# Two numbers are in the ratio 3:5. If the sum of numbers is 144, then what is the value of the smallest number?

$

{\text{A}}{\text{. 54}} \\

{\text{B}}{\text{. 72}} \\

{\text{C}}{\text{. 90}} \\

{\text{D}}{\text{. 48}} \\

$

Last updated date: 15th Mar 2023

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Answer

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Hint: Here, we will proceed by assuming the two numbers as x and y and then according to the problem statements we will obtain some equations which will be solved by using the substitution method.

Complete step-by-step answer:

Let the two numbers be x and y.

Given the sum of the two numbers is 144 i.e., $

x + y = 144 \\

\Rightarrow y = 144 - x{\text{ }} \to {\text{(1)}} \\

$

Also, the ratio of these two numbers is 3:5 i.e., $\dfrac{x}{y} = \dfrac{3}{5}$

By cross multiplying the above equation, we get

$5x = 3y{\text{ }} \to {\text{(2)}}$

Putting the value of y from equation (1), equation (2) becomes

$

\Rightarrow 5x = 3\left( {144 - x} \right) \\

\Rightarrow 5x = 432 - 3x \\

\Rightarrow 8x = 432 \\

\Rightarrow x = 54 \\

$

Put x=54 in equation (1), the value of y is given by

$ \Rightarrow y = 144 - 54 = 90$

So, the assumed two numbers are 54 and 90. The smallest number is 54.

Hence, option A is correct.

Note: In this particular problem, we have developed two equations in two variables which can be easily solved by substitution method (used in the above solution) or elimination method (in this method we make the coefficient of all the variables same except one variable whose value will be eventually calculated by performing algebra to the given equation).

Complete step-by-step answer:

Let the two numbers be x and y.

Given the sum of the two numbers is 144 i.e., $

x + y = 144 \\

\Rightarrow y = 144 - x{\text{ }} \to {\text{(1)}} \\

$

Also, the ratio of these two numbers is 3:5 i.e., $\dfrac{x}{y} = \dfrac{3}{5}$

By cross multiplying the above equation, we get

$5x = 3y{\text{ }} \to {\text{(2)}}$

Putting the value of y from equation (1), equation (2) becomes

$

\Rightarrow 5x = 3\left( {144 - x} \right) \\

\Rightarrow 5x = 432 - 3x \\

\Rightarrow 8x = 432 \\

\Rightarrow x = 54 \\

$

Put x=54 in equation (1), the value of y is given by

$ \Rightarrow y = 144 - 54 = 90$

So, the assumed two numbers are 54 and 90. The smallest number is 54.

Hence, option A is correct.

Note: In this particular problem, we have developed two equations in two variables which can be easily solved by substitution method (used in the above solution) or elimination method (in this method we make the coefficient of all the variables same except one variable whose value will be eventually calculated by performing algebra to the given equation).

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