Barges from ports X and Y went to cities A and B. X sent 32 barges and Y sent 8. City A needs 22 barges and B needs 18. Shipping costs $220 from X to A, $300 from X to B, $400 from Y to A, and $180 from Y to B. $9280 was spent. How many barges went where?
Well, it looks like we have a barge-ain of a problem here! Let's get into the barge-math-ics and solve it.
Let's start by figuring out how many barges went from port X to city A. Each trip from X to A costs $220, and since a total of $9280 was spent, we can divide $9280 by $220 to get the total number of trips. That gives us 9280 / 220 = 42 trips.
Since there were 32 barges sent from port X, it means that 32 barges went to city A. Now, let's move on to city B.
Each trip from X to B costs $300, so if we divide $9280 by $300, we'll find out how many trips were made. 9280 / 300 = 30.9333...
Since we can't have a fraction of a trip, we'll round down to the nearest whole number. That means there were 30 trips from X to B.
Now it's time to figure out how many barges went from Y to city A and B. From port Y, 8 barges were sent, and the cost to ship to A is $400 and to B is $180.
Dividing the total amount spent, $9280, by the cost to ship to A, $400, we get 9280 / 400 = 23 trips from Y to A.
Dividing the total amount spent, $9280, by the cost to ship to B, $180, we get 9280 / 180 = 51.5555...
Rounding down to the nearest whole number, we get 51 trips from Y to B.
So, to summarize:
- 32 barges went from port X to city A.
- 30 barges went from port X to city B.
- 23 barges went from port Y to city A.
- 51 barges went from port Y to city B.
I hope that helps clear things up! If not, I'm here to entertain with more barge puns.
Let's solve the problem step-by-step:
Step 1: Determine the number of barges sent from X and Y to cities A and B.
Let's assume the number of barges sent from X to A as "x", the number of barges sent from X to B as "y", the number of barges sent from Y to A as "z", and the number of barges sent from Y to B as "w".
From the given information, we can create the following equations:
x + z = 32 ... (Equation 1)
y + w = 8 ... (Equation 2)
x + y = 22 ... (Equation 3)
z + w = 18 ... (Equation 4)
Step 2: Solve the system of equations.
From Equations 1 and 3, we can solve for x and z by subtracting Equation 3 from Equation 1:
(x + z) - (x + y) = 32 - 22
z - y = 10 ... (Equation 5)
From Equations 2 and 4, we can solve for w and z by subtracting Equation 2 from Equation 4:
(z + w) - (y + w) = 18 - 8
z - y = 10 ... (Equation 6)
From Equations 5 and 6, we can see that z - y = 10. This means that the number of barges sent from X to A will be equal to the number of barges sent from Y to B. Let's assume this common value as "a".
z = a
y = a
Substituting these values in Equations 1 and 2:
x + a = 32 ... (Equation 7)
a + w = 8 ... (Equation 8)
From Equation 7, we get:
x = 32 - a ... (Equation 9)
Substituting the values of x, y, and z in Equation 8:
a + w = 8
Substituting the value of z = a:
a + w = 8 ... (Equation 10)
Step 3: Find the values of x, y, z, and w.
From Equation 9, substitute the value of x back into Equation 3:
(32 - a) + y = 22
Simplifying the equation:
y = 22 - 32 + a
y = a - 10 ... (Equation 11)
From Equation 10, we can substitute the value of w as:
w = 8 - a ... (Equation 12)
Step 4: Calculate the total cost.
The total cost is given as $9280. We can calculate the total cost using the shipping costs between ports and cities:
$220 * x (X to A) + $300 * y (X to B) + $400 * z (Y to A) + $180 * w (Y to B) = $9280
Substituting the values of x, y, z, and w:
$220 * (32 - a) + $300 * (a - 10) + $400 * a + $180 * (8 - a) = $9280
Simplifying and solving for a:
$7040 - $220a + $300a - $3000 + $400a + $1440 - $180a = $9280
$7040 + $300a = $9280
$300a = $9280 - $7040
$300a = $2240
a = $2240 / $300
a = 7
Step 5: Calculate the values of x, y, z, and w using the value of a.
From Equation 11, substitute the value of a:
y = 7 - 10
y = -3
From Equation 12, substitute the value of a:
w = 8 - 7
w = 1
Substituting the values of a, x, y, z, and w into Equations 7 and 9:
x + a = 32 ... (Equation 7)
(32 - a) + a = 32
x = 32 - 7
x = 25
Substituting the values into Equation 1 and 2:
25 + z = 32 ... (Equation 1)
z = 32 - 25
z = 7
Now we have determined the values of x, y, z, and w:
x = 25
y = -3
z = 7
w = 1
Step 6: Answer the question.
Based on our calculations, the number of barges that went where are as follows:
From X to A: 25 barges
From X to B: -3 barges (which indicates that 3 barges were sent from B to X)
From Y to A: 7 barges
From Y to B: 1 barge
Therefore, 25 barges went from X to A, 3 barges went from B to X, 7 barges went from Y to A, and 1 barge went from Y to B.
To solve this problem, let's break it down step by step.
1. Let's denote the number of barges sent from port X to city A as XA, the number of barges sent from port X to city B as XB, the number of barges sent from port Y to city A as YA, and the number of barges sent from port Y to city B as YB.
2. We are given that XA + YA = 32 (the total number of barges sent from port X and port Y is 32) and XB + YB = 8 (the total number of barges sent from port X and port Y is 8). These are our first two equations.
3. We are also given that XA + XB = 22 (city A needs 22 barges) and YA + YB = 18 (city B needs 18 barges). These are our next two equations.
4. We have the shipping costs as $220 from X to A, $300 from X to B, $400 from Y to A, and $180 from Y to B. We can use this information to create another two equations:
- 220 * XA + 400 * YA = total cost for shipping from X to A
- 300 * XB + 180 * YB = total cost for shipping from X to B
5. We have the total cost for shipping as $9280, so we can create our last equation:
220 * XA + 400 * YA + 300 * XB + 180 * YB = 9280
Now we have a system of linear equations that we can solve to find the values of XA, XB, YA, and YB:
XA + YA = 32 ---(1)
XB + YB = 8 ---(2)
XA + XB = 22 ---(3)
YA + YB = 18 ---(4)
220 * XA + 400 * YA = Total cost for shipping from X to A ---(5)
300 * XB + 180 * YB = Total cost for shipping from X to B ---(6)
220 * XA + 400 * YA + 300 * XB + 180 * YB = 9280 ---(7)
Solving this system of equations will give us the number of barges that went from each port to each city.
Please give me a moment to solve the equations for you.