 1.

Now, let me ask you to find the missing numbers. First question: 5 glasses of juice shared equally among 4 friends is the same as how many glasses of juice shared equally among 8 friends? If we double the number of friends, we need to double the amount of juice to give each person the same share. So 5/4 = 10/8. Second question: 4 kg of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided equally in how many bags? If we triple the amount of potatoes, we need to triple the number of bags to keep the share the same. So 4/3 = 12/9. Third question: 7 rotis divided among 5 children is the same as how many rotis divided among how many children? There are many possibilities. For example, if we double both numbers, we get 14/10. If we triple both, we get 21/15. Any equivalent fraction would work.

Now, let me ask you an interesting question. In which group will each child get more chikki? Group 1: 1 chikki divided between 2 children, or Group 2: 5 chikkis divided among 8 children. We must compare 1/2 and 5/8. We have seen that 1/2 = 4/8, and clearly 4/8 < 5/8. So the children in Group 2 will get more chikki each.

What about the following groups? In which group will each child get more? Group 1: 1 chikki divided between 2 children, or Group 2: 4 chikkis divided among 7 children. We must compare 1/2 and 4/7. Now, 1/2 = 4/8. But we want to compare with 4/7. When 4 chikkis are divided equally among 7 children, each one will get 4/7 chikki. When 4 chikkis are divided equally among 8 children, each one will get 4/8 chikki. So 4/7 > 4/8. And since 4/8 = 1/2, we can conclude that 4/7 > 1/2. So the children in Group 2 get more.

This shows us an important rule: If the number of units that are shared is the same, but the number of children among whom the units are shared is more, then the share is less. Therefore, 4/7 > 4/8 and 4/8 = 1/2, so 4/7 > 1/2.

Now, suppose the number of children is kept the same, but the number of units that are being shared is increased. What can you say about each child's share now? The share will increase! This reasoning explains why 1/5 < 2/5, 3/7 < 4/7, and 1/2 < 5/8. When the number of children is the same, if we increase the amount being shared, each child gets more.

Now, let us decide in which of the two groups will each child get a larger share. First pair: Group 1 has 3 glasses of sugarcane juice divided equally among 4 children, so each gets 3/4. Group 2 has 7 glasses divided equally among 10 children, so each gets 7/10. Which is more, 3/4 or 7/10? Second pair: Group 1 has 4 glasses divided equally among 7 children, so each gets 4/7. Group 2 has 5 glasses divided equally among 7 children, so each gets 5/7. Which is more, 4/7 or 5/7? Obviously, 5/7 > 4/7 because the denominators are the same. So the groups in the second part were easier to compare because the number of children was the same.

Now, when the denominators are not the same, we need to find equivalent fractions so that the fractional units are the same. For example, to compare 3/4 and 7/10, we can find fractions equivalent to both with the same denominator. The product of the two denominators (4 and 10) is 40. So 3/4 = 30/40 and 7/10 = 28/40. Since 30/40 > 28/40, we conclude that 3/4 > 7/10.

This is exactly what we are going to learn next - how to compare fractions!

Now, let us learn about expressing a fraction in lowest terms, or in its simplest form. 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. In other words, a fraction is said to be in lowest terms if its numerator and denominator are as small as possible.

Any fraction can be expressed in lowest terms by finding an equivalent fraction whose numerator and denominator are as small as possible. Let us see how to express fractions in lowest terms.

Example: Is the fraction 16/20 in lowest terms? No, because 4 is a common factor of 16 and 20. Let us reduce 16/20 to lowest terms. We know that both 16 and 20 are divisible by 4. So, 16 ÷ 4 = 4 and 20 ÷ 4 = 5. Therefore, 16/20 = 4/5. Now, there is no common factor between 4 and 5. Hence, 16/20 expressed in lowest terms is 4/5. So, 4/5 is called the simplest form of 16/20, since 4 and 5 have no common factor other than 1.

Any fraction can be converted to lowest terms by dividing both the numerator and denominator by the highest common factor between them.

Expressing a fraction in lowest terms can also be done in steps. Suppose we want to express 36/60 in lowest terms. First, we notice that both the numerator and denominator are even. So, we divide both by 2, and see that 36/60 = 18/30. Both the numerator and denominator are even again, so we can divide them each by 2 again; we get 18/30 = 9/15. We now notice that 9 and 15 are both multiples of 3, so we divide both by 3 to get 9/15 = 3/5. Now, 3 and 5 have no common factor other than 1, so 36/60 in lowest terms is 3/5. Alternatively, we could have noticed that in 36/60, both the numerator and denominator are multiples of 12: we see that 36 = 3 × 12 and 60 = 5 × 12. Therefore, we could have concluded that 36/60 = 3/5 straight away. Either method works and will give the same answer! But sometimes it can be easier to go in steps.

Now, let us learn about comparing fractions. Which is greater, 4/5 or 7/9? It can be difficult to compare two such fractions directly. However, we know how to find fractions equivalent to two fractions with the same denominator. Let us see how we can use it: 4/5 = (4 × 9)/(5 × 9) = 36/45, and 7/9 = (7 × 5)/(9 × 5) = 35/45. 45 is a common multiple of 5 and 9, so we can use 45 as a common denominator. Clearly, 36/45 > 35/45. So, 4/5 > 7/9!

Let us try this for another pair: 7/9 and 17/21. 63 is a common multiple of 9 and 21. We can then write: 7/9 = (7 × 7)/(9 × 7) = 49/63, and 17/21 = (17 × 3)/(21 × 3) = 51/63. Clearly, 49/63 < 51/63. So, 7/9 < 17/21!

Let us summarize the steps to compare the sizes of two or more given fractions:

Step 1: Change the given fractions to equivalent fractions so that they all are expressed with the same denominator or same fractional unit.

Step 2: Now, compare the equivalent fractions by simply comparing the numerators, i.e., the number of fractional units each has.

Now, let us learn about addition and subtraction of fractions. Meena's father made some chikki. Meena ate 1/2 of it and her younger brother ate 1/4 of it. How much of the total chikki did Meena and her brother eat together?

We can arrive at the answer by visualizing it. Let us take a piece of chikki and divide it into two halves first. Meena ate 1/2 of it as shown in the picture. Let us now divide the remaining half into two further halves as shown. Each of these pieces is 1/4 of the whole chikki. Meena's brother ate 1/4 of the whole chikki, as is shown in the picture. The total chikki eaten is 1/2 (by Meena) and 1/4 (by her brother). The total chikki eaten = 1/2 + 1/4 = 1/4 + 1/4 + 1/4 = 3 × 1/4 = 3/4. How much of the total chikki is remaining? The whole is 1, and they ate 3/4, so 1 - 3/4 = 1/4 is remaining.

Now, let us learn about adding fractions with the same fractional unit or denominator. Example: Find the sum of 2/5 and 1/5. Let us represent both using the rectangular strips. In both fractions, the fractional unit is the same 1/5, so each strip will be divided into 5 equal parts. So 2/5 will be represented as 2 shaded parts out of 5, and 1/5 will be represented as 1 shaded part out of 5. Adding the two given fractions is the same as finding out the total number of shaded parts, each of which represent the same fractional unit 1/5. In this case, the total number of shaded parts is 3. Since each shaded part represents the fractional unit 1/5, we see that the 3 shaded parts together represent the fraction 3/5. Therefore, 2/5 + 1/5 = 3/5.

Example: Find the sum of 4/7 and 6/7. Let us represent both again using the rectangular strip model. Here in both fractions, the fractional unit is the same, i.e., 1/7, so each strip will be divided into 7 equal parts. Then 4/7 will be represented as 4 shaded parts, and 6/7 will be represented as 6 shaded parts. In this case, the total number of shaded parts is 10, and each shaded part represents the fractional unit 1/7, so the 10 shaded parts together represent the fraction 10/7. While adding fractions with the same fractional unit, just add the number of fractional units from each fraction. Therefore, 4/7 + 6/7 = 10/7 = 1 + 3/7 = 1 3/7.

Now, let us learn about adding fractions with different fractional units or denominators. Example: Find the sum of 1/4 and 1/3. To add fractions with different fractional units, first convert the fractions into equivalent fractions with the same denominator or fractional unit. In this case, the common denominator can be made 3 × 4 = 12, i.e., we can find equivalent fractions with fractional unit 1/12. Let us write the equivalent fraction for each given fraction: 1/4 = (1 × 3)/(4 × 3) = 3/12, and 1/3 = (1 × 4)/(3 × 4) = 4/12. Now, 3/12 and 4/12 have the same fractional unit, i.e., 1/12. Therefore, 1/4 + 1/3 = 3/12 + 4/12 = 7/12.

This method of addition, which works for adding any number of fractions, was first explicitly described in general by Brahmagupta in the year 628 CE! We will describe the history of the development of fractions in more detail later in the chapter. For now, we simply summarize the steps in Brahmagupta's method for addition of fractions.

Brahmagupta's method for adding fractions has three steps:

Step 1: Find equivalent fractions so that the fractional unit is common for all fractions. This can be done by finding a common multiple of the denominators (e.g., the product of the denominators, or the smallest common multiple of the denominators).

Step 2: Add these equivalent fractions with the same fractional units. This can be done by adding the numerators and keeping the same denominator.

Step 3: Express the result in lowest terms if needed.

Let us carry out another example of Brahmagupta's method. Example: Find the sum of 2/3 and 1/5. The denominators of the given fractions are 3 and 5. The lowest common multiple of 3 and 5 is 15. Then we see that 2/3 = (2 × 5)/(3 × 5) = 10/15, and 1/5 = (1 × 3)/(5 × 3) = 3/15. Therefore, 2/3 + 1/5 = 10/15 + 3/15 = 13/15.

Example: Find the sum of 1/6 and 1/3. The smallest common multiple of 6 and 3 is 6. 1/6 will remain 1/6. 1/3 = (1 × 2)/(3 × 2) = 2/6. Therefore, 1/6 + 1/3 = 1/6 + 2/6 = 3/6. The fraction 3/6 can now be re-expressed in lowest terms, if desired. This can be done by dividing both the numerator and denominator by 3 (the biggest common factor of 3 and 6): 3/6 = (3 ÷ 3)/(6 ÷ 3) = 1/2. Therefore, 1/6 + 1/3 = 1/2.

Now, let us learn about subtraction of fractions with the same fractional unit or denominator. Brahmagupta's method also applies when subtracting fractions! Let us start with the problem of subtracting 4/7 from 6/7, i.e., what is 6/7 − 4/7? To solve this problem, we can again use the rectangular strips. In both fractions, the fractional unit is the same, i.e., 1/7. Let us first represent the bigger fraction using a rectangular strip model as shown: 6/7. Each shaded part represents 1/7. Now, we need to subtract 4/7. To do this, let us remove 4 of the shaded parts. We can do this here directly because both fractions have the same fractional units. So, we are left with 2 shaded parts, i.e., 6/7 − 4/7 = 2/7.

Now, let us learn about subtraction of fractions with different fractional units or denominators. Example: What is 3/4 − 2/3? As we already know the procedure for subtraction of fractions with the same fractional units, let us convert each of the given fractions into equivalent fractions with the same fractional units. 3/4 = (3 × 3)/(4 × 3) = 9/12. And similarly, 2/3 = (2 × 4)/(3 × 4) = 8/12. Therefore, 3/4 − 2/3 = 9/12 − 8/12 = 1/12.

Brahmagupta's method for subtracting two fractions has the same three steps:

Step 1: Convert the given fractions into equivalent fractions with the same fractional unit, i.e., the same denominator.

Step 2: Carry out the subtraction of fractions having the same fractional units. This can be done by subtracting the numerators and keeping the same denominator.

Step 3: Simplify the result into lowest terms if needed.

Now, let me share with you 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, globally, originated in India. In ancient Indian mathematical texts, such as the Bakshali manuscript (from around the year 300 CE), when they wanted to write 1/2, they wrote it as 1/2, which is indeed very similar to the way we write it today! This method of writing and working with fractions continued to be used in India for the next several centuries, including by Aryabhata (499 CE), Brahmagupta (628 CE), Sridharacharya (c. 750 CE), and Mahaviracharya (c. 850 CE), among others. The line segment between the numerator and denominator in "1/2," and in other fractions was later introduced by the Moroccan mathematician Al-Hassar (in the 12th century). Over the next few centuries, the notation then spread to Europe and around the world.

Fractions had also been used in other cultures such as the ancient Egyptian and Babylonian civilizations, but they primarily used only fractional units, that is, fractions with a 1 in the numerator. More general fractions were expressed as sums of fractional units, now called "Egyptian fractions." Writing numbers as the sum of fractional units, e.g., 19/24 = 1/2 + 1/6 + 1/8, can be quite an art and leads to beautiful puzzles.

General fractions (where the numerator is not necessarily 1) were first introduced in India, along with their rules of arithmetic operations like addition, subtraction, multiplication, and even division of fractions. The ancient Indian treatises called the "Sulba-sutras" show that even during Vedic times, Indians had discovered the rules for operations with fractions. General rules and procedures for working with and computing with fractions were first codified formally and in a modern form by Brahmagupta.

Brahmagupta's methods for working with and computing with fractions are still what we use today. For example, Brahmagupta described how to add and subtract fractions as follows: "By the multiplication of the numerator and the denominator of each of the fractions by the other denominators, the fractions are reduced to a common denominator. Then, in case of addition, the numerators (obtained after the above reduction) are added. In case of subtraction, their difference is taken." (Brahmagupta, Brahmasphutasiddhanta, Verse 12.2, 628 CE)

The Indian concepts and methods involving fractions were transmitted to Europe via the Arabs over the next few centuries, and they came into general use in Europe in around the 17th century and then spread worldwide.

Now, let me give you a fun puzzle! It is easy to add up fractional units to obtain the sum 1 if one uses the same fractional unit, for example, 1/2 + 1/2 = 1, 1/3 + 1/3 + 1/3 = 1, 1/4 + 1/4 + 1/4 + 1/4 = 1, etc. However, can you think of a way to add fractional units that are all different to get 1? It is not possible to add two different fractional units to get 1. The reason is that 1/2 is the largest fractional unit, and 1/2 + 1/2 = 1. To get different fractional units, we would have to replace at least one of the 1/2's with some smaller fractional unit - but then the sum would be less than 1! Therefore, it is not possible for two different fractional units to add up to 1.

We can try to look instead for a way to write 1 as the sum of three different fractional units. Can you find three different fractional units that add up to 1? It turns out there is only one solution to this problem (up to changing the order of the 3 fractions)! Here is a systematic way to find the solution. We know that 1/3 + 1/3 + 1/3 = 1. To get the fractional units to be different, we will have to increase at least one of the 1/3's, and decrease at least one of the other 1/3's to compensate for that increase. The only way to increase 1/3 to another fractional unit is to replace it by 1/2. So 1/2 must be one of the fractional units. Now 1/2 + 1/4 + 1/4 = 1. To get the fractional units to be different, we will have to increase one of the 1/4's and decrease the other 1/4 to compensate for that increase. Now the only way to increase 1/4 to another fractional unit, that is different from 1/2, is to replace it by 1/3. So two of the fractions must be 1/2 and 1/3! What must be the third fraction then, so that the three fractions add up to 1? This explains why there is only one solution to the above problem: 1/2 + 1/3 + 1/6 = 1.

What if we look for 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? You can try using similar reasoning as in the cases of two and three fractional units, or find your own method!

Now, let me summarize everything we have learned in this chapter.

First, we learned that when a whole number of units is divided into equal parts and shared equally, a fraction results. We call this "fraction as equal share."

We learned that when one whole basic unit is divided into equal parts, each part is called a fractional unit. These include 1/2, 1/3, 1/4, 1/5, and so on.

We learned how to read fractions. In a fraction such as 5/6, 5 is called the numerator and 6 is called the denominator.

We learned about mixed fractions, which contain a whole number part and a fractional part, such as 2 1/2 or 3 2/3.

We learned that fractions can be shown on a number line, and every fraction has a point associated with it on the number line.

We learned about equivalent fractions - when two or more fractions represent the same share or number, they are called equivalent fractions. For example, 1/2 = 2/4 = 3/6.

We learned about lowest terms - a fraction whose numerator and denominator have no common factor other than 1 is said to be in lowest terms or in its simplest form. For example, 4/5 is the simplest form of 16/20.

We learned Brahmagupta's method for adding fractions: when adding fractions, convert them into equivalent fractions with the same fractional unit (i.e., the same denominator), and then add the number of fractional units in each fraction to obtain the sum. This is accomplished by adding the numerators while keeping the same denominator.

We learned Brahmagupta's method for subtracting fractions: when subtracting fractions, convert them into equivalent fractions with the same fractional unit (i.e., the same denominator), and then subtract the number of fractional units. This is accomplished by subtracting the numerators while keeping the same denominator.

We also learned about the rich history of fractions in India, dating back to the Vedic period, and the contributions of great Indian mathematicians like Brahmagupta.

And finally, we explored a fun puzzle about writing 1 as a sum of different fractional units.

This concludes our lesson on Chapter 7: Fractions. I hope you have understood all the concepts clearly. Remember, fractions are all around us in our daily life - when we share food, measure ingredients for cooking, or divide our time. Keep practicing, and you will become very comfortable with fractions. Thank you for being such wonderful students! See you in the next lesson!