Arithmetic Expressions ⇒⇒ Exercise 2

Question 1

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

Solution :

Let us choose the number 5

Here are a few expressions that evaluate to 5:

5 × 1

2 + 3

9 − 4

( 6 + 4)/ 2

( 8 − 3) × 1

( 7 − 2 ) × ( 6 − 5)

Question 1

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

Solution :

(a) 13 + 4 = ____ + 6

First, calculate the left side of the equation:

13 + 4 = 17

Now, we know that 17 = ____ + 6

To find the missing number, subtract 6 from 17

17 − 6 = 11

The blank is 11 .

(b) 22 + ______ = 6 × 5

First, calculate the right side of the equation:

6 × 5 = 30

Now, we know that 22 + ______ = 30

To find the missing number, subtract 22 from 30

30 − 22 = 8

The blank is 8 .

(c) 8 × ______ = 64 ÷ 2

First, calculate the right side of the equation:

64 ÷ 2 = 32

Now, we know that 8 × ______ = 32

To find the missing number, divide 32 by 8

32 ÷ 8 = 4

The blank is 4 .

(d) 34 - ______ = 25

We need to find a number that, when subtracted from 34, gives 25

To do this, we can subtract 25 from 34

34 − 25 = 9

The blank is 9 .

Question 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

Solution :

Here are the values of each expression

(a) 67 - 19 = 48
(b) 67 - 20 = 47
(c) 35 + 25 = 60
(d) 5 × 11 = 55
(e) 120 ÷ 3 = 40

Arranging them in ascending (increasing) order of their values:

(e) 120 ÷ 3 = 40
(b) 67 - 20 = 47
(a) 67 - 19 = 48
(d) 5 × 11 = 55
(c) 35 + 25 = 60

Therefore, 120 ÷ 3 < 67 – 20 < 67 – 19 < 5 × 11 < 35 + 25

Question 3

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.

Solution :

(a) 245 + 289 ____ 246 + 285

The key is to look at how the numbers change on each side of the comparison.

On the left side, we have 245 and 285 increased by 4 to become 289 and right side 245 increased by 1 to become 246 and 289 .

Now,

245 + 285 + 4 ____ 245 + 1 + 285

Removing the common terms,

4 _>_1

Therefore, 245 + 289 > 246 + 285

(b) 273 – 145 ____ 272 – 144

Let's compare the first numbers: 273 on the left, 272 on the right. The right side is 1 less

Let's compare the second numbers (the ones being subtracted): 145 on the left, 144 on the right. The right side is also 1 less.

If you start with a number that is 1 less, and you subtract a number that is also 1 less, the difference will remain the same.

Therefore, 273 – 145 = 272 – 144

(c) 364 + 587 ____ 363 + 589

On the left side, we have 363 increased by 1 to become 364 and right side 587 increased by 2 to become 589.

Now,

363 + 1 + 587 ____ 363 + 587 + 2

Removing the common terms,

1 _<_2

Therefore, 364 + 587 < 363 + 589

(d) 124 + 245 ____ 129 + 245

On the right side, we have 124 increased by 4 to become 129 .

Now,

124 + 245____ 124 + 4 + 245

Removing the common terms,

0 _<_4

Therefore, 124 + 245 < 129 + 245

(e) 213 – 77 ____ 214 – 76

Let's compare the first numbers: 213 on the left, 214 on the right. The right side is 1 more.

Let's compare the second numbers (the ones being subtracted): 77 on the left, 76 on the right. The right side is 1 less .

If you start with a number that is 1 more, and you subtract a number that is 1 less, the difference will be larger. The right side will be greater.

Therefore, 213 – 77 < 214 – 76

Question 1

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

Solution :

Example 1: Let us take numbers 20 and 7

(Rule a − b = a + (−b) )

Expression: 20 − 7 = 13

20 + (−7) = 13

Example 2: Let us take numbers 10 and -3

Expression: 10 − (−3) = 10 + 3 = 13

(Subtracting a negative is equivalent to adding a positive)

10 + (−(−3)) = 10 + 3 = 13

In all these cases, replacing subtraction with the addition of the negative of the second number yields the exact same result.

Question 2

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?

Solution :

Understanding a − b = a + ( −b ) using the Token Model

The Token Model

Positive Tokens: Represent positive units

Negative Tokens: Represent negative units

Key Rule: One positive token and one negative token together form a "zero pair." They cancel each other out, meaning their combined value is zero.

Let's take an example: 3 − 2 and then compare it to 3 + ( −2).

Representing Subtraction : 3 − 2

First number : Represent 3 with three positive (+) tokens.

+1 +1 +1

Subtract 2 means removing 2 positive tokens: You need to "take away" two positive (+) tokens.

+1 +1

After taking away two positive tokens, you are left with one positive tokens.

+1

So, 3 − 2 = 1

Representing Subtraction : 3 + ( −2)

First number : Represent 3 with three positive (+) tokens.

+1 +1 +1

Add 2 negative tokens means adding 2 negative (-) tokens : You need to "take away" two positive (+) tokens.

-1 -1

Combine them: (+1 and -1 cancel each other). Group the positive and negative tokens into zero pairs:

(+1 -1) (+1 -1) +1

The two zero pairs vanish, leaving you with one positive tokens.

+1

So, 3 + ( −2) = 1

Conclusion : This model clearly demonstrates that subtraction is just a specific type of addition – the addition of an inverse.

Question 1

Complete the table:

Solution :

Question 2

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

Solution :

No, changing the order in which terms are added does not give different values.
This fundamental property in mathematics is called the commutative property of addition.

Question 3

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

Solution :

Yes, the commutative property of addition also holds true when there are negative numbers involved.
The order in which you add terms, whether positive or negative, does not change the sum.

Example : One positive and one negative number

Expression 1:

7 + (−3) = 7 − 3 = 4

Expression 2:

(−3) + 7 = −3 + 7 = 4.

Question 1

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

Solution :

The Token Model

Positive Tokens: Represent positive units

Negative Tokens: Represent negative units

Key Rule: One positive token and one negative token together form a "zero pair." They cancel each other out, meaning their combined value is zero.

When you add numbers using the token model, you're essentially combining collections of tokens.

Example 1 : Consider 3 + ( −2)

Start with 3 positive tokens .

+1 +1 +1

Add 2 negative tokens

-1 -1

Combine them: (+1 and -1 cancel each other). Group the positive and negative tokens into zero pairs:

(+1 -1) (+1 -1) +1

The two zero pairs vanish, leaving you with one positive tokens.

+1

What's left? 1 positive token. So, , 3 + ( −2) = 1

Now, Consider (−2) + 3

Start with 2 negative tokens .

-1 -1

Add 3 positive tokens

+1 +1 +1

Combine them: (+1 and -1 cancel each other). Group the positive and negative tokens into zero pairs:

(-1 +1) (-1 +1) +1

The two zero pairs vanish, leaving you with one positive tokens.

+1

What's left? 1 positive token. So, , (−2) + 3 = 1

Conclusion : When you add numbers using the token model, you're essentially combining collections of tokens.
The key reason the order doesn't matter is that combining collections of objects doesn't depend on the order in which you pick up those collections.

Question 2

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

Solution :

Yes, the commutative property of addition also holds true when there are negative numbers involved.
The order in which you add terms, whether positive or negative, does not change the sum.

Example : Adding multiple terms, including negative numbers:

Expression 1:

4 + ( −10) + 6

= ( 4 + (−10) ) + 6

−6 + 6 = 0

Now, change the order:

(−10) + 4 + 6

(−10) + ( 4 + 6 )

−10 + 10 = 0

And another order :

6 + (−10) + 4

(6 + (−10)) + 4

-4 + 4 = 0

Question 3

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

Solution :

Expression 1: Consider 4 + ( −10) + 6

Start with 4 positive tokens .

+1 +1 +1 +1

Add 10 negative tokens

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1

Combine them: (+1 and -1 cancel each other). Group the positive and negative tokens into zero pairs:

(+1 -1) (+1 -1) (+1 -1) (+1 -1) -1 -1 -1 -1 -1 -1

What's left? 6 negative tokens -1 -1 -1 -1 -1 -1

Now, Add 6 positive tokens to the result

+1 +1 +1 +1 +1 +1

Combine them: (+1 and -1 cancel each other). Group the positive and negative tokens into zero pairs:

(-1 +1) (-1 +1) (-1 +1) (-1 +1) (-1 +1)(-1 +1)

What's left? Zero tokens. So, 4 + (−10) + 6 =0

If you were to change the order (e.g., 6 + (−10) + 4),
the process of combining collections and cancelling pairs would ultimately lead to the same final state of zero tokens.

Question 1

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.

Solution :

Yes, adding the terms of an expression in any order gives the same value, but this is specifically true for addition.

Example : Adding multiple terms :

Expression 1:

4 + ( −10) + 6

= ( 4 + (−10) ) + 6

−6 + 6 = 0

Now, change the order:

(−10) + 4 + 6

(−10) + ( 4 + 6 )

−10 + 10 = 0

And another order :

6 + (−10) + 4

(6 + (−10)) + 4

-4 + 4 = 0

These examples demonstrate that for expressions involving only addition (even with negative numbers ), the order in which you add them does not change the final value.

Question 3

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?
(Numbers list: 1342, 774, 8611, 9055,1022) 

Solution :

No, Manasa does not have to start all over again.

Since addition is commutative and associative, the order in which numbers are added doesn't change the final sum.

This means Manasa can simply add the forgotten number to her previous total.

Original Sum: Manasa's current sum is 11749.

Forgotten Number: 9055

Corrected Sum: 11749 + 9055 = 20804

Question 1

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

Solution :

Cost of each dosa: ₹ 23

Number of dosas: 4

cost of the dosas:

4 × 23 = 92

Add the tip to the cost of the dosas: 92 + 5 = 97

Expression : (4 × 23 ) + 5

Question 2

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.

Solution :

Cost of each dosa: ₹ 23

Number of dosas: 7

cost of the dosas:

7 × 23 = 161

Add the tip to the cost of the dosas: 161 + 5 = 166

Expression : (7 × 23 )+ 5

Identifying the Terms

The first term is (7 × 23). This represents the total cost of the dosas, which is ₹161

The second term is 5. This represents the fixed tip amount

Question 3

Children playing “Fire in the mountain”. 33 students were playing and ruby didn’t play. Teacher called out ‘5’. Students formed groups of 5. Ruby wrote 6 x 5 + 3. Think and discuss why she wrote this.

Solution :

Given : Total students : 33 students

Teacher called out '5': This means students need to form groups of 5.

Ruby's expression : 6 × 5 + 3

6 × 5: This part represents the total number of students ( 6 × 5 = 30, ) who successfully formed complete groups of 5

+ 3: This part represents the remainder – the number of students ( 33 − 30 = 3 ) left over after forming as many complete groups of 5 as possible

Ruby's expression is essentially a representation of the division algorithm (dividend = quotient × divisor + remainder)

Question 1

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.

Solution :

(i) If the teacher had called out ‘4’

Ruby would write: 33 ÷ 4

Ruby's expression : 8 × 4 + 1

(ii) If the teacher had called out ‘7’

Ruby would write: 33 ÷ 7

Ruby's expression : 4 × 7 + 5

(iii) Let's assume a hypothetical class size of 29 students

(i) If the teacher had called out ‘4’

Ruby would write: 29 ÷ 4

Ruby's expression : 7 × 4 + 1

(i) If the teacher had called out ‘7’

Ruby would write: 29 ÷ 7

Ruby's expression : 4 × 7 + 1

Question 2

Identify the terms in the two expressions for ₹432

Solution :

(i) 432 = 4 × 100 + 1 × 20 + 1 × 10 + 2 × 1

The terms in the expression: ( 4 × 100)( 1 × 20)( 1 × 10 )( 2 × 1 )

(ii) 432 = 8 × 50 + 1 × 10 + 4 × 5 + 2 × 1

The terms in the expression: ( 8 × 50 )( 1 × 10 )( 4 × 5 )( 2 × 1 )

Question 3

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

Solution :

There are many ways to give ₹432 to someone,

(i) 432 = 2 × 200 + 1 × 20 + 1 × 10 + 2 × 1

The terms in the expression: ( 2 × 200)( 1 × 20)( 1 × 10 )( 2 × 1 )

(ii) 432 = 4 × 100 + 32 × 1

The terms in the expression: ( 4 × 100 )( 32 × 1 )

(iii) 432 = 1 × 100 + 16 × 20 + 1 × 10 + 2 × 1

The terms in the expression: ( 1 × 100 )( 16 × 20)( 1 × 10 )( 2 × 1 )

Question 1

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

Solution :

( a ) 28 – 7 + 8

The terms in this expression are: 28, -7, and 8

To find value: We perform the operations from left to right

28 + (-7) + 8

21 + 8 = 29

Therefore, the value of 28 − 7 + 8 is 29.

( b ) 39 - 2 × 6 + 11

The terms in this expression are: 39, ( -2 × 6), and 11

Value : 39 + (-2 × 6) + 11

39 - 12 + 11

27 + 11 = 28

Therefore, the value of 39 - 2 × 6 + 11 is 28

(c) 40 - 10 + 10 + 10

The terms in this expression are: 40, -10, and 10, 10

Value: 40 + (-10) + 10 + 10

30 + 10 + 10 = 50

Therefore, the value of 40 - 10 + 10 + 10 is 50.

(d) 48 - 10 × 2 + 16 ÷ 2

The terms in this expression are: 48, ( -10 × 2) and ( 16 ÷ 2 )

Value 48 + (-10 × 2) + ( 16 ÷ 2)

48 - 20 + 8 = 56

Therefore, the value of 48 - 10 × 2 + 16 ÷ 2 is 56

(e) 6 × 3 - 4 × 8 × 5

The terms in this expression are: ( 6 × 3) and ( -4 × 8 × 5)

Value 6 × 3 + ( -4 × 8 × 5)

18 - 160 = -142

Therefore, the value of 6 × 3 - 4 × 8 × 5 is -142

Question 2

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

Solution :

( a ) 89 + 21 - 10

A librarian was counting books in the children's section. She found 89 storybooks on one shelf and 21 picture books on another.
Later that day, 10 books were checked out by students.

How many books remain in the children's section?

Value: 89 + 21 − 10

110 − 10 = 100

There are 100 books remaining in the children's section.

( b ) 5 × 12 - 6

At a local bakery, a baker prepared 5 trays of cookies. Each tray had exactly 12 cookies.
Unfortunately, while moving them to the cooling rack, 6 cookies accidentally slipped off one of the trays and broke.

How many cookies were still perfect and ready to be sold?

Value: 5 × 12 -6

60 − 6 = 54

There were 54 perfect cookies ready to be sold.

(c) 4 × 9 + 2 × 6

For a charity event, volunteers were packing gift bags. They packed 4 large boxes, and each box contained 9 small toys.
They also packed 2 smaller boxes, with 6 candy bars in each.

What's the total number of items (toys and candy bars) they packed for the gift bags?

Value: 4 × 9 + 2 × 6

36 + 12 = 48

They packed a total of 48 items for the gift bags.

Question 3

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

Solution :

( 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.

The total number of gold coins Princess Elsa have : ( 100 × 2 )

The total number of gold coins Princess Anna have : ( 100 ÷ 2 )

The expression describing the total number of gold coins :

( 100 × 2 ) + ( 100 ÷ 2 )

Value of the Expression :

200 + 50 = 250

Therefore, Princess Elsa and Princess Anna together have 250 gold 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?
(ii) for two groups having three adults each?

(i) For four adults and three children :

Expression: (4 × 40 ) + ( 3 × 20)

Value of the Expression :

160 + 60 = 220

The total cost of tickets for four adults and three children is ₹220

(ii) For two groups having three adults each

Expression: ( 2 × 3 × 40)

Value of the Expression :

6 × 40 = 240

The total cost of tickets for two groups having three adults each is ₹240.

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

By observing the given picture:

Border = 3 cm

Grill = 2 cm

Gap = 5 cm

the total height of the window = number of gaps × 5 cm + number of grills × 2 cm + number of borders × 3 cm

Expression: 7 × 5 + 6 × 2 + 2 × 3

The terms in this expression are: (7 × 5) ( 6 × 2) ( 2 × 3)

Value: 35 + 12 + 6

= 47 + 6 = 53

Therefore, total height of the window is 53 cm.

Question 1

Go through the example (in the first column) and fill the blanks, doing as little computation as possible.

Solution :

Question 1

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

Solution :

(a) 24 + (6 – 4) = 24 + 6 [_] __

24 + (6 – 4) = 24 + 6 - 4

(b) 38 + ( _ [_] _ ) = 38 + 9 – 4

38 + ( 9 – 4) = 38 + 9 – 4

(c) 24 - (6 + 4) = 24 [_] 6 – 4

24 – (6 + 4) = 24 – 6 – 4

(d) 24 – 6 – 4 = 24 – 6 [_] __

24 – 6 – 4 = 24 – 6 – 4

(e) 27 – (8 + 3) = 27 [_] 8 [_] 3

27 - (8 + 3) = 27 - 8 - 3

(f ) 27– (____[_]______) = 27 – 8 + 3

27 - (8 - 3) = 27 - 8 + 3

Question 2

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

Solution :

(a) 14 + (12 + 10)

( When a plus sign precedes a bracket, the signs of the terms inside remain unchanged )

= 14 + 12 + 10

= 14 + 22 = 36

(b) 14 – ( 12 + 10 )

( When a minus sign precedes a bracket, the signs of the terms inside are reversed )

= 14 - 12 - 10

= 14 - 22 = -8

(c) 14 + ( 12 – 10)

( When a plus sign precedes a bracket, the signs of the terms inside remain unchanged )

= 14 + 12 - 10

= 14 + 2 = 16

(d) 14 – ( 12 – 10)

( When a minus sign precedes a bracket, the signs of the terms inside are reversed )

= 14 - 12 + 10

= 14 - 2 = 12

(e) -14 + 12 – 10

( This expression does not contain any brackets, so it remains the same )

= - 2 - 10

= -12

(f) 14 – (-12 – 10)

( When a minus sign precedes a bracket, the signs of the terms inside are reversed )

= 14 + 12 + 10

= 14 + 22 = 36

Question 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?

Solution :

(a)(6 + 10) - 2 and 6 + (10 - 2)

Initial Guess: these expressions will have the same value. This looks like an illustration of the associative property for addition.

(6 + 10) - 2 = 16 - 2 = 14

6 + (10 - 2) = 6 + 8 = 14

( When a plus sign precedes a bracket, the signs of the terms inside remain unchanged )

Yes, both expressions are equal to 14

(b) 16 - ( 8 - 3) and ( 16 - 8) - 3

Initial Guess: these expressions will not have the same value. Subtraction is not associative.

16 - ( 8 - 3)
= 16 - 8 + 3 = 16 - 5 = 11

( When a minus sign precedes a bracket, the signs of the terms inside are reversed )

( 16 - 8) - 3 = 8 - 3 = 5

No, the expressions are not equal (11 ≠ 5).

(c) 27 - (18 + 4) and 27 + (-18 - 4)

Initial Guess: these expressions will have the same value. The second expression looks like the result of distributing the negative sign in the first expression.

27 - (18 + 4)
= 27 - 18 - 4 = 27 - 22 = 5

( When a minus sign precedes a bracket, the signs of the terms inside are reversed )

27 + (-18 - 4) = 27 -22 = 5

Yes, both expressions are equal to 5.

Question 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.

Solution :

(a) 319 + 537, 319 – 537, –537 + 319, 537 – 319

319 + 537 is a sum of two positive numbers.

319 –537 is equivalent to 319 + (−537). This will be a negative value.

−537 + 319 is the same as 319 + (−537) due to the commutative property of addition .

(the order of addends doesn't change the sum)

537 – 319 is a difference of two positive numbers, resulting in a positive value.

Therefore, the expressions with the same value are:
319 – 537 and −537 + 319

(b) 87 + 46 – 109, 87 + 46 – 109,
87 + 46 – 109, 87 – 46 + 109,
87 – (46 + 109), (87 – 46) +109

87 + 46 – 109.

87 + 46 – 109.

87 + 46 – 109.

These three expressions are identical expressions, thus they certainly have the same value.

87–( 46 + 109)

87 −46 −109.

( When a minus sign precedes a bracket, the signs of the terms inside are reversed )

This is distinct from 87 + 46 − 109.

87 – (46 + 109)

This expression is equivalent to 87 −46 + 109

(87 – 46) + 109

This expression is equivalent to 87 −46 + 109

Here, it just specifies the order of operations, but it results in the same sequence of operations as 87−46 + 109.

Therefore, the expressions with the same value are:
87 + 46 – 109 (all three occurrences)

87 – 46 + 109 and (87 – 46) + 109

Question 5

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

Solution :

(a) 34 – 9 + 12 = 13

We need to group the addition of 9 and 12 first.

34 – ( 9 + 12)

(First, calculate the sum inside the brackets then substitute this value back into the expression)

34 – 21 = 13

Therefore, the correct expression is 34 – (9 + 12) = 13

(b) 56 – 14 – 8 = 34

We need to group the substraction of 56 and 14 first.

( 56 – 14 ) – 8

42 – 8 = 34

Therefore, the correct expression is (56 – 14) – 8 = 34

(c) –22 – 12 + 10 + 22 = – 22

We need to group some terms to make the subsequent operations result in -22

–22 – ( 12 + 10 ) + 22

(First, calculate the sum inside the brackets then substitute this value back into the expression)

–22 – 22 + 22

–44 + 22 = –22

Therefore, the correct expression is –22 – (12 + 10) + 22 = –22.

Question 6

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

Solution :

(a) 423 + ____ = 419 + ____

This is a great exercise in understanding how changes to one part of an equation must be balanced by changes to another part to maintain equality.

The first term on the left, $423$, is $4$ more than the first term on the right, $419$ ($423 - 419 = 4$).

To keep the sums equal, the second term on the left must be $4$ less than the second term on the right.

If we choose $\mathbf{1}$ for the first blank, we must choose $1 + 4 = \mathbf{5}$ for the second blank.

$( 423 + 1 = 424 )$ = $ ( 419 + 5 = 424 )$

(b) 207 – 68 = 210 – ____

The first term on the right, $210$, is $3$ more than the first term on the left, $207$ ($210 - 207 = 3$).

Since we added $3$ to the starting number ($207 \rightarrow 210$), we must also add $3$ to the number being subtracted to keep the final result the same.

Therefore, the blank must be $68 + 3 = \mathbf{71}$.

$( 207 – 68 = 139 )$ = $ ( 210 – 71 = 139)$

Question 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.

Solution :

Let’s find all possible different values that can be made using 2, 3, and 5, the operators ‘+’ and ‘–’, and brackets wherever needed.

Try simple linear forms (no brackets)

$ 3 − 2 + 5 = 6 $

$ 2 −3 − 5 = −6 $

$ 3 − 2 −5 = −4 $

$ 2 + 3 + 5 = 10$

$ 2 + 3 − 5 = 0 $

$ 2 − 3 + 5 = 4 $

$ 5 − 2 + 3 = 6 $ (duplicate)

$ 5 + 2 + 3 = 10 $ (duplicate)

$ 5 - 2 - 3 = 0 $ (duplicate)

So far, distinct values =$ {10, 0, 4, -6, 6, -4} $

Use brackets to change order of operations

$2 - (3 + 5) = -6 $

$3 - (2 + 5) = -4 $

$ 2 + ( 3 − 5)= 0 $

$ 3 − ( 2 − 5) = 6 $

$ ( 5 + 3) + 2 = 10$

Using 2, 3, 5 with +, –, and brackets, you can generate 6 different values:

=$ {10, 0, 4, -6, 6, -4} $

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

Solution :

(a) Does she always get the correct answer?

Yes, Jasoda always gets the correct answer.

Reason: Subtracting 9 is the same as subtracting 10 and then adding 1.

$ N - 9$ = $ N - 10 + 1 $

Example:

$$36 - 9 = 27$$

and

$$(36 - 10) + 1 = 27$$

$$\text{L.H.S} = \text{R.H.S}$$

(b) Other similar strategies

Here are some other similar strategies using addition, subtraction, and multiplication:

1: Adding a Near-Multiple of 10

Reason: To add $9$, you can add $10$ and subtract $1$.

Example:

$47 + 9$

Strategy: $47 + 10 - 1$

Calculation: $57 - 1 = \mathbf{56}$

2. Subtracting a Near-Multiple of 10

Reason: To subtract $19$, you can subtract $20$ and add $1$.

Example:

$72 - 19$

Strategy: $72 - 20 + 1$

Calculation: $52 + 1 = \mathbf{53}$

3. Multiplying by a Near-Multiple of 10

Reason: To multiply by $9$, you can multiply by $10$ and subtract the original number once

Example:

$14 \times 9$

Strategy: $14 \times 10 - 14 \times 1$

Calculation: $140 - 14 = \mathbf{126}$

Multiplying by a Near-Multiple of 10

Reason: To multiply by $11$, you can multiply by $10$ and add the original number once

Example:

$14 \times 11$

Strategy: $14 \times 11 + 14 \times 1$

Calculation: $140 + 14 = \mathbf{154}$

Question 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)
(b) 73 – (14 – 1).
(c) 73 + (–14 + 1)
(d) 73 + (–14 – 1)

Solution :

1. Expressions Equal to : $73 - 14 + 1$

The expression $73 - 14 + 1$ simplifies to $59 + 1 = \mathbf{60}$.

The following collection expressions are equal to it:

(b) $73 - (14 - 1)$

Reasoning: $14 - 1 = 13$, so the expression is $73 - 13 = \mathbf{60}$.

(c) $73 + (-14 + 1)$

Reasoning: Adding a sum, $+(-14 + 1)$, is the same as removing the parentheses: $73 - 14 + 1$. The value is $\mathbf{60}$.

2. Expressions Equal to : $73 - 14 - 1$

The expression $73 - 14 - 1$ simplifies to $59 - 1 = \mathbf{58}$.

The following collection expressions are equal to it:

(a) $73 - (14 + 1)$

Reasoning: $14 + 1 = 15$, so the expression is $73 - 15 = \mathbf{58}$..

(d) $73 + (-14 - 1)$

Reasoning: Adding a sum, $+(-14 - 1)$, is the same as removing the parentheses: $73 - 14 - 1$. The value is $\mathbf{58}$.

Question 1

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 :

The expression for the amount Lhamo and Norbu will have to pay is:

$2 \times (\text{Cost of a veg.cutlet}$ + $ \text{Cost of a rasgulla})$

Using the given prices:

$${2 \times (43 + 24)}$$

$$ \text{ Total Cost =} \text{ ₹134}.$$

Question 2

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

Solution :

For two people, total cost :

$2 \times (\text{Cost of a veg.cutlet}$ + $ \text{Cost of a rasgulla})$

Using the given prices:

$${2 \times (43 + 24)}$$

$$ \text{ Total Cost =} \text{ ₹134}.$$

No, the expression $2 \times 43 + 24$ does not correctly describe the total amount they have to pay.

This expression calculates:

The cost of two vegetable cutlets ($2 \times 43 = \text{₹86}$)

The cost of one rasgulla ($\text{₹24}$).

$\text{Incorrect Calculation} = \text{₹86} + \text{₹24} = \text{₹110}$

The correct expression must account for two cutlets and two rasgullas. This can be written in two equivalent ways:

Grouping by Person: $2 \times (43 + 24)$

Grouping by Item: $(2 \times 43) + (2 \times 24)$

Question 3

If another friend, Sangmu, joins them and orders the same items, what will be the expression for the total amount to be paid? [Expression for Lhamo and Norbu who each ordered a vegetable cutlet costing ₹43 and a rasgulla costing ₹24 is 2 × (43 + 24)]

Solution :

When Sangmu joins Lhamo and Norbu, there are now 3 people in total, and each one orders the same items (one vegetable cutlet and one rasgulla).

$3 \times (\text{Cost of a veg.cutlet}$ + $ \text{Cost of a rasgulla})$

Using the given prices:

$$ \text{ Expression = } {3 \times (43 + 24)}$$

here, $(43 + 24)$ is the total cost for one person ($\text{₹43}$ for the cutlet and $\text{₹24}$ for the rasgulla).

$$ \text{ Total Cost =} \text{ ₹ 201}.$$

Question 1

Use this method to find the following products:

(а) 95 × 8
(b) 104 × 15
(c) 49 × 50

Is this quicker than the multiplication procedure you use generally?

Solution :

(а) 95 × 8

We can write $95$ as $(100 - 5)$.

Using the given prices:

$$ {95 \times 8 }= {(100 - 5) \times 8 }$$

$$ {= (100 \times 8)} - {(5 \times 8) }$$

( Distributive Property )

$$ {= 800 - 40 }$$

$$ \text{ = 760}$$

(b) 104 × 15

We can write $104$ as $(100 + 4)$.

Using the given prices:

$$ {104 \times 15 }= {(100 + 4) \times 15 }$$

$$ {= (100 \times 15)} + {(4 \times 15) }$$

( Distributive Property )

$$ {= 1500 + 60 }$$

$$ \text{ = 1560}$$

(c) 49 × 50

We can write $49$ as $(50 - 1)$.

Using the given prices:

$$ {49 \times 50 }= {(50 - 1) \times 50 }$$

$$ {= (50 \times 50)} - {(1 \times 50) }$$

( Distributive Property )

$$ {= 2500 - 50 }$$

$$ \text{ = 2450}$$

Question 2

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

Solution :

Products that are quicker to find using the distributive property (the method of breaking up numbers) are those where one of the factors is close to a power of ten (like 10, 100, 1000) or a multiple of ten (like 20, 50, 200).

Example 1:

$99 \times 12$

$(100 - 1) \times 12 = (100 \times 12) - (1 \times 12)$

$ \text{ = 1188}.$

Example 2:

$19 \times 35$

$(20 - 1) \times 35 = (20 \times 35) - (1 \times 35)$

$700 - 35$

$ \text{ = 665}.$

Question 1

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

(а) 3 × (6 + 7) = 3 × 6 + 3 × 7
(b) (8 + 3) × 4 = 8 × 4 + 3 × 4
(c) 3 × (5 + 8) = 3 × 5 ☐ 3 × ____
(d) (9 + 2) × 4 = 9 × 4 ☐2 × ____
(e) 3 × (____ + 4) = 3 ____ + ____
(f) (____ + 6) × 4 = 13 × 4 + ____
(g) 3 × ( ____ + ____ ) = 3 × 5 + 3 × 2
(h) ( ____ + ____ ) × ____ = 2 × 4 + 3 × 4
(i) 5 × (9 – 2) = 5 × 9 – 5 × ____
(j) (5 – 2) × 7 = 5 × 7 – 2 × ____
(k) 5 × (8 – 3) = 5 × 8 ☐ 5 × ____
(l) (8 – 3) × 7 = 8 × 7 ☐ 3 × 7
(m) 5 × (12 – ____ ) = ____ ☐ 5 × ____
(n) (15 – ____ ) × 7 = ____ ☐ 6 × 7
(o) 5 × ( ____ + ____ ) = 5 × 9 – 5 × 4
(p) ( ____ + ____ ) × ___ = 17 × 7 – 9 × 7

Solution :

The blanks and boxes are filled based on the Distributive Property of multiplication over addition and subtraction.

The Distributive Property of Multiplication over Addition states:

$$a \times (b + c) = (a \times b) + (a \times c)$$

The Distributive Property of Multiplication over Subtraction states:

$$a \times (b - c) = (a \times b) - (a \times c)$$

Distributive Property over Addition

(а) $3 \times (6 + 7) = 3 \times 6 + 3 \times 7$
(b) $(8 + 3) \times 4 = 8 \times 4 + 3 \times 4$
(c) $3 \times (5 + 8) = 3 \times 5 \mathbf{+} 3 \times \mathbf{8}$
(d) $(9 + 2) \times 4 = 9 \times 4 \mathbf{+} 2 \times \mathbf{4}$
(e) $3 \times (\mathbf{5} + 4) = 3 \mathbf{\times} \mathbf{5} + \mathbf{3 \times 4}$
(f) $(\mathbf{13} + 6) \times 4 = 13 \times 4 + \mathbf{6 \times 4}$
(g) $3 \times (\mathbf{5} + \mathbf{2}) = 3 \times 5 + 3 \times 2$
(h) $(\mathbf{2} + \mathbf{3}) \times \mathbf{4} = 2 \times 4 + 3 \times 4$

Distributive Property over Subtraction

(i)$5 \times (9 - 2) = 5 \times 9 - 5 \times \mathbf{2}$
(j) $(5 - 2) \times 7 = 5 \times 7 - 2 \times \mathbf{7}$
(k) $5 \times (8 - 3) = 5 \times 8 \mathbf{-} 5 \times \mathbf{3}$
(l) $(8 - 3) \times 7 = 8 \times 7 \mathbf{-} 3 \times 7$
(m) $5 \times (12 - \mathbf{4}) = \mathbf{5 \times 12} \mathbf{-} 5 \times \mathbf{4}$
(n) $(15 - \mathbf{6}) \times 7 = \mathbf{15 \times 7} \mathbf{-} 6 \times 7$
(o) $5 \times (\mathbf{9} - \mathbf{4}) = 5 \times 9 - 5 \times 4$
(p) $(\mathbf{17} - \mathbf{9}) \times \mathbf{7} = 17 \times 7 - 9 \times 7$

Question 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) x 29 ____ (3 – 8) x 29
(b) 15 + 9 x 18 ____ (15 + 9) x 18
(c) 23 x (17 – 9) ____ 23 x 17 + 23 x 9
(d) (34 – 28) x 42 ____ 34 x 42 – 28 x 42

Solution :

(a) (8 – 3) × 29 ____ (3 – 8) × 29

Reasoning :

Expression (LHS) $\mathbf{(8 - 3)}$ is positive ($+5$).

Expression (RHS) $\mathbf{(3 - 8)}$ is negative ($-5$).

Since $29$ is positive, a positive number multiplied by $29$ will be greater than a negative number multiplied by $29$.

∴ (8 – 3) × 29 (>) (3 – 8) × 29.

(b) 15 + 9 × 18 ____ (15 + 9) × 18

Reasoning :

Order of Operations (PEMDAS/BODMAS) dictates that in the LHS, multiplication ($9 \times 18$) is performed before addition (adding $15$).

In the RHS, the parentheses force the addition ($\mathbf{15 + 9}$) to be done first, and the entire sum is then multiplied by $18$.

Since $15$ is positive, multiplying the entire sum by $18$ will yield a larger result than adding $15$ to only one of the factors multiplied by $18$.

∴ 15 + 9 × 18 (<) (15 + 9) × 18.

(c) 23 × (17 – 9) ____ 23 × 17 + 23 × 9

Reasoning :

This relates to the Distributive Property. The LHS, $23 \times (17 – 9)$, is equal to $23 \times 17 \mathbf{-} 23 \times 9$.

The RHS, $23 \times 17 \mathbf{+} 23 \times 9$, is an addition,

which will be larger than the subtraction.

∴ 23 × (17 – 9)(<) 23 × 17 + 23 × 9

(d) (34 – 28) × 42 ____ 34 × 42 – 28 × 42

Reasoning :

This is the Distributive Property of multiplication over subtraction. The property states that $a \times (b - c) = a \times b - a \times c$.

Here, the LHS is in the form $ (b - c) \times a$ and the RHS is $b \times a - c \times a$, which are equal due to the Commutative Property of Multiplication and the Distributive Property.

∴ (34 – 28) × 42 = 34 × 42 – 28 × 42

Question 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
(b) ________ × (________ + ________) = 14
(c) ________ × (________ + ________) = 14
(d) ________ × (________ + ________) = 14

Solution :

The key is to use the factors of 14, which are 1, 2, 7, and 14, for the term outside the parentheses.

(a) $\mathbf{1} \times (\mathbf{7} + \mathbf{7}) = 14$

Reasoning :

$7 + 7 = 14$, and $1 \times 14 = 14$.

(b) $\mathbf{1} \times (\mathbf{10} + \mathbf{4}) = 14$

Reasoning :

$10 + 4 = 14$, and $1 \times 14 = 14$. (You can use any two numbers that add up to 14 here, like $8+6$ or $5+9$).

(c) $\mathbf{2} \times (\mathbf{5} + \mathbf{2}) = 14$

Reasoning :

$5 + 2 = 7$, and $2 \times 7 = 14$. (The number in the parentheses must sum to $14 \div 2 = 7$).

(d) $\mathbf{7} \times (\mathbf{1} + \mathbf{1}) = 14$

Reasoning :

$1 + 1 = 2$, and $7 \times 2 = 14$. (The number in the parentheses must sum to $14 \div 7 = 2$).

Question 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.

Solution :

(i) Sum:: The numbers are: five 4s and four 8s.

Sum = $ (5 \times 4) + (4 \times 8) = 20 + 32 = \mathbf{52} $

Also, Sum = $(5 + 8) \times 4 = 13 \times 4 = \mathbf{52}$

(ii) Sum:: The numbers are: Eight 5s and Eight 6s.

Sum = $ (8 \times 5) + (8 \times 6) = 40 + 48 = \mathbf{88} $

Also, Sum = $ 8 \times (5 + 6) = 8 \times (11) = \mathbf{88} $

Question 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 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.

(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 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 top. In how many days will the snail get the treat?

Solution :

(a) Mango Supply in a Week

First, Find the total daily supply, then Multiply by Days

Total Daily Supply = $9 \text{ kg} + 11 \text{ kg} = 20 \text{ kg}$.

Multiply the daily supply by the number of days in a week:

$20 \times 7 = \mathbf{140 \text{ kg}}$.

The total number of mangoes supplied by them in a week is $\mathbf{140 \text{ kg}}$.

(b) Binu's Yearly Savings

First, Calculate Monthly Savings, then Multiply by Months

Binu’s total monthly expenditures =

$\text{₹} 5,000 \text{ (rent)} + \text{₹} 5,000 \text{ (food)}$ $+ \text{₹} 2,000 \text{ (others)} = \text{₹} 12,000$.

Monthly Savings:

$\text{₹} 20,000 \text{ (income)} - \text{₹} 12,000 \text{ (expenses)} = \text{₹} 8,000$

Yearly Savings:

$\text{₹} 8,000 \text{/month} \times 12 \text{ months} = \mathbf{\text{₹} 96,000}$.

The amount Binu will save by the end of the year is ₹96,000.

(c) The Climbing Snail

The snail's net progress each full day/night cycle is $3 \text{ cm}$ up $- 2 \text{ cm}$ down $= 1 \text{ cm}$.

∴ The distance climbed in 7 days = 7 cm

The height of the post is 10 cm.

The final day's climb (Day 8): On the morning of Day 8, the snail climbs $3 \text{ cm}$

$7 \text{ cm} + 3 \text{ cm} = 10 \text{ cm}$. The snail reaches the treat.

The snail will get the treat in $\mathbf{8 \text{ days}}$.

Question 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

Solution :

Total Stories in 8 Weeks

Melvin Reads per Week

Reading days per week except Tuesday and Saturday:

$7 - 2 = 5$ days

He reads one story (which is two pages) per reading day

Stories per week: $5 \text{ days/week} \times 1 \text{ story/day} = 5$ stories.

Total Stories in 8 Weeks

$5 \text{ stories/week} \times 8 \text{ weeks} = \mathbf{40}$ stories

Stories completed in 8 weeks are represented by the expressions given below:

(b) (7 – 2) × 8

(g) 7 × 8 – 2 × 8.

Question 3

Find different ways of evaluating the following expressions:

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

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

Solution :

There are several different ways to evaluate the given expressions by grouping terms.

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

Method 1: Direct Sequential Calculation

$$\begin{aligned} &1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10 \\ &= (1 - 2) + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10 \\ &= -1 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10 \\ &= (-1 + 3) - 4 + 5 - 6 + 7 - 8 + 9 - 10 \\ &= 2 - 4 + 5 - 6 + 7 - 8 + 9 - 10 \\ &= -2 + 5 - 6 + 7 - 8 + 9 - 10 \\ &= 3 - 6 + 7 - 8 + 9 - 10 \\ &= -3 + 7 - 8 + 9 - 10 \\ &= 4 - 8 + 9 - 10 \\ &= -4 + 9 - 10 \\ &= 5 - 10 \\ &= \mathbf{-5}\end{aligned}$$

Method 2: Grouping into Pairs

Group the consecutive positive and negative terms into pairs.

$$\begin{aligned} &(1 - 2) + (3 - 4) + (5 - 6) + (7 - 8) + (9 - 10) \\ &= (-1) + (-1) + (-1) + (-1) + (-1) \\ &= 5 \times (-1) \\ &= \mathbf{-5}\end{aligned}$$

Method 3: Grouping Positive and Negative Terms

Separate the positive terms and the negative terms, sum each group, and then subtract

$(1 + 3 + 5 + 7 + 9) - (2 + 4 + 6 + 8 + 10)$

$$\begin{aligned} & (1 + 3 + 5 + 7 + 9) - (2 + 4 + 6 + 8 + 10) \\ &= (25) - (30) \\ &= \mathbf{-5}\end{aligned}$$

(b) $1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1$

Method 1: Direct Sequential Calculation

$$\begin{aligned} &(1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) \\ &= 0 + 0 + 0 + 0 + 0 \\ &= \mathbf{0}\end{aligned}$$

Method 2: Grouping into Pairs

Group the consecutive positive and negative terms into pairs.

$$\begin{aligned} & (1 + 1 + 1 + 1 + 1) - (1 + 1 + 1 + 1 + 1) \\ &= (5) - (5) \\ &= \mathbf{0}\end{aligned}$$

Method 3: Using Multiplicative Grouping

Represent the sums of the identical terms using multiplication

$$\begin{aligned} & (5 \times 1) - (5 \times 1) \\ &= 5 - 5 \\ &= \mathbf{0}\end{aligned}$$

Question 4

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

(a) 49 – 7 + 8 ____ 49 – 7 + 8
(b) 83 x 42 – 18 ____ 83 x 40 – 18
(c) 145 – 17 x 8 ____ 145 – 17 x 6
(d) 23 x 48 – 35 ____ 23 x (48 – 35)
(e) (16 – 11) x 12 ____ -11 x 12 + 16 x 12
(f) (76 – 53) x 88 ____ 88 x (53 – 76)
(g) 25 x (42 + 16) ____ 25 x (43 + 15)
(h) 36 x (28 – 16) ____ 35 x (27 – 15)

Solution :

The expressions are compared below using $<$, $>$, or $=$ along with the reasoning.

(a) $49 - 7 + 8$ vs $49 - 7 + 8$

$$49 - 7 + 8 = 42 + 8 = 50$$

Both expressions are identical, so they are equal.

$$49 - 7 + 8 \mathbf{=} 49 - 7 + 8$$

(b) $83 \times 42 - 18$ vs $83 \times 40 - 18$

Reasoning:

We are comparing two expressions that are both in the form (Some Number) $\times$ (Factor) $- 18$.

Since $83 \times 42$ will be greater than $83 \times 40$ (because $42 > 40$ and $83$ is positive), and we are subtracting the same number ($18$) from both products, the first expression must be greater.

$$83 \times 42 - 18 \mathbf{>} 83 \times 40 - 18$$

(c) $145 - 17 \times 8$ vs $145 - 17 \times 6$

Reasoning:(Order of Operations - PEMDAS/BODMAS)

We are comparing two expressions that are both in the form $145 -$ (Some Product).

The product $\mathbf{17 \times 8}$ is greater than the product $\mathbf{17 \times 6}$ (since $8 > 6$ and $17$ is positive).

When you subtract a larger number ($17 \times 8$) from $145$, the result will be smaller than if you subtract a smaller number ($17 \times 6$) from $145$.

$$145 - 17 \times 8 \mathbf{<} 145 - 17 \times 6$$

(d) $23 \times 48 - 35$ vs $23 \times (48 - 35)$

Reasoning (Distributive Property):

The Distributive Property states that $a \times (b - c) = a \times b - a \times c$.

We are comparing $23 \times 48 - \mathbf{35}$ with $23 \times 48 - \mathbf{(23 \times 35)}$.

Since $23 \times 35$ is clearly much larger than $35$ ($23 \times 35 = 805$), subtracting $35$ will result in a larger number than subtracting $23 \times 35$.

$$23 \times 48 - 35 \mathbf{>} 23 \times (48 - 35)$$

(e) $(16 - 11) \times 12$ vs $-11 \times 12 + 16 \times 12$

Reasoning (Distributive Property):

The second expression, $-11 \times 12 + 16 \times 12$, can be factored using the Distributive Property, $a \times c + b \times c = (a + b) \times c$.

Factoring out $12$ gives: $(-11 + 16) \times 12$.

Since $-11 + 16 = 16 - 11$, the second expression is equal to $(16 - 11) \times 12$.

$(16 - 11) \times 12 \mathbf{=} -11 \times 12 + 16 \times 12$

(f) $(76 - 53) \times 88$ vs $88 \times (53 - 76)$

Reasoning :

Let $A = (76 - 53)$ and $B = (53 - 76)$

$A = 76 - 53 = 23$.

$B = 53 - 76 = -(76 - 53) = -23$.

Since $23 \times 88$ is a positive number and $88 \times (-23)$ is a negative number, the first expression must be greater than the second.

$(76 - 53) \times 88 \mathbf{>} 88 \times (53 - 76)$

(g) $25 \times (42 + 16)$ vs $25 \times (43 + 15)$

Reasoning :

We are comparing two expressions that are both in the form $25 \times (\text{sum})$. Since $25$ is positive, we only need to compare the two sums inside the parentheses.

First sum: $\mathbf{42 + 16} = 58$

Second sum: $\mathbf{43 + 15} = 58$

Since $42 + 16 = 43 + 15$, the products must be equal.

$25 \times (42 + 16) \mathbf{=} 25 \times (43 + 15)$

(h)$36 \times (28 - 16)$ vs $35 \times (27 - 15)$

Reasoning :

We will first calculate the value inside the parentheses for each expression.

First expression: $36 \times \mathbf{(28 - 16)} = 36 \times \mathbf{12}$.

Second expression: $35 \times \mathbf{(27 - 15)} = 35 \times \mathbf{12}$.

Since $36 > 35$ and we are multiplying by the same positive factor ($12$), the first expression must be greater.

$$36 \times (28 - 16) \mathbf{>} 35 \times (27 - 15)$$

Question 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 x 44 + 76
(ii) 93 + 37 x 76 + 44
(iii) (93 + 37) x (44 + 76)
(iv) 37 x 44 + 93 + 76

Solution :

(a) Given Expression: $83 - 37 - 12$

(i) $84 - 38 - 12$

We can rewrite the terms $84$ and $-38$ to see the relationship:

$84 - 38 - 12 = (83 + 1) - (37 + 1) - 12$

$ = 83 + 1 - 37 - 1 - 12$

$$\text{Result: } \mathbf{83 - 37 - 12}$$

Conclusion: (i) is equal to the given expression.

(ii) $84 – (37 + 12)$

We can rewrite the terms $84$ to see the relationship:

$84 – (37 + 12) = (83 + 1) - (37 + 12) $

$ = 83 + 1 - 37 - 12$

$$\text{Result: } = (83 - 37 - 12) + 1$$

Conclusion: Option (ii) is not equal to the given expression $83 - 37 - 12$..

(iii) $83 - 38 - 13$:

We can rewrite the terms $-13$ and $-38$to see the relationship:

$83 - (37 + 1) - (12 + 1) = 83 - 37 - 1 - 12 - 1$

$ = 83 - 37 - 12 - 2$

$$\text{Result: } = (83 - 37 - 12) - 2$$

Conclusion: Option (iii) is not equal to the given expression $83 - 37 - 12$..

(iv) $-37 + 83 - 12$:

By the Commutative Property of Addition, we can rearrange the terms:

$83 + (-37) + (-12) = 83 - 37 - 12$.

This expression is equal to the given expression.

Expressions equal to $83 - 37 - 12$ are (i) and (iv).

(b) Given Expression: $93 + 37 \times 44 + 76$

We must strictly follow the Order of Operations (PEMDAS/BODMAS): Multiplication and Division come before Addition and Subtraction.

Given Expression: $93 + (37 \times 44) + 76$. The terms being added are $93$, $(37 \times 44)$, and $76$.

(i) $37 + 93 \times 44 + 76$:

We can rewrite the terms :

$37 + (93 \times 44) + 76$

Since $37 \times 44$ is not equal to $93 \times 44$, and the rest of the terms are different, this expression is not equal.

(ii) $93 + 37 \times 76 + 44$:

We can rewrite the terms:

This expression is $93 + (37 \times 76) + 44$

Since $37 \times 44$ is not equal to $37 \times 76$, this expression is not equal.

(iii) $(93 + 37) \times (44 + 76)$:

The grouping by parentheses forces the additions to occur before the multiplication.

This expression is not equal (it dramatically changes the operation).

(iv) $37 \times 44 + 93 + 76$:

We can rewrite the terms :

$(37 \times 44) + 93 + 76$.

Since addition is Commutative (terms can be added in any order), we can rearrange the terms from the given expression: $93 + (37 \times 44) + 76 = (37 \times 44) + 93 + 76$.

This expression is equal to the given expression.

The expression equal to $93 + 37 \times 44 + 76$ is (iv).

Question 6

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

Solution :

Let’s choose the number 15.

Here are ten different expressions that give the value 15:

$3 \times 5$

Basic Multiplication

$30 \div 2$

Basic Division

$10 + 5$

Basic Addition

$25 - 10$

Basic Subtraction

$4 \times 4 - 1$

Multiplication and Subtraction

$(2 + 1) \times 5$

Order of Operations (Parentheses first)

$100 \div 10 + 5$

Division and Addition

$\sqrt{169} + 2$

Square Root and Addition

$5 \times (1 + 2)$

Distributive Property (or Parentheses first)

$10 + \sqrt{25}$

Square Root and Addition