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?

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Super-Hard SAT Math Question 185 | Free SAT Prep | Aniko

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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: Co‌ntеn‍t by Аnikο.aі

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?

For mixture problems, write one equation for total mass and one equation for total amount of the key ingredient (here, copper). If a component amount is described in terms of another (like “4 more than 20% of $x$ ”), translate it into an algebraic expression immediately, substitute into both equations to get one variable, and solve the resulting linear equation. Prepа‌red‌ b⁠у Аn‍іko.ai

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Translate the relationship into an expression

If $x$ is the mass of the 55% alloy, write the pure copper mass using “4 more than 20% of $x$ .” аnikο.⁠аі/sаt

Use total mass to get the 25% alloy amount

The three component masses add to 300. Substitute your expressions for the 55% alloy and pure copper to express the 25% alloy mass in terms of $x$ .

Write a copper-balance equation

Set (copper from 55% alloy) + (copper from 25% alloy) + (pure copper) equal to $0.44\cdot 300$ .

Desmos Guide

Enter the copper-total expression

In Desmos, let $x$ be the mass of the 55% alloy, then enter

y=0.55x+0.25(296-1.2x)+(0.2x+4).

Enter the target copper amount

Enter

y=132

because 44% of 300 grams is 132 grams of copper.

Find the intersection

Click the intersection point. The $x$ -coordinate is the mass of the 55% copper alloy (in grams). Then match it to the answer choices. Wr⁠іttеn bу A‍nik‍о

Step-by-step Explanation

Define the variable and write each mass in terms of it

Let $x$ be the mass (in grams) of the 55% copper alloy.

Pure copper mass is 4 grams more than 20% of $x$ , so it is $0.2x+4$ .

Then the 25% alloy must make up the rest of the 300 grams: ani‍кo.a⁠i/ѕ‌аt

\text{25% alloy mass}=300-x-(0.2x+4)=296-1.2x.

Write the total copper equation

Total copper in the final alloy is

0.44(300)=132\text{ grams}.

Copper from each component is:

55% alloy: $0.55x$
25% alloy: $0.25(296-1.2x)$
pure copper: $0.2x+4$

So,

0.55x+0.25(296-1.2x)+(0.2x+4)=132.

Solve and interpret

Simplify the equation:

\begin{aligned} 0.55x+0.25(296-1.2x)+0.2x+4 &= 132 \\ 0.55x+74-0.3x+0.2x+4 &= 132 \\ 0.45x+78 &= 132 \\ 0.45x &= 54 \\ x &= 120 \end{aligned}

So, the mass of the 55% copper alloy used is 120 grams.

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Step-by-step Explanation

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