Hint: Apply Euclid Division Algorithm in order to obtain HCF of 2 given numbers in the context of the given problem.

Question.1. A worker needs to pack 350 kg of rice and 150 kg of wheat in bags such that each bag weighs the same.
Each bag should either contain rice or wheat. Which option shows the correct steps to find the greatest amount of rice/wheat the worker can pack in each bag?

(a)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 3(50) + 0
Step 3: Greatest amount = 50 kg
(b)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 2(50) + 0
Step 3: Greatest amount = 50 kg
(c)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 3(50) + 0
Step 3: Greatest amount = 150 kg
(d)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 2(50) + 0
Step 3: Greatest amount = 150 kg

Question.2. Pranay wants to stack few one-rupee coins and some five-rupee coins in such a way that:

a. Each stack has the same number of coins.
b. There is least number of stacks.
c. Each stack either has one rupee or five-rupee coins.
d. No coins are left over after creating stacks.

His first step to find the number of coins that should be in each stack is 195=1(180)+15. Given that he has more of five-rupee coins than one-rupee coins, how many one-rupee coins stack can he make?

(a) 12
(b) 13
(c) 15
(d) 25

Ans.1. (a)
Step 1: 350 = 2(150) + 50
Step 2: 150 = 3(50) + 0
Step 3: Greatest amount = 50 kg
Ans.2. (a) 12

Hint: Apply Euclid Division Algorithm in order to prove results of positive integers in the form of ax+b where a and b are constants.

Question.3. Given that p is a non-negative integer, which of these gives positive integers that are multiple of 5?

(a) 10p and 10p+2 .
(b) 10p and 10p+3 .
(c) 10p and 10p+4 .
(d) 10p and 10p+5 .

Question.4. In the equation below, a, b, q, r are integers, 0 \leq r <b , a is a multiple of 3 and b=9.
a=bq+r
Which of the following forms represent a?

(a) Only 9q and 9q+3 , as only these forms when divided by 3 gives r=0 .
(b) Only 9q+1 and 9q+4 , as only these forms when divided by 3 gives r=1 .
(c) Only 9q , 9q+3 , and 9q+6 , as only these forms when divided by 3 gives r=0 .
(d) Only 9q+1 , 9q+4 , and 9q+7 , as only these forms when divided by 3 gives r=1 .

Ans.3. (d) 10p and 10p+5 .
Ans.4. (c) Only 9q , 9q+3 , and 9q+6 , as only these forms when divided by 3 gives r=0.

Hint: Use the Fundamental Theorem of Arithmetic in order to calculate HCF and LCM of the given numbers in the context of the given problem.

Question.5. Rahul has 40 cm long red and 84 cm long blue ribbon. He cuts each ribbon into pieces such that all pieces are of equal length. What is the length of each piece?

(a) 4 cm as it is the LCM of 40 and 84
(b) 4 cm as it is the HCF of 40 and 84
(c) 8 cm as it is the LCM of 40 and 84
(d) 8 cm as it is the HCF of 40 and 84

Question.6. Three bulbs red, green and yellow flash at intervals of 80 seconds, 90 seconds and 110 seconds. All three flash together at 8:00 am. At what time will the three bulbs flash altogether again?

(a) 9:00 am
(b) 9:12 am
(c) 10:00 am
(d) 10:12 am

Ans.5. (b) 4 cm as it is the HCF of 40 and 84
Ans.6. (d) 10:12 am

Hint: Recall the properties of irrational number in order to prove that whether the sum / difference / product / quotient of 2 numbers is irrational or not.

Question.7. Which of the following is an irrational number?

(a) \frac{\sqrt{2}}{\sqrt{8}}
(b) \frac{\sqrt{3}}{3\sqrt{5}}
(c) \frac{\sqrt{5}}{\sqrt{20}}
(d) \frac{\sqrt{63}}{\sqrt{7}}

Question.8. A teacher creates the question “Which of the following could be the sum of two rational numbers?”.
She now needs to create three incorrect choices and one correct answer. Which option shows the choices that the teacher should create?

(a) First choice: \pi ; Second choice: 20+16; Third choice: 50-1; Correct Answer: 49
(b) First choice: 227; Second choice: 25+16; Third choice: 64; Correct Answer: 5
(c) First choice: 125; Second choice: 36+42; Third choice: 81; Correct Answer: 169
(d) None of them

Ans.7. (b) \frac{\sqrt{3}}{3\sqrt{5}}
Ans.8. (a) First choice: \pi ; Second choice: 20+16; Third choice: 50-1; Correct Answer: 49

Hint: Apply theorems of irrational number in order to prove whether a given number is irrational or not.

Question.9. Which of the following is NOT an irrational number?

(a) 2 \times \frac{1}{\sqrt{2}}
(b) \sqrt{2} \times \frac{1}{{2}}
(c) \sqrt{2} \times \frac{1}{{\sqrt{3}}}
(d) \sqrt{2} \times \frac{1}{\sqrt{2}}

Question.10. Is 9+\sqrt{2} an irrational number?

(a) No, because if 9+\sqrt{2}=\frac{a}{b} where a and b are integers and b\neq 0 , then \sqrt{2}=\frac{9b-a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .
(b) No, because if 9+\sqrt{2}=\frac{a}{b} , where a and b are integers and b\neq 0 , then \sqrt{2}=\frac{9b+a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .
(c) Yes, because if 9+\sqrt{2}=\frac{a}{b} , where a and b are integers and b\neq 0 , then \sqrt{2}=\frac{9b-a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .
(d) Yes, because if 9+\sqrt{2}=\frac{a}{b} , where a and b are integers and b\neq 0 , then \sqrt{2}=\frac{9b+a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .

Ans.9. (d) \sqrt{2} \times \frac{1}{\sqrt{2}}
Ans.10. (b) No, because if 9+\sqrt{2}=\frac{a}{b} , where a and b are integers and b\neq 0 , then \sqrt{2}=\frac{9b+a}{b} , but \sqrt{2} is an irrational number. So, 9+\sqrt{2} \neq \frac{a}{b} .

Hint: Apply theorems of rational numbers in order to find out about the nature of their decimal representation and their factors.

Question.11. Which of the following is equivalent to a decimal that terminates?

(a) \frac{1}{5^{2}2^{2}}
(b) \frac{1}{2^{2}3}
(c) \frac{1}{5^{2}7}
(d) \frac{1}{5^{2}11^{2}}

Question.12. The fractions \frac{3}{a} and \frac{7}{b} are equivalent to decimals that terminate. Which best describes the product of a and b?

(a) It is a prime number.
(b) It cannot be an odd number.
(c) It is of the form 21 k , where k could be multiples of 2 or 5.
(d) It is of the form 21 k , where k could be multiples of 7 or 9.

Ans.11. (a) (a) \frac{1}{5^{2}2^{2}}
Ans.12. (c) It is of the form 21 k , where k could be multiples of 2 or 5.