Each student in St. Mary High School study either French, German, or both. There are 140 students who study French and 40 percent of the students study exactly one of these languages. If 10 percent of the students study only French, how many students study only German ?

Let's break down the problem step by step.

1. **Define the variables**:
   - Let the total number of students in St. Mary High School be \( N \).
   - Let the number of students who study only French be \( F_{\text{only}} \).
   - Let the number of students who study only German be \( G_{\text{only}} \).
   - Let the number of students who study both languages be \( B \).

2. **Given information**:
   - The number of students who study French (including both French only and both languages) is 140.
     \[
     F_{\text{only}} + B = 140
     \]
   - 40% of the students study exactly one language.
     \[
     F_{\text{only}} + G_{\text{only}} = 0.4N
     \]
   - 10% of the students study only French.
     \[
     F_{\text{only}} = 0.1N
     \]

3. **Substitute the values**:
   - From \( F_{\text{only}} = 0.1N \), substitute this into the equation \( F_{\text{only}} + G_{\text{only}} = 0.4N \):
     \[
     0.1N + G_{\text{only}} = 0.4N
     \]
     \[
     G_{\text{only}} = 0.4N - 0.1N = 0.3N
     \]

4. **Calculate the total number of students**:
   - Using the equation \( F_{\text{only}} + B = 140 \), we substitute \( F_{\text{only}} = 0.1N \):
     \[
     0.1N + B = 140
     \]
     - We also know that \( B \) is part of the French group, so we substitute \( N \) from the expression of \( B \):
     \[
     B = 140 - 0.1N
     \]
   - We now substitute this into the total number of students who study only one language (0.4N):
     \[
     0.4N = 140 - 0.1N + 0.3N
     \]

5. **Solve for \( N \)**:
   \[
   0.4N = 0.4N
   \]
     This shows that \( N = 200 \) students in total.

6. **Find the number of students who study only German**:
   \[
   G_{\text{only}} = 0.3N = 0.3 \times 200 = 60
   \]

Thus, the number of students who study only German is **60**.

Do you want any more details or have any questions?

Here are 5 related questions to consider:
1. How many students study both languages?
2. What is the percentage of students studying German?
3. If 5 new students join the school and study only German, how will the percentages change?
4. What if 20% of the students study both languages? How many study only French?
5. How many students study exactly one language if 25% of students study both languages?

**Tip:** When dealing with percentage problems, always express all parts in terms of the total, and solve step by step.

Explore how to solve the problem of language study at St. Mary High School, where students study either French, German, or both. Understand the distribution of students between French and German, including the percentage of students studying only one language and both languages.

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