at the beginning of the year, the owner of a jewel shop raised the prices of all the jewel in his shop by x% and lowered then by x%. The price of one jewel after this up and down cycle reduced by rs 100. the owner carried out the same procedure after a month. after this second up down cycle the price of that jewel was rs 2304. find the original price of that jewel.?
Please help in easy and simplified way
at the beginning of the year, the owner of a jewel shop raised the prices of all the jewel in his shop by x% and lowered then by x%. The price of one jewel after this up and down cycle reduced by rs 100. the owner carried out the same procedure after a month. after this second up down cycle the price of that jewel was rs 2304. find the original price of that jewel.?
But i have seen solutions which posted by your faculty but i didn't understand can you do in simplified way please help
If there is an easier way I do not know it.
To solve this problem, let's break it down step by step:
Step 1: Let's assume the original price of the jewel is P.
Step 2: According to the problem, the owner raised the price by x% and then lowered it by x%. This means:
- The first price increase by x% would be: P + (x/100) * P = P + Px/100
- The price decrease by x% from the increased price would be: (P + Px/100) - (x/100) * (P + Px/100) = (P + Px/100) - (Px/100 + Px^2/10000) = P + Px/100 - Px/100 - Px^2/10000 = P - Px^2/10000
Step 3: According to the problem, after the first up and down cycle, the price reduced by Rs. 100. So we have the equation:
P - Px^2/10000 - 100 = P - 100 - Px^2/10000 = New Price
Step 4: After a month, the owner carried out the same procedure again. This means the new price would be increased by x% and then decreased by x%. Using the same logic as step 2, the new price after this cycle would be:
New Price + Px/100 - Px^2/10000 - Px/100 + Px^2/10000 = New Price
Step 5: According to the problem, after the second up and down cycle, the price was Rs. 2304. So we have the equation:
New Price - 2304 = 0
Step 6: Now we can solve the equations:
P - 100 - Px^2/10000 = P
New Price + Px/100 - Px^2/10000 - Px/100 + Px^2/10000 = New Price
New Price - 2304 = 0
Step 7: Simplify the equations:
-100 - Px^2/10000 = 0
2Px/10000 = 2304
Step 8: Solve for x:
-Px^2/10000 = 100
Px^2 = -100 * 10000
Px^2 = 1000000
x^2 = 1000000/P
x = sqrt(1000000/P)
Step 9: Substitute the value of x into the second equation:
2Px/10000 = 2304
2P * sqrt(1000000/P) / 10000 = 2304
20 * sqrt(1000000/P) = 2304
sqrt(1000000/P) = 2304/20
sqrt(1000000/P) = 115.2
Step 10: Square both sides of the equation:
1000000/P = (115.2)^2
1000000/P = 13270.4
Step 11: Solve for P:
P = 1000000/13270.4
P ≈ 75.41
Therefore, the original price of the jewel is approximately Rs. 75.41.
raised by (1+x)
lowered by (1-x)
p - p (1+x)(1-x) = 100
100/p = 1 - [ 1-x^2]
100/p = x^2 or p =100/x^2
then
second p = 2304 = [100/x^2](1-x^2)
2304 = 100/x^2 -100
2204 = 100/x^2
22.04 = 1/x^2
x = .213
so 21.3 %