# The ages of A and B are in the ratio 3 : 5; eight years later their ages will be in the ratio 5 : 7. Find their current ages.

Last updated date: 29th Mar 2023

•

Total views: 306.3k

•

Views today: 3.83k

Answer

Verified

306.3k+ views

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 ‘x’ and the current age of B as ‘y’. 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,

$\dfrac{x}{y}=\dfrac{3}{5}$ …………………………………………………………….. (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,

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

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

$\Rightarrow x=\dfrac{3}{5}y$ ………………………………………………………. (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 …………………………………………………. (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 ‘x’ and the current age of B as ‘y’. 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,

$\dfrac{x}{y}=\dfrac{3}{5}$ …………………………………………………………….. (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,

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

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

$\Rightarrow x=\dfrac{3}{5}y$ ………………………………………………………. (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 …………………………………………………. (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.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India