Let's solve each of these questions step by step.

Question 14:

Problem Statement: A wizard named Nepo says: "I am only three times my son's age. My father is 40 years more than twice my age. Together the three of us are a mere 1240 years old." How old is Nepo?

Solution: Let's define:

• $S$ = Son's age
• $N$ = Nepo's age
• $F$ = Father's age

Given:

1. Nepo’s age is three times his son's age: $N = 3S$

2. Nepo's father is 40 years more than twice Nepo's age: $F = 2N + 40$

3. The sum of their ages is 1240: $S + N + F = 1240$

Step 1: Substitute $N = 3S$ into the other equations

Using $N = 3S$ in equation 3: $S + 3S + F = 1240$ $4S + F = 1240$

Step 2: Substitute $F = 2N + 40$ into the equation

Since $N = 3S$, then $F = 2(3S) + 40$: $F = 6S + 40$

Substitute $F = 6S + 40$ into the equation $4S + F = 1240$: $4S + 6S + 40 = 1240$ $10S + 40 = 1240$ $10S = 1200$ $S = 120$

Step 3: Find Nepo's age $N$ and his father’s age $F$

1. $N = 3S = 3 \times 120 = 360$
2. $F = 6S + 40 = 6 \times 120 + 40 = 720$

Answer: Nepo's age is 360 years.

Question 15:

Problem Statement: A is as much younger than B as he is older than C. If the sum of the ages of B and C is 50 years, what is definitely the difference between ages of A and B?

Solution: Let:

• $A$ = Age of A
• $B$ = Age of B
• $C$ = Age of C

Given:

1. A is as much younger than B as he is older than C. This implies that: $B - A = A - C$

2. The sum of ages of B and C is 50: $B + C = 50$

Step 1: Solve the age difference equation

Rearrange $B - A = A - C$: $B + C = 2A$

Step 2: Substitute $B + C = 50$

Since $B + C = 50$, we have: $2A = 50$ $A = 25$

Step 3: Substitute $A = 25$ back to find $B - A$

Since $B - A = A - C$ and $B + C = 50$:

• $B - A = 25 - C$
• The difference