To solve the problem related to the transport of grapes in boxes, we will begin by analyzing the relationship between workers and the time taken to fold the boxes. The relationship is an inverse proportion because as the number of workers increases, the time taken to complete the task decreases.
Given Data:
- One worker folds 30 boxes in 60 minutes.
- 1 worker (X = 1) takes Y = 60 minutes.
- 3 workers (X = 3) takes Y = A minutes.
- 4 workers (X = 4) takes Y = 20 minutes.
- 5 workers (X = 5) takes Y = B minutes.
- 12 minutes is provided for Y when X = 5.
Step 1: Understanding Inverse Proportion
If \( X \) is the number of workers and \( Y \) is the time taken, the relationship can be represented as: \[ X \times Y = k \] Where \( k \) is a constant.
Step 2: Calculate the Constant \( k \)
Using data for one worker: \[ 1 \times 60 = k \implies k = 60 \]
Step 3: Derive Values for A and B
Using the constant \( k = 60 \):
-
For 3 workers: \[ 3 \times A = 60 \implies A = \frac{60}{3} = 20 \text{ minutes} \]
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For 5 workers: \[ 5 \times B = 60 \implies B = \frac{60}{5} = 12 \text{ minutes} \]
Summary:
- Number of workers \( X \): 1, 3, 4, 5
- Time required \( Y \): 60, 20, 20, 12 (where A = 20 and B = 12)
Step 4: Identifying Variables
- Independent Variable: Number of workers (X)
- Dependent Variable: Time taken (Y)
Step 5: Suitable Formula
The relationship derived can be stated in the formula: \[ X \times Y = 60 \] Or solved for \( Y \): \[ Y = \frac{60}{X} \]
Final Results:
- Time taken by 3 workers (A) = 20 minutes
- Time taken by 5 workers (B) = 12 minutes
- Independent Variable: Number of workers
- Dependent Variable: Time taken
- Formula: \( Y = \frac{60}{X} \)
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