algebra word problem?
United Home investments makes 20% profit on it's good grade of vinyl siding, 30% of it's profit on it's better grade, and 60% profeit on it's best grade. So far this year , United Home Investments had $40,000 in sales of good siding and $50,000 in sales of better siding. The companies goal s to have at least an overall profit of 50% of the total sales. What would the sales figures be to reach it's goal.
The answer must be detailed here to help me with my real homework assignments. A table would be great-thanks in advance for your help.
Comments
Let n be the amount of sales (in thousands) of best grade siding.
$40k * 20% = $8k
$50k * 30% = $15k
n * 60% = 0.6n
Total profit will be 8 + 15 + 0.6n
Total sales will be 40 + 50 + n
The profit on the siding should equal 50% of the total sales.
8 + 15 + 0.60n = 0.50 (40 + 50 + n)
Now solve for n:
Simplifying:
23 + 0.6n = 45 + 0.5n
Subtract 0.5n from both sides:
23 + 0.1n = 45
Subtract 23 from both sides:
0.1n = 22
Multiply both sides by 10:
n = 220
So they must sell $220,000 of their 'best grade' siding to make a overall profit of 50%.
Here's a double check:
$40k * 20% = $8k
$50k * 30% = $15k
$220k * 60% = $132k
Total profit = 8 + 15 + 132 = 155
Total sales = 40 + 50 + 220 = 310
155 / 310 = 0.5 = 50%
So the answer is they must sell $220,000 of best grade siding.
Sales % Profit
GoodGrade $40,000 20 $8,000
Better Grade $50,000 30 $15,000
Best Grade x 60 $)0.6x
Totals $90.000+x 50 $23,000+.6x
So .5(90,000+x) =23,000 +.6x
45,000 +.5x =23,000 +.6x
22,000 =.1x
220,000 = x
Total Sales = $310,000
Total profit = 23,000 + 132,000 = $155,000 which is 50% of sales.
$40,000 * .2 = $8000 (profit so far from good grade)
$50,000 * .3 = $15000 (profit so far from better grade)
E * .6 = ? (profit so far from best grade)
$8000 + $15000 + (E * .6) = .5 ($40000 + $50000 + E)
or re-written as
$23000 + .6E = .5 ($90000 + E)
Derived from the fact that the sum of the profits must = 1/2 the total sales. So if we divide both sides by .5 we now have:
$46000 + 1.2E = $90000 + E
Subtract $46000 from both sides:
1.2E = $44000 + E
Subtract 1E from both sides:
.2E = $44000
Divide both sides by .2:
E= $220000
To check:
$40000 * .2 = $8000
$50000 * .3 = $15000
$220000 * .6 = $132000
+ ____________________
total $310000 $155000
and this works since $155000 is 1/2 of $310000
I couldn't work it out, but I'd say the first answer is right