Working With Fractions

Namaste, 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 explore a very important chapter – Chapter 8: Working with Fractions. Now, I know all of you have learned about fractions in your earlier classes, but in this chapter, we are going to go much deeper. We will learn how to multiply fractions, how to divide fractions, and how to solve real-life problems using these operations. Are you ready? Let us begin!

Let me start with a story. Imagine there is a boy named Aaron who loves walking. He walks 3 kilometres in just 1 hour. Now, if he walks for 5 hours, how far do you think he will go? This is simple, isn't it? We just multiply 5 hours by 3 kilometres per hour, and we get 15 kilometres. That is 3 + 3 + 3 + 3 + 3, five times three equals fifteen. Easy enough!

Now, here is something interesting. Aaron has a pet tortoise. This tortoise walks very slowly – it can walk only 1/4 kilometre in 1 hour. Can you imagine how slow that is? Now, the question is: if the tortoise walks for 3 hours, how far will it go? Here, the distance covered in one hour is a fraction, 1/4 kilometre. But that does not change our method. We still multiply! So, distance covered in 3 hours equals 3 times 1/4 kilometre. That is 1/4 + 1/4 + 1/4, which equals 3/4 kilometre. So the tortoise can walk 3/4 kilometre in 3 hours. Not bad for a slowpoke, right?

Now, let us make this a little more interesting. What if Aaron himself walks for only 1/5 of an hour? That is just 12 minutes, you know. How far will he walk in that short time? We need to find 1/5 multiplied by 3 kilometres. Let us think about this carefully. In 1 hour, Aaron walks 3 kilometres. So in 1/5 of an hour, he will walk 1/5 of that distance. That means we divide 3 kilometres into 5 equal parts, and take one part. So the distance is 3/5 kilometre. This tells us that 1/5 multiplied by 3 equals 3/5. Simple, isn't it?

Now, what if Aaron walks for 2/5 of an hour? How far will he go? We need to find 2/5 multiplied by 3 kilometres. Here is how we can think about it. First, find the distance covered in 1/5 of an hour, which we already know is 3/5 kilometre. Now, since 2/5 is twice 1/5, we just double that distance. So we multiply 3/5 by 2, which gives us 6/5 kilometre. And that is equal to 1 and 1/5 kilometres. So we can say that 2/5 multiplied by 3 equals 6/5.

Now, students, let me explain what we did here. When we multiply a fraction by a whole number, we follow two simple steps. First, we divide the whole number by the denominator of the fraction. Then, we multiply the result by the numerator of the fraction. In our example, we divided 3 by 5 to get 3/5, and then multiplied by 2 to get 6/5. This is the method we use every time we multiply a fraction by a whole number. Remember this!

Now, let us look at some examples to make this clearer.

Example 1: A farmer had 5 grandchildren. She distributed 2/3 acre of land to each of her grandchildren. How much land in all did she give to her grandchildren?

So we need to find 5 multiplied by 2/3. We can write this as 2/3 added five times: 2/3 + 2/3 + 2/3 + 2/3 + 2/3. That gives us 10/3 acres. In mixed form, that is 3 and 1/3 acres. So the farmer gave 10/3 or 3 1/3 acres of land in total to her grandchildren.

Example 2: 1 hour of internet time costs ₹8. How much will 1 1/4 hours of internet time cost?

First, we need to convert 1 1/4 hours into an improper fraction. 1 1/4 is the same as 5/4 hours. Now we multiply 5/4 by ₹8. We can write this as 5 multiplied by 8/4. Now 8 divided by 4 is 2, so we get 5 times 2, which equals ₹10. So 1 1/4 hours of internet time costs ₹10.

Great! Now we have learned how to multiply a whole number with a fraction, and a fraction with a whole number. But what happens when both numbers in the multiplication are fractions? That is what we are going to learn next.

Let us go back to our tortoise. The tortoise can walk 1/4 kilometre in 1 hour. Now, how far will it walk in half an hour? We need to find 1/2 multiplied by 1/4 kilometre. Let us think about this carefully. In 1 hour, the tortoise walks 1/4 kilometre. So in half an hour, it will walk half of that distance. To find this, we can use a unit square. Imagine a square that represents 1 whole kilometre. We divide this square into 4 equal parts because the denominator of 1/4 is 4. One of those parts is shaded, representing 1/4. Now, we need to divide this 1/4 into 2 equal parts because the denominator of 1/2 is 2. When we do that, we get 8 equal parts in total, and only 1 part is shaded. So the distance covered is 1/8 kilometre. This tells us that 1/2 multiplied by 1/4 equals 1/8.

Now, let us try a more complicated example. Suppose the tortoise walks faster and can cover 2/5 kilometre in 1 hour. How far will it walk in 3/4 of an hour? We need to find 3/4 multiplied by 2/5.

Here is how we solve it. First, find the distance covered in 1/4 of an hour. That is 2/5 divided by 4, which equals 2/20. Then, multiply this by 3 to get the distance covered in 3/4 of an hour. So 3 multiplied by 2/20 equals 6/20, which simplifies to 3/10. So 3/4 multiplied by 2/5 equals 3/10 kilometre.

Now, students, let me explain what we did. When we multiply two fractions, we follow a similar method to what we did with a fraction and a whole number. First, we divide the multiplicand by the denominator of the multiplier. Then, we multiply the result by the numerator of the multiplier. In this case, we divided 2/5 by 4 to get 2/20, and then multiplied by 3 to get 6/20, which simplifies to 3/10.

But there is an even simpler way to multiply fractions. We can just multiply the numerators together and multiply the denominators together! That is what we call Brahmagupta's formula. Brahmagupta was a famous Indian mathematician who lived in the 7th century. He gave us this beautiful formula: a/b multiplied by c/d equals (a × c) divided by (b × d).

Let me show you how this works. For 3/4 multiplied by 2/5, we multiply the numerators: 3 × 2 = 6. Then we multiply the denominators: 4 × 5 = 20. So we get 6/20, which simplifies to 3/10. Exactly what we got before!

This formula works for any two fractions. Let us try another example. Multiply 5/4 by 3/2. Using the formula, we multiply 5 × 3 = 15 for the numerator, and 4 × 2 = 8 for the denominator. So we get 15/8. In mixed form, that is 1 and 7/8.

And guess what, students? This formula also works when one of the numbers is a whole number. We just write the whole number as a fraction with denominator 1. For example, 3 multiplied by 3/4 is the same as 3/1 multiplied by 3/4. That gives us (3 × 3) divided by (1 × 4), which equals 9/4 or 2 and 1/4. Similarly, 3/5 multiplied by 4 is the same as 3/5 multiplied by 4/1, which gives us (3 × 4) divided by (5 × 1), which equals 12/5 or 2 and 2/5.

Now, let me tell you about something very important – simplifying fractions before we multiply. When we multiply fractions, we can cancel out common factors between the numerators and denominators before multiplying. This makes our calculation much easier.

For example, let us multiply 12/7 by 5/24. Instead of multiplying 12 × 5 and 7 × 24 first, we can look for common factors. We see that 12 and 24 have a common factor of 12. We can divide both by 12. So 12 divided by 12 is 1, and 24 divided by 12 is 2. Now our multiplication becomes (1 × 5) divided by (7 × 2), which equals 5/14. This is much simpler than multiplying 12 × 5 = 60 and 7 × 24 = 168, and then simplifying 60/168 to 5/14. Both methods give the same answer, but cancelling first makes it easier!

Let us try one more example. Multiply 14/15 by 25/42. We can see that 14 and 42 have a common factor of 14. Also, 15 and 25 have a common factor of 5. Let us cancel these common factors. 14 divided by 14 is 1, and 42 divided by 14 is 3. 15 divided by 5 is 3, and 25 divided by 5 is 5. Now our multiplication becomes (1 × 5) divided by (3 × 3), which equals 5/9. Easy, right?

This process of cancelling common factors before multiplying is called cancelling the common factors. It is a very useful technique, and I want you to practice it.

Now, students, I want to tell you something interesting about the product of fractions. Have you ever wondered whether the product of two fractions is always greater than the numbers we are multiplying? Let us explore this together.

We know that when we multiply a number by 1, the product remains the same. So let us see what happens when we multiply pairs of numbers where neither of them is 1.

When we multiply two counting numbers greater than 1, say 3 and 5, the product is 15, which is greater than both 3 and 5. That makes sense, doesn't it?

But what happens when we multiply 1/4 and 8? We get 1/4 × 8 = 2. Here, the product 2 is greater than 1/4, but it is less than 8. Interesting!

What happens when we multiply 3/4 and 2/5? We get 3/4 × 2/5 = 6/20 = 3/10. Let us compare this with 3/4 and 2/5. If we express 3/4 as 15/20 and 2/5 as 8/20, we can see that 3/10 or 6/20 is less than both 15/20 and 8/20. So the product is less than both the numbers!

So here is what we can conclude: When both numbers being multiplied are greater than 1, the product is greater than both the numbers. When both numbers are between 0 and 1, the product is less than both the numbers. And when one number is between 0 and 1 and the other is greater than 1, the product is less than the number greater than 1 but greater than the number between 0 and 1. This is a very important observation, and it will help you understand fractions better.

Now, let me tell you about another property of multiplication. We know that 1/2 × 1/4 = 1/8. Now, what is 1/4 × 1/2? That is also 1/8! So the order of multiplication does not matter. This is called the commutative property of multiplication. In general, a/b × c/d = c/d × a/b. This makes sense because the area of a rectangle remains the same even if we swap its length and breadth.

Now, students, we have learned how to multiply fractions. But what about dividing fractions? That is what we are going to learn next.

Let us start with a simple question. What is 12 ÷ 4? You know this already – it is 3. But can we restate this as a multiplication problem? What should be multiplied by 4 to get 12? That is, 4 × ? = 12. The answer is 3, because 4 × 3 = 12. So 12 ÷ 4 is the same as finding what number multiplied by 4 gives 12.

We can use this same technique to divide fractions. Let me show you how.

What is 1 ÷ 2/3? Let us rewrite this as a multiplication problem: 2/3 × ? = 1. What should be multiplied by 2/3 to get 1? If we multiply 2/3 by 3/2, we get 1, because 2/3 × 3/2 = 6/6 = 1. So 1 ÷ 2/3 = 3/2.

Let us try another one: 3 ÷ 2/3. This is the same as 2/3 × ? = 3. We already know that 2/3 × 3/2 = 1. So to get 3, we just multiply by 3. So 2/3 × 3/2 × 3 = 3. Therefore, 3 ÷ 2/3 = 3/2 × 3 = 9/2.

What about 1/5 ÷ 1/2? Rewriting it as a multiplication problem, we have 1/2 × ? = 1/5. How do we solve this? We know that 1/2 × 2 = 1. So if we multiply 1/5 by 2, we get 2/5. And 1/2 × 2/5 = 2/10 = 1/5. So the answer is 2/5. Therefore, 1/5 ÷ 1/2 = 2/5.

Now, let us try 2/3 ÷ 3/5. Rewriting this as multiplication, we have 3/5 × ? = 2/3. We need to find what to multiply 3/5 by to get 2/3. We know that 3/5 × 5/3 = 1. So if we multiply 2/3 by 5/3, we get 10/9. And 3/5 × 10/9 = 30/45 = 2/3. So the answer is 10/9. Therefore, 2/3 ÷ 3/5 = 10/9.

Now, students, let me summarize what we did in these division problems. In every division problem, we have a dividend, a divisor, and a quotient. The technique we used was:

First, find the number which gives 1 when multiplied by the divisor. We call this the reciprocal of the divisor. For example, the reciprocal of 3/5 is 5/3. When we multiply a fraction by its reciprocal, we get 1.

Then, we multiply the dividend by this reciprocal to get the quotient.

So, to divide two fractions, we find the reciprocal of the divisor and multiply it by the dividend. In other words, a/b ÷ c/d = a/b × d/c = (a × d) / (b × c).

This is Brahmagupta's formula for division of fractions, which he gave us in 628 CE in his book Brāhmasphuṭasiddhānta.

Now, let me tell you something interesting about division. When we divide two whole numbers, say 6 ÷ 3, we get the quotient 2, which is less than the dividend 6. But what happens when we divide 6 by 1/4? We get 6 ÷ 1/4 = 24. Here, the quotient is greater than the dividend! And what happens when we divide 1/8 by 1/4? We get 1/8 ÷ 1/4 = 1/2. Here too, the quotient is greater than the dividend.

So here is what we can conclude: When the divisor is between 0 and 1, the quotient is greater than the dividend. When the divisor is greater than 1, the quotient is less than the dividend. This is the opposite of what we saw with multiplication!

Now, students, let me show you some real-life problems involving fractions. This will help you understand how useful these concepts are.

Example 3: Leena made 5 cups of tea. She used 1/4 litre of milk for this. How much milk is there in each cup of tea?

Leena used 1/4 litres of milk in 5 cups of tea. So in 1 cup of tea, the volume of milk should be 1/4 ÷ 5. Writing this as a multiplication problem, we have 5 × (milk per cup) = 1/4. Using Brahmagupta's method, the reciprocal of 5 is 1/5. Multiplying this reciprocal by the dividend 1/4, we get 1/5 × 1/4 = 1/20. So each cup of tea has 1/20 litre of milk.

Example 4: This is a problem from an ancient Indian text called the Śulbasūtra, which is one of the oldest geometry texts in the world, dating back to around 800 BCE. The problem is: Cover an area of 7 1/2 square units with square bricks each of whose sides is 1/5 units. How many such square bricks are needed?

Each square brick has an area of 1/5 × 1/5 = 1/25 square units. The total area to be covered is 7 1/2 square units, which is 15/2 square units. As (Number of bricks) × (Area of a brick) = Total Area, we have Number of bricks = 15/2 ÷ 1/25. The reciprocal of the divisor is 25. Multiplying the reciprocal by the dividend, we get 25 × 15/2 = 375/2 = 187.5. So we need 187.5 bricks? That does not make sense! Wait, let me recalculate. 25 × 15/2 = (25 × 15) / 2 = 375/2 = 187.5. But we cannot have half a brick! Let me think about this again. Actually, 7 1/2 square units is 15/2, and 1/25 is the area of one brick. So 15/2 ÷ 1/25 = 15/2 × 25 = 375/2 = 187.5. This means we need 188 bricks to cover the area, with some overlap or cutting. But mathematically, the answer is 375/2 or 187.5.

Example 5: This problem was posed by an Indian mathematician named Chaturveda Pṛthūdakasvāmī in the 9th century CE. Here it is: Four fountains fill a cistern. The first fountain can fill the cistern in a day. The second can fill it in half a day. The third can fill it in a quarter of a day. The fourth can fill the cistern in one fifth of a day. If they all flow together, in how much time will they fill the cistern?

Let us solve this step by step. In a day, the number of times the first fountain will fill the cistern is 1 ÷ 1 = 1. The second fountain will fill it 1 ÷ 1/2 = 2 times. The third will fill it 1 ÷ 1/4 = 4 times. The fourth will fill it 1 ÷ 1/5 = 5 times. So together, they will fill the cistern 1 + 2 + 4 + 5 = 12 times in a day. Therefore, the time needed to fill the cistern together is 1/12 of a day.

Now, students, let me tell you about a fascinating problem involving fractional relations. Here is a square with some lines drawn inside. We need to find what fraction of the area of the whole square the shaded region occupies.

Let the area of the whole square be 1 square unit. We can see that the top right square occupies 1/4 of the area of the whole square. Now, look at this red square. The area of the triangle inside it, coloured yellow, is half the area of the red square. So the area of the yellow triangle is 1/2 × 1/4 = 1/8 square units.

Now, what fraction of this yellow triangle is shaded? The shaded region occupies 3/4 of the area of the yellow triangle. So the area of the shaded part is 3/4 × 1/8 = 3/32 square units. Thus, the shaded region occupies 3/32 of the area of the whole square.

Now, I want to share with you a very interesting problem from Bhāskarācārya's book Līlāvatī, written in 1150 CE. Here it is: "O wise one! A miser gave to a beggar 1/5 of 1/16 of 1/4 of 1/2 of 2/3 of 3/4 of a dramma. If you know the mathematics of fractions well, tell me O child, how many cowrie shells were given by the miser to the beggar."

Dramma refers to a silver coin used in those times. The tale says that 1 dramma was equivalent to 1280 cowrie shells. Let us see what fraction of a dramma the person gave. We need to multiply these fractions: 1/2 × 2/3 × 3/4 × 1/5 × 1/16 × 1/4. Let us calculate this step by step.

First, 1/2 × 2/3 = 2/6 = 1/3. Then, 1/3 × 3/4 = 3/12 = 1/4. Then, 1/4 × 1/5 = 1/20. Then, 1/20 × 1/16 = 1/320. Then, 1/320 × 1/4 = 1/1280.

So the miser gave 1/1280 of a dramma. Since 1 dramma = 1280 cowrie shells, this is equal to 1 cowrie shell. You can see Bhāskarācārya's humour in this problem! The miser gave only one coin of the least value to the beggar.

Now, students, I want to tell you a little about the history of fractions. As you have seen, fractions are an important type of number, playing a critical role in a variety of everyday problems that involve sharing and dividing quantities equally. The general notion of non-unit fractions as we use them today – equipped with the arithmetic operations of addition, subtraction, multiplication, and division – developed largely in India.

The ancient Indian geometry texts called the Śulbasūtra, which go back as far as 800 BCE, used general non-unit fractions extensively, including performing division of such fractions. Fractions even became commonplace in the popular culture of India as far back as 150 BCE, as evidenced by an offhand reference to the reduction of fractions to lowest terms in the philosophical work of the revered Jain scholar Umasvati.

General rules for performing arithmetic operations on fractions – in essentially the modern form in which we carry them out today – were first codified by Brahmagupta in his Brāhmasphuṭasiddhānta in 628 CE. He wrote: "Multiplication of two or more fractions is obtained by taking the product of the numerators divided by the product of the denominators." And for division, he wrote: "The division of fractions is performed by interchanging the numerator and denominator of the divisor; the numerator of the dividend is then multiplied by the (new) numerator, and the denominator by the (new) denominator."

Bhāskara II in his book Līlāvatī in 1150 CE clarified Brahmagupta's statement further in terms of the notion of reciprocal: "Division of one fraction by another is equivalent to multiplication of the first fraction by the reciprocal of the second."

Many other Indian mathematicians, such as Śhrīdharāchārya, Mahāvīrāchārya, Caturveda Pṛthūdakasvāmī, and Bhāskara II, developed the usage of arithmetic of fractions significantly further.

The Indian theory of fractions and arithmetic operations on them was transmitted to, and its usage developed further, by Arab and African mathematicians. The theory was then transmitted to Europe via the Arabs over the next few centuries, and came into general use in Europe in only around the 17th century, after which it spread worldwide. The theory is indeed indispensable today in modern mathematics.

Now, students, let me give you a quick recap of everything we have learned in this chapter.

First, we learned about multiplication of fractions. We learned that to multiply a fraction by a whole number, we divide the whole number by the denominator and then multiply by the numerator. We also learned Brahmagupta's formula for multiplying two fractions: a/b × c/d = (a × c) / (b × d). We learned that we can cancel common factors before multiplying to simplify our calculations. We discovered that when both numbers being multiplied are greater than 1, the product is greater than both the numbers. When both numbers are between 0 and 1, the product is less than both the numbers. And when one number is between 0 and 1 and the other is greater than 1, the product is between the two numbers. We also learned that multiplication of fractions is commutative, meaning the order does not matter.

Then we learned about division of fractions. We learned that to divide two fractions, we find the reciprocal of the divisor and multiply it by the dividend. We learned that the reciprocal of a fraction a/b is b/a. We discovered that when the divisor is between 0 and 1, the quotient is greater than the dividend, and when the divisor is greater than 1, the quotient is less than the dividend.

We also solved several real-life problems involving fractions, including problems from ancient Indian mathematics.

And finally, we learned about the rich history of fractions in India, and how Indian mathematicians contributed to the development of this important branch of mathematics.

This, my dear students, is what we call Working with Fractions. I hope you have understood all the concepts clearly. Remember, practice is the key to mastering fractions. So, keep practicing, and you will become experts in no time!

Thank you for listening so attentively. Until next time, goodbye and keep learning!