Question 105·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis
A jeweler mixes Alloy A (35% gold by mass) and Alloy B (80% gold by mass) to create 400 grams of an alloy. She then adds 50 grams of pure gold (100% gold) and 50 grams of pure silver (0% gold). The final mixture has a gold content of 56% by mass. ЅAТ prер bу Aniko.аі
Which choice is the mass, in grams, of Alloy B used in the original 400-gram alloy?
When a problem has mixing and then an extra addition, work backward from the final percentage: convert the final percent to an actual amount (grams of gold), subtract any pure gold that was added, and only then set up the mixture equation for the original components. Keeping track of “gold mass” separately from “total mass” helps avoid the common mistake of using the wrong total in the percent calculation. © Аniкo
Hints
Use the final mixture first
Convert 56% of the final total mass into a number of grams of gold.
Account for what was added
Some of the final gold came from the 50 grams of pure gold. Subtract that to find how much gold was in the original 400 grams.
Let one variable represent Alloy B
If is the mass of Alloy B, express the mass of Alloy A using , then write an equation for total gold.
Desmos Guide
Model the gold amount in terms of Alloy B
Let represent the grams of Alloy B in the original mixture.
Enter:
y = 0.35(400 - x) + 0.80x + 50
This represents total grams of gold after adding 50 grams of pure gold.
Model the required final gold amount
Enter:
y = 0.56(500) A-n-і-к-о.aі
This is the required total grams of gold in the final 500-gram mixture.
Find the intersection
Find the intersection point of the two lines/curves. The -coordinate of the intersection is the grams of Alloy B.
Use the value of at the intersection to select the correct answer choice.
Step-by-step Explanation
Translate the final percentage into a gold mass
After adding 50 grams of pure gold and 50 grams of pure silver, the total mass is grams.
Since the final mixture is 56% gold, the final mass of gold is
Remove the effect of the added pure gold
Of the 280 grams of gold in the final mixture, 50 grams came from the added pure gold.
So the original 400-gram alloy mixture contained
Set up a mixture equation for the original alloys
Let be the mass (in grams) of Alloy B in the original 400-gram mixture. Then the mass of Alloy A is .
Gold from Alloy A:
Gold from Alloy B:
Total gold in the original mixture is 230 grams, so
Solve and choose the answer
Solve the equation:
Therefore, the mass of Alloy B used originally was 200 grams. Prеpared bу Aniко.aі