Questions: Question 15 Fiona and Alice went shopping and spent the same amount of money. They each purchased a necklace and a pair of earrings. Fiona's earrings cost 2/3 as much as Alice's earrings. Alice paid 35 for her necklace, while Fiona paid 45 for hers. Choose the response that gives an equation that represents the situation and then gives the cost of Alice's earrings. Let x represent the cost of Alice's earrings. A) x+35=2/3 x+45; cost of Alice's earrings is 30 B) x+35=2/3(x+45); cost of Alice's earrings is 15 C) 2/3 x+35=x+45; cost of Alice's earrings is 30 D) x+2/3(35)=1/3 x+45; cost of Alice's earrings is 32.50

Question 15 Fiona and Alice went shopping and spent the same amount of money. They each purchased a necklace and a pair of earrings. Fiona's earrings cost 2/3 as much as Alice's earrings. Alice paid 35 for her necklace, while Fiona paid 45 for hers. Choose the response that gives an equation that represents the situation and then gives the cost of Alice's earrings. Let x represent the cost of Alice's earrings. A) x+35=2/3 x+45; cost of Alice's earrings is 30 B) x+35=2/3(x+45); cost of Alice's earrings is 15 C) 2/3 x+35=x+45; cost of Alice's earrings is 30 D) x+2/3(35)=1/3 x+45; cost of Alice's earrings is 32.50
Transcript text: Question 15 Fiona and Alice went shopping and spent the same amount of money. They each purchased a necklace and a pair of earrings. Fiona's earrings cost $\frac{2}{3}$ as much as Alice's earrings. Alice paid $\$ 35$ for her necklace, while Fiona paid $\$ 45$ for hers. Choose the response that gives an equation that represents the situation and then gives the cost of Alice's earrings. Let $\times$ represent the cost of Alice's earrings. A) $x+35=\frac{2}{3} x+45$; cost of Alice's earrings is $\$ 30$ B) $x+35=\frac{2}{3}(x+45)$; cost of Alice's earrings is $\$ 15$ C) $\frac{2}{3} x+35=x+45$; cost of Alice's earrings is $\$ 30$ D) $x+\frac{2}{3}(35)=\frac{1}{3} x+45$; cost of Alice's earrings is $\$ 32.50$
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Solution

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Solution Steps

To solve this problem, we need to set up an equation based on the given information. Fiona and Alice spent the same amount of money, and we know the costs of their necklaces and the relationship between the costs of their earrings. Let \( x \) represent the cost of Alice's earrings. Fiona's earrings cost \( \frac{2}{3} \) of Alice's earrings. We can set up the equation based on the total spending of both Fiona and Alice and solve for \( x \).

Solution Approach
  1. Let \( x \) be the cost of Alice's earrings.
  2. Fiona's earrings cost \( \frac{2}{3} x \).
  3. Set up the equation based on their total spending: \( x + 35 = \frac{2}{3} x + 45 \).
  4. Solve the equation for \( x \).
Step 1: Define the Variables

Let \( x \) represent the cost of Alice's earrings. According to the problem, Fiona's earrings cost \( \frac{2}{3} x \).

Step 2: Set Up the Equation

Since Fiona and Alice spent the same amount of money, we can set up the equation based on their total expenditures: \[ x + 35 = \frac{2}{3} x + 45 \]

Step 3: Solve the Equation

To solve for \( x \), we rearrange the equation: \[ x + 35 - \frac{2}{3} x = 45 \] This simplifies to: \[ \frac{1}{3} x + 35 = 45 \] Subtracting 35 from both sides gives: \[ \frac{1}{3} x = 10 \] Multiplying both sides by 3 results in: \[ x = 30 \]

Final Answer

The cost of Alice's earrings is \( \boxed{x = 30} \).

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