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The problem involves a system of gears with the number of teeth given for each gear. Here's the relevant data and setup:

- Gear A has 20 teeth.
- Gear B has 30 teeth.
- Gear C has 10 teeth.
- Gear D has 30 teeth.
- Gear E has 20 teeth.

The question asks how many turns Gear C will make in 2 minutes if all the gears make 416 turns in 4 minutes. To solve this, we must consider the gear ratio relationships between gears that are connected, and how the rotations propagate through the system.

### Step-by-Step Solution:

1. **Understanding Gear Ratios**:
   - The rotation of a pair of connected gears is inversely proportional to their number of teeth. This means that if gear \( i \) has \( N_i \) teeth and gear \( j \) has \( N_j \) teeth, the number of rotations \( R \) for each is related by:
     \[
     \frac{R_i}{R_j} = \frac{N_j}{N_i}
     \]
   - So, for each pair of gears, we will use this relationship.

2. **Gear A and B**:
   - Gears A and B are directly connected. The ratio of their teeth is \( \frac{N_B}{N_A} = \frac{30}{20} = 1.5 \).
   - Thus, if gear A makes \( x \) turns, gear B will make \( \frac{x}{1.5} \) turns.

3. **Gear B and C**:
   - Gears B and C are connected, with a ratio of \( \frac{N_C}{N_B} = \frac{10}{30} = \frac{1}{3} \).
   - If gear B makes \( y \) turns, gear C will make \( \frac{y}{\frac{1}{3}} = 3y \) turns.

4. **Gear D and C**:
   - Gears C and D are connected, with the ratio \( \frac{N_D}{N_C} = \frac{30}{10} = 3 \).
   - If gear C makes \( z \) turns, gear D will make \( \frac{z}{3} \) turns.

5. **Gears D and E**:
   - Gears D and E are connected, with the ratio \( \frac{N_E}{N_D} = \frac{20}{30} = \frac{2}{3} \).
   - If gear D makes \( w \) turns, gear E will make \( \frac{w}{\frac{2}{3}} = \frac{3}{2}w \) turns.

6. **Calculating Rotations in 2 Minutes**:
   - We know that in 4 minutes all gears make a total of 416 turns.
   - In 2 minutes, the system would make half the number of turns, which is \( 416 / 2 = 208 \) turns.

7. **Finding Gear C's Turns**:
   - We need to propagate the rotations through the system to determine how many turns gear C will make.

Let me calculate this step and find out the number of turns gear C will make in 2 minutes.Gear C will make 416 turns in 2 minutes.

Would you like more details or have any questions? Here are some related questions to explore:

1. How do gear ratios affect rotational speed in a system of gears?
2. If the teeth counts of two gears were swapped, how would it affect the system's behavior?
3. How would the system behave if an additional gear was added?
4. Can the direction of rotation be influenced by gear connections?
5. How does the efficiency of a gear system change with the size and number of gears?

**Tip**: In gear systems, the larger the number of teeth a gear has, the slower it turns compared to a smaller gear, assuming they are connected directly.

En el sistema de engranajes que se muestra, la cantidad de dientes de las ruedas A, B, C, D y E son 20; 30; 10; 30 y 20, respectivamente. Si en 4 minutos todas ellas dan 416 vueltas, ¿cuántas vueltas dará la rueda C en 2 minutos?

Descubre cómo calcular cuántas vueltas dará una rueda en un sistema de engranajes dado el número de dientes de las ruedas conectadas. Este problema de engranajes incluye conceptos clave de proporciones y movimiento circular. Ideal para estudiantes de Secundaria (Grados 9-12).

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