So students, welcome to today's mathematics lesson. I am so happy to be here with you to learn about Chapter 2, which is all about Arithmetic Expressions. This is a very important chapter because it will help you understand how to read, write, and evaluate mathematical expressions properly. You have been using arithmetic expressions since your early school days, but here we will go much deeper and understand the rules and properties that govern these expressions. Let us begin our journey.
So students, let us first understand what we mean by an arithmetic expression. You may have seen mathematical phrases like 13 plus 2, 20 minus 4, 12 multiplied by 5, and 18 divided by 3. Such phrases are called arithmetic expressions. An arithmetic expression is simply a combination of numbers and mathematical operations like addition, subtraction, multiplication, and division.
Every arithmetic expression has a value, which is the number it evaluates to. For example, the value of the expression 13 plus 2 is 15. We use the equality sign to denote the relationship between an arithmetic expression and its value. So we write 13 plus 2 equals 15. This expression can be read as 13 plus 2 or the sum of 13 and 2.
Now students, let me give you a real-life example. Imagine Mallika spends 25 rupees every day for lunch at school. Write the expression for the total amount she spends on lunch in a week from Monday to Friday. There are 5 school days, so the expression for the total amount is 5 multiplied by 25. We read this as 5 times 25 or the product of 5 and 25. The value of this expression is 125 rupees.
Now here is something interesting, students. Different expressions can have the same value. For example, can you think of multiple ways to express the number 12 using two numbers and any of the four operations? Let me give you some examples. We can write 10 plus 2, which equals 12. We can write 15 minus 3, which also equals 12. We can write 3 multiplied by 4, which is 12. And we can write 24 divided by 2, which is also 12. So you see, there are many different expressions that can represent the same number. I want you to choose your favourite number and write as many expressions as you can having that value. This is a fun exercise that will help you understand how flexible mathematical expressions can be.
Now students, let us learn about comparing expressions. Just as we compare numbers using equal to, less than, and greater than signs, we can also compare expressions. We compare expressions based on their values and write the appropriate sign accordingly. For example, let us compare 10 plus 2 and 7 plus 1. The value of 10 plus 2 is 12, and the value of 7 plus 1 is 8. Since 12 is greater than 8, we write 10 plus 2 is greater than 7 plus 1. Similarly, let us compare 13 minus 2 and 4 multiplied by 3. The value of 13 minus 2 is 11, and the value of 4 multiplied by 3 is 12. Since 11 is less than 12, we write 13 minus 2 is less than 4 multiplied by 3.
Now students, here is an important skill. Sometimes we can compare expressions without actually calculating their values. Let me show you how. Which is greater, 1023 plus 125 or 1022 plus 128? Let us imagine a situation to understand this. Imagine Raja had 1023 marbles and got 125 more today. So he has 1023 plus 125 marbles. Imagine Joy had 1022 marbles and got 128 more today. So he has 1022 plus 128 marbles. Who has more? Let us think carefully. To begin with, Raja had 1 more marble than Joy. But Joy got 3 more marbles than Raja today. So the difference is that Joy got 2 more marbles than Raja in total. Therefore, Joy has more marbles. So we can write 1023 plus 125 is less than 1022 plus 128. We did this without actually adding the numbers!
Let us try another one. Which is greater, 113 minus 25 or 112 minus 24? Imagine Raja had 113 marbles and lost 25 of them. He has 113 minus 25 marbles. Joy had 112 marbles and lost 24 today. He has 112 minus 24 marbles. Who has more marbles left? Raja had 1 marble more than Joy to begin with. But he also lost 1 marble more than Joy did. So they have an equal number of marbles now. Therefore, 113 minus 25 equals 112 minus 24.
So students, you see that by thinking about the differences between the numbers, we can compare expressions without doing complicated calculations. This is a very useful skill.
Now students, let us move on to a more complex topic. Sometimes, when an expression is not accompanied by a context, there can be more than one way of evaluating its value. In such cases, we need some tools and rules to specify how exactly the expression has to be evaluated.
Let me give you an example from language. Look at these two sentences: "Shalini sat next to a friend with toys" and "Shalini sat next to a friend, with toys". In the first sentence, without the comma, it seems the friend has toys and Shalini sat next to her. In the second sentence, with the comma, it seems Shalini has the toys and she sat with them next to her friend. The same words, but different punctuation, give completely different meanings. Just as punctuation marks are used to resolve confusion in language, brackets and the notion of terms are used in mathematics to resolve confusion in evaluating expressions.
Now let us see an expression that can be evaluated in more than one way. Here is Example 4 from your textbook. Mallesh brought 30 marbles to the playground. Arun brought 5 bags of marbles with 4 marbles in each bag. How many marbles did Mallesh and Arun bring to the playground?
Mallesh summarized this by writing the mathematical expression 30 plus 5 multiplied by 4. Now, without knowing the context behind this expression, Purna found the value to be 140. He added 30 and 5 first to get 35, and then multiplied 35 by 4 to get 140. But Mallesh found the value to be 50. He multiplied 5 and 4 first to get 20, and then added 20 to 30 to get 50. In this case, Mallesh is right. But why did Purna get it wrong?
Just looking at the expression 30 plus 5 multiplied by 4, it is not clear whether we should do the addition first or multiplication first. This is where brackets come to our rescue.
In the expression to find the number of marbles, which is 30 plus 5 multiplied by 4, we had to first multiply 5 and 4, and then add this product to 30. This order of operations is clarified by the use of brackets as follows: 30 plus open bracket 5 multiplied by 4 close bracket. When evaluating an expression having brackets, we need to first find the values of the expressions inside the brackets before performing other operations. So in the above expression, we first find the value of 5 multiplied by 4, which is 20, and then do the addition. Thus, this expression describes the number of marbles: 30 plus open bracket 5 multiplied by 4 close bracket equals 30 plus 20 equals 50.
Let me give you another example. Irfan bought a pack of biscuits for 15 rupees and a packet of toor dal for 56 rupees. He gave the shopkeeper 100 rupees. Write an expression that can help us calculate the change Irfan will get back from the shopkeeper.
Irfan spent 15 rupees on a biscuit packet and 56 rupees on toor dal. So the total cost is 15 plus 56. He gave 100 rupees to the shopkeeper. So he should get back 100 minus the total cost. Can we write that expression as 100 minus 15 plus 56? Let us check. If we first subtract 15 from 100 and then add 56 to the result, we get 85 plus 56 equals 141. It is absurd that he gets more money than he paid the shopkeeper! So this is wrong.
We can use brackets in this case: 100 minus open bracket 15 plus 56 close bracket. Evaluating the expression within the brackets first, we get 15 plus 56 equals 71. Then 100 minus 71 equals 29. So Irfan will get back 29 rupees. This makes sense!
So students, you see how important brackets are in clarifying the order of operations. Now let us learn about terms in expressions.
Suppose we have the expression 30 plus 5 multiplied by 4 without any brackets. Does it have no meaning? When there are expressions having multiple operations and the order of operations is not specified by the brackets, we use the notion of terms to determine the order.
Terms are the parts of an expression separated by a plus sign. For example, in 12 plus 7, the terms are 12 and 7. We will keep marking each term of an expression like this. This way of marking the terms is not a usual practice, but we will do it until you become familiar with the concept.
Now, what are the terms in 83 minus 14? We know that subtracting a number is the same as adding the inverse of the number. Recall that the inverse of a given number has the sign opposite to it. For example, the inverse of 14 is minus 14, and the inverse of minus 14 is 14. Thus, subtracting 14 from 83 is the same as adding minus 14 to 83. That is, 83 minus 14 equals 83 plus minus 14. So the terms of the expression 83 minus 14 are 83 and minus 14.
All subtractions in an expression are converted to additions in this manner to identify the terms. Here are some more examples. The expression minus 18 minus 3 can be written as minus 18 plus minus 3. The expression 6 multiplied by 5 plus 3 has terms 6 multiplied by 5 and 3. The expression 2 minus 10 plus 4 multiplied by 6 can be written as 2 plus minus 10 plus 4 multiplied by 6. Notice that 6 multiplied by 5 and 4 multiplied by 6 are single terms as they do not have any plus sign.
Now students, we will see how terms are used to determine the order of operations to find the value of an expression. We will start with expressions having only additions, with all the subtractions suitably converted into additions.
Does changing the order in which the terms are added give different values? Let us consider a simple expression having only two terms. Imagine Madhu is flying a drone from a terrace. The drone goes 6 meters up and then 4 meters down. Write an expression to show how high the final position of the drone is from the terrace.
The drone is 6 minus 4 equals 2 meters above the terrace. Writing it as sum of terms: 6 plus minus 4 equals 2. Will the sum change if we swap the terms? Minus 4 plus 6 also equals 2. It doesn't change in this case.
We already know that swapping the terms does not change the sum when both the terms are positive numbers. Will this also hold when there are terms having negative numbers as well? Take some more expressions and check. You will find that it does!
So students, in an expression having two terms, swapping them does not change the value. We can write this as Term 1 plus Term 2 equals Term 2 plus Term 1. This is called the commutative property of addition.
Now consider an expression having three terms: minus 7 plus 10 plus minus 11. Let us add these terms in two different ways. First, let us add the first two terms and then add their sum to the third term. So minus 7 plus 10 equals 3, and 3 plus minus 11 equals minus 8. Now let us add the last two terms first and then add their sum to the first term. So 10 plus minus 11 equals minus 1, and minus 7 plus minus 1 equals minus 8. We get the same sum in both cases!
Again, we know that while adding positive numbers, grouping them in any of the above two ways gives the same sum. Will this also hold when there are terms having negative numbers as well? Take some more expressions and check. You will find that it does!
So students, grouping the terms of an expression in either of the following ways gives the same value: Term 1 plus Term 2 plus Term 3 equals Term 1 plus Term 2 plus Term 3. This is called the associative property of addition.
Let us consider the expression minus 7 plus 10 plus minus 11 again. What happens when we change the order and add minus 7 and minus 11 first, and then add this sum to 10? We get minus 7 plus minus 11 equals minus 18, and minus 18 plus 10 equals minus 8. We get the same sum as before!
Thus, the addition of terms in an expression in any order gives the same value. Therefore, in an expression having only additions, it does not matter in what order the terms are added: they all give the same value.
Now students, let us consider expressions having multiplication and division also, without the order of operations specified by the brackets. The values of such expressions are found by first evaluating the terms. Once all the terms are evaluated, they are added.
For example, the expression 30 plus 5 multiplied by 4 is evaluated as follows. First we evaluate the term 5 multiplied by 4, which is 20. Then we have 30 plus 20, which equals 50.
Another example: the expression 5 multiplied by open bracket 3 plus 2 close bracket plus 7 multiplied by 8 plus 3 is evaluated as follows. First we evaluate the bracket: 3 plus 2 equals 5, and 5 multiplied by 5 equals 25. Then we evaluate 7 multiplied by 8, which is 56. So we have 25 plus 56 plus 3, which equals 84.
Now students, let me tell you about Manasa. Manasa is adding a long list of numbers. It took her five minutes to add them all and she got the answer 11749. Then she realised that she had forgotten to include the fourth number 9055. Does she have to start all over again? No! She can simply add 9055 to the sum she already got. This is because of the associative property of addition. She does not need to start from scratch.
In mathematics we use the phrase commutative property of addition instead of saying swapping terms does not change the sum. Similarly, grouping does not change the sum is called the associative property of addition.
Now students, let us look at some more examples to understand terms better.
Example 7: Amu, Charan, Madhu, and John went to a hotel and ordered four dosas. Each dosa costs 23 rupees, and they wish to thank the waiter by tipping 5 rupees. Write an expression describing the total cost.
Cost of 4 dosas is 4 multiplied by 23. Can the total amount with tip be written as 4 multiplied by 23 plus 5? Let us evaluate it. 4 multiplied by 23 is 92, and 92 plus 5 equals 97. So 4 multiplied by 23 plus 5 is a correct way of writing the expression. The terms in this expression are 4 multiplied by 23 and 5.
If the total number of friends goes up to 7 and the tip remains the same, how much will they have to pay? The expression would be 7 multiplied by 23 plus 5. Let us evaluate it: 7 multiplied by 23 is 161, and 161 plus 5 equals 166 rupees.
Example 8: Children in a class are playing Fire in the mountain, run, run, run! Whenever the teacher calls out a number, students are supposed to arrange themselves in groups of that number. Whoever is not part of the announced group size is out.
Ruby wanted to rest and sat on one side. The other 33 students were playing the game in the class. The teacher called out 5. Once children settled, Ruby wrote 6 multiplied by 5 plus 3, which she understood as 3 more than 6 multiplied by 5. Why did she write this? Because when 33 students are divided into groups of 5, we get 6 complete groups of 5 students each, which is 30 students, and 3 students are left out. So the expression is 6 multiplied by 5 plus 3.
If the teacher had called out 4, Ruby would write 8 multiplied by 4 plus 1, because 33 divided by 4 gives 8 groups of 4 students each, which is 32 students, and 1 student is left out.
If the teacher had called out 7, Ruby would write 4 multiplied by 7 plus 5, because 33 divided by 7 gives 4 groups of 7 students each, which is 28 students, and 5 students are left out.
Example 9: Raghu bought 100 kilograms of rice from the wholesale market and packed them into 2 kilogram packets. He already had four 2 kilogram packets. Write an expression for the number of 2 kilogram packets of rice he has now and identify the terms.
He had 4 packets already. The number of new 2 kilogram packets of rice is 100 divided by 2, which we also write as 100 over 2. The number of 2 kilogram packets he has now is 4 plus 100 over 2. The terms are 4 and 100 over 2.
Example 10: Kannan has to pay 432 rupees to a shopkeeper using coins of 1 rupee and 5 rupees, and notes of 10 rupees, 20 rupees, 50 rupees and 100 rupees. How can he do it?
There is more than one possibility. For example, 432 equals 4 multiplied by 100 plus 1 multiplied by 20 plus 1 multiplied by 10 plus 2 multiplied by 1. This means 4 notes of 100 rupees, 1 note of 20 rupees, 1 note of 10 rupees and 2 coins of 1 rupee.
Another way is 432 equals 8 multiplied by 50 plus 1 multiplied by 10 plus 4 multiplied by 5 plus 2 multiplied by 1. This means 8 notes of 50 rupees, 1 note of 10 rupees, 4 notes of 5 rupees and 2 coins of 1 rupee.
Can you think of some more ways of giving 432 rupees to someone?
Example 11: Here are two pictures. Which of these two arrangements matches with the expression 5 multiplied by 2 plus 3?
Let us write this expression as a sum of terms. 5 multiplied by 2 plus 3 equals 10 plus 3 equals 13. This expression 5 multiplied by 2 plus 3 can be understood as 3 more than 5 multiplied by 2, which describes the arrangement on the left.
What is the expression for the arrangement in the right making use of the number of yellow and blue squares? We need to use brackets for this. The expression is 2 multiplied by open bracket 5 plus 3 close bracket. Notice that this arrangement can also be described using 5 plus 3 plus 5 plus 3, or 5 multiplied by 2 plus 3 multiplied by 2.
Now students, let us learn about removing brackets. This is a very important concept.
Let us find the value of this expression: 200 minus open bracket 40 plus 3 close bracket. We first evaluate the expression inside the bracket to 43 and then subtract it from 200. But it is simpler to first subtract 40 from 200: 200 minus 40 equals 160. And then subtract 3 from 160: 160 minus 3 equals 157. What we did here was 200 minus 40 minus 3. Notice that we did not do 200 minus 40 plus 3. So 200 minus open bracket 40 plus 3 close bracket equals 200 minus 40 minus 3.
We also saw this earlier in the case of Irfan purchasing a biscuit packet for 15 rupees and a toor dal packet for 56 rupees. When he paid 100 rupees, the change he gets is 100 minus open bracket 15 plus 56 close bracket, which equals 29 rupees.
The change could also have been calculated as follows. First subtract the cost of the biscuit packet, which is 15, from 100: 100 minus 15 equals 85. This is the amount the shopkeeper owes Irfan if he had purchased only the biscuits. As he has purchased toor dal also, its cost is taken from this remaining amount of 85. So to find the change, we need to subtract the cost of toor dal, which is 56, from 85. 85 minus 56 equals 29. What we have done here is 100 minus 15 minus 56. So 100 minus open bracket 15 plus 56 close bracket equals 100 minus 15 minus 56.
Notice how upon removing the brackets preceded by a negative sign, the signs of the terms inside the brackets change. Observe the signs of 40 and 3 in the first example, and that of 15 and 56 in the second.
Example 13: Consider the expression 500 minus open bracket 250 minus 100 close bracket. Is it possible to write this expression without the brackets?
To evaluate this expression, we need to subtract 250 minus 100, which is 150, from 500: 500 minus 150 equals 350.
If we were to directly subtract 250 from 500, then we would have subtracted 100 more than what we needed to. So we should add back that 100 to 500 minus 250 to make the expression take the same value as 500 minus open bracket 250 minus 100 close bracket. This sequence of operations is 500 minus 250 plus 100. Thus, 500 minus open bracket 250 minus 100 close bracket equals 500 minus 250 plus 100.
Check that 500 minus open bracket 250 minus 100 close bracket is not equal to 500 minus 250 minus 100.
Notice again that when the brackets preceded by a negative sign are removed, the signs of the terms inside the brackets change. In this case, the signs of 250 and minus 100 change to minus 250 and plus 100.
Example 14: Hira has a rare coin collection. She has 28 coins in one bag and 35 coins in another. She gifts her friend 10 coins from the second bag. Write an expression for the number of coins left with Hira.
This can be expressed by 28 plus open bracket 35 minus 10 close bracket. We know that this is the same as 28 plus open bracket 35 plus minus 10 close bracket. Since the terms can be added in any order, this expression can simply be written as 28 plus 35 plus minus 10, or 28 plus 35 minus 10. Thus, 28 plus open bracket 35 minus 10 close bracket equals 28 plus 35 minus 10 equals 53.
When the brackets are NOT preceded by a negative sign, the terms within them do not change their signs upon removing the brackets. Notice the sign of the terms 35 and minus 10 in the above expression.
Rather than simply remembering rules for when to change the sign and when not to, you can figure it out for yourself by thinking about the meanings of the expressions.
Now students, let us learn about a very important property called the distributive property. This is Example 15.
Lhamo and Norbu went to a hotel. Each of them ordered a vegetable cutlet and a rasgulla. A vegetable cutlet costs 43 rupees and a rasgulla costs 24 rupees. Write an expression for the amount they will have to pay.
As each of them had one vegetable cutlet and one rasgulla, each of their shares can be represented by 43 plus 24. What about the total amount they have to pay? Can it be described by the expression 2 multiplied by 43 plus 24? Writing it as sum of terms gives 2 multiplied by 43 plus 24. This expression means 24 more than 2 multiplied by 43. But we want an expression which means twice or double of 43 plus 24.
We can make use of brackets to write such an expression: 2 multiplied by open bracket 43 plus 24 close bracket. So we can say that together they have to pay 2 multiplied by open bracket 43 plus 24 close bracket. This is also the same as paying for two vegetable cutlets and two rasgullas: 2 multiplied by 43 plus 2 multiplied by 24. Therefore, 2 multiplied by open bracket 43 plus 24 close bracket equals 2 multiplied by 43 plus 2 multiplied by 24.
If another friend, Sangmu, joins them and orders the same items, what will be the expression for the total amount to be paid? It would be 3 multiplied by open bracket 43 plus 24 close bracket.
Example 16: In the Republic Day parade, there are boy scouts and girl guides marching together. The scouts march in 4 rows with 5 scouts in each row. The guides march in 3 rows with 5 guides in each row. How many scouts and guides are marching in this parade?
The number of boy scouts marching is 4 multiplied by 5. The number of girl guides marching is 3 multiplied by 5. The total number of scouts and guides will be 4 multiplied by 5 plus 3 multiplied by 5.
This can also be found by first finding the total number of rows, which is 4 plus 3, and then multiplying their sum by the number of children in each row. Thus, the number of boys and girls can be found by open bracket 4 plus 3 close bracket multiplied by 5.
Therefore, 4 multiplied by 5 plus 3 multiplied by 5 equals open bracket 4 plus 3 close bracket multiplied by 5.
Computing these expressions, we get 4 multiplied by 5 plus 3 multiplied by 5 equals 20 plus 15 equals 35. And open bracket 4 plus 3 close bracket multiplied by 5 equals 7 multiplied by 5 equals 35.
Now students, notice that 5 multiplied by 4 plus 3 is NOT equal to 5 multiplied by open bracket 4 plus 3 close bracket. Can you explain why? 5 multiplied by 4 plus 3 equals 5 multiplied by 4 plus 3 equals 20 plus 3 equals 23. But 5 multiplied by open bracket 4 plus 3 close bracket equals 5 multiplied by 7 equals 35. So they are different!
Is 5 multiplied by open bracket 4 plus 3 close bracket equal to 5 multiplied by open bracket 3 plus 4 close bracket equal to open bracket 3 plus 4 close bracket multiplied by 5? Yes, all three expressions are equal.
The observations that we have made in the previous two examples can be seen in a general way as follows.
Consider 10 multiplied by 98 plus 3 multiplied by 98. This means taking the sum of 10 times 98 and 3 times 98. Clearly, this is the same as 10 plus 3 equals 13 times 98. Thus, 10 multiplied by 98 plus 3 multiplied by 98 equals open bracket 10 plus 3 close bracket multiplied by 98.
Writing this equality the other way, we get open bracket 10 plus 3 close bracket 98 equals 10 multiplied by 98 plus 3 multiplied by 98.
Swapping the numbers in the products above, this property can be seen in the following form: 98 multiplied by 10 plus 98 multiplied by 3 equals 98 multiplied by open bracket 10 plus 3 close bracket, and 98 multiplied by open bracket 10 plus 3 close bracket equals 98 multiplied by 10 plus 98 multiplied by 3.
Similarly, let us consider the expression 14 multiplied by 10 minus 6 multiplied by 10. This means subtracting 6 times 10 from 14 times 10. Clearly, this is 14 minus 6 equals 8 times 10. Thus, 14 multiplied by 10 minus 6 multiplied by 10 equals open bracket 14 minus 6 close bracket multiplied by 10, or open bracket 14 minus 6 close bracket multiplied by 10 equals 14 multiplied by 10 minus 6 multiplied by 10.
This property can be nicely summed up as follows: The multiple of a sum or difference is the same as the sum or difference of the multiples. This is called the distributive property.
Now students, let us look at some examples of using this property to make calculations easier.
Example 17: Given 53 multiplied by 18 equals 954. Find out 63 multiplied by 18.
As 63 multiplied by 18 means 63 times 18, we can write 63 multiplied by 18 as open bracket 53 plus 10 close bracket multiplied by 18. Using the distributive property, this equals 53 multiplied by 18 plus 10 multiplied by 18, which equals 954 plus 180, which equals 1134.
Example 18: Find an effective way of evaluating 97 multiplied by 25.
97 multiplied by 25 means 97 times 25. We can write it as open bracket 100 minus 3 close bracket multiplied by 25. We know that this is the same as the difference of 100 times 25 and 3 times 25: 97 multiplied by 25 equals 100 multiplied by 25 minus 3 multiplied by 25. Let us calculate this: 100 multiplied by 25 is 2500, and 3 multiplied by 25 is 75. So 2500 minus 75 equals 2425. So 97 multiplied by 25 equals 2425.
This method is quicker than the multiplication procedure we generally use. We can use this method to find products like 95 multiplied by 8, 104 multiplied by 15, and 49 multiplied by 50. This method is useful when one of the numbers is close to a multiple of 10, 50, 100, 1000, and so on.
Now students, let us summarize what we have learned in this chapter.
We started by revising the meaning of some simple expressions and their values. We learned how to compare certain expressions through reasoning instead of bluntly evaluating them.
To help read and evaluate complex expressions without confusion, we use terms and brackets. Terms are the parts of an expression separated by a plus sign. When an expression is written as a sum of terms, changing the order of the terms or grouping the terms does not change the value of the expression. This is because of the commutative property of addition and the associative property of addition, respectively.
We saw that when we remove brackets preceded by a negative sign, the terms within the bracket change their sign. We also learned about the distributive property, which says that multiplying a number with an expression inside brackets is equal to multiplying the number with each term in the bracket and then adding or subtracting the results.
These are very powerful tools that will help you not only in this chapter but throughout your mathematical journey. The skills you have learned here about understanding terms, using brackets, and applying the properties of addition and multiplication will be used again and again in algebra and other branches of mathematics.
So students, always remember these key ideas: always look for brackets to know which operation to do first, identify the terms in an expression, use the commutative and associative properties to rearrange terms for easier calculation, and use the distributive property to simplify complex multiplications.
Thank you for listening so attentively. Keep practicing these concepts, and you will become very comfortable with arithmetic expressions. Good luck with your studies!