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For a certain frequency table which has...

For a certain frequency table which has been partly reproduced here, the arithmetic mean was found to be Rs. 28.07
`{:("Income (in Rs.)",15,20,25,30,35,40),("No. of workers",8,12,?,16,?,10 ):}`
If the total number of workers is 75, then the missing frequencies are

A

14,15

B

15,14

C

13,16

D

12,17

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the missing frequencies in the given frequency table based on the provided information. Here’s a step-by-step solution: ### Step 1: Set up the frequency table We have the following income and corresponding number of workers (frequencies): | Income (in Rs.) | 15 | 20 | 25 | 30 | 35 | 40 | |------------------|----|----|----|----|----|----| | No. of workers | 8 | 12 | F1 | 16 | F2 | 10 | ### Step 2: Define the total number of workers The total number of workers is given as 75. Therefore, we can write the equation for the total frequency: \[ 8 + 12 + F1 + 16 + F2 + 10 = 75 \] ### Step 3: Simplify the equation Calculating the known frequencies: \[ 8 + 12 + 16 + 10 = 46 \] Now substituting this back into the equation gives: \[ 46 + F1 + F2 = 75 \] ### Step 4: Solve for F1 + F2 Rearranging the equation: \[ F1 + F2 = 75 - 46 \] \[ F1 + F2 = 29 \quad \text{(Equation 1)} \] ### Step 5: Use the formula for the arithmetic mean The arithmetic mean (x̄) is given as Rs. 28.07. The formula for the mean is: \[ \bar{x} = \frac{\sum (F_i \cdot x_i)}{\sum F_i} \] Substituting the known values: \[ 28.07 = \frac{(15 \cdot 8) + (20 \cdot 12) + (25 \cdot F1) + (30 \cdot 16) + (35 \cdot F2) + (40 \cdot 10)}{75} \] ### Step 6: Calculate the known products Calculating the known products: - \( 15 \cdot 8 = 120 \) - \( 20 \cdot 12 = 240 \) - \( 30 \cdot 16 = 480 \) - \( 40 \cdot 10 = 400 \) Now substituting these values into the equation: \[ 28.07 = \frac{120 + 240 + 25F1 + 480 + 35F2 + 400}{75} \] ### Step 7: Simplify the equation Combining the known sums: \[ 28.07 = \frac{1240 + 25F1 + 35F2}{75} \] ### Step 8: Multiply both sides by 75 To eliminate the denominator, multiply both sides by 75: \[ 28.07 \cdot 75 = 1240 + 25F1 + 35F2 \] Calculating \( 28.07 \cdot 75 \): \[ 2105.25 = 1240 + 25F1 + 35F2 \] ### Step 9: Rearranging the equation Rearranging gives: \[ 25F1 + 35F2 = 2105.25 - 1240 \] \[ 25F1 + 35F2 = 865.25 \quad \text{(Equation 2)} \] ### Step 10: Solve the system of equations Now we have two equations: 1. \( F1 + F2 = 29 \) (Equation 1) 2. \( 25F1 + 35F2 = 865.25 \) (Equation 2) ### Step 11: Express F1 in terms of F2 From Equation 1: \[ F1 = 29 - F2 \] ### Step 12: Substitute into Equation 2 Substituting \( F1 \) into Equation 2: \[ 25(29 - F2) + 35F2 = 865.25 \] ### Step 13: Expand and simplify Expanding gives: \[ 725 - 25F2 + 35F2 = 865.25 \] Combining like terms: \[ 725 + 10F2 = 865.25 \] ### Step 14: Solve for F2 Subtracting 725 from both sides: \[ 10F2 = 865.25 - 725 \] \[ 10F2 = 140.25 \] Dividing by 10: \[ F2 = 14.025 \approx 14 \] ### Step 15: Find F1 Substituting \( F2 \) back into Equation 1: \[ F1 + 14 = 29 \] \[ F1 = 29 - 14 = 15 \] ### Final Answer The missing frequencies are: - \( F1 = 15 \) - \( F2 = 14 \)
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