NCERT Solutions for Class 7 Maths Chapter 2 Arithmetic Expressions

Page No. 24 

Simple Expressions

Q: Choose your favourite number and write as many expressions as you can having that value.
Ans: Let’s choose the number 12. Expressions with the value 12 are:

• 10 + 2 = 12
• 15 - 3  = 12
• 3 × 4 = 12
• 24 ÷ 2 = 12 etc.

Page No. 25 

Figure it Out

Q1: 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.

Q2: 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:

• 67 - 19 = 48
• 67 - 20 = 47
• 35 + 25 = 60
• 5 × 11 = 55
• 120 ÷ 3 = 40

Ascending order: 40 < 47 < 48 < 55 < 60
So, the order is: 120 ÷ 3 < 67 - 20 < 67 - 19 < 5 × 11 < 35 + 25.
Thus, (e) < (b) < (a) < (d) < (c).

Page No. 26 

Q: 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.

Ans:

(a) 245 + 289 > 246 + 285

245 is 1 less than 246
But 289 is 4 more than 285
So overall, the left side is 3 more than the right side.

(b) 273 − 145 = 272 − 144

273 is 1 more than 272
145 is also 1 more than 144
So both changes cancel each other.

(c) 364 + 587 < 363 + 589

364 is 1 more than 363
But 587 is 2 less than 589
So overall, the left side is 1 less.

(d) 124 + 245 < 129 + 245

124 is 5 less than 129
245 is the same on both sides
So the left side is 5 less.

(e) 213 − 77 < 214 − 76

213 is 1 less than 214But 77 is 1 more than 76
So the left side becomes 2 less.

Page No. 28 

Terms in Expressions

Suppose we have the expression 30 + 5 × 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 ‘+’ sign. For example, in 12+7, the terms are 12 and 7, as marked below.

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,

Thus, the terms of the expression 83 – 14 are 83 and –14.

Q: 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.

Q: 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 of Integers:

• +1 token = positive number

• −1 token = negative number

Subtracting a number means removing positive tokens.
But removing +1 tokens is the same as adding −1 tokens.

Example:

• 5 − 3 → Remove 3 positive tokens from 5 → Left with 2

• 5 + (−3) → Add 3 negative tokens to 5 → They cancel with 3 positives → Left with 2

Conclusion:
Subtracting a number is the same as adding its opposite.
So, 5 − 3 = 5 + (−3).

Page No. 29 & 30 

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

Ans:

Q: 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.

Swapping and Grouping

Q: 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.

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

Q: Will this also hold when there are terms having negative numbers as well? Take some more expressions and check.
Ans:
Yes, while adding the terms having negative numbers, grouping them in any order gives the same result.
As,

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

Ans: Yes

Example:

Start with 0.
Add 5 positive and 5 negative tokens:
They cancel each other in pairs → Final total = 0

Why it matters:
Positive and negative tokens balance each other.
The final value depends on which type is more.

Example:
3 − 5 → 3 positives, 5 negatives
Cancel 3 pairs → Left with 2 negatives → Answer: −2

This shows how subtraction can lead to negative numbers using the token model.

Page No. 31 & 32

Q: 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, adding the terms of an expression in any order gives the same value. This is because addition is commutative and associative, which means:

• Commutative: Changing the order of numbers doesn’t change the sum.
  (Example: 4 + 5 = 5 + 4 = 9)

• Associative: Grouping numbers differently doesn’t change the sum.
  (Example: (2 + 3) + 4 = 2 + (3 + 4) = 9)

Let’s check with more terms:

Example 1:
7 + (-2) + 5 + (-3)

Try adding in different orders:

• (7 + 5) + (-2) + (-3) = 12 - 2 - 3 = 7

• 7 + (-2) + (-3) + 5 = 7 - 2 - 3 + 5 = 7

• (-2 + 5) + (-3 + 7) = 3 + 4 = 7

All give the same result.

Q: 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:

• Positive numbers are shown as positive tokens (+1 each)

• Negative numbers are shown as negative tokens (–1 each)

• A positive token and a negative token together cancel out (because +1 + (–1) = 0)

Example:

Suppose you have 3 positive tokens (+3) and you add 2 more positive tokens (+2):

+3 + +2 = +5
You now have 5 positive tokens.

If you change the order:
+2 + +3 = +5
You still get 5 tokens. So, the order doesn't matter.

Q: 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.

Page no. 32

Example 7: Amu, Charan, Madhu, and John went to a hotel and ordered four dosas. Each dosa cost ₹23, and they wish to thank the waiter by tipping ₹5. Write an expression describing the total cost. 

Ans: Cost of 4 dosas = 4 × 23          

 Can the total amount with tip be written as 4 × 23 + 5? Evaluating it, we get

Thus, 4 × 23 + 5 is a correct way of writing the expression.

Q: 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.

The terms in the expression 7 × 23 + 5 are 7 × 23, 5.

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 × 5 + 3 (understood as 3 more than 6 × 5)

Q: Think and discuss why she wrote this. The expression written as a sum of terms is— 

Ans:  Ruby observed what happened in the game:

• The children had to form groups of 5.

• She noticed that there were 6 full groups of 5 students each.

  • That makes 6 × 5 = 30 students.

• But the total number of children playing was 33.

• So, 3 children were left out who could not fit into a full group of 5.

Therefore, Ruby wrote:
6 × 5 + 3,
which means:

• 30 students are in complete groups (6 groups of 5),

• And 3 students are left out.

Final Expression:

6 × 5 + 3 — written as a sum of terms:

(6 × 5) + 3

Page No. 33

Q: 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 ____________ 
Ans: 

• If the teacher had called out ‘4’, Ruby would write
  Expression: 8 × 4 + 1. 
  Terms: 8 × 4, 1.
• If the teacher had called out ‘7’, Ruby would write
   Expression: 4 × 7 + 5. 
  Terms: 4 × 7, 5.

Write expressions like the above for your class size.

Ans: Do it Yourself!

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? 

Ans: 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

Q: 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.

Q: Can you think of some more ways of giving ₹432 to someone?
Ans: Some more ways to give ₹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.

Page No. 34 

Figure it Out

Q1: 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 
= 27 + 11 = 38

(c) 40 - 10 + 10 + 10
Ans: Terms: 40, -10, 10, 10. 
Expression: 40 + (-10) + 10 + 10 
= 30 + 10 + 10 
= 40 + 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

Q2: 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.

Q3: 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: Number of gold coins Princess Elsa got = 100
Number of gold coins Princess Anna got = 100
Princess Elsa used the coins to start the business and doubled her coins.
So, the number of coins Princess Elsa has = 2 × 100
Princess Anna bought jewellery and has only half of the coins left.
So, the number of coins Princess Anna has = 100/2
Therefore, the total number of gold coins Princess Elsa and Princess Anna have together = 2 × 100 + 100/2
Thus, the expression describing the above situation is 2 × 100 + 100/2
Terms: 2 × 100, 100/2
Now, 2 × 100 + 100/2 = 200 + 50 = 250

(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: (i) Metro train ticket for an adult = ₹ 40
So, the metro train ticket for four adults = 4 × 40
Metro train ticket for a child = ₹ 20
So, the metro train ticket for three children = 3 × 20
Therefore, the expression describing the total cost of tickets for four adults and three children is 4 × 40 + 3 × 20
Terms: 4 × 40, 3 × 20
Now, 4 × 40 + 3 × 20 = 160 + 60 = 220

(ii) for two groups having three adults each?
Ans: Metro train ticket for an adult = ₹ 40
So, the metro train ticket for a group of three adults = 3 × 40
Therefore, the expression describing the total cost of tickets for the two groups having three adults each is 2 × (3 × 40).
Terms: 2 × (3 × 40)
Now, 2 × (3 × 40) = 2 × 120 = 240

(c) Find the total height of the window by writing an expression describing the relationship among the measurements shown in the picture.
Ans: By observing the given picture, the total height of the window = number of gaps × 5 cm + number of grills × 2 cm + number of borders × 3 cm = 7 × 5 + 6 × 2 + 2 × 3
Terms: 7 × 5, 6 × 2, 2 × 3
Now, 7 × 5 + 6 × 2 + 2 × 3 = 35 + 12 + 6 = 47 + 6 = 53 

Page No. 37, 38 & 39 

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 in the blanks, doing as little computation as possible.
Ans:

Is −15 one more or one less than −16?

Ans: Yes, –15 is one more than –16, so the value will be 1 more than 37.

Is −17 one more or one less than −16?

Ans: Yes, –17 is one less than –16, so the value will be 1 less than 37.

Figure it Out

Q1: 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____
Ans: 24 + (6 - 4) = 24 + 6 - 4 = 26

(b) 38 + (_____  _____) = 38 + 9 – 4
Ans: 38 + (9 - 4) = 38 + 9 - 4 = 43

(c) 24 – (6 +4) = 246 – 4
Ans: 24 - (6 + 4) = 24 - 6 - 4 = 14

(d) 24 – 6 – 4 = 24 – 6_____
Ans: 24 - 6 - 4 = 24 - 6 - 4 = 14

(e) 27 – (8 + 3) = 278 3
Ans: 27 - (8 + 3) = 27 - 8 - 3 = 16

(f ) 27– (__________) = 27 – 8 + 3
Ans: 27 - (8 - 3) = 27 - 8 + 3 = 22

Q2: Remove the brackets and write the expression having the same value.
(a) 14 + (12 + 10)
Ans: 14 + (12 + 10) 
= 14 + 12 + 10 
= 14 + 22 = 36

(b) 14 - (12 + 10)
Ans: 14 - (12 + 10) 
= 14 - 12 - 10 
= 14 - 22 = -8

(c) 14 + (12 - 10)
Ans: 14 + (12 - 10) 
= 14 + 12 - 10 
= 14 + 2 = 16

(d) 14 - (12 - 10)
Ans: 14 - (12 - 10) 
= 14 - 12 + 10 
= 14 - 2 = 12

(e) -14 + 12 - 10
Ans: No brackets to remove. 
Expression: -14 + 12 - 10 
= -14 + 2 = -12

(f) 14 - (-12 - 10)
Ans: 14 - (-12 - 10) 
= 14 + 12 + 10 
= 14 + 22 
= 36

Q3: 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.

Q4: 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: 

Expressions having the same terms have the same value. Therefore, 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: 

Expressions having the same terms have equal values.
Therefore, 87 + 46 – 109, 87 + 46 – 109, 87 + 46 – 109 have the same value.
Also, 87 – 46 + 109 and (87 – 46) + 109 have the same value.

Q5: 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.

Q6: 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: 210 is 3 more than 207

So to keep both sides equal, we also need to make the second number on the right side 3 more than 68.

207 − 68 = 210 − 71

Because increasing the first number by 3 and also increasing the second number by 3 keeps the value the same.

Q7: 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,
• 5 + 3 – 2 = 6, etc.

Q8: 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.

Q9: 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

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.

Ans:  As each of them had one vegetable cutlet and one rasagulla, each of their shares can be represented by 43 + 24.

Q: What about the total amount they have to pay? Can it be described by the expression: 2 × 43 + 24? 
Ans: Each person had 1 vegetable cutlet and 1 rasgulla.
Cost of one vegetable cutlet = ₹43
Cost of one rasgulla = ₹24
Per person cost = ₹43 + ₹24
For two people, total cost = 2 × (₹43 + ₹24) = 2×43 + 2×24 = ₹86 + ₹48 = ₹134
So, the expression 2 × (43 + 24) is correct.

Correct expression is: 2 × (43 + 24)

Q: If another friend, Sangmu, joins them and orders the same items, what will be the expression for the total amount to be paid?
Ans: Cost of one vegetable cutlet = ₹43
Cost of one rasgulla = ₹24
Total cost for one person = 43 + 24 = ₹67

Now, there are 3 people (Lhamo, Norbu, and Sangmu).
So, the total amount to be paid = 3 × (43 + 24)
= 3 × 67 = ₹201

Expression: 3 × (43 + 24)

Page No. 41 & 42 

Tinker the Terms II

Example 17: Given 53 × 18 = 954. Find out 63 × 18. 

Ans: As 63 × 18 means 63 times 18, 
63 × 18 = (53 + 10) × 18
                  = 53 ×18 + 10×18 
                  = 954 + 180 
                  = 1134.

Q: 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

Q: 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.

Q: 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

Q1: Fill in the blanks with numbers and boxes by signs, so that the expressions on both sides are equal.

Ans: (a) 3 × (6 + 7) = 3 × 6 + 3 × 7
(b) (8 + 3) × 4 = 8 × 4 + 3 × 4
(c) 3 × (5 + 8) = 3 × 5 + 3 × 8
(d) (9 + 2) × 4 = 9 × 4 + 2 × 4
(e) 3 × (10 + 4) = 30 + 12
(f) (13 + 6) × 4 = 13 × 4 + 24
(g) 3 × (5 + 2) = 3 × 5 + 3 × 2
(h) (2 + 3) × 4 = 2 × 4 + 3 × 4
(i) 5 × (9 – 2) = 5 × 9 – 5 × 2
(j) (5 – 2) × 7 = 5 × 7 – 2 × 7
(k) 5 × (8 – 3) = 5 × 8 – 5 × 3
(l) (8 – 3) × 7 = 8 × 7 – 3 × 7
(m) 5 × (12 – 3) = 60 – 5 × 3
(n) (15 – 6) × 7 = 105 – 6 × 7
(o) 5 × (9 – 4) = 5 × 9 – 5 × 4
(p) (17 – 9) × 7 = 17 × 7 – 9 × 7

Q2: 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:  (8 – 3) × 29 > (3 – 8) × 29

Explanation:

• (8 – 3) = 5, so the left side is 5 × 29 = 145

• (3 – 8) = –5, so right side is –5 × 29 = –145
  Since 145 is greater than –145,
  145 > –145 

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

Explanation:

• Follow BODMAS:
  Left side → 9 × 18 = 162 → 15 + 162 = 177
  Right side → (15 + 9) × 18 = 24 × 18 = 432
  ⇒  177 < 432

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

Explanation:
This shows the distributive property:
Left side: 23 × (17 – 9) = 23 × 8 = 184
Right side: 23 × 17 = 391, 23 × 9 = 207
391 + 207 = 598

⇒  184 < 598

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

Explanation:
Left side: (34 – 28) × 42 = 6 × 42 = 252
Right side: 34 × 42 = 1428, 28 × 42 = 1176 → 1428 – 1176 = 252
Both sides are equal.

Q3: 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

Q4: 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: 5 × 4 + 4 × 8 = 20 + 32 = 52
or 4 × (4 + 8) + 4 = 4 × 12 + 4 = 52

Ans: 8 × (5 + 6) = 8 × 11 = 88
or 8 × 5 + 8 × 6 = 40 + 48 = 88

Figure it Out

Q1: 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 the 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 number of mangoes supplied by them in a week to the local district market.
Ans: Supplies of mangoes by Rahim in the market each day = 9 kg
Supplies of mangoes by Shyam in the market each day = 11 kg
Total supplies of mangoes in the market on each day = (9 + 11) kg
Therefore, total supplies of mangoes in the market in all 7 days = 7 × (9 + 11) kg
= 7 × 20
= 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: Binu’s per month earning = ₹ 20,000
Binu’s total monthly expenditures = ₹ 5,000 on rent + ₹ 5,000 on food + ₹ 2,000 on other expenses
= 5,000 + 5,000 + 2,000
Therefore, Binu’s monthly savings = ₹ 20,000 – ₹(5,000 + 5,000 + 2,000)
= ₹ 20,000 – ₹ 12,000
= ₹ 8,000
Thus, Binu’s total yearly savings = 12 × 8000 = 96000
Hence, Binu will save ₹ 96000 by the end of the year.

(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: Since the snail climbs 3 cm up the post in daytime and slips down by 2 cm at night.
So, the distance climbed by the snail of the post = 3 – 2 = 1 cm in a day.
∴ The distance climbed in 7 days = 7 cm
The height of the post is 10 cm.
The distance climbed on the 8th day before slipping = 7 + 3 = 10 cm
So, the snail will take 8 days to reach the top of the post and get the delicious treat.

Q2: 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: Number of days in a week except Tuesday and Saturday = 7 – 2
Since Melvin reads a two-page story every day except Tuesday and Saturday.
Therefore, number of stories read in a week = 1 × (7 – 2)
So, number of stories read in 8 weeks = 8 × 1 × (7 – 2)
= 8 × (7 – 2) or (7 – 2) × 8 [Expression (b)]
or 7 × 8 – 2 × 8 [Expression (g)]
Only expressions (b) and (g) describe this scenario.

Q3: Find different ways of evaluating the following expressions:
(a) 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10
Ans: 1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 + 9 – 10
= (1 + 3 + 5 + 7 + 9) + (-2 – 4 – 6 – 8 – 10)
= 25 + (-30)
= -5
or
1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 + 9 – 10
= (1 – 2) + (3 – 4) + (5 – 6) + (7 – 8) +(9 – 10)
= (-1) + (-1) + (-1) + (-1) + (-1)
= -5

(b) 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1
Ans:  1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1
= (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1)
= 0 + 0 + 0 + 0 + 0
= 0
or
1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1
= (1 + 1 + 1 + 1 + 1) + (-1 – 1 – 1 – 1 – 1)
= 5 + (-5)
= 0

Q4: Compare the following pairs of expressions using '<', '>', or '=' by reasoning.
(a) 49 - 7 + 8 __ 49 - 7 + 8
Ans: 49 - 7 + 8 = 49 - 7 + 8.

(∵ All the terms on both sides are the same)

(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).

Page No. 44

Q5: 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) (a) 83 – 37 – 12 = 83 – 37 – 12 + (1 – 1)
= (83 + 1) – 37 – 1 – 12
= 84 – 38 – 12 (option (i))
= 34
Or
83 – 37 – 12 = -37 + 83 – 12
= 46 – 12
= 34 (option (iv))
Hence, (i) and (iv) are equal to the given expression 83 – 37 – 12.
(b) The expressions equal to 93 + 37 × 44 + 76 are:
Rearrange the terms, and we get 93 + 37 × 44 + 76, which is equal to the given expression.
Hence, (iv) is equal to the given expression 93 + 37 × 44 + 76.

Q6: 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