A father is 3 times as old as his son. After 12 years, his age will be twice as that of his son then. Find their present ages.

1. A father is 3 times as old as his son. After 12 years, his age will be twice as that of his son then. Find their present ages. Step 1: Define variables. Let the father's present age be F years. Let the son's present age be S years. Step 2: Formulate equations. From the first statement: "A father is 3 times as old as his son." F = 3S (1) After 12 years: Father's age will be F+12. Son's age will be S+12. From the second statement: "After 12 years, his age will be twice as that of his son then." F+12 = 2(S+12) (2) Step 3: Solve the system of equations. Substitute equation (1) into equation (2): 3S + 12 = 2(S+12) 3S + 12 = 2S + 24 Subtract 2S from both sides: 3S - 2S + 12 = 24 S + 12 = 24 Subtract 12 from both sides: S = 24 - 12 S = 12 Now substitute the value of S back into equation (1) to find F: F = 3S F = 3 × 12 F = 36 Step 4: State the present ages. The father's present age is 36 years. The son's present age is 12 years. The present ages are: Father: 36 years Son: 12 years 2. The ages of 2 friends Annie and Niyu differ by 3 years. Annie's father Durram is twice as old as Annie and Niyu is twice as old as his sister Kathy. The ages of Kathy and Durram differ by 30 years. Find the ages of Annie and Niyu. Step 1: Define variables. Let Annie's age be A. Let Niyu's age be N. Let Durram's age be D. Let Kathy's age be K. Step 2: Formulate equations. "The ages of 2 friends Annie and Niyu differ by 3 years." This means |A - N| = 3. So, either A - N = 3 or N - A = 3. "Annie's father Durram is twice as old as Annie." D = 2A (1) "Niyu is twice as old as his sister Kathy." N = 2K K = (N)/(2) (2) "The ages of Kathy and Durram differ by 30 years." Since Durram is a father and Kathy is a sister (likely younger than Niyu), Durram is expected to be older than Kathy. D - K = 30 (3) Step 3: Solve the system of equations. Substitute equations (1) and (2) into equation (3): 2A - (N)/(2) = 30 Multiply the entire equation by 2 to eliminate the fraction: 4A - N = 60 (4) Now we have two possible cases for the age difference between Annie and Niyu: Case 1: Annie is 3 years older than Niyu (A - N = 3) From A - N = 3, we can write N = A - 3. Substitute this into equation (4): 4A - (A - 3) = 60 4A - A + 3 = 60 3A + 3 = 60 3A = 57 A = (57)/(3) A = 19 Now find Niyu's age: N = A - 3 = 19 - 3 = 16 Let's check this solution: If A=19, D = 2 × 19 = 38. If N=16, K = (16)/(2) = 8. D - K = 38 - 8 = 30. This is consistent with the problem statement. Case 2: Niyu is 3 years older than Annie (N - A = 3) From N - A = 3, we can write N = A + 3. Substitute this into equation (4): 4A - (A + 3) = 60 4A - A - 3 = 60 3A - 3 = 60 3A = 63 A = (63)/(3) A = 21 Now find Niyu's age: N = A + 3 = 21 + 3 = 24 Let's check this solution: If A=21, D = 2 × 21 = 42. If N=24, $K = \