Question 1

In the boxes below, fill in ‘<’, ‘>’ 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) $(8 - 3)	imes 29$ \quad $(3 - 8)	imes 29$
(b) $15 + 9 	imes 18$ \quad $(15 + 9) 	imes 18$
(c) $23 	imes (17 - 9)$ \quad $23 	imes 17 + 23 	imes 9$
(d) $(34 - 28)	imes 42$ \quad $34	imes 42 - 28	imes 42$

Accepted Answer

(a) > : since 8 - 3 > 3 - 8 (b) < (c) < (d) = — For (a), 8 - 3 = 5 and 3 - 8 = -5, so 5 × 29 > -5 × 29. For (b), 15 + 9 × 18 means 15 + (9 × 18) = 15 + 162 = 177, while (15 + 9) × 18 = 24 × 18 = 432, so 177 < 432. For (c), 23 × (17 - 9) = 23 × 8 = 184, while 23 × 17 + 23 × 9 = 391 + 207 = 598, so 184 < 598. For (d), (34 - 28) × 42 = 6 × 42 = 252, and 34 × 42 - 28 × 42 = (34 - 28) × 42 = 252, so they are equal.

Question 2

Here is one way to make 14: $2 	imes (1 + 6) = 14$. Are there other ways of getting 14? Fill them out below:

(a) $	imes (\quad + \quad)$ = 14
(b) $	imes (\quad + \quad)$ = 14
(c) $	imes (\quad + \quad)$ = 14
(d) $	imes (\quad + \quad)$ = 14

Accepted Answer

(a) $2 	imes (5 + 2) = 14$
(b) $2 	imes (3 + 4) = 14$
(c) $7 	imes (1 + 1) = 14$
(d) $2 	imes (6 + 1) = 14$ — By choosing different pairs of numbers inside the brackets and multiplying by the number outside, we get 14:
(a) 2 × (5 + 2) = 2 × 7 = 14
(b) 2 × (3 + 4) = 2 × 7 = 14
(c) 7 × (1 + 1) = 7 × 2 = 14
(d) 2 × (6 + 1) = 2 × 7 = 14

Question 3

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.

(I) [Image of a figure with numbers]

(II) [Image of a figure with numbers]

Accepted Answer

For I: Way 1:
(5 × 4) + (4 × 8) = 20 + 32 = 52
Way 2:
2 × (4 + 8 + 4) + (8 + 4 + 8) = 2 × 16 + 20 = 32 + 20 = 52

For II: Way 1:
(8 × 5) + (8 × 6) = 40 + 48 = 88
Way 2:
8 × (5 + 6) = 8 × 11 = 88

Students are encouraged to find more ways. — By grouping numbers differently and using distributive property, the sum can be calculated in multiple ways as shown above.

Question 4

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?

Accepted Answer

(a) Expression: 7 × (9 + 11) kg
Value: 7 × 20 = 140 kg

(b) Expression: 12 × 20,000 - 12 × (5,000 + 5,000 + 2,000)
= 240,000 - 12 × 12,000 = 240,000 - 144,000 = ₹96,000

(c) The snail climbs 3 cm during the day and slips 2 cm at night, so net climb per day = 1 cm.
In 7 days, it climbs 7 cm.
On the 8th day, it climbs 3 cm more, reaching 10 cm.
Therefore, the snail reaches the top on the 8th day. — For (a), total mangoes per day = 9 + 11 = 20 kg, for 7 days = 140 kg.
For (b), yearly income = 20,000 × 12 = 240,000; yearly expenses = (5,000 + 5,000 + 2,000) × 12 = 144,000; savings = 240,000 - 144,000 = 96,000.
For (c), net climb per day is 1 cm except on the last day when it reaches the top without slipping back.

Question 5

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) $5 	imes 2 	imes 8$
(b) $(7 - 2) 	imes 8$
(c) $8 	imes 7$
(d) $7 	imes 2 	imes 8$
(e) $7 	imes 5 - 2$
(f) $(7 + 2) 	imes 8$
(g) $7 	imes 8 - 2 	imes 8$
(h) $(7 - 5) 	imes 8$

Accepted Answer

Expressions describing the scenario (total stories) = (b) and (g).

Explanation:
Melvin reads on 7 - 2 = 5 days per week (excluding Tuesday and Saturday).
Number of stories in 8 weeks = 5 × 2 × 8 = 80 stories.
Expression (b) is (7 - 2) × 8 = 5 × 8 = 40 (number of reading days, not stories).
Expression (g) is 7 × 8 - 2 × 8 = 56 - 16 = 40 (number of reading days).
Since he reads 2 pages per story, total stories = 40 stories (if counting days), but the question asks for stories completed reading, so expressions (b) and (g) describe the scenario best. — Melvin reads 2 pages per day on 5 days a week (7 - 2 days). Over 8 weeks, total reading days = 5 × 8 = 40 days. Total stories = 40 stories (2 pages per story). Expressions (b) and (g) represent the number of reading days.

Question 6

Find different ways of evaluating the following expressions:

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

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

Accepted Answer

(a) Way 1:
$1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10 = (1 + 3 + 5 + 7 + 9) + (-2 - 4 - 6 - 8 - 10) = 25 - 30 = -5$

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

(b) Way 1:
$1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 = (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) = 0$

Way 2:
$= (1 + 1 + 1 + 1 + 1) + (-1 - 1 - 1 - 1 - 1) = 5 - 5 = 0$ — By grouping positive and negative terms differently, the expressions can be evaluated in multiple ways leading to the same result.

Question 7

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

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

Accepted Answer

(a) = (b) > (c) < (d) > (e) = (f) > (g) = (h) > — (a) Both sides are the same expression. (b) 83 × 42 - 18 > 83 × 40 - 18 because 42 > 40. (c) 145 - 17 × 8 < 145 - 17 × 6 because 17 × 8 > 17 × 6. (d) 23 × 48 - 35 > 23 × (48 - 35) because 23 × 48 - 35 = (23 × 48) - 35, which is greater than 23 × 13. (e) (16 - 11) × 12 = 5 × 12 = 60; -11 × 12 + 16 × 12 = (-132) + 192 = 60. (f) (76 - 53) × 88 = 23 × 88 = 2024; 88 × (53 - 76) = 88 × (-23) = -2024; so first > second. (g) 25 × (42 + 16) = 25 × 58 = 1450; 25 × (43 + 15) = 25 × 58 = 1450; equal. (h) 36 × (28 - 16) = 36 × 12 = 432; 35 × (27 - 15) = 35 × 12 = 420; so first > second.

Question 8

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 that 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) $93 + 37 	imes 44 + 76$
(i) $37 + 93 	imes 44 + 76$
(ii) $93 + 37 	imes 76 + 44$
(iii) $(93 + 37) 	imes (44 + 76)$
(iv) $37 	imes 44 + 93 + 76$

Accepted Answer

(a) Expressions equal to $83 - 37 - 12$ are (i) and (iv).
(i) $84 - 38 - 12$ is equal because 84 - 38 = 46, 46 - 12 = 34; 83 - 37 = 46, 46 - 12 = 34.
(iv) $-37 + 83 - 12$ rearranged is $83 - 37 - 12$.

(b) Expression equal to $93 + 37 	imes 44 + 76$ is (iv).
(iv) $37 	imes 44 + 93 + 76$ is just rearranged terms of the original expression. — By rearranging terms and using associative and commutative properties, expressions (i) and (iv) in (a) and (iv) in (b) are equal to the given expressions.

Question 9

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

Accepted Answer

Let's choose the number 26. Here are ten different expressions with the value 26:

1. $10 + 16$
2. $30 - 4$
3. $13 	imes 2$
4. $\frac{52}{2}$
5. $5 + (3 	imes 7)$
6. $(5 	imes 5) + 1$

Students are encouraged to try for other numbers and create their own expressions. — By using addition, subtraction, multiplication, and division in different combinations, multiple expressions can be formed that equal the chosen number.

Question 10

What is arithmetic in mathematics?

Accepted Answer

The branch of mathematics dealing with numbers and basic operations like addition, subtraction, multiplication, and division — Arithmetic is a branch of mathematics that focuses on numbers and the basic operations performed on them, such as addition, subtraction, multiplication, and division. It forms the foundation for many other mathematical topics.

Question 11

Which of the following is NOT a property of arithmetic operations?

Accepted Answer

Reflective property — The main properties of arithmetic operations are commutative, associative, distributive, identity, and inverse properties. Reflective property is not a property of arithmetic operations.

Question 12

Which property of addition states that changing the order of numbers does not change the sum?

Accepted Answer

Commutative property — The commutative property of addition states that a + b = b + a, meaning the order of numbers can be changed without affecting the sum.

Question 13

Explain the associative property of multiplication with an example.

Accepted Answer

The associative property of multiplication states that the way numbers are grouped does not affect the product. For example, (2 × 3) × 4 = 2 × (3 × 4). Both equal 24. — The associative property means that when multiplying three or more numbers, the grouping of numbers does not change the result. For instance, (2 × 3) × 4 = 6 × 4 = 24, and 2 × (3 × 4) = 2 × 12 = 24, so both sides are equal.

Question 14

State and explain the distributive property of multiplication over addition.

Accepted Answer

The distributive property states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the results. For example, a × (b + c) = a × b + a × c. — The distributive property connects multiplication and addition. It shows that to multiply a number by a sum, you can multiply the number separately by each addend and then add the products. For instance, 3 × (4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27.

Question 15

Which number is the identity element for multiplication?

Accepted Answer

1 — The identity element for multiplication is 1 because any number multiplied by 1 remains unchanged, e.g., a × 1 = a.

Arithmetic

Arithmetic — Study Notes

2.1 Simple Expressions

2.2 Reading and Evaluating Complex Expressions

Terms in Expressions and Properties of Addition

Practice Questions — Arithmetic

All 8 Chapters in Ganita Prakash