# Rs 6500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got Rs 30 less. Find the original number of persons.

Answer

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Hint: Find the share of each individual before and after decreasing the share by adding 15 persons. Use it with the given information and factorize to get the final answer.

Complete step-by-step answer:

Total amount of money = Rs 6500

Let the original number of people be x

Then the share of an individual would be $\dfrac{{6500}}{x}$

After adding 15 people , total number of people = x + 15

Share of an individual after increasing 15 people = $\dfrac{{6500}}{{x + 15}}$

But it is given that when 15 people were added each individual gets Rs 30 less

Therefore ,

$\dfrac{{6500}}{x} - \dfrac{{6500}}{{x + 15}} = 30$ ( subtracting both the equations )

$ \Rightarrow 6500\left( {\dfrac{{x + 16 - x}}{{x\left( {x + 15} \right)}}} \right) = 30$

$ \Rightarrow 650\left( {15} \right) = 3\left( {{x^2} + 15x} \right)$

$ \Rightarrow {x^2} + 15x - 3250 = 0$

$ \Rightarrow {x^2} + 65x - 50x - 3250 = 0$ (factorizing)

$ \Rightarrow x\left( {x + 65} \right) - 50\left( {x + 65} \right)$

$\left( {x + 65} \right)\left( {x - 50} \right) = 0$

Taking positive value of x , we get x = 50

Therefore,

Initial number of people were 50.

Note: The negative value that comes after factorization should be ignored in such problems. Simplification while solving lengthy equations saves time and decreases the chances of errors.

Complete step-by-step answer:

Total amount of money = Rs 6500

Let the original number of people be x

Then the share of an individual would be $\dfrac{{6500}}{x}$

After adding 15 people , total number of people = x + 15

Share of an individual after increasing 15 people = $\dfrac{{6500}}{{x + 15}}$

But it is given that when 15 people were added each individual gets Rs 30 less

Therefore ,

$\dfrac{{6500}}{x} - \dfrac{{6500}}{{x + 15}} = 30$ ( subtracting both the equations )

$ \Rightarrow 6500\left( {\dfrac{{x + 16 - x}}{{x\left( {x + 15} \right)}}} \right) = 30$

$ \Rightarrow 650\left( {15} \right) = 3\left( {{x^2} + 15x} \right)$

$ \Rightarrow {x^2} + 15x - 3250 = 0$

$ \Rightarrow {x^2} + 65x - 50x - 3250 = 0$ (factorizing)

$ \Rightarrow x\left( {x + 65} \right) - 50\left( {x + 65} \right)$

$\left( {x + 65} \right)\left( {x - 50} \right) = 0$

Taking positive value of x , we get x = 50

Therefore,

Initial number of people were 50.

Note: The negative value that comes after factorization should be ignored in such problems. Simplification while solving lengthy equations saves time and decreases the chances of errors.

Last updated date: 03rd Oct 2023

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