The Party Company produce two types of party pack. Their “Ghastly” party pack contains 10 balloons and 64 sweets. Their “Devilish” party pack contains 20 balloons and 16 sweets. Each week, the company must use 3000 balloons and 8000 sweets in their party packs.
How many “Ghastly” and “Devilish” party packs can the company make each week?
The company sells each “Ghastly” party pack at a profit of £1.20 and each “Devilish” party pack at a profit of £1.80. How much profit can they expect to make each week?
To find out how many "Ghastly" and "Devilish" party packs the company can make each week, we can set up a system of equations based on the given information.
Let's assume the number of "Ghastly" party packs produced is represented by the variable G, and the number of "Devilish" party packs produced is represented by the variable D.
Based on the number of balloons used in each pack, we can write the equation:
10G + 20D = 3000 --- Equation 1
Similarly, based on the number of sweets used in each pack, we can write the equation:
64G + 16D = 8000 --- Equation 2
Now, we can solve this system of equations to find the values of G and D.
Multiply Equation 1 by 6 and Equation 2 by 4 to make the coefficient of D the same in both equations:
60G + 120D = 18000 --- Equation 3
64G + 16D = 8000 --- Equation 2
Now, subtract Equation 2 from Equation 3:
(60G + 120D) - (64G + 16D) = 18000 - 8000
-4G + 104D = 10000 --- Equation 4
To simplify Equation 4, divide by 4:
-G + 26D = 2500
Now, multiply Equation 1 by 3 and Equation 2 by 8 to make the coefficient of G the same in both equations:
30G + 60D = 9000 --- Equation 5
64G + 16D = 8000 --- Equation 2
Subtract Equation 2 from Equation 5:
(30G + 60D) - (64G + 16D) = 9000 - 8000
-34G + 44D = 1000 --- Equation 6
Now, solve this system of equations consisting of Equation 4 and Equation 6. There are a few methods to solve this system, such as substitution or elimination. Let's use the elimination method.
Multiply Equation 6 by 2:
-68G + 88D = 2000 --- Equation 7
Add Equation 7 to Equation 4:
(-G + 26D) + (-68G + 88D) = 2500 + 2000
-69G + 114D = 4500 --- Equation 8
Now, multiply Equation 8 by 4:
-276G + 456D = 18000
Now, subtract this equation from Equation 6 to eliminate D:
(-276G + 456D) - (-34G + 44D) = 18000 - 1000
-242G + 412D = 17000
Simplify the equation:
-121G + 206D = 8500 --- Equation 9
Now, solve Equations 8 and 9 using the elimination method. Multiply Equation 9 by 69 and Equation 8 by 121 to make the coefficient of G the same:
(69)(-121G + 206D) = (69)(8500)
(121)(-69G + 114D) = (121)(4500)
-8349G + 14214D = 586500
-8349G + 13854D = 544500
Now, subtract these two equations to eliminate G:
(-8349G + 14214D) - (-8349G + 13854D) = 586500 - 544500
360D = 42000
Divide both sides of the equation by 360:
D = 42000/360
D = 116.67
Since the number of packs must be a whole number, we can round D to the nearest whole number: D ≈ 117
Now, substitute the value of D into Equation 4 to solve for G:
-G + 26(117) = 2500
-G + 3042 = 2500
-G = 2500 - 3042
-G = -542
G ≈ 542
Therefore, the company can make approximately 542 "Ghastly" party packs and 117 "Devilish" party packs each week.
To find the profit, we can multiply the number of packs of each type by the profit per pack and sum them up.
Profit from Ghastly party packs = G * £1.20 = 542 * £1.20
Profit from Devilish party packs = D * £1.80 = 117 * £1.80
Total profit = Profit from Ghastly party packs + Profit from Devilish party packs
So, to calculate the profit, simply multiply the quantities and prices:
Total profit = (542 * £1.20) + (117 * £1.80)