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
To solve this problem, let's break it down step-by-step.
Step 1: Find the total number of goats at the beginning of 2002.
Since Sunil had four dozen goats, which is 4 x 12 = 48 goats.
Step 2: Calculate the increase in stock in 2002.
Sunil increased his stock by x% in 2002, which means he added x/100 * 48 = 0.48x goats to his stock.
Step 3: Find the number of goats at the end of 2002.
The total number of goats at the end of 2002 is 48 + 0.48x goats.
Step 4: Calculate the amount of goats sold in 2002.
Sunil sold y% of his stock, which is y/100 * (48 + 0.48x) = 0.48yx + 0.12y goats.
Step 5: Find the number of goats remaining at the beginning of 2003.
The number of goats at the beginning of 2003 is (48 + 0.48x) - (0.48yx + 0.12y) = 48 - 0.48yx + 0.48x - 0.12y = 48 + 0.48(x - yx) - 0.12y
Step 6: Calculate the increase in stock in 2003.
Sunil increased his stock by x% in 2003, which means he added x/100 * (48 + 0.48(x - yx) - 0.12y) = (0.48x - 0.48yx + 0.12y) * x/100 goats.
Step 7: Find the number of goats at the end of 2003.
The total number of goats at the end of 2003 is (48 + 0.48(x - yx) - 0.12y) + (0.48x - 0.48yx + 0.12y) * x/100 goats.
Step 8: Set up the equation for the number of goats at the end of 2003.
The equation becomes (48 + 0.48(x - yx) - 0.12y) + (0.48x - 0.48yx + 0.12y) * x/100 = 5 x 12.
Step 9: Simplify the equation.
48 + 0.48(x - yx) - 0.12y + (0.48x - 0.48yx + 0.12y) * x/100 = 60.
48 + 0.48x - 0.48yx - 0.12y + (0.0048x - 0.0048yx + 0.0012y) = 60.
48 + 0.48x - 0.48yx - 0.12y + 0.0048x - 0.0048yx + 0.0012y = 60.
Step 10: Combine like terms.
0.48x + 0.0048x - 0.48yx - 0.0048yx - 0.12y + 0.0012y = 12.
0.4848x - 0.4848yx - 0.1188y + 0.0012y = 12.
0.4848x - 0.6060yx - 0.1176y = 12.
Step 11: Divide both sides by 0.4848.
x - 1.25yx - 0.2424y ≈ 24.79.
Now let's analyze the given options:
1 - X > 100y / 100 - y.
This option is not always true as it depends on the values of x and y.
2 - 100y / 100 - y > x > y.
This option is also not always true as it depends on the values of x and y.
3 - X < y.
This option cannot be determined based on the given information. It may or may not be true.
4 - 100y / 100-y < x < y.
This option is also not always true as it depends on the values of x and y.
Therefore, none of the options always hold true based on the given information.
To solve this problem, let's break it down step by step:
Step 1: Convert four dozen goats to units
Since a dozen is equal to 12, four dozen goats would be equal to 4 x 12 = 48 goats.
Step 2: Calculate the increase in stock by x%
The increase in stock would be given by:
New stock = 48 + (x/100) * 48
Step 3: Calculate the stock after selling off y%
The stock after selling off y% would be given by:
Final stock = New stock - (y/100) * New stock
Step 4: Calculate the stock at the beginning of 2003 after increasing stock by x%
The stock at the beginning of 2003 after the increase would be given by:
New stock = Final stock + (x/100) * Final stock
Step 5: Calculate the stock at the end of 2003 after selling off y%
The stock at the end of 2003 after selling off y% would be given by:
Final stock = New stock - (y/100) * New stock
Step 6: Convert the final stock to dozens
To convert the final stock to dozens, divide the total number of goats by 12:
Final stock in dozens = Final stock / 12
Now, let's analyze the given options:
1 - X > 100y / 100 -y
This equation does not accurately represent the relationship between x and y based on the given problem, so it is not always true.
2 - 100y / 100-y > x > y
This equation also does not accurately represent the relationship between x and y based on the given problem, so it is not always true.
3 - X < y
This equation states that x is always less than y. Since x represents the increase in stock percentage and y represents the stock sold percentage, it is not always the case that the increase in stock will be less than the stock sold. Therefore, this option is not always true.
4 - 100y / 100-y < x < y
This equation represents the correct relationship between x and y. Since x represents the increase in stock percentage and y represents the stock sold percentage, it is always true that the increase in stock percentage will be greater than the percentage of stock sold and less than the total stock sold percentage. Therefore, option 4 is always true.
So, the answer is option 4 - 100y / 100-y < x < y.
consider an example: a 10% gain and then a 10% loss is
1.1 * 0.9 = .99 = a 1% loss
so, you need x > y
Only one of the choices above includes that assertion