The ratio of guppies to angelfish in a shop was 2:3. The number of angelfish was 2/3 the number of goldfish. The shopkeeper decided to replace 1/4 of the guppies with some angelfish. The number of angelfish she added was equal to the number of guppies she replaced. There were 30 more goldfish than angelfish in the end. How many guppies were in the shop in the beginning? PLEASE SHOW ALL YOUR WORK!!!!

Question

The ratio of guppies to angelfish in a shop was 2:3. The number of angelfish was 2/3 the number of goldfish. The shopkeeper decided to replace 1/4 of the guppies with some angelfish. The number of angelfish she added was equal to the number of guppies she replaced. There were 30 more goldfish than angelfish in the end. How many guppies were in the shop in the beginning? PLEASE SHOW ALL YOUR WORK!!!!

Answer 1

you cannot have both

the ratio of guppies to angelfish in a shop was 2:3. ==> g/a = 2/3 ==> g = 2/3 a
and
The number of angelfish was 2/3 the number of goldfish. a = 2/3 g

Answer 2

To solve this problem, we'll break it down into steps and derive equations for each step. Let's proceed step by step:

Step 1: Understanding the initial ratio of guppies to angelfish.
The ratio of guppies to angelfish is given as 2:3. Let's assume the number of guppies is 2x and the number of angelfish is 3x, where x is a common factor.

Step 2: Relationship between angelfish and goldfish.
The problem states that the number of angelfish is 2/3 the number of goldfish. Let's assume the number of goldfish is y. Therefore, the equation becomes:
3x = 2/3 * y

Step 3: Replacement of guppies.
The shopkeeper decided to replace 1/4 of the guppies with an equal number of angelfish. Let's find the number of guppies replaced first.
1/4 * 2x = x/2 (guppies replaced)
Now, the number of angelfish added is also x/2.

Step 4: Final number of angelfish and goldfish.
After the replacement, the number of angelfish becomes:
3x + x/2 = 6x/2 + x/2 = 7x/2

The problem mentions that there are 30 more goldfish than angelfish at the end. This can be represented as:
y = 7x/2 + 30

Now, we have two equations with two variables (x and y) that we can solve.

Equation 1: 3x = 2/3 * y
Equation 2: y = 7x/2 + 30

To simplify the process, we can multiply both sides of equation 1 by 3 to remove the fraction:
9x = 2y

Substituting equation 2 into equation 1, we have:
9x = 2*(7x/2 + 30)
9x = 7x + 60
9x - 7x = 60
2x = 60
x = 30

Now, we know that x = 30, and we can find the number of guppies in the shop initially by substituting it into 2x:
2 * 30 = 60

Therefore, there were 60 guppies in the shop in the beginning.