Question

Tina, Ed, and Lauren agree to share the cost of a gift and to make their contributions in proportion to their ages. Ed's age is \(\frac{1}{2}\) of Tina's age, and Lauren's age is \(\frac{1}{3}\) of Ed's age. If Lauren's share of the cost is $2.50, what is the cost of the gift?

• A. $25
• B. $20
• C. $15
• D. $12
• E. $10

Hint:

When dealing with ratios based on fractional relationships, it's often helpful to express all quantities in terms of the smallest unit. Here, Lauren is the youngest, so expressing Ed's and Tina's ages in terms of Lauren's age simplifies finding the integer ratio.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a problem about ratios and proportions. The cost is shared in the same ratio as the ages of the three individuals. We need to determine this ratio and then use the known share of one person to find the total cost.
Step 2: Detailed Explanation:
1. Establish the ratio of the ages.
Let T, E, and L be the ages of Tina, Ed, and Lauren, respectively.
We are
\[ E = \frac{1}{2} T \implies T = 2E \]
\[ L = \frac{1}{3} E \implies E = 3L \]
To find the ratio, let's express everyone's age in terms of one person's age. Using L as the base is easiest.
We know \(E = 3L\).
We also know \(T = 2E\). Substitute \(E = 3L\) into this equation:
\[ T = 2(3L) = 6L \]
So, the ratio of their ages is L : E : T, which is \(L : 3L : 6L\).
Dividing by L, the numerical ratio is \(1 : 3 : 6\).
2. Use the ratio to find the total cost.
The shares of the cost are in the same proportion, \(1 : 3 : 6\).
The total number of "parts" in this ratio is \(1 + 3 + 6 = 10\) parts.
Lauren's share corresponds to the '1' part of the ratio. This means her share is \(\frac{1}{10}\) of the total cost.
We are given that Lauren's share is $2.50.
\[ \frac{1}{10} \times (\text{Total Cost}) = $2.50 \]
To find the total cost, we multiply Lauren's share by 10:
\[ \text{Total Cost} = $2.50 \times 10 = $25.00 \]
Step 3: Final Answer:
The total cost of the gift is $25.