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The image contains a problem related to fish in an aquarium. Here’s a transcription of the problem:

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There are a total of 164 fish in an aquarium. If the number of angelfish is 5 times the number of guppies, the ratio of the number of goldfish to the number of angelfish is 2:3.

**(a)** Find the ratio of the number of angelfish to the number of goldfish and the number of guppies.  
**(b)** How many more angelfish than goldfish are there in the aquarium?

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Let’s break this down step by step.

Let:
- \( A \) be the number of angelfish,
- \( G_u \) be the number of guppies,
- \( G_o \) be the number of goldfish.

From the problem:
1. \( A = 5 \cdot G_u \) (the number of angelfish is 5 times the number of guppies).
2. The ratio of goldfish to angelfish is \( 2:3 \), so \( \frac{G_o}{A} = \frac{2}{3} \), or \( G_o = \frac{2}{3}A \).

The total number of fish is 164, so:
\[
A + G_u + G_o = 164.
\]

Substituting the relations \( A = 5G_u \) and \( G_o = \frac{2}{3}A \) into this equation, we get:
\[
5G_u + G_u + \frac{2}{3}(5G_u) = 164.
\]

Let's solve this equation.

From the calculations:

- The number of guppies \( G_u \) is approximately \( 17.57 \), or about 18 guppies.
- The number of angelfish \( A \) is approximately \( 87.86 \), or about 88 angelfish.
- The number of goldfish \( G_o \) is approximately \( 58.57 \), or about 59 goldfish.

### Answer to the questions:
**(a)** The ratio of the number of angelfish to the number of goldfish is \( \frac{88}{59} \).

**(b)** The number of angelfish exceeds the number of goldfish by \( 88 - 59 = 29 \).

Would you like more details on any of these calculations?

Here are 5 questions to expand on this:

1. How do we simplify ratios like \( 88:59 \)?
2. What happens if the ratio of goldfish to angelfish changes?
3. How would you solve this problem with non-integer solutions?
4. Can we create an equation for more complex ratios involving multiple species?
5. How can we interpret the significance of fish ratios in ecology?

**Tip:** When working with ratios, always check that your final numbers fit within any total constraints (like the total number of fish in the aquarium).

There are a total of 164 fish in an aquarium. If the number of angelfish is 5 times the number of guppies, the ratio of the number of goldfish to the number of angelfish is 2:3. 
(a) Find the ratio of the number of angelfish to the number of goldfish and the number of guppies. 
(b) How many more angelfish than goldfish are there in the aquarium?

Learn how to solve a ratio-based word problem involving 164 fish in an aquarium, with different numbers of angelfish, guppies, and goldfish. This problem introduces algebraic equations and proportions, suitable for students in Grades 6-8.

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