https://file.aiquickdraw.com/mathbot/file/1727170235903p8k79sfa.jpg

The problem presented in the image is as follows:

A box contains oranges and apples in a ratio of 7:11, respectively. When ten apples are removed, the ratio becomes 3:4. How many apples were there initially?

Let's define:
- Let the number of oranges be \( 7x \).
- Let the number of apples be \( 11x \).

Initially, the ratio of oranges to apples is \( 7:11 \), so the total number of oranges and apples can be expressed in terms of \( x \).

### Step 1: Setting up the equation for the new ratio
After 10 apples are removed, the number of apples becomes \( 11x - 10 \), and the new ratio of oranges to apples is given as \( 3:4 \).

Thus, the new ratio gives the equation:

\[
\frac{7x}{11x - 10} = \frac{3}{4}
\]

### Step 2: Solving for \( x \)
We can cross-multiply to solve for \( x \):

\[
4 \times 7x = 3 \times (11x - 10)
\]

Expanding both sides:

\[
28x = 33x - 30
\]

Subtracting \( 28x \) from both sides:

\[
0 = 5x - 30
\]

Adding 30 to both sides:

\[
30 = 5x
\]

Dividing by 5:

\[
x = 6
\]

### Step 3: Finding the initial number of apples
Since \( x = 6 \), the initial number of apples is:

\[
11x = 11 \times 6 = 66
\]

So, the initial number of apples was **66**.

Would you like further details or explanations? Here are some related questions to expand your understanding:

1. How do you verify the solution using the new ratio after removing apples?
2. What would the total number of fruits be initially?
3. Can you change the ratio of oranges to apples and analyze a similar situation?
4. How would the solution change if 15 apples were removed instead of 10?
5. What happens to the ratio if you add a certain number of oranges instead of removing apples?

**Tip**: Ratios are often converted to equations by introducing a common factor for easy solving!

A box contains oranges and apples in a ratio of 7:11, respectively. When ten apples are removed, the ratio becomes 3:4. How many apples were there initially?

Learn how to solve a ratio problem where a box contains oranges and apples in a ratio of 7:11, and 10 apples are removed, altering the ratio to 3:4. Detailed solution with algebraic steps. Suitable for Grades 7-10.

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