Certainly! Let's break down the problem step-by-step: --- ## Problem Restatement: - For every334 turns of A, Gear turns 80. - Gear B's teeth need to be calculated to make this possible. - Known: - Number of teeth on Gear A = 14 - Number of teeth on Gear C = 120 (output gear) - Question: Find the number of teeth on Gear B. --- ## Step 1: Understand gear relationships Gears connected in a gear train transfer rotation, and the number of teeth affects the rotation ratio. - The **gear ratio** is given by: \[ \text{Gear Ratio} = \frac{\text{Number of teeth on driven gear}}{\text{Number of teeth on driving gear}} \] - When gears are in contact, their rotation ratios multiply across the train. --- ## Step 2: Identify the gear train Based on the diagram and description: - Gear A (input gear) drives Gear B. - Gear B then drives Gear C (output gear). So, the overall gear ratio: \[ \text{Input rotations of Gear A} \rightarrow \text{Gear B} \rightarrow \text{Gear C} \] --- ## Step 3: Set up the ratio - Gear A turns **334 times** for every **80 turns** of Gear C: \[ \frac{\text{Turns of Gear C}}{\text{Turns of Gear A}} = \frac{80}{334} \] - The gear ratio between Gear A and Gear C: \[ \text{Gear ratio} = \frac{\text{Teeth of Gear C}}{\text{Teeth of Gear A}} = \frac{120}{14} \approx 8.57 \] But this is not the entire ratio because of the intermediate gear B. --- ## Step 4: Find the gear ratio involving Gear B - Let \( T_B \) be the number of teeth on Gear B. - The total gear ratio from Gear A to Gear C: \[ \text{Total ratio} = \left(\frac{T_B}{14}\right) \times \left(\frac{120}{T_B}\right) = \frac{120}{14} \approx 8.57 \] - However, the actual turns ratio (from the problem): \[ \frac{\text{Turns of Gear C}}{\text{Turns of Gear A}} = \frac{80}{334} \approx .239 \] - The gear ratio from A to C via B must match this ratio: \[ \left(\frac{T_B}{14}\right) \times \left(\frac{120}{T_B}\right) = 8.57 \] But this simplifies to: \[ \frac{120}{14} \approx 8.57 \] which confirms the ratio between Gear A and Gear C directly. --- ## Step 5: Incorporate the turns ratio Since the gear train involves Gear B, the actual ratio of rotations is: \[ \text{Rotation ratio} = \left(\frac{T_B}{14}\right) \times \left(\frac{120}{T_B}\right) = 8.57 \] But the actual turns ratio (from the problem) is: \[ \frac{80}{334} \approx .239 \] This indicates that Gear B must modify the ratio to match the observed turns. --- ## Step 6: Find \( T_B \) The ratio between Gear A and Gear C is: \[ \frac{\text{Teeth of Gear C}}{\text{Teeth of Gear A}} = \frac{120}{14} \approx 8.57 \] The actual turns ratio: \[ \frac{80}{334} \approx .239 \] The gear B's teeth should create this overall ratio. The key is: \[ \text{Gear B ratio} = \frac{T_B}{T_A} \] and \[ \text{Gear C ratio} = \frac{120}{T_B} \] The overall ratio: \[ \frac{\text{Turns of Gear C}}{\text{Turns of Gear A}} = \frac{T_B}{14} \times \frac{120}{T_B} = \frac{120}{14} \approx 8.57 \] but this conflicts with the actual ratio (.239). --- ## **Conclusion:** The ratio of gears should satisfy: \[ \text{Total ratio} = \frac{\text{Teeth on Gear C}}{\text{Teeth on Gear A}} \times \frac{1}{T_B} \text{ (adjusted for gear B)} \] Given the approximate ratios, the calculations suggest that: \[ T_B = \frac{T_A \times \text{desired ratio}}{\text{actual ratio}} \] or more straightforwardly: \[ T_B = \text{Number of teeth on Gear B} \text{ that adjusts the ratio to } .239 \] --- ## **Final Calculation:** Since the problem asks for an approximate value and the gear ratio should match the turns ratio: \[ \boxed{ T_B \approx \frac{14 \times 80}{334} \approx \frac{112}{334} \approx 3.36 } \] **But since gear teeth must be whole numbers, the approximate number of teeth on Gear B is:** \[ \boxed{ T_B \approx 3 \text{ or } 4 } \] --- ## **Final Answer:** **Number of teeth on Gear B ≈ 4** --- Let me know if you'd like me to clarify any step!