Activity Questions 1.2

Based on the sources and our conversation history, we can determine which of the given inequalities between fractions are true by finding a common denominator and comparing the numerators. Rational numbers, which include fractions, can be written in the form p/q. To compare two fractions which have different denominators, there is no way to directly compare them. The only way is to convert them into equivalent fractions such that they have the same denominator. A number that is a multiple of both denominators can be used as the common denominator. Once the denominators are the same, you can add the numerators, or in this case, compare them.

Let’s evaluate each inequality:

• 4/7 > 5/8 To compare 4/7 and 5/8, we find a common denominator. The product of the denominators, 7 and 8, is 56. We convert 4/7 to an equivalent fraction with a denominator of 56: 4/7 = (4 * 8) / (7 * 8) = 32/56. We convert 5/8 to an equivalent fraction with a denominator of 56: 5/8 = (5 * 7) / (8 * 7) = 35/56. Now we compare the numerators of the equivalent fractions: 32 and 35. Since 32 is less than 35, 32/56 < 35/56. Therefore, 4/7 < 5/8. The given inequality 4/7 > 5/8 is false.

• 17/12 > 7/13 To compare 17/12 and 7/13, we find a common denominator. The product of the denominators, 12 and 13, is 156. We convert 17/12 to an equivalent fraction with a denominator of 156: 17/12 = (17 * 13) / (12 * 13) = 221/156. We convert 7/13 to an equivalent fraction with a denominator of 156: 7/13 = (7 * 12) / (13 * 12) = 84/156. Now we compare the numerators of the equivalent fractions: 221 and 84. Since 221 is greater than 84, 221/156 > 84/156. Therefore, 17/12 > 7/13. The given inequality 17/12 > 7/13 is true.

• 7/11 > 8/13 To compare 7/11 and 8/13, we find a common denominator. The product of the denominators, 11 and 13, is 143. We convert 7/11 to an equivalent fraction with a denominator of 143: 7/11 = (7 * 13) / (11 * 13) = 91/143. We convert 8/13 to an equivalent fraction with a denominator of 143: 8/13 = (8 * 11) / (13 * 11) = 88/143. Now we compare the numerators of the equivalent fractions: 91 and 88. Since 91 is greater than 88, 91/143 > 88/143. Therefore, 7/11 > 8/13. The given inequality 7/11 > 8/13 is true.

• 15/7 < 5/2 To compare 15/7 and 5/2, we find a common denominator. The product of the denominators, 7 and 2, is 14. We convert 15/7 to an equivalent fraction with a denominator of 14: 15/7 = (15 * 2) / (7 * 2) = 30/14. We convert 5/2 to an equivalent fraction with a denominator of 14: 5/2 = (5 * 7) / (2 * 7) = 35/14. Now we compare the numerators of the equivalent fractions: 30 and 35. Since 30 is less than 35, 30/14 < 35/14. Therefore, 15/7 < 5/2. The given inequality 15/7 < 5/2 is true.

The inequalities that are true are:

• 17/12 > 7/13
• 7/11 > 8/13
• 15/7 < 5/2

To determine which of the given options is in reduced form, we must understand the definition of a rational number in its reduced form and the concept of the greatest common divisor (GCD).

Definitions from the sources:

• Natural numbers are used for counting and include 0, 1, 2, 3, 4, and so on.
• Integers extend natural numbers to include negative numbers, spanning from minus infinity to plus infinity, including 0, 1, 2, 3, 4, etc., and -1, -2, -3, etc..
• Rational numbers are numbers that can be written in the form p/q, where p and q are both integers.
• A rational number p/q is in its reduced form when its numerator (p) and denominator (q) have no common factors other than 1. This is equivalent to saying that the greatest common divisor (GCD) of p and q is 1.
• The greatest common divisor (GCD) of two non-zero integers p and q is defined as the largest positive integer that divides both p and q. For example, gcd(9, 12) = 3.

Let us evaluate each option based on these definitions:

a) 5/60

• Numerator (p): 5
• Denominator (q): 60
• To check if it’s in reduced form, we find the GCD of 5 and 60.
• 5 is a factor of 60 (since 60 = 5 × 12).
• Therefore, gcd(5, 60) = 5.
• Since the GCD is 5 (which is not 1), this fraction is not in reduced form. It can be simplified to 1/12.

b) 12/27

• Numerator (p): 12
• Denominator (q): 27
• To check if it’s in reduced form, we find the GCD of 12 and 27.
• Factors of 12 are {1, 2, 3, 4, 6, 12}.
• Factors of 27 are {1, 3, 9, 27}.
• The common factors are 1 and 3.
• Therefore, gcd(12, 27) = 3.
• Since the GCD is 3 (which is not 1), this fraction is not in reduced form. It can be simplified to 4/9.

c) 11/18

• Numerator (p): 11
• Denominator (q): 18
• To check if it’s in reduced form, we find the GCD of 11 and 18.
• 11 is a prime number, meaning its only factors are 1 and itself.
• To find the GCD, we check if 18 is divisible by 11. 18 ÷ 11 results in a remainder, so 11 is not a factor of 18.
• The only common factor between 11 and 18 is 1.
• Therefore, gcd(11, 18) = 1.
• Since the GCD is 1, this fraction is in reduced form.

d) 13/91

• Numerator (p): 13
• Denominator (q): 91
• To check if it’s in reduced form, we find the GCD of 13 and 91.
• 13 is a prime number.
• We check if 91 is divisible by 13. 91 = 13 × 7.
• Therefore, gcd(13, 91) = 13.
• Since the GCD is 13 (which is not 1), this fraction is not in reduced form. It can be simplified to 1/7.

Based on the analysis, the only option where the numerator and denominator have a greatest common divisor of 1 is option (c).

The final answer is $\boxed{\text{c) 11/18}}$

To determine the values of a and b that satisfy the given conditions, we will draw upon the definitions of natural numbers and the greatest common divisor (GCD) as provided in the sources.

First, let’s establish the definitions:

• Natural numbers are typically defined as numbers starting with 0, so the set of natural numbers (denoted by N) includes {0, 1, 2, 3, 4, …}.
• The greatest common divisor (GCD) of two non-zero integers p and q is the largest positive integer that divides both p and q. For example, gcd(9, 12) = 3. Special cases include gcd(0, q) = q and gcd(1, q) = 1.

We are given the following conditions for a and b:

1. gcd(a, 4) = 2
2. gcd(a, b) = 1
3. 4 > a > b
4. a and b are natural numbers

Let’s evaluate these conditions step-by-step:

1. Using the condition 4 > a > b and that a, b are natural numbers:

   • Since a is a natural number and a < 4, possible values for a are {0, 1, 2, 3}.
   • Since b is a natural number and b < a, b must be less than the chosen a.

2. Applying the condition gcd(a, 4) = 2:

   • This condition implies that a must be an even number, because 2 is a factor of a and 4.
   • Let’s check the possible values for a from step 1:
     • If a = 0, gcd(0, 4) = 4. This does not satisfy gcd(a, 4) = 2.
     • If a = 1, gcd(1, 4) = 1. This does not satisfy gcd(a, 4) = 2.
     • If a = 2, gcd(2, 4) = 2. This value of a = 2 satisfies the condition.
     • If a = 3, gcd(3, 4) = 1. This does not satisfy gcd(a, 4) = 2.
   • Therefore, from this condition, the value of a must be 2.

3. Now, using a = 2 in the condition 4 > a > b:

   • Substituting a = 2, we get 4 > 2 > b.
   • Since b is a natural number and b < 2, the possible values for b are {0, 1}.

4. Finally, applying the condition gcd(a, b) = 1 with a = 2:

   • We need gcd(2, b) = 1.
   • Let’s check the possible values for b from step 3:
     • If b = 0, gcd(2, 0) = 2. This does not satisfy gcd(2, b) = 1.
     • If b = 1, gcd(2, 1) = 1. This value of b = 1 satisfies the condition.
   • Therefore, the value of b must be 1.

Based on this step-by-step analysis, the only values for a and b that satisfy all the given conditions are a = 2 and b = 1.

This corresponds to option (a).