Question

John's grandfather was five times older to him 5 years ago. He would be two times of his age after 25 years from now. What is the ratio of John's age to that of his grandfather?

• A. 7 : 11
• B. 5 : 11
• C. 3 : 11
• D. 4 : 11

Hint:

Let John's age 5 years ago be x, set up two age equations for 5 years ago and 25 years from now, then solve for x.

Answer 1

The Correct Option is C

To solve the problem of determining the ratio of John's age to that of his grandfather's, we start by setting up the problem with algebraic expressions based on the information provided. Let's outline the step-by-step approach:

1. Assume that John's current age is \(x\) years.
2. Assume that John's grandfather's current age is \(y\) years.
3. According to the problem, John's grandfather was five times older than John 5 years ago. This translates to:
   • \(y - 5 = 5(x - 5)\)
   • Expanding and solving the equation gives: \(y - 5 = 5x - 25 \Rightarrow y = 5x - 20 + 5 \Rightarrow y = 5x - 20\)
4. The problem also states that John's grandfather will be twice John's age 25 years from now. This can be written as:
   • \(y + 25 = 2(x + 25)\)
   • Solving this equation gives: \(y + 25 = 2x + 50 \Rightarrow y = 2x + 50 - 25 \Rightarrow y = 2x + 25\)
5. We now have two equations:
   • Equation 1: \(y = 5x - 20\)
   • Equation 2: \(y = 2x + 25\)
6. Setting these equations equal to solve for \(x\):
   • \(5x - 20 = 2x + 25\)
   • \(5x - 2x = 25 + 20\)
   • \(3x = 45 \Rightarrow x = 15\)
7. Substitute \(x = 15\) back into either equation for \(y\). Using Equation 1:
   • \(y = 5(15) - 20 \Rightarrow y = 75 - 20 = 55\)
8. Therefore, John's current age is 15 years, and his grandfather's current age is 55 years.
9. The ratio of John's age to his grandfather's age is \(\frac{15}{55}\).
10. Simplifying the ratio: \(\frac{15}{55} = \frac{3}{11}\).

Thus, the correct answer is that the ratio of John's age to his grandfather's age is \(3 : 11\), which matches option

3 : 11

Answer 2

Step 1: Set up variables for 5 years ago.
Let John's age 5 years ago be \(x\) years. Since the grandfather was five times as old at that time, his age 5 years ago was \(5x\) years.

Step 2: Write both ages 25 years from now.
From '5 years ago' to '25 years from now' is a gap of 30 years. So John's age 25 years from now is \(x + 30\), and the grandfather's age 25 years from now is \(5x + 30\).

Step 3: Use the second condition to form an equation.
The grandfather's age then is twice John's age then, so \(5x + 30 = 2(x + 30)\). Expanding, \(5x + 30 = 2x + 60\), so \(3x = 30\), giving \(x = 10\).

Step 4: Find present ages.
John's present age \(= x + 5 = 15\) years. Grandfather's present age \(= 5x + 5 = 55\) years.

Step 5: Form the ratio.
Ratio of John's age to grandfather's age \(= 15 : 55 = 3 : 11\).

Final Answer:
The ratio of John's age to his grandfather's age is 3:11. \[ \boxed{3:11} \]