Question 185·200 Super-Hard SAT Math Questions·Algebra

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A 300-gram metal alloy is 44% copper by mass. The alloy is made by combining three materials:

some amount of an alloy that is 25% copper,
some amount of an alloy that is 55% copper, and
some amount of pure copper.

The mass of pure copper used was 4 grams more than 20% of the mass of the 55% copper alloy used.

Which choice is the mass, in grams, of the 55% copper alloy used?

A mixture equation tracks how many grams of copper each material contributes. Let the requested alloy’s mass be the unknown; the given comparison determines the pure copper mass, and the total determines the remaining mass. Then use a Desmos regression to solve the copper equation. Don’t add the percentages: each applies to a different mass.

Hints

Hint 1
Let $x$ be the mass of the 55% copper alloy. The pure copper weighs 4 grams more than 20% of that mass, so write its mass using $x$ before working out the third material.
Hint 2
The three material masses add to 300 grams. Once you’ve written the 55% alloy and pure copper masses, subtract both from 300 to get the mass of the 25% alloy.
Hint 3
For each alloy, its copper mass is its percent copper multiplied by its mass. Pure copper contributes all of its mass. What must those three copper masses add up to?

Step-by-step

Approach 1: Track the copper in each material
Step 1
Express the pure copper mass
Let $x$ be the mass, in grams, of the 55% copper alloy. The pure copper weighs 20% of that mass, plus 4 grams, so its mass is:
$0.20x+4.$
Step 2
Find the remaining alloy mass
All three materials together weigh 300 grams. Subtract the 55% alloy and the pure copper to get the 25% alloy’s mass:
$300-x-(0.20x+4).$
Step 3
Balance the grams of copper
An alloy’s copper percentage applies to that alloy’s mass. Pure copper is entirely copper, so it contributes $0.20x+4$ grams. The finished alloy contains 44% of all 300 grams as copper. Add grams of copper, not percentages of different materials. Set the contributions equal to the finished amount:
$0.25[300-x-(0.20x+4)]+0.55x+(0.20x+4)=0.44(300).$
Step 4
Solve for the requested mass
Type the equation into Desmos with $x_1$ in place of $x$ and $\sim$ in place of $=$ . Desmos uses $x_1$ as the value to find and reports $x_1=120$ under PARAMETERS. Since $x$ is the 55% alloy’s mass, that alloy weighs 120 grams. Choice C.

Approach 2: Count copper above a 25% baseline
Step 1
Find the baseline copper mass
Imagine that all 300 grams were 25% copper. Type $0.25(300)$ into Desmos; it prints $75$ . That’s the baseline copper mass before accounting for the materials that contain more copper.
Step 2
Account for copper above the baseline
Each gram of the 55% alloy has $0.55-0.25$ grams more copper than the baseline. Each gram of pure copper has $1-0.25$ grams more. Add those extras to $75$ and match the finished alloy’s copper mass:
$75+(0.55-0.25)x+(1-0.25)(0.20x+4)=0.44(300).$
Step 3
Solve the excess-copper equation
Keep the first Desmos line and type this equation with $x_1$ and $\sim$ . Under PARAMETERS, Desmos reports $x_1=120$ , so the 55% copper alloy weighs 120 grams. Choice C.

Question 185·200 Super-Hard SAT Math Questions·Algebra

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