# Three times one number is equal to five times the other. If the sum of two numbers is 80, find the numbers.

A.25 and 55

B.50 and 30

C.45 and 35

D.60 and 20

Answer

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**Hint**: First we have to convert the word statement into a mathematical equation, here we get two simultaneous linear equations in two unknowns, solving these equations gives the values of the two numbers.

**Complete step by step solution:**

Let the first number be $ x $ and the second number be $ y $ .

Three times the first number is $ 3x $ .

Five times the second number is $ 5y $ .

It is given that three times first number is equal to five times the second number i.e. $ 3x = 5y $

Taking $ 5y $ to the left side we have, $ 3x - 5y = 0 $ ----- (1)

Also given that sum of both numbers is $ 80 $

i.e. $ x + y = 80 $ -----(2)

Equations (1) and (2) are Simultaneous linear equations with two unknowns $ x $ and $ y $ .

Now we will solve for $ x $ and $ y $ ,

First we will write both equations one below the other, and use method of elimination by equating coefficients to solve these linear equations,

$

3x - 5y = 0 \\

x + y = 80 \;

$

Multiply the first equation by the coefficient of $ x $ of the second equation and multiply the second equation by the coefficient of $ x $ of the first equation to make the coefficients of $ x $ as equal.

Means multiply first equation by 1 and second equation by 3 then we get,

$

3x - 5y = 0 \\

3x + 3y = 3 \times 80 \;

$

To cancel the variable $ x $ we need to subtract first equation from the second

$

(3x - 5y = 0) - (3x + 3y = 240) \\

\Rightarrow 0 - 5y - 3y = - 240 \\

\Rightarrow - 8y = - 240 \\

\Rightarrow y = \dfrac{{ - 240}}{{ - 8}} \;

$

Solving for $ y $ we get $ y = 30 $

Substituting the value of $ y $ in any one of the equations above we get the value of $ x $ .

Substituting $ y = 30 $ in $ x + y = 80 $ we get

$

x + 30 = 80 \\

\Rightarrow x = 80 - 30 \\

\Rightarrow x = 50 \;

$

Therefore the required two numbers are $ 50 $ and $ 30 $ .

Therefore the correct answer is option B.

**So, the correct answer is “Option B”.**

**Note**: Conversion of statements into mathematical equations is the very important step in solving these kinds of problems. Once we have obtained the set of equations we can use the above method of equating the coefficients or the method of substitution or the method of cross multiplication also. When the options are given for the problem we can use the trial and error method and substitute each option in the obtained equations to select the right option which can save time.

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