Question

a metalsmith is mixing two molten metals, each containing different percentages of silver. the table shows the amount of each molten metal used.
silver

	total mass (g)	silver percent	total
90% silver	$x$	$0.90$	$0.9x$
mixture	$15$	$0.75$	$t$

which two expressions are both equivalent to $t$, the total number of grams of silver in the mixture?

• $(15)(0.75)$ and $0.7(15 - x) + 0.9x$
• $(15)(0.75)$ and $0.7(15 - x)(0.9x)$
• $15 + 0.75$ and $0.7(15 - x) + 0.9x$
• $15 + 0.75$ and $0.7(15 - x)(0.9x)$

Explanation:

Step1: Analyze total silver from mixture

The mixture has a total mass of 15 g and 75% silver. So total silver \( t = 15\times0.75 \).

Step2: Analyze total silver from components

The two molten metals contribute \( 0.7(15 - x) \) (from 70% silver metal) and \( 0.9x \) (from 90% silver metal). So total silver \( t = 0.7(15 - x)+0.9x \).

Step3: Match with options

Check each option:

• First option: \( (15)(0.75) \) (from mixture) and \( 0.7(15 - x)+0.9x \) (from components) match.
• Second option: \( 0.7(15 - x)(0.9x) \) is incorrect (should be sum, not product).
• Third option: \( 15 + 0.75 \) is wrong (total silver is mass×percent, not mass + percent).
• Fourth option: Both expressions are wrong (sum error and product error).

Answer:

(15)(0.75) and \( 0.7(15 - x) + 0.9x \)