Question

One year ago the ratio of raj and ramana's age was 4:5 .one year hence the ratio of ages will be 5:6 .the present age of ramana is in arithmetic problems on ages

Answer 1

let ages of Raj and Raman be 4x and 5x
then their ages one year will be 4x+1 and 5x+1
then according to question, their ratio is 5:6
$\frac{4x + 1}{5x + 1} = \frac{5}{6 } \\ 24x + 6 = 25x + 5 \\ 25x - 24x = 6 - 5 \\ x = 1$
the present age of Raj and Raman will be 4 and 5 years respectively

Answer 2

Correct Question: One year ago, the ratio of Raj and Ramana's age was 4:5. One year hence the ratio of ages will be 5:6. Find the present ages of Raj and Ramana using arithmetic problems on ages.

Given:  Ratio of Raj and Ramana's age one year ago = 4 : 5

            The ratio of their ages one year hence = 5 : 6

To find: Their present ages

Let: Raj and Ramana's present ages be $x$ and $y$ years respectively.

Solution: According to the given question,

One year ago, the ratio of their ages was 4 : 5 and one year hence, the ratio will be 5: 6.

Now, if their present ages are $x$ and $y$ then their ages one year ago must have been $(x-1)$ and $(y-1)$.

Similarly, their ages one year hence will be $(x+1)$ and $(y+1)$

∵ $\frac{(x-1)}{(y-1)}=\frac{4}{5}$        [Given in question]

Using cross multiplication,

⇒ $5(x-1) = 4(y-1)$

⇒ $5x - 5 = 4y - 4$

⇒ $5x - 4y - 1 = 0$   ...(1)

also,

∵ $\frac{(x+1)}{(y+1)} = \frac{5}{6}$        [Given in question]

Using cross multiplication,

⇒ $6(x+1) = 5(y+1)$

⇒ $6x + 6 = 5y + 5$

⇒ $6x - 5y +1 = 0$   ...(2)

Using the elimination method to solve the equations:

Multiplying equation (1) by 5

⇒ $5(5x - 4y - 1) = 5 *0$

⇒ $25x - 20y - 5 = 0$   ...(3)

Multiplying equation (2) by 4

⇒ $4(6x - 5y +1) = 4*0$

⇒ $24x - 20y + 4 = 0$   ...(4)

Subtracting equation (4) from equation (3)

⇒ $(25x - 20y - 5) - (24x - 20y + 4) = 0$

⇒ $25x - 20y -5 - 24x + 20y - 4 = 0$

⇒ $x - 9 = 0$

⇒ $x = 9$

Putting $x = 9$ in equation (1)

⇒ $5(9) - 4y - 1 = 0$

⇒ $45 - 4y - 1 = 0$

⇒ $4y = 44$

⇒ $y = \frac{44}{4} = 11$

Hence, the present age of Raj is 9 years and the present age of Ramana is 11 years.