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.
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.
