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# There are three numbers in the ratio $2:3:5$ and the sum of the numbers is $800$. Find the three numbers.

Last updated date: 20th Mar 2023
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Hint: In the given question, we are provided with the ratio of three numbers. The sum of these three numbers is also given to us in the question itself. We first assume these numbers in terms of a single variable with the help of ratio given to us and then add up these numbers to match the sum. Solving the equation thus formed helps us to find the value of the variable and the three numbers.
Method of transposition involves doing the exact same mathematical thing on both sides of an equation with the aim of simplification in mind. This method can be used to solve various algebraic equations like the one given in question with ease.

Complete step by step solution:
The ratio of the three numbers is $2:3:5$.
Let us assume the three numbers to be $2x$, $3x$ and $5x$. This is also in accordance with the ratio of the three numbers provided to us in the question itself.
Also, we are given that the sum of the three numbers is $800$.
So, we first add up all the three numbers and then equate the result with the sum of the numbers given to us.
So, we get, $2x + 3x + 5x = 800$
Simplifying the equation further, we get,
$\Rightarrow 10x = 800$
Now, In order to find the value of variable x, we need to isolate x from the rest of the parameters.
Hence, dividing both sides of the equation by $10$, we get,
$\Rightarrow x = 80$
Hence, we get the value of variable x as $80$.
Now, we assume the three numbers to be $2x$, $3x$ and $5x$. The numbers can be found out easily by substituting the value of the variable x in the expressions.
Therefore, we get the three numbers as: $160$, $240$ and $400$.

Note:
There is no fixed way of solving a given algebraic equation. Algebraic equations can be solved in various ways. Method of transposition involves doing the exact same thing on both sides of an equation with the aim of bringing like terms together and isolating the variable or the unknown term in order to simplify the equation and finding the value of the required parameter.