In a mixture problem, percentages tell you how much of one ingredient is in each batch. Track the aluminum in grams: work backward from the final alloy to find how much was in the earlier batch. Then write an equation for the two alloys that made that batch and solve it with a Desmos regression. Don't mistake grams of aluminum for grams of alloy.
Hints
- Hint 1
The final alloy contains aluminum from two sources: the earlier batch and the third alloy. Subtract the aluminum added by the third alloy, not its entire mass, to find what was in the earlier batch.
- Hint 2
If is the mass of the alloy, the alloy makes up the rest of the grams. How can you write that remaining mass using ?
- Hint 3
A mixture equation adds the aluminum from each starting alloy. Multiply each alloy's mass by its own aluminum percentage, then set their sum equal to the aluminum you found in the earlier batch.
Step-by-step
Work backward and balance the aluminum
Step 1Find the aluminum in the earlier batch
A percent means a share out of . The final alloy has of grams of aluminum, and the third alloy added of grams. Melting preserves the aluminum, so subtract the amount added. Type in Desmos; it gives grams in the earlier batch. Balance grams of aluminum, not grams of alloy.
- Step 2
Write the other alloy's mass
Let be the mass in grams of the alloy. Both starting alloys together weigh grams, so the alloy weighs grams. Using for both would wrongly make their masses equal.
- Step 3
Write an equation for the aluminum
The alloy supplies grams of aluminum because means of its mass. The alloy supplies grams. Together they supply the grams found in the earlier batch, so the mixture equation is
- Step 4
Solve for the mass requested
Type in Desmos. Replace with and with so Desmos finds the value that makes both sides equal. Under PARAMETERS, it reports . Since is the mass of the alloy, the metalworker used grams of it. Choice C.