Two moonflower vines are growing on a trellis in Mallory's backward. She bought the first vine when it was 11 inches long and found that it grows at a rate of 0.25 inches per day. Exactly 20 days later, Mallory bought the second vine, which started at 24 inches long and has a growth rate of 0.125 inches per day. How many days will Mallory have had the first vine when the lengths of the two vines are the same?

Explanation

To solve the problem step by step, we will first define the growth of each vine and then set up an equation to find out when their lengths will be equal.

### Step 1: Define the initial conditions and growth rates
- The first vine was bought at a length of 11 inches and grows at a rate of 0.25 inches per day.
- The second vine was bought 20 days later at a length of 24 inches and grows at a rate of 0.125 inches per day.

### Step 2: Calculate the length of the first vine after 20 days
After 20 days, the length of the first vine can be calculated as follows:
\[ \text{Length of first vine after 20 days} = 11 + (20 \times 0.25) \]
\[ = 11 + 5 = 16 \text{ inches} \]

### Step 3: Define the lengths of the vines after 'd' days from the purchase of the second vine
Let \( d \) be the number of days after the second vine is bought. The lengths of the vines can be expressed as:
- Length of the first vine after \( d \) days from the purchase of the second vine:
\[ \text{Length of first vine} = 16 + (0.25 \times d) \]
- Length of the second vine after \( d \) days:
\[ \text{Length of second vine} = 24 + (0.125 \times d) \]

### Step 4: Set up the equation for when the lengths are equal
We want to find \( d \) when the lengths of both vines are equal:
\[ 16 + 0.25d = 24 + 0.125d \]

### Step 5: Solve the equation
Rearranging the equation:
\[ 0.25d - 0.125d = 24 - 16 \]
\[ 0.125d = 8 \]
Dividing both sides by 0.125:
\[ d = \frac{8}{0.125} = 64 \]

### Step 6: Calculate the total days Mallory has had the first vine
Since Mallory bought the second vine 20 days after the first vine, the total days she has had the first vine when the lengths are equal is:
\[ \text{Total days} = 64 + 20 = 84 \]

### Final Answer
Mallory will have had the first vine for **84 days** when the lengths of the two vines are the same.

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