(1) A business shipped 108 packages one day. Customers are charged $3.50 for each standard-delivery package and $7.50 for each express-delivery package. Total shipping charges for the day were $506.00. How many of each kind of package were shipped?
Standard:
Express:
(2) A department store sold 41 shirts one day. All short-sleeved shirts cost $12.00 each and all long-sleeved shirts cost $17.00 each. Total receipts for the day were $582.00. How many of each kind of shirt were sold?
Short Sleeve:
Long Sleeve:
x+y=108
3.5x+7.5y=506
1st eq * 3.5 =
3.5x+3.5y = 378
subtract from second equation
4y = 128
y=32
x=76 (32+y=108 | 108-32 = 76)
#2 done the same way as the first
x+y = 41 ---#1
12x + 17y = 582 ---> #2
#1 times 12
12x + 12y = 492
12x + 17y = 582
subtract them:
5y = 90
y = 18
then x = 23
23 short-sleeved and 18 long-sleeved shirts
To solve these types of problems, we can use a system of equations. Let's start with the first question:
(1) Let's assume the number of standard-delivery packages is represented by 'S' and the number of express-delivery packages is represented by 'E.'
We can set up two equations based on the given information:
Equation 1: S + E = 108 (since the total number of packages shipped is 108)
Equation 2: 3.50S + 7.50E = 506 (since the total shipping charges for the day were $506)
To solve this system of equations, we can use the method of substitution or elimination. Let's use substitution in this case.
From equation 1, we can solve for S in terms of E:
S = 108 - E
Now substitute this value of S into equation 2:
3.50(108 - E) + 7.50E = 506
Multiply out the terms:
378 - 3.50E + 7.50E = 506
Combine like terms:
4E = 128
Divide both sides by 4:
E = 32
Now we can substitute this value of E back into equation 1 to find S:
S + 32 = 108
Subtract 32 from both sides:
S = 76
So, 76 standard-delivery packages were shipped, and 32 express-delivery packages were shipped.
Standard: 76
Express: 32
Now let's move on to the second question:
(2) Similarly, let's assume the number of short-sleeved shirts is represented by 'SS' and the number of long-sleeved shirts is represented by 'LS.'
We can set up two equations based on the given information:
Equation 1: SS + LS = 41 (since the total number of shirts sold is 41)
Equation 2: 12(SS) + 17(LS) = 582 (since the total receipts for the day were $582)
Again, we can solve this system of equations using substitution.
From equation 1, we can solve for SS in terms of LS:
SS = 41 - LS
Now substitute this value of SS into equation 2:
12(41 - LS) + 17(LS) = 582
Multiply out the terms:
492 - 12LS + 17LS = 582
Combine like terms:
5LS = 90
Divide both sides by 5:
LS = 18
Now substitute this value of LS back into equation 1 to find SS:
SS + 18 = 41
Subtract 18 from both sides:
SS = 23
So, 23 short-sleeved shirts were sold, and 18 long-sleeved shirts were sold.
Short Sleeve: 23
Long Sleeve: 18