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Hint: Assume the ages of A and B as variables and apply the two conditions to get the system equations with two variables. Solve these equations and you will get the current ages of A and B.

Complete step-by-step answer:

To solve the given problem we will assume the current age of A as â€˜xâ€™ and the current age of B as â€˜yâ€™. By using the notations we will write the given data as follows,

The current ages of A and B are in the ratio 3 : 5

Therefore, x : y = 3 : 5

The above equation can also be written as,

$\dfrac{x}{y}=\dfrac{3}{5}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (1)

Also, After eight years the ratio of ages of A and B will become 5 : 7

Therefore, x + 8 : y + 8 = 5 : 7

The above equation can also be written as,

$\dfrac{x+8}{y+8}=\dfrac{5}{7}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (2)

As we have written the given data therefore we will write the equation (1) and equation (2) one by one,

Therefore equation (1) will become,

$\dfrac{x}{y}=\dfrac{3}{5}$

By cross multiplication in the above equation we will get,

$\Rightarrow 5\times x=3\times y$

If we shift 5 on the right hand side of the equation we will get,

$\Rightarrow x=\dfrac{3\times y}{5}$

$\Rightarrow x=\dfrac{3}{5}y$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (3)

Also equation (2) will become,

$\dfrac{x+8}{y+8}=\dfrac{5}{7}$

By cross multiplication in the above equation we will get,

$\Rightarrow 7\times \left( x+8 \right)=5\times \left( y+8 \right)$

If we multiply the constants inside the bracket we will get,

$\Rightarrow 7\times x+7\times 8=5\times y+5\times 8$

$\Rightarrow 7x+56=5y+40$

$\Rightarrow 56-40=5y-7x$

$\Rightarrow 16=5y-7x$

By rearranging the above equation we will get,

$\Rightarrow 5y-7x=16$

Now we will put the value of equation (3) in the above equation therefore we will get,

$\Rightarrow 5y-7\times \left( \dfrac{3}{5}y \right)=16$

$\Rightarrow 5y-\dfrac{21}{5}y=16$

If we multiply by 5 on both sides of the equation we will get,

$\Rightarrow 5\times 5y-5\times \dfrac{21}{5}y=5\times 16$

$\Rightarrow 25y-21y=80$

$\Rightarrow 4y=80$

$\Rightarrow y=\dfrac{80}{4}$

Therefore, y = 20 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (4)

Therefore the current age of B is 20 years.

Now we will put the value of equation (4) in equation (3), therefore we will get,

\[\Rightarrow x=\dfrac{3}{5}\times 20\]

\[\Rightarrow x=3\times 4\]

Therefore, x = 12

Therefore the current age of A is 12 years.

Therefore the ages of A and B are 12 years and 20 years respectively.

Note: You can use A and B as the ages directly in place of using the variables so that you can get the direct answers in terms of A and B.

Complete step-by-step answer:

To solve the given problem we will assume the current age of A as â€˜xâ€™ and the current age of B as â€˜yâ€™. By using the notations we will write the given data as follows,

The current ages of A and B are in the ratio 3 : 5

Therefore, x : y = 3 : 5

The above equation can also be written as,

$\dfrac{x}{y}=\dfrac{3}{5}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.. (1)

Also, After eight years the ratio of ages of A and B will become 5 : 7

Therefore, x + 8 : y + 8 = 5 : 7

The above equation can also be written as,

$\dfrac{x+8}{y+8}=\dfrac{5}{7}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦ (2)

As we have written the given data therefore we will write the equation (1) and equation (2) one by one,

Therefore equation (1) will become,

$\dfrac{x}{y}=\dfrac{3}{5}$

By cross multiplication in the above equation we will get,

$\Rightarrow 5\times x=3\times y$

If we shift 5 on the right hand side of the equation we will get,

$\Rightarrow x=\dfrac{3\times y}{5}$

$\Rightarrow x=\dfrac{3}{5}y$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (3)

Also equation (2) will become,

$\dfrac{x+8}{y+8}=\dfrac{5}{7}$

By cross multiplication in the above equation we will get,

$\Rightarrow 7\times \left( x+8 \right)=5\times \left( y+8 \right)$

If we multiply the constants inside the bracket we will get,

$\Rightarrow 7\times x+7\times 8=5\times y+5\times 8$

$\Rightarrow 7x+56=5y+40$

$\Rightarrow 56-40=5y-7x$

$\Rightarrow 16=5y-7x$

By rearranging the above equation we will get,

$\Rightarrow 5y-7x=16$

Now we will put the value of equation (3) in the above equation therefore we will get,

$\Rightarrow 5y-7\times \left( \dfrac{3}{5}y \right)=16$

$\Rightarrow 5y-\dfrac{21}{5}y=16$

If we multiply by 5 on both sides of the equation we will get,

$\Rightarrow 5\times 5y-5\times \dfrac{21}{5}y=5\times 16$

$\Rightarrow 25y-21y=80$

$\Rightarrow 4y=80$

$\Rightarrow y=\dfrac{80}{4}$

Therefore, y = 20 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. (4)

Therefore the current age of B is 20 years.

Now we will put the value of equation (4) in equation (3), therefore we will get,

\[\Rightarrow x=\dfrac{3}{5}\times 20\]

\[\Rightarrow x=3\times 4\]

Therefore, x = 12

Therefore the current age of A is 12 years.

Therefore the ages of A and B are 12 years and 20 years respectively.

Note: You can use A and B as the ages directly in place of using the variables so that you can get the direct answers in terms of A and B.

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