The net profit margin ratio for two companies is modelled by 5x/3x-3 for company A and by 8x/3x-1 for company B, where x is the number of items sold in thousands. Sam is trying to decide in which company he should buy stocks, so he is going to compare the ratios. He has researched both companies, and he know that each provides over 25000 items per month.
a) Would the two companies ever have the same ratio, and is so, at what levels of sales?
b) Which company has a high ratio for 30 000 items sold? Find the two ratios for both companies to the nearest thousand of a unit.
c) which company should he invest in if he wants to go with the company that he has a higher ratio most of the time?
a)
if A = B then
5x/(3x-3) = 8x/(3x-1)
24x^2 - 24x = 15x^2 - 5x
9x^2 - 19x = 0
x(9x - 19) = 0
x = 0 , reject since no sales is not an option
or
x = 19/9 thousands = appr 2111
at 2111 sales, the ratios would be the same
b) sub x = 30000 into each ratio to see which gives the higher value
c) what do you think??
a) To find the levels of sales at which the two companies have the same ratio, we set the two ratios equal to each other and solve for x:
5x/(3x-3) = 8x/(3x-1)
Cross multiplying:
5x(3x-1) = 8x(3x-3)
Expanding:
15x^2 - 5x = 24x^2 - 24x
Combining like terms:
9x^2 - 19x = 0
Factoring out common factor x:
x(9x - 19) = 0
Setting each factor equal to zero:
x = 0 or 9x - 19 = 0
Since x represents the number of items sold in thousands, x cannot be zero. So, we solve the second equation:
9x - 19 = 0
9x = 19
x ≈ 19/9
Therefore, the two companies will have the same ratio when approximately 19,000 items are sold.
b) To find the ratios for both companies when 30,000 items are sold:
For Company A:
Ratio = (5x)/(3x - 3)
Ratio = (5(30))/(3(30) - 3)
Ratio ≈ 150/87
For Company B:
Ratio = (8x)/(3x - 1)
Ratio = (8(30))/(3(30) - 1)
Ratio ≈ 240/89
To the nearest thousandth of a unit, the ratios of Company A and Company B for 30,000 items sold are approximately 1.724 and 2.697, respectively.
c) To determine which company has a higher ratio most of the time, we need to compare the ratios for different levels of sales. Since the ratios are dependent on x, we'll analyze the ratios for x > 19/9, as that's the level of sales where the ratios are equal.
For values of x > 19/9 (approximately 2.111), the ratio for Company B (8x/(3x-1)) is greater than the ratio for Company A (5x/(3x-3)). Therefore, if Sam wants to go with the company that has a higher ratio most of the time, he should invest in Company B.
a) To find the levels of sales at which the two companies have the same ratio, we need to set their ratios equal to each other and solve for x.
For company A: 5x / (3x - 3)
For company B: 8x / (3x - 1)
Setting them equal: 5x / (3x - 3) = 8x / (3x - 1)
To solve this equation, we can cross-multiply:
5x(3x - 1) = 8x(3x - 3)
Expanding the terms:
15x^2 - 5x = 24x^2 - 24x
Rearranging the equation by moving all terms to one side:
0 = 24x^2 - 15x^2 - 24x + 5x
Combining like terms:
0 = 9x^2 - 19x
Factoring out an x:
0 = x(9x - 19)
Setting each factor equal to zero:
x = 0 or 9x - 19 = 0
For x = 0, it means there are no items sold, which is not a meaningful scenario. So, we discard x = 0.
Solving 9x - 19 = 0 for x:
9x = 19
x = 19/9 ≈ 2.111
Therefore, the two companies have the same ratio at a sales level of approximately 2,111 items sold.
b) To find which company has a higher ratio for 30,000 items sold, we can substitute x = 30 into the given expressions for both companies.
For company A: Ratio = 5x / (3x - 3)
Substituting x = 30: Ratio_A = 5(30) / (3(30) - 3)
Ratio_A = 150 / (90 - 3)
Ratio_A = 150 / 87 ≈ 1.724
For company B: Ratio = 8x / (3x - 1)
Substituting x = 30: Ratio_B = 8(30) / (3(30) - 1)
Ratio_B = 240 / (90 - 1)
Ratio_B = 240 / 89 ≈ 2.697
To the nearest thousandth, the ratio for company A is approximately 1.724, and for company B is approximately 2.697.
c) If Sam wants to invest in the company that has a higher ratio most of the time, he should look at the historical sales data and compare the ratios for a range of sales levels for both companies. He can then choose the company that consistently has the higher ratio across various sales levels.