Question

Five years ago, Adil was thrice as old as Bharat. Ten years later Adil shall be twice as old as Bharat. To know the present ages of Adil and Bharat: (i) form the linear equations representing the above information. (ii) show that the system of equations is consistent with unique solution. (iii) find the present ages of Adil and Bharat.

Hint:

Always double check your answers by substituting the computed ages back into the original word problem:
- Five years ago, Adil was 45 and Bharat was 15. \( 45 = 3 \times 15 \) (True).
- Ten years later, Adil will be 60 and Bharat will be 30. \( 60 = 2 \times 30 \) (True).
This self-check ensures absolute accuracy.

Answer 1

Step 1: Understanding the Question:
The topic of this question is Pair of Linear Equations in Two Variables (Age Word Problems).
We are given the relationship between the ages of Adil and Bharat at two different times: five years ago and ten years in the future.
We need to model this mathematically, prove the consistency of the equations, and find their current ages.

Step 2: Key Formula or Approach:
- Let Adil's present age be \( x \) years and Bharat's present age be \( y \) years.
- Formulate linear equations based on the past and future relationships.
- Check consistency using the condition for a unique solution:
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]

Step 3: Detailed Explanation:
1. Part (i): Formulate the Linear Equations:
Let Adil's present age be \( x \) years and Bharat's present age be \( y \) years.
- Five years ago:
Adil's age was \( x - 5 \).
Bharat's age was \( y - 5 \).
According to the condition:
\[ x - 5 = 3(y - 5) \]
\[ x - 5 = 3y - 15 \]
\[ x - 3y = -10 \quad \text{(Equation 1)} \]
- Ten years later:
Adil's age will be \( x + 10 \).
Bharat's age will be \( y + 10 \).
According to the condition:
\[ x + 10 = 2(y + 10) \]
\[ x + 10 = 2y + 20 \]
\[ x - 2y = 10 \quad \text{(Equation 2)} \]
2. Part (ii): Show that the system is consistent with a unique solution:
Let us write the equations in standard form:
- Equation 1: \( 1x - 3y + 10 = 0 \implies a_1 = 1, b_1 = -3, c_1 = 10 \)
- Equation 2: \( 1x - 2y - 10 = 0 \implies a_2 = 1, b_2 = -2, c_2 = -10 \)
Compare the ratios of the coefficients:
\[ \frac{a_1}{a_2} = \frac{1}{1} = 1 \]
\[ \frac{b_1}{b_2} = \frac{-3}{-2} = \frac{3}{2} \]
Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the system represents two intersecting lines.
Therefore, the system is consistent and has a unique solution.
3. Part (iii): Find the present ages:
Subtract Equation 1 from Equation 2:
\[ (x - 2y) - (x - 3y) = 10 - (-10) \]
\[ y = 20 \]
Substitute \( y = 20 \) in Equation 2:
\[ x - 2(20) = 10 \]
\[ x - 40 = 10 \]
\[ x = 50 \]
Therefore, Adil's present age is 50 years and Bharat's present age is 20 years.

Step 4: Final Answer:
(i) The linear equations are \(x - 3y = -10\) and \(x - 2y = 10\).
(ii) The ratios are unequal (\(1 \neq 1.5\)), proving the system is consistent.
(iii) Adil's present age is 50 years and Bharat's present age is 20 years.