Maths Solutions Class 7 Chapter 2 Ganita Prakash NCERT

Arithmetic Expressions

Page No. 24

Question: Choose your favourite number and write as many expressions as you can having that value.

Solution: Let’s choose the number 12. Expressions with the value 12 are:

• 10 + 2
• 15 – 3
• 3 × 4
• 24 ÷ 2

Figure it Out

Page No. 25

1. Fill in the blanks to make the expressions equal on both sides of the = sign:

(a) 13 + 4 = ______ + 6

Ans: 13 + 4 = 17, so 11 + 6 = 17. The blank is 11.

(b) 22 + ______ = 6 × 5

Ans: 6 × 5 = 30, so 22 + 8 = 30. The blank is 8.

(c) 8 × ______ = 64 ÷ 2

Ans: 64 ÷ 2 = 32, so 8 × 4 = 32. The blank is 4.

(d) 34 – ______ = 25

Ans: 34 – 9 = 25. The blank is 9.

2. Arrange the following expressions in ascending (increasing) order of their values.

(a) 67 – 19

(b) 67 – 20

(c) 35 + 25

(d) 5 × 11

(e) 120 ÷ 3

Ans: Calculate each expression:

(a) 67 – 19 = 48

(b) 67 – 20 = 47

(c) 35 + 25 = 60

(d) 5 × 11 = 55

(e) 120 ÷ 3 = 40

Ascending order: 40, 47, 48, 55, 60
So, the order is: 120 ÷ 3, 67 – 20, 67 – 19, 5 × 11, 35 + 25.

Question: Which is greater? 1023 + 125 or 1022 + 128?

Ans: Raja had 1023 marbles and got 125 more (1023 + 125). Joy had 1022 marbles and got 128 more (1022 + 128).
Raja started with 1 more marble than Joy, but Joy got 3 more marbles than Raja today.
So, Joy has 2 more marbles. Therefore, 1023 + 125 < 1022 + 128.

Question: Which is greater? 113 – 25 or 112 – 24?

Ans: Raja had 113 marbles and lost 25 (113 – 25). Joy had 112 marbles and lost 24 (112 – 24).
Raja started with 1 more marble but lost 1 more than Joy.
So, they have the same number of marbles now. Therefore, 113 – 25 = 112 – 24.

Page No. 26

Question: Use ‘>’ or ‘<’ or ‘=’ in each of the following expressions to compare them. Can you do it without complicated calculations? Explain your thinking in each case.

(a) 245 + 289 _____ 246 + 285

(b) 273 − 144 _____ 272 − 144

(c) 364 + 587 _____ 363 + 589

(d) 124 + 245 _____ 129 + 245

(e) 213 − 77 _____ 214 − 76

Ans:

(a) 245 + 289 > 246 + 285

(b) 273 − 144 > 272 − 144

(c) 364 + 587 < 363 + 589

(d) 124 + 245 < 129 + 245

(e) 213 − 77 < 214 − 76

Page No. 27

Question: 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?

Ans: No, the expression 100 – 15 + 56 is not correct.
The correct expression for the change Irfan should get back is 100 – (15 + 56).
This is because we need to subtract the total cost (15 + 56 = 71) from 100, which gives 100 – 71 = 29 rupees.

Page No. 28

Question: Check if replacing subtraction by addition in this way does not change the value of the expression, by taking different examples.

Ans: Subtraction can be written as adding the inverse. For example:

• 83 – 14 = 83 + (-14). Calculate: 83 – 14 = 69, and 83 + (-14) = 69.
• 20 – 5 = 20 + (-5). Calculate: 20 – 5 = 15, and 20 + (-5) = 15.
  In both cases, the value remains the same.

Question: Can you explain why subtracting a number is the same as adding its inverse, using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

Ans: In the Token Model, positive numbers are represented by positive tokens, and negative numbers by negative tokens.
Subtracting a number means removing that many positive tokens.
If you remove positive tokens, it’s like adding the same number of negative tokens.
For example, 5 – 3 means removing 3 positive tokens from 5, leaving 2.
This is the same as 5 + (-3), where adding -3 tokens reduces the total to 2.

Page No. 29 & 30

Question: In the following table, some expressions are given. Complete the table.

Ans:

Question: Does changing the order in which the terms are added give different values?

Ans: No, changing the order of addition does not change the value. For example, 6 + (-4) = 2, and (-4) + 6 = 2.
This is due to the commutative property of addition.

Question: Will this also hold when there are terms having negative numbers as well? Take some more expressions and check.

Ans: Yes, it holds. For example:

• (-5) + 7 = 2, and 7 + (-5) = 2.
• (-3) + (-2) = -5, and (-2) + (-3) = -5.

Swapping terms does not change the value.

Question: Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

Ans: In the Token Model, adding tokens (positive or negative) in any order doesn’t change the total. For example, 6 + (-4) means 6 positive tokens plus 4 negative tokens, which cancel to 2 positive tokens. If we do (-4) + 6, we still cancel 4 pairs, leaving 2 positive tokens.

Page No. 31 & 32

Question: Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than 3 terms also.

Ans: Yes, it holds. For example:

• (-7) + 10 + (-11) + 5 = -3 (in any order).
• 3 + (-2) + 4 + (-5) = 0 (in any order).

This is due to the commutative property of addition.

Question: 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?

Ans: No, she doesn’t need to start over.
She can add 9055 to 11749: 11749 + 9055 = 20804.
This is because addition is commutative and associative.

Question: If the total number of friends goes up to 7 and the tip remains the same, how much will they have to pay? Write an expression for this situation and identify its terms.

Ans: Cost of 7 dosas = 7 × 23. Total cost with tip = 7 × 23 + 5.
Terms: 7 × 23, 5.
Evaluate: 7 × 23 = 161, 161 + 5 = 166. They pay ₹166.

Page No. 33

Question: For each of the cases below, write the expression and identify its terms:

If the teacher had called out ‘4’, Ruby would write _______
If the teacher had called out ‘7’, Ruby would write _______
Write expressions like the above for your class size.

Ans:

• For 4: 33 ÷ 4 = 8 groups of 4 with 1 left (33 – 32 = 1).
  If the teacher had called out ‘4’, Ruby would write Expression: 8 × 4 + 1. Terms: 8 × 4, 1.
• For 7: 33 ÷ 7 = 4 groups of 7 with 5 left (33 – 28 = 5).
  If the teacher had called out ‘4’, Ruby would write Expression: 4 × 7 + 5. Terms: 4 × 7, 5.
• For a class of 30: For 5, 30 ÷ 5 = 6 groups.
  If the teacher had called out ‘4’, Ruby would write Expression: 6 × 5. Terms: 6 × 5.

Question: Identify the terms in the two expressions above.

Ans:

• 432 = 4 × 100 + 1 × 20 + 1 × 10 + 2 × 1. Terms: 4 × 100, 1 × 20, 1 × 10, 2 × 1.
• 432 = 8 × 50 + 1 × 10 + 4 × 5 + 2 × 1. Terms: 8 × 50, 1 × 10, 4 × 5, 2 × 1.

Question: Can you think of some more ways of giving ₹432 to someone?

Ans: Some more ways to gave ₹432 to someone are as follows:

• 4 × 100 + 3 × 10 + 2 × 1 = 400 + 30 + 2 = 432.
  Terms: 4 × 100, 3 × 10, 2 × 1.
• 2 × 100 + 4 × 50 + 3 × 10 + 2 × 1 = 200 + 200 + 30 + 2 = 432.
  Terms: 2 × 100, 4 × 50, 3 × 10, 2 × 1.

Question: What is the expression for the arrangement in the right making use of the number of yellow and blue squares?

Ans: The right arrangement has 2 groups of (5 yellow + 3 blue). Expression: 2 × (5 + 3).

• Other forms: 5 + 3 + 5 + 3 or 5 × 2 + 3 × 2.
• Evaluate: 2 × (5 + 3) = 2 × 8 = 16.

Figure it Out

Page No. 34 & 35

1. Find the values of the following expressions by writing the terms in each case.

(a) 28 – 7 + 8
Ans:

• Terms: 28, -7, 8.
• Expression: 28 + (-7) + 8 = 21 + 8 = 29.

(b) 39 – 2 × 6 + 11
Ans:

• Terms: 39, -2 × 6, 11.
• Expression: 39 + (-2 × 6) + 11 = 39 – 12 + 11 = 38.

(c) 40 – 10 + 10 + 10
Ans:

• Terms: 40, -10, 10, 10.
• Expression: 40 + (-10) + 10 + 10 = 30 + 10 + 10 = 50.

(d) 48 – 10 × 2 + 16 ÷ 2
Ans:

• Terms: 48, -10 × 2, 16 ÷ 2.
• Expression: 48 + (-10 × 2) + (16 ÷ 2) = 48 – 20 + 8 = 36.

(e) 6 × 3 – 4 × 8 × 5
Ans:

• Terms: 6 × 3, -4 × 8 × 5.
• Expression: 6 × 3 + (-4 × 8 × 5) = 18 – 160 = -142.

2. Write a story/situation for each of the following expressions and find their values.

(a) 89 + 21 – 10
Ans:

• Story: Ria had 89 candies, got 21 more, and gave 10 to her friend. How many candies does she have now?
• Value: 89 + 21 – 10 = 100.

(b) 5 × 12 – 6
Ans:

• Story: A shop sells 5 packs of 12 pencils each but removes 6 defective ones. How many pencils are left?
• Value: 5 × 12 – 6 = 60 – 6 = 54.

(c) 4 × 9 + 2 × 6

Ans:

• Story: A family buys 4 pizzas costing ₹9 each and 2 drinks costing ₹6 each. What is the total cost?
• Value: 4 × 9 + 2 × 6 = 36 + 12 = 48.

3. For each of the following situations, write the expression describing the situation, identify its terms and find the value of the expression.

(a) Queen Alia gave 100 gold coins to Princess Elsa and 100 gold coins to Princess Anna last year. Princess Elsa used the coins to start a business and doubled her coins. Princess Anna bought jewellery and has only half of the coins left. Write an expression describing how many gold coins Princess Elsa and Princess Anna together have.

Ans:

• Elsa’s coins: 2 × 100. Anna’s coins: 100 ÷ 2.
• Expression: 2 × 100 + 100 ÷ 2.
• Terms: 2 × 100, 100 ÷ 2.
• Value: 2 × 100 + 100 ÷ 2 = 200 + 50 = 250 coins.

(b) A metro train ticket between two stations is ₹40 for an adult and ₹20 for a child. What is the total cost of tickets:

(i) for four adults and three children?

Ans:

• Expression: 4 × 40 + 3 × 20. Terms: 4 × 40, 3 × 20.
• Value: 4 × 40 + 3 × 20 = 160 + 60 = 220. Total cost is ₹220.

(ii) for two groups having three adults each?

Ans:

• Expression: 2 × (3 × 40). Terms: 2 × (3 × 40).
• Value: 2 × (3 × 40) = 2 × 120 = 240. Total cost is ₹240.

(c) Find the total height of the window by writing an expression describing the relationship among the measurements shown in the picture.

Ans: Total height of the window is as follows:
Expression: 5 X 7 + 2 X 6 + 3 X 2
Terms: 5 X 7, 2 X 6, 3 X 2
Value: 35 + 12 + 6 = 53 cm.

Calculation: 5 X 7 + 2 X 6 + 3 X 2 = 35 + 12 + 6 = 53 cm

Figure it Out

Page No. 37, 38 & 39

Question: 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 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.

Ans:

Figure it Out

1. Fill in the blanks with numbers, and boxes with operation signs such that the expressions on both sides are equal.

(a) 24 + (6 – 4) = 24 + 6 – 4 = 26

(b) 38 + (5 – 4) = 38 + 9 – 4 = 39

(c) 24 – (6 + 4) = 24 – 6 – 4 = 14

(d) 24 – 6 – 4 = 24 – 6 – 4 = 14

(e) 27 – (8 + 3) = 27 – 8 + 3 = 16

(f) 27 – (5 + 3) = 27 – 8 + 3 = 19

2. Remove the brackets and write the expression having the same value.

(a) 14 + (12 + 10)
Ans: 14 + (12 + 10) = 14 + 12 + 10.

(b) 14 – (12 + 10)
Ans: 14 – (12 + 10) = 14 – 12 – 10.

(c) 14 + (12 – 10)
Ans: 14 + (12 – 10) = 14 + 12 – 10.

(d) 14 – (12 – 10)
Ans: 14 – (12 – 10) = 14 – 12 + 10.

(e) -14 + 12 – 10
Ans: No brackets to remove. Expression: -14 + 12 – 10.

(f) 14 – (-12 – 10)
Ans: 14 – (-12 – 10) = 14 + 12 + 10.

3. Find the values of the following expressions. For each pair, first try to guess whether they have the same value. When are the two expressions equal?

(a) (6 + 10) – 2 and 6 + (10 – 2)
Ans:

• Initial Guess: Different, due to bracket placement.
• Values: (6 + 10) – 2 = 16 – 2 = 14. 6 + (10 – 2) = 6 + 8 = 14.
• Conclusion: They are equal because (a + b) – c = a + (b – c).

(b) 16 – (8 – 3) and (16 – 8) – 3
Ans:

• Initial Guess: Different.
• Values: 16 – (8 – 3) = 16 – 5 = 11. (16 – 8) – 3 = 8 – 3 = 5.
• Conclusion: They are not equal.

(c) 27 – (18 + 4) and 27 + (-18 – 4)
Ans:

• Initial Guess: Same, as – (a + b) = -a – b.
• Values: 27 – (18 + 4) = 27 – 22 = 5. 27 + (-18 – 4) = 27 – 18 – 4 = 5.
• Conclusion: They are equal.

4. In each of the sets of expressions below, identify those that have the same value. Do not evaluate them, but rather use your understanding of terms.

(a) 319 + 537, 319 – 537, – 537 + 319, 537 – 319
Ans: Identifications as follows:

319 – 537 and –537 + 319 are the same, because changing the order in addition of a negative number results in subtraction.
319 + 537 and 537 – 319 are not the same, but they are both positive.
319 – 537 and –537 + 319 are negative and equal to each other.
The expressions that have the same value are:

319 – 537
–537 + 319

(b) 87 + 46 – 109, 87 + 46 – 109, 87 + 46 – 109, 87 – 46 + 109, 87 – (46 + 109), (87 – 46) + 109
Ans: The first three are identical: 87 + 46 – 109.
87 – 46 + 109 = (87 – 46) + 109 (associative).
87 – (46 + 109) = 87 – 46 – 109 (different).
Equal: 87 + 46 – 109 (three times), 87 – 46 + 109, (87 – 46) + 109.

5. Add brackets at appropriate places in the expressions such that they lead to the values indicated.

(a) 34 – 9 + 12 = 13

Ans: To get 13, we need to first subtract 9 from 34, then add 12.
So, (34 – 9) + 12 = 25 + 12 = 37 (not correct).
Instead, try 34 – (9 + 12):
34 – (9 + 12) = 34 – 21 = 13.
So, the expression is 34 – (9 + 12) = 13.

(b) 56 – 14 – 8 = 34

Ans: To get 34, we need to subtract 14 and 8 from 56 in the correct order.
(56 – 14) – 8 = 42 – 8 = 34.
So, the expression is (56 – 14) – 8 = 34.

(c) –22 – 12 + 10 + 22 = – 22

Ans: To get –22, we need to group the terms carefully.
–22 – (12 + 10) + 22 = –22 – 22 + 22 = –22.
So, the expression is –22 – (12 + 10) + 22 = –22.

6. Using only reasoning of how terms change their values, fill the blanks to make the expressions on either side of the equality (=) equal.

(a) 423 + ______= 419 + ______

Ans: We need to make both sides equal.
423 + ___ = 419 + ___.
423 is 4 more than 419 (423 – 419 = 4).
So, the number on the right side should be 4 more than the number on the left side.
If we put 0 on the left, then 0 + 4 = 4 on the right.
423 + 0 = 419 + 4.
So, the blanks are 0 and 4.

(b) 207 – 68 = 210 – ______

Ans: We need to make both sides equal.
207 – 68 = 210 – ___.
First, calculate 207 – 68 = 139.
So, 210 – ___ = 139.
210 – 139 = 71.
So, the blank is 71.

7. Using the numbers 2, 3 and 5, and the operators ‘+’ and ‘–’, and brackets, as necessary, generate expressions to give as many different values as possible. For example, 2 – 3 + 5 = 4 and 3 – (5 – 2) = 0.

Ans: Let’s use 2, 3, and 5 with + and – to get different values:

• 2 + 3 + 5 = 10
• 2 + 3 – 5 = 0
• 2 – 3 + 5 = 4
• 2 – (3 + 5) = –6
• (2 + 3) – 5 = 0
• 3 – (5 – 2) = 0
• 5 – (2 + 3) = 0
• 5 + 3 – 2 = 6

So, the different values are –6, 0, 4, 6, 10.

8. Whenever Jasoda has to subtract 9 from a number, she subtracts 10 and adds 1 to it. For example, 36 – 9 = 26 + 1.

(a) Do you think she always gets the correct answer? Why?

Ans: Yes, Jasoda always gets the correct answer.
Subtracting 10 and adding 1 is the same as subtracting 9 because 10 – 1 = 9.
For example, 36 – 9 = 27.
Jasoda does 36 – 10 + 1 = 26 + 1 = 27, which is correct.

(b) Can you think of other similar strategies? Give some examples.

Ans: Yes, we can use other strategies:
Instead of subtracting 9, subtract 8 and subtract 1 more: 36 – 9 = 36 – 8 – 1 = 28 – 1 = 27.
Or, subtract 5 and subtract 4 more: 36 – 9 = 36 – 5 – 4 = 31 – 4 = 27.
Both give the correct answer.

9. Consider the two expressions: a) 73 – 14 + 1, b) 73 – 14 – 1. For each of these expressions, identify the expressions from the following collection that are equal to it.

(a) 73 – (14 + 1)

Ans: 73 – (14 + 1) = 73 – 15 = 58.
Expression a) 73 – 14 + 1 = 59 + 1 = 60 (not equal).
Expression b) 73 – 14 – 1 = 59 – 1 = 58 (equal).
So, it matches expression b).

(b) 73 – (14 – 1)

Ans: 73 – (14 – 1) = 73 – 13 = 60.
Expression a) 73 – 14 + 1 = 60 (equal).
Expression b) 73 – 14 – 1 = 58 (not equal).
So, it matches expression a).

(c) 73 + (–14 + 1)

Ans: 73 + (–14 + 1) = 73 + (–13) = 73 – 13 = 60.
Expression a) 73 – 14 + 1 = 60 (equal).
Expression b) 73 – 14 – 1 = 58 (not equal).
So, it matches expression a).

(d) 73 + (–14 – 1)

Ans: 73 + (–14 – 1) = 73 + (–15) = 73 – 15 = 58.
Expression a) 73 – 14 + 1 = 60 (not equal).
Expression b) 73 – 14 – 1 = 58 (equal).
So, it matches expression b).

Removing Brackets—II

Question: What about the total amount they have to pay? Can it be described by the expression: 2 × 43 + 24?

Ans: Let’s calculate the expression:
2 × 43 = 86
86 + 24 = 110
The total amount from the expression is 110.

This can be the total amount they have to pay if:

• They buy 2 things that cost 43 each (2 × 43 = 86), and
• They add 24 more (like a fee or tax), so 86 + 24 = 110.
• Yes, the expression 2 × 43 + 24 can describe the total amount if this matches their payment situation.

Question: If another friend, Sangmu, joins them and orders the same items, what will be the expression for the total amount to be paid?

Ans: First, one person pays for 2 vegetable cutlets and 2 rasgullas.
The cost is 2 × (43 + 24).
If Sangmu joins and orders the same, there will be 2 sets of items.
So, the total amount is 2 × (43 + 24) + 2 × (43 + 24).
This can be written as 2 × (43 + 24) + 2 × (43 + 24).
Since both are the same, it is 2 × (43 + 24) + 2 × (43 + 24) = 4 × (43 + 24).
So, the expression is 4 × (43 + 24).

Page No. 41 & 42

Question: Use this method to find the following products:

(a) 95 × 8

Ans: Write 95 as (100 – 5): 100 × 8 = 800
5 × 8 = 40
800 – 40 = 760
So, 95 × 8 = 760.

(b) 104 × 15

Ans: Write 104 as (100 + 4): 100 × 15 = 1500
4 × 15 = 60
1500 + 60 = 1560
So, 104 × 15 = 1560.

(c) 49 × 50

Ans: Write 49 as (50 – 1): 50 × 50 = 2500
1 × 50 = 50
2500 – 50 = 2450
So, 49 × 50 = 2450.

Question: Is this quicker than the multiplication procedure you use generally?

Ans: Yes, this method can be quicker because it uses easier numbers like 100 or 50 and then adjusts with simple subtraction or addition, instead of doing long multiplication step by step.

Question: Which other products might be quicker to find like the ones above?

Ans: Products like 98 × 25, 99 × 10, 103 × 15, or 51 × 50 might be quicker because they can be written as (100 – 2) × 25, (100 – 1) × 10, (100 + 3) × 15, or (50 + 1) × 50, and solved using the same easy method.

Figure it Out

1. Fill in the blanks with numbers, and boxes by signs, so that the expressions on both sides are equal.

(a) 3 × (6 + 7) = 3 × 6 + 3 × 7

Ans: This is already correct.
3 × (6 + 7) = 3 × 13 = 39
3 × 6 + 3 × 7 = 18 + 21 = 39
No blanks to fill.

(b) (8 + 3) × 4 = 8 × 4 + 3 × 4

Ans: This is already correct.
(8 + 3) × 4 = 11 × 4 = 44
8 × 4 + 3 × 4 = 32 + 12 = 44
No blanks to fill.

(c) 3 × (5 + 8) = 3 × 5 ___ 3 × ___

Ans: 3 × (5 + 8) = 3 × 13 = 39
3 × 5 = 15, so we need 3 × ___ to make 39.
39 – 15 = 24, so 3 × 8 = 24.
The sign is +.
So, 3 × (5 + 8) = 3 × 5 + 3 × 8.

(d) (9 + 2) × 4 = 9 × 4 ___ 2 × ___

Ans: (9 + 2) × 4 = 11 × 4 = 44
9 × 4 = 36, so we need 2 × ___ to make 44.
44 – 36 = 8, so 2 × 4 = 8.
The sign is +.
So, (9 + 2) × 4 = 9 × 4 + 2 × 4.

(e) 3 × (___ + 4) = 3 ___ + ___

Ans: Let’s find the number on the left.
The right side should be 3 × ___ + 3 × 4 (because of the pattern).
If we put 7 in the first blank:
3 × (7 + 4) = 3 × 11 = 33
3 × 7 + 3 × 4 = 21 + 12 = 33
The sign is +.
So, 3 × (7 + 4) = 3 × 7 + 12.

(f) (___ + 6) × 4 = 13 × 4 + ___

Ans: 13 × 4 = 52, so the left side should also be 52.
(___ + 6) × 4 = 52
___ + 6 = 52 ÷ 4 = 13
___ = 13 – 6 = 7
(7 + 6) × 4 = 13 × 4 + 6 × 4
6 × 4 = 24
So, (7 + 6) × 4 = 13 × 4 + 24.

(g) 3 × (___ + ___) = 3 × 5 + 3 × 2

Ans: 3 × 5 + 3 × 2 = 15 + 6 = 21
So, 3 × (___ + ___) = 21
___ + ___ = 21 ÷ 3 = 7
We can use 5 and 2 (since they are on the right).
So, 3 × (5 + 2) = 3 × 5 + 3 × 2.

(h) (___ + ___) × ___ = 2 × 4 + 3 × 4

Ans: 2 × 4 + 3 × 4 = 8 + 12 = 20
So, (___ + ___) × ___ = 20
Use numbers from the right: 2 + 3 = 5, and 20 ÷ 5 = 4
(2 + 3) × 4 = 20
So, (2 + 3) × 4 = 2 × 4 + 3 × 4.

(i) 5 × (9 – 2) = 5 × 9 – 5 × ___

Ans: 5 × (9 – 2) = 5 × 7 = 35
5 × 9 = 45, so 45 – 5 × ___ = 35
5 × ___ = 45 – 35 = 10
5 × 2 = 10
So, 5 × (9 – 2) = 5 × 9 – 5 × 2.

(j) (5 – 2) × 7 = 5 × 7 – 2 × ___

Ans: (5 – 2) × 7 = 3 × 7 = 21
5 × 7 = 35, so 35 – 2 × ___ = 21
2 × ___ = 35 – 21 = 14
2 × 7 = 14
So, (5 – 2) × 7 = 5 × 7 – 2 × 7.

(k) 5 × (8 – 3) = 5 × 8 ___ 5 × ___

Ans: 5 × (8 – 3) = 5 × 5 = 25
5 × 8 = 40, so 40 ___ 5 × ___ = 25
5 × ___ = 40 – 25 = 15
5 × 3 = 15
The sign is –.
So, 5 × (8 – 3) = 5 × 8 – 5 × 3.

(l) (8 – 3) × 7 = 8 × 7 ___ 3 × 7

Ans: (8 – 3) × 7 = 5 × 7 = 35
8 × 7 = 56, so 56 ___ 3 × 7 = 35
3 × 7 = 21
56 – 21 = 35
The sign is –.
So, (8 – 3) × 7 = 8 × 7 – 3 × 7.

(m) 5 × (12 – ___) = ___ 5 × ___

Ans: Let’s use 8 in the first blank:
5 × (12 – 8) = 5 × 4 = 20
On the right, 5 × 8 = 40, so 40 ___ 5 × ___ = 20
5 × ___ = 40 – 20 = 20
5 × 4 = 20
The sign is –.
So, 5 × (12 – 8) = 5 × 8 – 5 × 4.

(n) (15 – ___) × 7 = ___ 6 × 7

Ans: Let’s use 9 in the first blank:
(15 – 9) × 7 = 6 × 7 = 42
On the right, 6 × 7 = 42
15 × 7 = 105, so 105 ___ 42 = 42
105 – 63 = 42, so 9 × 7 = 63
The sign is –.
So, (15 – 9) × 7 = 15 × 7 – 6 × 7.

(o) 5 × (___ – ___) = 5 × 9 – 5 × 4

Ans: 5 × 9 – 5 × 4 = 45 – 20 = 25
So, 5 × (___ – ___) = 25
___ – ___ = 25 ÷ 5 = 5
Use 9 – 4 = 5
So, 5 × (9 – 4) = 5 × 9 – 5 × 4.

(p) (___ – ___) × ___ = 17 × 7 – 9 × 7

Ans: 17 × 7 – 9 × 7 = (17 – 9) × 7 = 8 × 7 = 56
So, (___ – ___) × ___ = 56
Use 17 – 9 = 8, and 8 × 7 = 56
So, (17 – 9) × 7 = 17 × 7 – 9 × 7.

2.In the boxes below, fill ‘<’, ‘>’ or ‘=’ after analysing the expressions on the LHS and RHS. Use reasoning and understanding of terms and brackets to figure this out and not by evaluating the expressions.

(a) (8 – 3) × 29 ___ (3 – 8) × 29
Ans: >

(b) 15 + 9 × 18 ___ (15 + 9) × 18
Ans: <

(c) 23 × (17 – 9) ___ 23 × 17 + 23 × 9
Ans: =

(d) (34 – 28) × 42 ___ 34 × 42 – 28 × 42
Ans: =

3. Here is one way to make 14: _2_ × ( _1_ + _6_ ) = 14. Are there other ways of getting 14? Fill them out below:

(a) ___ × (___ + ___ ) = 14
Ans: 7 × (1 + 1) = 14

(b) ___ × (___ + ___ ) = 14
Ans: 2 × (5 + 2) = 14

(c) ___ × (___ + ___ ) = 14
Ans: 1 × (10 + 4) = 14

(d) ___ × (___ + ___ ) = 14
Ans: 14 × (1 + 0) = 14

4. Find out the sum of the numbers given in each picture below in at least two different ways. Describe how you solved it through expressions.

Ans: Numbers: 4, 8, 4, 8, 4, 8, 4, 8, 4 (in a 3 × 3 grid)
First Way: Add all numbers one by one:
4 + 8 + 4 + 8 + 4 + 8 + 4 + 8 + 4 = 52
Second Way: Count how many 4s and 8s:

• There are 5 fours: 5 × 4 = 20
• There are 4 eights: 4 × 8 = 32
• Add them: 20 + 32 = 52

So, the sum is 52.

Ans: Numbers: 5, 6, 6, 5, 6, 5, 5, 6, 6, 5, 5, 6, 5, 6, 6, 5 (in a 4 × 4 grid)
First Way: Add all numbers one by one:
5 + 6 + 6 + 5 + 6 + 5 + 5 + 6 + 6 + 5 + 5 + 6 + 5 + 6 + 6 + 5 = 93
Second Way: Count how many 5s and 6s:

• There are 8 fives: 8 × 5 = 40
• There are 8 sixes: 8 × 6 = 48
• Add them: 40 + 48 = 88

Now add the remaining numbers (if any, but here we counted all):
Total is 88 (I made a mistake in the first way, let’s correct by counting properly).

Let’s recheck the first way correctly:
5 + 6 + 6 + 5 + 6 + 5 + 5 + 6 + 6 + 5 + 5 + 6 + 5 + 6 + 6 + 5 = 88
So, the sum is 88.

Figure it Out

1. Read the situations given below. Write appropriate expressions for each of them and find their values.

(a) The district market in Begur operates on all seven days of a week. Rahim supplies 9 kg of mangoes each day from his orchard and Shyam supplies 11 kg of mangoes each day from his orchard to this market. Find the amount of mangoes supplied by them in a week to the local district market.

Ans:

• Expression for Rahim: 9 kg × 7 days
• Value for Rahim: 9 × 7 = 63 kg
• Expression for Shyam: 11 kg × 7 days
• Value for Shyam: 11 × 7 = 77 kg
• Total amount: 63 kg + 77 kg = 140 kg

(b) Binu earns ₹20,000 per month. She spends ₹5,000 on rent, ₹5,000 on food, and ₹2,000 on other expenses every month. What is the amount Binu will save by the end of a year?

Ans:

• Expression for monthly savings: ₹20,000 – (₹5,000 + ₹5,000 + ₹2,000)
• Monthly savings: ₹20,000 – ₹12,000 = ₹8,000
• Expression for yearly savings: ₹8,000 × 12 months
• Value for yearly savings: ₹8,000 × 12 = ₹96,000

(c) During the daytime a snail climbs 3 cm up a post, and during the night while asleep, accidentally slips down by 2 cm. The post is 10 cm high, and a delicious treat is on its top. In how many days will the snail get the treat?

Ans:

• Expression for daily progress: 3 cm – 2 cm = 1 cm
• Total height to climb: 10 cm
• Number of days: 10 cm ÷ 1 cm = 10 days
• Value: The snail will get the treat in 10 days

2. Melvin reads a two-page story every day except on Tuesdays and Saturdays. How many stories would he complete reading in 8 weeks? Which of the expressions below describes this scenario?

(a) 5 × 2 × 8
(b) (7 – 2) × 8
(c) 8 × 7
(d) 7 × 2 × 8
(e) 7 × 5 – 2
(f) (7 + 2) × 8
(g) 7 × 8 – 2 × 8
(h) (7 – 5) × 8

Ans: Melvin reads 2 pages daily, 5 days a week (7 – 2). In 8 weeks: (7 – 2) × 8 = 5 × 8 = 40 stories.
Correct expression: (b) (7 – 2) × 8.

3. Find different ways of evaluating the following expressions:

(a) 1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 + 9 – 10

Ans:

• Way 1: Pair terms: (1 – 2) + (3 – 4) + (5 – 6) + (7 – 8) + (9 – 10) = -1 – 1 – 1 – 1 – 1 = -5.
• Way 2: Add sequentially: 1 – 2 = -1, -1 + 3 = 2, 2 – 4 = -2, -2 + 5 = 3, 3 – 6 = -3, -3 + 7 = 4, 4 – 8 = -4, -4 + 9 = 5, 5 – 10 = -5.

Value: -5.

(b) 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1

Ans:

• Way 1: Pair terms: (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) = 0 + 0 + 0 + 0 + 0 = 0.
• Way 2: Add sequentially: 1 – 1 = 0, 0 + 1 = 1, 1 – 1 = 0, …, ending at 0.

Value: 0.

4. Compare the following pairs of expressions using ‘<‘, ‘>’ or ‘=’ by reasoning.

(a) 49 – 7 + 8 __ 49 – 7 + 8

Ans: Same expression.
So, 49 – 7 + 8 = 49 – 7 + 8.

(b) 83 × 42 – 18 __ 83 × 40 – 18

Ans: 83 × 42 = 83 × (40 + 2) = 83 × 40 + 83 × 2.
So, 83 × 42 – 18 > 83 × 40 – 18.

(c) 145 – 17 × 8 __ 145 – 17 × 6

Ans: 17 × 8 = 136, 17 × 6 = 102. 145 – 136 < 145 – 102.
So, 145 – 17 × 8 < 145 – 17 × 6.

(d) 23 × 48 – 35 __ 23 × (48 – 35)

Ans: 23 × 48 – 35 > 23 × 13 (since 48 > 13).
So, 23 × 48 – 35 > 23 × (48 – 35).

(e) (16 – 11) × 12 __ -11 × 12 + 16 × 12

Ans: (16 – 11) × 12 = 5 × 12. -11 × 12 + 16 × 12 = (16 – 11) × 12 = 5 × 12.
So, (16 – 11) × 12 = -11 × 12 + 16 × 12.

(f) (76 – 53) × 88 __ 88 × (53 – 76)

Ans: (76 – 53) = 23, (53 – 76) = -23.
So, 23 × 88 > -23 × 88.
Therefore, (76 – 53) × 88 > 88 × (53 – 76).

(g) 25 × (42 + 16) __ 25 × (43 + 15)

Ans: 42 + 16 = 58, 43 + 15 = 58.
So, 25 × (42 + 16) = 25 × (43 + 15).

(h) 36 × (28 – 16) __ 35 × (27 – 15)

Ans: 28 – 16 = 12, 27 – 15 = 12.
But 36 > 35.
So, 36 × (28 – 16) > 35 × (27 – 15).

5. Identify which of the following expressions are equal to the given expression without computation. You may rewrite the expressions using terms or removing brackets. There can be more than one expression which is equal to the given expression.

(a) 83 – 37 – 12

(i) 84 – 38 – 12

(ii) 84 – (37 + 12)

(iii) 83 – 38 – 13

(iv) – 37 + 83 –12

(b) 93 + 37 × 44 + 76

(i) 37 + 93 × 44 + 76

(ii) 93 + 37 × 76 + 44

(iii) (93 + 37) × (44 + 76)

(iv) 37 × 44 + 93 + 76

Ans: (a) The expressions equal to 83 – 37 – 12 are:

(i) 84 – 38 – 12 (because 84 – 38 = 46 and 46 – 12 = 34, same as 83 – 37 – 12 = 34)
(ii) 84 – (37 + 12) (because 37 + 12 = 49, and 84 – 49 = 35,
but this is incorrect; correct check: 84 – 49 = 35, not 34, so only (i) and (iv) work)
(iv) –37 + 83 – 12 (because –37 + 83 = 46, and 46 – 12 = 34)

(b) The expressions equal to 93 + 37 × 44 + 76 are:

(i) 37 + 93 × 44 + 76 (same as 93 + 37 × 44 + 76)
(iv) 37 × 44 + 93 + 76 (same order of operations)

6. Choose a number and create ten different expressions having that value.

Ans: Let’s choose the number 10. Ten different expressions are:

• 5 + 5
• 20 – 10
• 2 × 5
• 50 ÷ 5
• 15 – 5
• 7 + 3
• 4 × 2 + 2
• 30 – 20
• 10 × 1
• 25 – 15