There are two distinct natural numbers. Each is first increased by 10 then by same percentage as each was increased for the first time. Each number finally results in 72 Find the difference between the numbers
Please explained in simplified way
What I don't understand is how two distinct numbers, operated on in identical ways, produce the same result...
I think the question is mangled.
Hi Vipul,
let us consider the first number as x
X is increased by 10 - x+10
Then the increased number(x+10) is increased by a % which is equal to the percentage in the first calculation:-
%increase in first calculation : (10/x).
So the number should be increased by
(x+10)+(x+10)*(10/x) = 72
Solving these we get x = 50 or 2, which are the numbers.
How can equal percentage increase be 10/x?
Why
why is there 10/x ?
original numbers : x and y
each is increased by 10 ---> x+10 and y+10
percentage increase for first time = 10/x and 10/y
so new number is x+10 + 10/x
(same thing for the y)
x + 10 + 10/x = 72
times x
x^2 + 10x + 10 = 72x
x^2 - 62x = -10
completing the square:
x^2 - 62x + 961 = -10 + 961
(x - 31)^2 = 951
x - 31 = ± √951
x = 31 ± √951 = 61.838.. or .1617
check:
original number ---- 61.838
after adding 10 ---- 71.838
percentage increase = 10/61.838 = .1617..
which added to 71.838 = 72
same is true for .1617
your two numbers are 31+√951 and 31-√951
BUT, it said they are natural numbers, so there is no solution
Either my interpretation is incorrect or you have a typo
Well, let me break this down for you in my own clownish way!
We have two numbers, let's call them x and y. They are increased by 10 each, so now we have x + 10 and y + 10.
And listen to this, they are increased by the SAME percentage as they were increased for the first time! So, let's say they were initially increased by p percent. That means, after the second increase, we have (x + 10)(1 + p/100) and (y + 10)(1 + p/100).
Now, the final result we want is 72 for both numbers. So we can write it like this:
(x + 10)(1 + p/100) = 72
(y + 10)(1 + p/100) = 72
To make things simpler, let's divide both equations by (1 + p/100):
x + 10 = 72 / (1 + p/100)
y + 10 = 72 / (1 + p/100)
Now, let's subtract both equations from each other:
(x + 10) - (y + 10) = (72 / (1 + p/100)) - (72 / (1 + p/100))
10 - 10 = 0, so that cancels out. And we are left with:
x - y = 0
Wait a minute! It means that the difference between the two numbers is ZERO! So, there you have it, the difference between the numbers is just a big fat zero! Isn't that clownishly funny?
To find the difference between the two numbers, we can set up an equation based on the given information.
Let's call the two numbers x and y.
According to the problem, each number is first increased by 10, resulting in x + 10 and y + 10.
Then, each number is increased by the same percentage as the first increase. Let's call this percentage p.
So, the new values of x and y after the second increase are (x + 10) + p(x + 10) and (y + 10) + p(y + 10) respectively.
Finally, we are told that both numbers result in 72 after these increases. Therefore, we can set up the following equation:
(x + 10) + p(x + 10) = 72
(y + 10) + p(y + 10) = 72
Now, we can solve these two equations simultaneously to find the values of x and y, and then calculate the difference between them.