Question 57·Hard·Ratios, Rates, Proportional Relationships, and Units
A manufacturer produces a 120-kilogram batch of a metal alloy that must contain metals A, B, and C in the weight ratio 5 : 3 : 2. Because of supply limits, only 80% of the required mass of metal A is available, and the remaining mass that would have been metal A is substituted with metal D.
The costs per kilogram are as follows:
| Metal | Cost per kilogram |
|---|---|
| A | $25 |
| B | $20 |
| C | $15 |
| D | $18 |
What is the total cost, in dollars, of producing the 120-kilogram batch? Round your answer to the nearest whole dollar.
For mixture and ratio problems with costs, first convert the ratio into actual amounts by dividing the total quantity by the sum of the ratio parts, then multiplying by each part. Next, carefully apply any percentage constraints (like “80% available”) to adjust those amounts and introduce any substituted materials, checking that the adjusted masses still add up to the given total. Finally, multiply each component’s mass by its cost per unit and add the results, watching your arithmetic and units so you do not lose or double-count any pieces of the mixture.
Hints
Turn the ratio into actual kilogram amounts
First, add the parts of the ratio and use the total mass of 120 kg to find how many kilograms correspond to one "part" of the ratio.
Figure out how much of metal A is missing
Once you know how many kilograms of A are needed from the ratio, use the 80% availability to find how many kilograms of A you actually have and how many kilograms must be replaced by metal D.
Translate masses into costs
For each metal, multiply the mass (in kilograms) by the cost per kilogram from the table, then add all those costs together to get the total.
Desmos Guide
Confirm the mass per ratio part
In Desmos, type 120/10 to verify that each part of the 5:3:2 ratio corresponds to 12 kilograms.
Check the original required masses of A, B, and C
Enter 5*12, 3*12, and 2*12 on separate lines to confirm the original masses of metals A, B, and C before any substitution.
Verify the available amount of metal A and the amount replaced by D
Type 0.8*60 to confirm the available mass of A, and then 60 - (0.8*60) to confirm how many kilograms must be replaced by metal D.
Compute the total cost
On a new line, enter 48*25 + 36*20 + 24*15 + 12*18. The value that Desmos displays for this expression is the total cost in dollars of producing the batch.
Step-by-step Explanation
Convert the ratio into actual masses of A, B, and C
The ratio of A:B:C is 5:3:2, so the total number of ratio parts is .
Each "part" of the alloy has mass kilograms.
So the original (desired) masses are:
- Metal A: kg
- Metal B: kg
- Metal C: kg
Apply the 80% availability to metal A and find the amount of D
Only 80% of the required mass of metal A is available.
The required mass of A was 60 kg, so the available amount is kg.
The missing mass of A is replaced by metal D:
- Missing A (and thus D added): kg
So the actual masses used are:
- A: 48 kg, B: 36 kg, C: 24 kg, D: 12 kg.
Compute the cost contributed by each metal
Multiply each metal's mass by its cost per kilogram:
- Metal A:
- Metal B:
- Metal C:
- Metal D:
Add the costs to find the total cost of the batch
Add all the individual costs:
Then .
So, the total cost of producing the 120-kilogram batch is \2{,}496$, which is already a whole dollar amount.