Understanding the Age Problem
This problem involves finding the current ages of a mother and her son based on information about their ages at different points in time (past and future) and then determining the ratio of their present ages.
Setting Up the Equations
Let's denote the present age of the mother as $M$ years and the present age of the son as $S$ years.
Information from the Past
- Eight years ago, the mother's age was $M - 8$.
- Eight years ago, the son's age was $S - 8$.
- The problem states that eight years ago, the mother's age was four times the son's age. This gives us the equation: $M - 8 = 4(S - 8)$
Information from the Future
- Eight years hence (in the future), the mother's age will be $M + 8$.
- Eight years hence, the son's age will be $S + 8$.
- The problem states that eight years hence, the mother's age will be two times the son's age. This gives us the equation: $M + 8 = 2(S + 8)$
Solving the System of Equations
We now have two equations:
- $M - 8 = 4(S - 8)$
- $M + 8 = 2(S + 8)$
Simplifying the Equations
Let's simplify the equations:
- From equation 1: $M - 8 = 4S - 32$ $M = 4S - 32 + 8$ $M = 4S - 24$ (Equation 3)
- From equation 2: $M + 8 = 2S + 16$ $M = 2S + 16 - 8$ $M = 2S + 8$ (Equation 4)
Finding the Present Ages
Now we can solve for $S$ by setting Equation 3 and Equation 4 equal to each other, since both represent $M$:
$4S - 24 = 2S + 8$Rearrange the terms to solve for $S$:
$4S - 2S = 8 + 24$ $2S = 32$ $S = \frac{32}{2}$ $S = 16$So, the son's present age is 16 years.
Now, substitute the value of $S$ (16) into either Equation 3 or Equation 4 to find $M$. Let's use Equation 4:
$M = 2S + 8$ $M = 2(16) + 8$ $M = 32 + 8$ $M = 40$So, the mother's present age is 40 years.
Calculating the Ratio of Present Ages
The question asks for the ratio of the present age of the mother to the age of the son.
Ratio = Mother's Present Age : Son's Present Age
Ratio = $M : S$
Ratio = $40 : 16$
To simplify the ratio, we find the greatest common divisor (GCD) of 40 and 16, which is 8.
Divide both parts of the ratio by 8:
$ \frac{40}{8} : \frac{16}{8} $ $ 5 : 2 $Therefore, the ratio of the present age of the mother to the age of the son is 5:2.
Important Questions from Problem on Ages (Notes)
- At present, Sharad is three times as old as his son, and his daughter is 3 years younger than the son. If the sum of the ages of these three people 3 years ago was 63 years, then Sharad's present age (in years) is:
A father said to his son, "I was as old as you are when I became your father." If the current age of father is 52 years, the age of son after 10 years will be___________.
- The sum of the age of A and 5 times the age of B is 55 years. When 3 times the age of A is added to 7 times the age of B, the result is 99 years. The sum of the ages (in years) of A and B is:
- The sum of the age of A and 3 times the age of B is 43 years. When 5 times the age of A is added to 2 times the age of B, the result is 98 years. The sum of the ages (in years) of A and B is:
- The sum of the age of A and 5 times the age of B is 36 years. When 3 times the age of A is added to 7 times the age of B, the result is 62 years. The sum of the ages (in years) of A and B is:
