Question

The ages of Raghu and Shyam are in the ratio 2:4. Five years from now the ratio of their ages will be 3:5. Find their present ages.

Answer 1

$\large\sf\underline{Given\::}$

• Ages of Raghu and Shyam are in the ratio = 2 : 4

• Five years from now the ratio will be = 3 : 5

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$\large\sf\underline{To\:find\::}$

• Present ages of Raghu and Shyam .

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$\large\sf\underline{Concept\::}$

In this question we are provided with the ratio of ages of Raghu and Shyam as 2 : 4 .

We are also given that the ratio of their ages after five years would be 3 : 5 .

We need to calculate the present ages .

Doing it is simple we would equate the given two ratios and calculate the value of common ratio and then their present age.

Let's do it :D

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$\large\sf\underline{Solution\::}$

Let the common ratio be x.

• Raghu's age = 2x

• Shyam's age = 4x

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Ratio of ages of Raghu and Shyam = $\sf\:\frac{2x}{4x}$

• After five years

Ratio of ages of Raghu and Shyam = $\sf\:\frac{2x+5}{4x+5}$ -- ( i )

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According to the question :

• After five years

Ratio of ages of Raghu and Shyam = $\sf\:\frac{3}{5}$ -- ( ii )

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• Equating ( i ) and ( ii )

$\sf\implies\:\frac{2x+5}{4x+5}=\frac{3}{5}$

• Cross multiplying

$\sf\implies\:3(4x+5)=5(2x+5)$

• Multiplying and removing the brackets

$\sf\implies\:12x+15=10x+25$

• Transposing +10x from RHS to LHS it becomes -10x

$\sf\implies\:12x-10x+15=25$

$\sf\implies\:2x+15=25$

• Transposing +15 from LHS to RHS it becomes -15

$\sf\implies\:2x=25-15$

$\sf\implies\:2x=10$

• Transposing 2 to RHS it goes to the denominator

$\sf\implies\:x=\frac{10}{2}$

• Reducing the fraction to the lower terms

$\sf\implies\:x=\cancel{\frac{10}{2}}$

$\small{\underline{\boxed{\mathrm\red{\implies\:x\:=\:5}}}}$ ★

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Now substituting the value of x in the ages of Raghu and Shyam we get :

• Age of Raghu = 2x = 2 × 5 = $\small\fbox\blue{10\:years}$ .

• Age of Shyam = 4x = 4 × 5 = $\small\fbox\blue{20\:years}$ .

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Verifying :

We got ages of Raghu and Shyam as 10 and 20 years respectively.

According to the question their ratio must be $\sf\:\frac{2}{4}$ .

Lets see if we get 2 : 4 after reducing 10 : 20 .

$\sf\:\frac{10}{20}$

$\sf\to\:\cancel{\frac{10}{20}}$

$\sf\to\:\frac{2}{4}$ [ by 5 ]

$\small\fbox\green{Hence~Verified~!! }$

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!! Hope it helps !!