6. There are between 50 and 60 eggs in a basket. When Roza counts by 3's, there are 2 eggs left over. When she counts by 5's, there are 4 left over. How many eggs are there in the basket?

7. The GCF of two numbers is 3 and their LCM is 180. If one of the numbers is 45, find the second number.

8. Which of the following statements are true and which are false?
a. The sum of any two rational numbers is rational.
b. The sum of any two irrational numbers is irrational.
c. The product of any two rational numbers is rational.
d. The product of any two irrational numbers is irrational.

9. Convert the decimal 3.25 to a fraction.

10. Determine whether the following are rational or irrational:
a. 2.75
b. 0.272727...



Answer :

Sure, let's work through each question step-by-step:

Question 6:
- Problem: There are between 50 and 60 number of eggs in a basket. When Roza counts by 3's, there are 2 eggs left over. When she counts by 5's, there are 4 left over. How many eggs are there in the basket?

- Step-by-step Solution:
- We need to find a number within the range of 50 to 60 that, when divided by 3, leaves a remainder of 2, and when divided by 5, leaves a remainder of 4.
- Consider the numbers in the range from 50 to 60: 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60.
- Checking divisibility:
- 50 % 3 = 2, 50 % 5 = 0
- 51 % 3 = 0, 51 % 5 = 1
- 52 % 3 = 1, 52 % 5 = 2
- 53 % 3 = 2, 53 % 5 = 3
- 54 % 3 = 0, 54 % 5 = 4
- 55 % 3 = 1, 55 % 5 = 0
- 56 % 3 = 2, 56 % 5 = 1
- 57 % 3 = 0, 57 % 5 = 2
- 58 % 3 = 1, 58 % 5 = 3
- 59 % 3 = 2, 59 % 5 = 4
- 60 % 3 = 0, 60 % 5 = 0
- Only 59 leaves remainders of 2 and 4 when divided by 3 and 5, respectively.
- Therefore, the number of eggs in the basket is 59.

Question 7:
- Problem: The GCF of two numbers is 3 and their LCM is 180. If one of the numbers is 45, then find the second number.

- Step-by-step Solution:
- Given values: GCF = 3, LCM = 180, and one of the numbers = 45.
- Using the relationship: \( \text{LCM} \times \text{GCF} = \text{First Number} \times \text{Second Number} \).
- Substituting the values: \( 180 \times 3 = 45 \times \text{Second Number} \).
- Simplify the equation: \( 540 = 45 \times \text{Second Number} \).
- Solve for the second number: \( \text{Second Number} = \frac{540}{45} = 12 \).
- Therefore, the second number is 12.

Question 8:
- Problem: Determine which of the following statements are true and which are false:
- (e) The sum of any two rational numbers is rational.
- (f) The sum of any two irrational numbers is irrational.
- (g) The product of any two rational numbers is rational.
- (h) The product of any two irrational numbers is irrational.

- Step-by-step Solution:
- (e) The sum of any two rational numbers is rational. This is true. Rational numbers are closed under addition.
- (f) The sum of any two irrational numbers is irrational. This is often true, but there exist counterexamples (e.g., \( \sqrt{2} + (-\sqrt{2}) = 0 \), which is rational).
- (g) The product of any two rational numbers is rational. This is true. Rational numbers are closed under multiplication.
- (h) The product of any two irrational numbers is irrational. This is false because there can be counterexamples (e.g., \( \sqrt{2} \times \sqrt{2} = 2 \), which is rational).

Question 9:
- Problem: Convert the decimal 3.25 to fractions.

- Step-by-step Solution:
- 3.25 can be written as \( 3 \frac{25}{100} = 3 \frac{1}{4} = \frac{13}{4} \).
- Therefore, the fraction form of 3.25 is \( \frac{13}{4} \).

Question 10:
- Problem: Determine whether the following numbers are rational or irrational:
- a) 2.75
- b) 0.272727...

- Step-by-step Solution:
- a) 2.75 is a terminating decimal, which can be expressed as a fraction (e.g., \( \frac{11}{4} \)). Thus, it is a rational number.
- b) 0.272727... is a repeating decimal, which means it can also be expressed as a fraction. Thus, it is a rational number.

So, summarizing the answers:
1. The number of eggs is 59.
2. The second number is 12.
3. Truth values of the statements: (e) True, (f) True (with counterexamples), (g) True, (h) False.
4. The decimal 3.25 as a fraction is \( \frac{13}{4} \).
5. Both 2.75 and 0.272727... are rational numbers.