Alloy X is produced by combining copper and zinc in a ratio by mass. Alloy Y is then made by mixing alloy X and additional pure copper in a ratio by mass (that is, 3 parts alloy X to 1 part copper). If alloy Y contains exactly grams of zinc, how many grams of copper does alloy Y contain?

Solution available with step-by-step explanation

SAT Ratios, Rates & Proportions Question 12 | Free SAT Prep | Aniko

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Question 12·Hard·Ratios, Rates, Proportional Relationships, and Units

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Alloy X is produced by combining copper and zinc in a $5:2$ ratio by mass.

Alloy Y is then made by mixing alloy X and additional pure copper in a $3:1$ ratio by mass (that is, 3 parts alloy X to 1 part copper).

If alloy Y contains exactly $210$ grams of zinc, how many grams of copper does alloy Y contain?

Your answer

(Express the answer as an integer)

For mixture and ratio problems, first convert each ratio into fractions of the whole (e.g., $5:2$ means $5/7$ copper and $2/7$ zinc) and carefully track where each component (like zinc) comes from. Use a variable for the common “part” size in layered ratios (such as 3:1 mixtures), set up an equation using the component whose total amount is given, solve for the part size, and then compute the amount of the quantity asked for by adding contributions from each source. This structured approach prevents mixing up ratios and keeps the algebra simple and fast on the SAT.

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Break down alloy X

Alloy X is in a $5:2$ ratio of copper to zinc. How many total parts is that, and what fraction of alloy X is zinc?

Track only the zinc first

In alloy Y, which ingredient(s) contain zinc: alloy X, pure copper, or both? Use a variable for the mass of one “part” in the 3:1 mixture, and write an expression for the zinc mass in alloy Y in terms of that variable.

Use the zinc equation to find the part size

Set your expression for zinc mass equal to 210 grams and solve for the size of one part. Once you know the part size, how many grams of copper come from the alloy X portion and how many from the added pure copper?

Desmos Guide

Use Desmos to find the mass of one part

In an expression line, enter 210*7/6. This corresponds to solving $\frac{6p}{7} = 210$ for $p$ (since $p = 210 \cdot \frac{7}{6}$ ). The output is the mass in grams of one “part” in the 3:1 mixture.

Use Desmos to compute total copper in alloy Y

Copper in alloy Y equals copper from alloy X plus copper from the extra pure copper part. That total is $\frac{5}{7}$ of the $3p$ grams from alloy X, plus $p$ grams from the pure copper part, which simplifies to $\frac{22}{7}p$ . In a new expression line, type (22/7) * (210*7/6) and read the numerical result, which is the total grams of copper in alloy Y.

Step-by-step Explanation

Understand the composition of alloy X

Alloy X is made by combining copper and zinc in a $5:2$ ratio by mass.

That means in every 7 total parts of alloy X, 5 parts are copper and 2 parts are zinc.
So the fraction of zinc in alloy X is $\dfrac{2}{7}$ , and the fraction of copper is $\dfrac{5}{7}$ .

Use zinc to find the size of one “part” in alloy Y’s mixture

Alloy Y is made from 3 parts alloy X and 1 part pure copper.

Let the mass of one part in this mixture be $p$ grams.

Then the mass of alloy X used is $3p$ grams.
All the zinc in alloy Y comes from alloy X (pure copper has no zinc).
Zinc in the $3p$ grams of alloy X is

\text{zinc mass} = \frac{2}{7} \cdot 3p = \frac{6p}{7}.

We are told alloy Y contains 210 grams of zinc, so

\frac{6p}{7} = 210.

Solve for $p$ :

6p = 210 \cdot 7 = 1470 \\[0.7em] p = \frac{1470}{6} = 245.

So each “part” in the alloy Y mixture has mass $p = 245$ grams.

Find the copper from alloy X inside alloy Y

Now find how much copper comes from the 3 parts of alloy X used in alloy Y.

The total mass of alloy X in alloy Y is $3p = 3 \cdot 245 = 735$ grams.
The fraction of copper in alloy X is $\dfrac{5}{7}$ .

So the mass of copper from alloy X in alloy Y is

\text{copper from alloy X} = \frac{5}{7} \cdot 735.

Compute this:

\frac{5}{7} \cdot 735 = 5 \cdot \frac{735}{7} = 5 \cdot 105 = 525.

So 525 grams of copper in alloy Y come from alloy X.

Add the extra pure copper to get total copper in alloy Y

Alloy Y also includes 1 part of pure copper, and each part has mass $p = 245$ grams.

Copper from alloy X: 525 grams
Copper from the extra pure copper part: 245 grams

Add them:

\text{total copper in alloy Y} = 525 + 245 = 770.

So alloy Y contains 770 grams of copper.

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Step-by-step Explanation

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Step-by-step Explanation