To find the total weight of Shipment 3, we need to set up equations based on the information given in the first two shipments.
Let:
- \( s \) = weight of each small crate (in pounds)
- \( l \) = weight of each large crate (in pounds)
From Shipment 1, we have the equation: \[ 50s + 32l = 4170 \quad \text{(1)} \]
From Shipment 2, we have the equation: \[ 25s + 40l = 3525 \quad \text{(2)} \]
Now, we can solve these two equations simultaneously to find the values of \( s \) and \( l \).
First, we can multiply Equation (2) by 2 to make the coefficients of \( s \) in the two equations equal: \[ 50s + 80l = 7050 \quad \text{(3)} \]
Next, we can subtract Equation (1) from Equation (3): \[ (50s + 80l) - (50s + 32l) = 7050 - 4170 \] This simplifies to: \[ 48l = 2880 \] Now, divide both sides by 48: \[ l = 60 \]
Now that we have \( l \), we can substitute \( l = 60 \) back into Equation (1) to find \( s \): \[ 50s + 32(60) = 4170 \] Calculating \( 32 \times 60 \): \[ 50s + 1920 = 4170 \] Now, subtract 1920 from both sides: \[ 50s = 4170 - 1920 \] \[ 50s = 2250 \] Now divide by 50: \[ s = 45 \]
Now that we have \( s = 45 \) and \( l = 60 \), we can find the weight of Shipment 3, which contains 30 small crates and 18 large crates: \[ \text{Weight of Shipment 3} = 30s + 18l \] Substituting in the values we found: \[ = 30(45) + 18(60) \] Calculating each term: \[ = 1350 + 1080 \] Now, add them together: \[ = 2430 \]
Thus, the total weight of Shipment 3 is 2430 pounds.
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