Class 7 Maths Chapter 2 Notes Arithmetic Expressions
Class 7 Maths Notes Chapter 2 – Class 7 Arithmetic Expressions Notes
→ We have been reading and evaluating simple expressions for quite some time now. Here we started by revising the meaning of some simple expressions and their values.
→ We learnt 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.
→ 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.
→ To evaluate expressions within brackets, we saw that when we remove brackets preceded by a negative sign, the terms within the brackets change their sign.
→ We also learnt about the “distributive property” — multiplying a number by an expression inside brackets is equal to multiplying the number by each term in the bracket.
Simple Expressions Class 7 Notes
You may have seen mathematical phrases like 13 + 2, 20 – 4, 12 × 5, and 18 ÷ 3. Such phrases are called arithmetic expressions.
Every arithmetic expression has a value, which is the number it evaluates to. For example, the value of the expression 13 + 2 is 15. This expression can be read as ‘13 plus 2’ or ‘the sum of 13 and 2’.
We use the equality sign ‘=’ to denote the relationship between an arithmetic expression and its value.
For example: 13 + 2 = 15.
Example 1.
Mallika spends ₹25 every day on lunch at school. Write the expression for the total amount she spends on lunch in a week from Monday to Friday.
Solution:
The expression for the total amount is 5 × 25.
5 × 25 is “5 times 25” or “the product of 5 and 25”.
Different expressions can have the same value. Here are multiple ways to express the number 12, using two numbers and any of the four operations +, –, ×, and ÷: 10 + 2, 15 – 3, 3 × 4, 24 ÷ 2.
Comparing Expressions
As we compare numbers using ‘=’, ‘<’, and ‘>’ signs, we can also compare expressions. We compare expressions based on their values and write the ‘equal to’, ‘greater than’, or ‘less than’ sign accordingly. For example, 10 + 2 > 7 + 1 because the value of 10 + 2 = 12 is greater than the value of 7 + 1 = 8. Similarly, 13 – 2 < 4 × 3.
Example 2.
Which is greater? 1023 + 125 or 1022 + 128?
Solution:
Imagining a situation could help us answer this without finding the values. Raja had 1023 marbles and got 125 more today. Now he has 1023 + 125 marbles. Joy had 1022 marbles and got 128 more today. Now he has 1022 + 128 marbles. Who has more?
This situation can be represented as shown in the picture on the right. To begin with, Raja had 1 more marble than Joy. But Joy got 3 more marbles than Raja today. We can see that Joy has (two) more marbles than Raja now. That is, 1023 + 125 < 1022 + 128.
Example 3.
Which is greater? 113 – 25 or 112 – 24?
Solution:
Imagine a situation, Raja had 113 marbles and lost 25 of them. He has 113 – 25 marbles. Joy had 112 marbles and lost 24 today. He has 112 – 24 marbles. Who has more marbles left with them?
Raja had 1 marble more than Joy. But he also lost 1 marble more than Joy did. Therefore, they have an equal number of marbles now. That is, 113 – 25 = 112 – 24.
Reading and Evaluating Complex Expressions Class 7 Notes
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.
To give an example with language, look at the following sentences:
(a) Sentence: “Shalini sat next to a friend with toys”.
Meaning: The friend has toys, and Shalini sat next to her.
(b) Sentence: “Shalini sat next to a friend, with toys”.
Meaning: Shalini has the toys, and she sat with them next to her friend.
This sentence without the punctuation could have been interpreted in two different ways. The appropriate use of a comma specifies how the sentence has to be understood. Let us see an expression that can be evaluated in more than one way.
Example 4.
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?
Solution:
Mallesh summarized this by writing the mathematical expression 30 + 5 × 4.
Without knowing the context behind this expression, Purna found the value of this expression to be 140.
He added 30 and 5 first, to get 35, and then multiplied 35 by 4 to get 140.
Mallesh found the value of this expression to be 50.
He multiplied 5 and 4 first to get 20 and 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 + 5 × 4, it is not clear whether we should do the addition fist or multiplication. Just as punctuation marks are used to resolve confusions in language, brackets and the notion of terms are used in mathematics to resolve confusions in evaluating expressions.
Brackets in Expressions
In the expression to fid the number of marbles — 30 + 5 × 4 — we had to first multiply 5 and 4, and then add this product to 30. This order of operations is clarifid by the use of brackets as follows: 30 + (5 × 4).
When evaluating an expression having brackets, we need to fi4st fid the values of the expressions inside the brackets before performing other operations. So, in the above expression, we first find the value of 5 × 4, and then do the addition. Thus, this expression describes the number of marbles: 30 + (5 × 4 ) = 30 + 20 = 50.
Example 5.
Irfan bought a pack of biscuits for ₹15 and a packet of toor dal for ₹56. He gave the shopkeeper ₹100. Write an expression that can help us calculate the change Irfan will get back from the shopkeeper.
Solution:
Irfan spent ₹15 on a biscuit packet and ₹56 on toor dal.
So, the total cost in rupees is 15 + 56. He gave ₹100 to the shopkeeper.
So, he should get back 100 minus the total cost.
Can we write that expression as 100 – 15 + 56?
Can we first subtract 15 from 100 and then add 56 to the result?
We will get 141. It is absurd that he gets more money than he paid the shopkeeper!
We can use brackets in this case: 100 – (15 + 56).
Evaluating the expression within the brackets first, we get 100 minus 71, which is 29. So, Irfan will get back ₹29.
Terms in Expressions
Suppose we have the expression 30 + 5 × 4 without any brackets. Does it have no meaning?
When expressions have 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 ‘+’ sign. For example, in 12+7, the terms are 12 and 7, as marked below.
12 + 7 = 12 + 7
We will keep marking each term of an expression as above. Note that this way of marking the terms is not a usual practice. This will be done until you become familiar with this concept.
Now, what are the terms in 83 – 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 –14, and the inverse of –14 is 14. Thus, subtracting 14 from 83 is the same as adding –14 to 83. That is, 83 – 14 = 83 + – 14. Thus, the terms of the expression 83 – 14 are 83 and –14.
All subtractions in an expression are converted to additions in this manner to identify the terms. Here are some more examples of expressions and their terms:
–18 – 3 = –18 + –3
6 × 5 + 3 = 6 × 5 + 3
2 – 10 + 4 × 6 = 2 + –10 + 4 × 6
Note that 6 × 5, 4 × 6 are single terms as they do not have any ‘+’ sign.
Now 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).
Swapping and Grouping
Let us consider a simple expression having only two terms.
Example 6.
Madhu is flying a drone from a terrace. The drone goes 6 m up and then 4 m down. Write an expression to show how high the final position of the drone is from the terrace.
Solution:
The drone is 6 – 4 = 2 m above the terrace.
Writing it as the sum of terms: 6 + –4 = 2
Will the sum change if we swap the terms?
–4 + 6 = 2
It doesn’t in this case.
We already know that swapping the terms does not change the sum when both terms are positive numbers.
Thus, in an expression having two terms, swapping them does not change the value.
Now consider an expression having three terms: (–7) + 10 + (–11).
Let us add these terms in the following two diffrent orders:
(adding the fist two terms and then adding their sum to the third term)
(adding the last two terms and then adding their sum to the first term)
What do you see? The sums are the same in both cases. Again, we know that while adding positive numbers, grouping them in any of the above two ways gives the same sum.
Thus, grouping the terms of an expression in either of the following ways gives the same value.
Let us consider the expression (–7) + 10 + (–11) again. What happens when we change the order and add -7 and -11 first, and then add this sum to 10? Will we get the same sum as before? We see that adding the terms of the expression (–7) + 10 + (–11) in any order gives the same sum of –8.
Thus, the addition of terms 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 let us consider expressions having multiplication and division also, without the order of operations specifid by the brackets. The values of such expressions are found by fist evaluating the terms. Once all the terms are evaluated, they are added.
For example, the expression 30 + 5 × 4 is evaluated as follows:
30 + 5 × 4 = 30 + 5 × 4 = 30 + 20 = 50
The expression 5 × (3 + 2) + 78 + 3 is evaluated as follows:
5 × (3 + 2) + 78 + 3 = 5 × (3 + 2) + 7 × 8 + 3
Where (3 + 2) is first evaluated, and this sum is multiplied by 5 (= 25).
The expression 7 × 8 is evaluated (= 56). This simplifies to 25 + 56 + 3 = 84.
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.
Swapping the Order of Things in Everyday Life
Manasa is going outside to play. Her mother says, “Wear your hat and shoes!” Which one should she wear fist? She can wear her hat first and then her shoes. Or she can wear her shoes first and then her hat.
Manasa will look exactly the same in both cases. Imagine a diffrent situation: Manasa’s mother says “Wear your socks and shoes!” Now the order matters. She should wear socks and then shoes. If she wears shoes and then socks, Manasa will feel very uncomfortable and look very different.
More Expressions and Their Terms
Example 7.
Amu, Charan, Madhu, and John went to a hotel and ordered four dosas. Each dosa costs ₹23, and they wish to thank the waiter by tipping ₹5. Write an expression describing the total cost.
Solution:
Cost of 4 dosas = 4 × 23
Can the total amount with tip be written as 4 × 23 + 5?
Evaluating it, we get 4 × 23 + 5 = 4 × 23 + 5 = 92 + 5 = 97
Thus, 4 × 23 + 5 is a correct way of writing the expression.
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.
Solution:
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 × 5 + 3 (understood as 3 more than 6 × 5)
Example 9.
Raghu bought 100 kg of rice from the wholesale market and packed them into 2 kg packets. He already had four 2 kg packets. Write an expression for the number of 2 kg packets of rice he has now and identify the terms.
Solution:
He had 4 packets.
The number of new 2 kg packets of rice is 100 ÷ 2, which we also write as \(\frac{100}{2}\).
The number of 2 kg packets he has now is 4 + \(\frac{100}{2}\).
The terms are 4 + \(\frac{100}{2}\)
Example 10.
Kannan has to pay ₹432 to a shopkeeper using coins of ₹1 and ₹5, and notes of ₹10, ₹20, ₹50 and ₹100. How can he do it?
Solution:
There is more than one possibility.
For example, 432 = 4 × 100 + 1 × 20 + 1 × 10 + 2 × 1
Meaning: 4 notes of ₹100, 1 note of ₹20, 1 note of ₹10 and 2 notes of ₹1
432 = 8 × 50 + 1 × 10 + 4 × 5 + 2 × 1
Meaning: 8 notes of ₹50, 1 note of ₹10, 4 notes of ₹5 and 2 notes of ₹1
Example 11.
Here are two pictures. Which of these two arrangements matches the expression 5 × 2 + 3?
Solution:
Let us write this expression as a sum of terms.
5 × 2 + 3 = 10 + 3 = 13
This expression, 5 × 2 + 3, can be understood as 3 more than 5 × 2, which describes the arrangement on the left.
What is the expression for the arrangement in the right,t making use of the number of yellow and blue squares?
Do you recall the use of brackets? We need to use brackets for this: 2 × (5 + 3)
Notice that this arrangement can also be described using 5 + 3 + 5 + 3 OR 5 × 2 + 3 × 2
Removing Brackets — I
Let us find the value of this expression, 200 – (40 + 3).
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 – 40 = 160.
And then subtract 3 from 160:
160 – 3 = 157.
What we did here was 200 – 40 – 3.
Notice that we did not do 200 – 40 + 3.
So, 200 – (40 + 3) = 200 – 40 – 3.
Example 12.
We also saw this earlier in the case of Irfan purchasing a biscuit packet (₹15) and a toor dal packet (₹56). When he paid ₹100, the change he got in rupees is:
100 – (15 + 56) = 29.
The change could also have been calculated as follows:
(a) First, subtract the cost of the biscuit packet (15) from 100:
100 – 15 = 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.
(b) So, to find the change, we need to subtract the cost of toor dal from 85.
85 – 56 = 29.
What we have done here is 100 – 15 – 56.
So, 100 – (15 + 56) = 100 – 15 – 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 those of 15 and 56 in the second.
Example 13.
Consider the expression 500 – (250 – 100). Is it possible to write this expression without the brackets?
Solution:
To evaluate this expression, we need to subtract 250 – 100 = 150 from 500:
500 – (250 – 100) = 500 – 150 = 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 – 250 to make the expression take the same value as 500 – (250 – 100).
This sequence of operations is 500 – 250 + 100.
Thus, 500 – (250 – 100) = 500 – 250 + 100.
Check that 500 – (250 – 100) is not equal to 500 – 250 – 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 –100 change to –250 and 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.
Solution:
This can be expressed by 28 + (35 – 10).
We know that this is the same as 28 + (35 + (–10)).
Since the terms can be added in any order, this expression can simply be written as 28 + 35 + (–10), or 28 + 35 – 10.
Thus, 28 + (35 – 10) = 28 + 35 – 10 = 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 –10 in the above expression.
Tinker the Terms I
What happens to the value of an expression if we increase or decrease the value of one of its terms? Some expressions are given in the following three columns. In each column, one or more terms are changed from the first expression. Go through the example (in the first column) and fill the blanks, doing as little computation as possible.
Removing Brackets II
Example 15.
Lhamo and Norbu went to a hotel. Each of them ordered a vegetable cutlet and a rasgulla. A vegetable cutlet costs ₹43 and a rasgulla costs ₹24. Write an expression for the amount they will have to pay.
Solution:
As each of them had one vegetable cutlet and one rasagulla, each of their shares can be represented by 43 + 24.
What about the total amount they have to pay? Can it be described by the expression: 2 × 43 + 24?
Writing it as the sum of terms gives: 2 × 43 + 24
This expression means 24 more than 2 × 43. But, we want an expression which means twice or double of 43 + 24.
We can make use of brackets to write such an expression: 2 × (43 + 24).
So, we can say that together they have to pay 2 × (43 + 24). This is also the same as paying for two vegetable cutlets and two rasgullas: 2 × 43 + 2 × 24.
Therefore, 2 × (43 + 24) = 2 × 43 + 2 × 24.
Example 16.
In the Republic Day parade, Boy Scouts and Girl Guides are 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 (see the figure below). How many scouts and guides are marching in this parade?
Solution:
The number of Boy Scouts marching is 4 × 5.
The number of Girl Guides marching is 3 × 5.
The total number of scouts and guides will be 4 × 5 + 3 × 5.
This can also be found by first finding the total number of rows, i.e., 4 + 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 (4 + 3) × 5.
Therefore, 4 × 5 + 3 × 5 = (4 + 3) × 5.
Computing these expressions, we get
4 × 5 + 3 × 5 = 4 × 5 + 3 × 5 = 20 + 15 = 35
(4 + 3) × 5 = 7 × 5 = 35
5 × 4 + 3 ≠ 5 × (4 + 3). Can you explain why?
Is 5 × (4 + 3) = 5 × (3 + 4) = (3 + 4) × 5?
The observations that we have made in the previous two examples can be seen in a general way as follows.
Consider 10 × 98 + 3 × 98. This means taking the sum of 10 times 98 and 3 times 98.
This is the same as 10 + 3 = 13 times 98.
Thus, 10 × 98 +3 × 98 = (10 + 3) × 98.
Writing this equality the other way, we get (10 + 3) 98 = 10 × 98 + 3 × 98.
Swapping the numbers in the products above, this property can be seen in the following form:
98 × 10 + 9 × 83 = 98 (10 + 3), and 98 (10 + 3) = 98 × 10 + 98 × 3.
Similarly, let us consider the expression 14 × 10 – 6 × 10. This means subtracting 6 times 10 from 14 times 10.
This is 14 – 6 = 8 times 10.
Thus, 14 × 10 – 6 × 10 = (14 – 6) × 10,
or (14 – 6) × 10 = 14 × 10 – 6 × 10
This property can be nicely summed up as follows:
The multiple of a sum (difference) is the same as the sum (difference) of the multiples.
Tinker the Terms II
Let us understand what happens when we change the numbers occurring in a product.
Example 17.
Given 53 × 18 = 954. Find out 63 × 18.
Solution:
As 63 × 18 means 63 times 18,
63 × 18 = (53 + 10) × 18
= 53 ×18 + 10 × 18
= 954 + 180
= 1134
Example 18.
Find an effective way of evaluating 97 × 25.
Solution:
97 × 25 means 97 times 25.
We can write it as (100 – 3) × 25
We know that this is the same as the difference of 100 times 25 and 3 times 25:
97 × 25 = 100 × 25 – 3 × 25
Class 7 Maths Notes
The post Arithmetic Expressions Class 7 Notes Maths Chapter 2 appeared first on Learn CBSE.
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