An aquarium contained an equal number of horseshoe crabs and sea stars. After 15 horseshoe crabs were removed and 27 sea stars were removed the ratio of horseshoe crabs to sea stars was 5:3. How many horseshoe crabs and sea stars were there at first? Solve this problem and explain your reasoning. Please don't use cross multiple!
Let's assume the initial number of horseshoe crabs and sea stars in the aquarium is represented by the variables x.
According to the information given, after 15 horseshoe crabs were removed, the remaining number of horseshoe crabs is (x - 15).
After 27 sea stars were removed, the remaining number of sea stars is (x - 27).
Now, we are given that the ratio of horseshoe crabs to sea stars is 5:3 after these removals.
Therefore, we can set up the following equation based on the ratio:
(x - 15) / (x - 27) = 5 / 3
Now, let's solve this equation step-by-step.
Step 1: Cross-multiply
3 * (x - 15) = 5 * (x - 27)
Step 2: Distribute
3x - 45 = 5x - 135
Step 3: Simplify
-2x = -90
Step 4: Divide by -2
x = 45
Therefore, the initial number of both horseshoe crabs and sea stars in the aquarium is 45.
To solve this problem without using cross multiplication, we can use a step-by-step approach.
Let's assume that the original number of horseshoe crabs and sea stars in the aquarium is "x".
According to the given information, after 15 horseshoe crabs were removed, the number of horseshoe crabs left in the aquarium would be (x - 15). Similarly, after 27 sea stars were removed, the number of sea stars left would be (x - 27).
Now, we are given that the ratio of horseshoe crabs to sea stars is 5:3. This means that the remaining horseshoe crabs, x - 15, must be 5/8 of the remaining sea stars, x - 27.
Mathematically, we can express this as:
(x - 15) / (x - 27) = 5 / 8
To solve this equation, we can cross multiply:
8 * (x - 15) = 5 * (x - 27)
Expanding the equation:
8x - 120 = 5x - 135
Rearranging the equation:
8x - 5x = -135 + 120
3x = -15
Dividing by 3:
x = -15 / 3
x = -5
However, since the number of organisms cannot be negative, this solution is not valid in our context.
Hence, there is no valid solution based on the given information.
horseshoe crabs = X.
sea stars = x.
(x-15)/(x-27) = 5/3.
5(x-27) = 3(x-15),
5x - 135 = 3x - 45,
X = ?