At the beginning of 2002, sunil had four dozen goats with him. he increased this stock by x %. at the end of 2002, he sold off y % of his stock. at the beginning of 2003 he again increased his stock by x % and and at the end of 2003, he again sold off y % of his stock. There were no other changes in the number of goats. At the end of 2003 he had five dozen goats with him after his sales which of the following always hold true ?
1 - X > 100y / 100 -y
2 - 100y / 100-y > x > y
3 - X < y
4 - 100y / 100-y < x < y
Question
Answer give m
To find out which of the given options always hold true, we need to analyze the situation step by step.
Let's break down the problem into the following steps:
Step 1: At the beginning of 2002, Sunil had four dozen goats, which means he had 4 x 12 = 48 goats.
Step 2: Sunil increased his stock by x %. To calculate the increase, we need to find x% of 48. To get the value, we multiply 48 by (x/100): (48 * x) / 100.
So now, Sunil has 48 + (48 * x / 100) goats.
Step 3: At the end of 2002, Sunil sold off y % of his stock. To calculate the sale, we need to find y% of the stock after the increase (from step 2). So we multiply the total stock from step 2 by (y/100): (48 + (48 * x / 100)) * (y / 100).
Now, the number of goats Sunil has at the beginning of 2003 is (48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100)).
Step 4: Sunil increased his stock by x % at the beginning of 2003. To calculate the increase, we multiply the stock from step 3 by (x/100): [(48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100))] * (x / 100).
So now, Sunil has [(48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100))] + [((48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100))) * (x / 100)] goats.
Step 5: At the end of 2003, Sunil sold off y % of his stock. To calculate the sale, we need to find y% of the stock after the second increase (from step 4). So we multiply the total stock from step 4 by (y/100): [((48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100))) + [((48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100))) * (x / 100)]] * (y/100).
Finally, we have the number of goats Sunil has at the end of 2003: [((48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100))) + [((48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100))) * (x / 100)]] - [((48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100))) + [((48 + (48 * x / 100)) - ((48 + (48 * x / 100)) * (y / 100))) * (x / 100)]] * (y/100).
Now, let's compare the options with this expression and see which ones always hold true.
Option 1: X > 100y / (100 - y).
Option 2: 100y / (100 - y) > x > y.
Option 3: X < y.
Option 4: 100y / (100 - y) < x < y.
After comparing the expressions, we find that Option 2, "100y / (100 - y) > x > y," always holds true. Therefore, that is the correct answer.
To solve this problem, we can break it down into smaller steps.
Step 1: Calculate the number of goats Sunil had at the end of 2002.
Since Sunil had four dozen goats at the beginning of 2002, which is equal to 4 * 12 = 48 goats.
Step 2: Calculate the increase in stock by x%.
To find the new stock at the end of 2002, we multiply the initial stock by (1 + x/100).
So the new stock at the end of 2002 is 48 * (1 + x/100).
Step 3: Calculate the stock sold off at the end of 2002, y%.
To find the stock sold off at the end of 2002, we multiply the new stock at the end of 2002 by y/100.
So the stock sold off at the end of 2002 is (48 * (1 + x/100)) * (y/100).
Step 4: Calculate the stock at the beginning of 2003.
To find the stock at the beginning of 2003, we subtract the stock sold off at the end of 2002 from the new stock at the end of 2002.
So the stock at the beginning of 2003 is (48 * (1 + x/100)) - ((48 * (1 + x/100)) * (y/100)).
Step 5: Calculate the new stock at the end of 2003.
To find the new stock at the end of 2003, we multiply the stock at the beginning of 2003 by (1 + x/100).
So the new stock at the end of 2003 is [(48 * (1 + x/100)) - ((48 * (1 + x/100)) * (y/100))] * (1 + x/100).
Step 6: Calculate the stock sold off at the end of 2003, y%.
To find the stock sold off at the end of 2003, we multiply the new stock at the end of 2003 by y/100.
So the stock sold off at the end of 2003 is [[(48 * (1 + x/100)) - ((48 * (1 + x/100)) * (y/100))] * (1 + x/100)] * (y/100).
Step 7: Calculate the stock at the end of 2003.
To find the stock at the end of 2003, we subtract the stock sold off at the end of 2003 from the new stock at the end of 2003.
So the stock at the end of 2003 is [[(48 * (1 + x/100)) - ((48 * (1 + x/100)) * (y/100))] * (1 + x/100)] - [[(48 * (1 + x/100)) - ((48 * (1 + x/100)) * (y/100))] * (1 + x/100)] * (y/100).
Step 8: Simplify the expression for the stock at the end of 2003.
By combining like terms and simplifying, we get:
48 * ((1 + x/100) - (1 + x/100) * (y/100) + (1 + x/100) * (y/100) - (1 + x/100) * (y^2/10000))
Step 9: Simplify further.
By further simplification, we get:
48 * (1 - (1 + x/100) * (y^2/10000))
To determine which statement always holds true, let's examine the simplified expression:
1 - X > 100y / 100 - y
This statement is not equivalent to the simplified expression.
2 - 100y / 100-y > x > y
This statement is not equivalent to the simplified expression.
3 - X < y
This statement is not equivalent to the simplified expression.
4 - 100y / 100-y < x < y
This statement is equivalent to the simplified expression.
Therefore, the correct answer is option 4 - 100y / 100-y < x < y.