Annie and Ravi have two candles. One of them is one inch longer than the other. Annie lights the longer candle at 7:00 and RAvi lights the shorter candle at 8:30. At 11:00 the two candles are the same length. The shorter candle burns out at 12:30 and the other at 1:00. What are the lengths of the two candles?

the two candles have height h and h+1

The shorter candle burns at the rate of h/4 in/hr
The longer burns at the rate of (h+1)/6 in/hr

So,

h-(h/4)(2.5) = (h+1) - (h+1)/6 * 4
h = 8

To solve this problem, we'll use the information given step by step. Let's start by assigning some variables:

Let's assume the length of the shorter candle is 'x' inches.
Since the longer candle is one inch longer, we can say its length is 'x + 1' inches.

According to the given information:
1. Annie lights the longer candle at 7:00, and it burns for 6 hours (until 1:00).
2. Ravi lights the shorter candle at 8:30, and it burns for 4 hours (until 12:30).
3. At 11:00, both candles are the same length.

Now we can calculate the lengths of the two candles.

Annie's candle burns for 6 hours, so the length that burns per hour is (x + 1) / 6 inches.
Ravi's candle burns for 4 hours, so the length that burns per hour is x / 4 inches.

Since both candles are the same length at 11:00, we can set up the following equation:
Length burned by Annie's candle = Length burned by Ravi's candle

((x + 1) / 6) * (11 - 7) = (x / 4) * (11 - 8.5)

Simplifying this equation, we get:
4(x + 1) = 6(x / 4)

Solving for x:
4x + 4 = 3x
x = 4

Therefore, the length of the shorter candle is 4 inches, and the length of the longer candle is 5 inches (4 + 1).

So, the lengths of the two candles are 4 inches and 5 inches, respectively.