Question

03. Two numbers are in the ratio 5:3 They differ by 18, what are the numbers.​

Answer 1

Answer:

$\boxed{\sf The two numbers = 45 \ and \ 27}$

Given:

Ratio of two numbers = 5:3

Difference of two numbers = 18

To Find:

The two numbers

Step-by-step explanation:

Let one number be 'x' and another number be 'y'.

$\therefore \\ \sf \frac{x}{y} = \frac{5}{3} \: \: \: \: \: \: \: \: \: \: \: .....Eq_{1} \\ \\ \sf x - y = 18 \: \: \: \: \: \: \: \: \: \: \: \: .....Eq_{2}$

$\sf From \ Eq_{1}: \\ \sf x = \frac{5}{3} y$

$\sf Substituting \ the \ value \ of \ x \ in \ Eq_{2}: \\ \sf \implies \frac{5}{3} y - y = 18 \\ \\ \sf \implies \frac{5}{3} y - y \times \frac{3}{3} = 18 \\ \\ \sf \implies \frac{5y}{3} - \frac{3y}{3} = 18 \\ \\ \sf \implies \frac{5y - 3y}{3} = 18 \\ \\ \sf \implies \frac{2y}{3} = 18 \\ \\ \sf \implies 2y = 18 \times 3 \\ \\ \sf \implies 2y = 54 \\ \\ \sf \implies y = \frac{54}{2} \\ \\ \sf \implies y = \frac{27 \times \cancel{2}}{ \cancel{2}} \\ \\ \sf \implies y = 27$

$\sf Substituting \ the \ value \ of \ y \ in \ Eq_{2} : \\ \sf \implies x - 27 = 18 \\ \\ \sf \implies x = 18 + 27 \\ \\ \sf \implies x = 45$

So,

The two numbers are 45 and 27.

Answer 2

Let the numbers be x and y respectively. ( x > y )

Given

Two numbers are in the ratio 5:3.

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

$\implies 3x=5y$

$\implies 3x-5y=0--(1)$

They differ by 18.

$\implies x-y=18--(2)$

Multiplying the second equation by 3, we get,

$\implies 3x-3y=54--(3)$

Solving (1) and (3), we get,

$3x-5y=0\\\underline{3x-3y=54}\\\underline{\underline{-2y=-54}}$

$\implies y =\dfrac{-54}{-2}$

$\implies y= 27$

Substituting the value of y in the second equation, we get,

$\implies x-27=18$

$\implies x = 18+27$

$\implies x = 45$

$\boxed{\boxed{\bold{Therefore, \ the \ numbers \ are \ 45 \ and \ 27 \ respectively}}}}}}}}}}}}$