Alright, let's solve the problem step by step.
### Part (a): Finding the ratio of the number of apples to oranges
We are given:
- 25% of the number of apples is equal to 75% of the number of oranges.
Let's denote:
- [tex]\( A \)[/tex] as the number of apples, and
- [tex]\( O \)[/tex] as the number of oranges.
According to the problem:
[tex]\[ 0.25A = 0.75O \][/tex]
To find the ratio of the number of apples to the number of oranges, let's first isolate [tex]\( A \)[/tex] on one side:
[tex]\[ 0.25A = 0.75O \][/tex]
Divide both sides by 0.25:
[tex]\[ A = 3O \][/tex]
So, the ratio of the number of apples to the number of oranges is:
[tex]\[ \frac{A}{O} = \frac{3O}{O} = 3 \][/tex]
Therefore, the ratio of apples to oranges is 3:1.
### Part (b): Finding the number of apples if there are 800 fruits in total
We are given that the total number of fruits (apples + oranges) is 800. So:
[tex]\[ A + O = 800 \][/tex]
From part (a), we know that [tex]\( A = 3O \)[/tex]. Substitute this into the total number of fruits equation:
[tex]\[ 3O + O = 800 \][/tex]
[tex]\[ 4O = 800 \][/tex]
Solve for [tex]\( O \)[/tex]:
[tex]\[ O = \frac{800}{4} = 200 \][/tex]
Now, substitute [tex]\( O = 200 \)[/tex] back into [tex]\( A = 3O \)[/tex] to find [tex]\( A \)[/tex]:
[tex]\[ A = 3 \times 200 = 600 \][/tex]
Therefore, if there are 800 apples and oranges in total, the number of apples is 600.
### Summary of Results
1. The ratio of the number of apples to the number of oranges is 3:1.
2. If the total number of fruits is 800, then there are 600 apples.
