Betty, Carly, Isabel, Lani, Alma, and Ursula ran an 800-meter race. Alma beat Isabel by 7 meters. Ursula beat Betty by 12 meters. Alma finished 5 meters ahead of Lani but 3 meters behind Ursula. Carly finished halfway between the first and last girls. In what order did the girls finish? What were the distances between each girl?

1. Ursula = y

2. Alma = y - 3 (3 meters behind)

3. Carly = y - 6 (3 meters behind Alma)

4. Lani = y - 8 (2 meters behind Carly)

5. Isabel = y - 10 (2 meters behind Lani)

6. Betty = y - 12 (2 meters behind Isabel)

Step-by-step explanation:

We are going to use different letters to represent the meters run by the girls:

Alma beat isabel by 7 meters, if A is x then Isabel is x - 7

A = x

I = x - 7

Ursula beat Betty be 12 meters. Let's call Ursula y, then:

U = y

B = y - 12

Alma finished 5 meters ahead of Lani but 3 meters behind Ursula. Let's call Lani z.

L = z

A = z + 5 = y - 3 (since Ursula was "y" and Alma finished 3 meters behind Ursula)

We're going to leave Carly until the end since she finished halfway between the first and the last girls.

Now we can see that Alma is represented by equations with 3 different letters:

A = x = z + 5 = y - 3

Therefore we know that

x = y - 3

⇒A = y - 3

And we'll proceed to express all the other girls distances in terms of "y"

We know that Isabel = x - 7. Therefore we can substitute x = y - 3 and we get that

Isabel = x - 7 = (y - 3) - 7 = y - 10

⇒I = y - 10

We already have Ursula and Betty expressed in terms of y.

⇒U = y

⇒B = y - 12

Now let's go back to Alma equation (A = x = z + 5 = y - 3) and we see that:

z + 5 = y - 3.

Therefore z = y - 3 - 5 = y - 8.

⇒z = y - 8

We have Lani left who was L = z = y - 8

⇒ L = y - 8

The first girl would be Ursula = y

And the last girl would be Betty = y - 12.

⇒Therefore, Carly finished halfway them both and that would be C = y - 6

So, the order in which they arrived was:

Ursula = y

Alma = y - 3 (3 meters behind)

Carly = y - 6 (3 meters behind Alma)

Lani = y - 8 (2 meters behind Carly)

Isabel = y - 10 (2 meters behind Lani)

Betty = y - 12 (2 meters behind Isabel)