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Out of 6 numbers, the sum of the first 5...

Out of 6 numbers, the sum of the first 5 numbers is 7 times the 6th number. If their average is 136, then the 6th number is:

A

116

B

102

C

84

D

96

Text Solution

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The correct Answer is:
To solve the problem step by step, we can follow these calculations: 1. **Understanding the Problem**: We have 6 numbers, and we need to find the 6th number based on the given conditions. The sum of the first 5 numbers is 7 times the 6th number, and the average of all 6 numbers is 136. 2. **Calculate the Total Sum of the Numbers**: Since the average of the 6 numbers is 136, we can find the total sum of these numbers. \[ \text{Total Sum} = \text{Average} \times \text{Number of Values} = 136 \times 6 = 816 \] 3. **Define Variables**: Let the 6th number be \( x \). Then, let the sum of the first 5 numbers be \( y \). 4. **Set Up the Equation**: According to the problem, the sum of the first 5 numbers is 7 times the 6th number: \[ y = 7x \] 5. **Express the Total Sum in Terms of \( x \)**: The total sum of the 6 numbers can be expressed as: \[ y + x = 816 \] Substituting \( y \) from the previous equation: \[ 7x + x = 816 \] This simplifies to: \[ 8x = 816 \] 6. **Solve for \( x \)**: Now, divide both sides by 8 to find the value of \( x \): \[ x = \frac{816}{8} = 102 \] 7. **Conclusion**: The 6th number is \( 102 \). ### Summary of Steps: 1. Calculate the total sum of the numbers using the average. 2. Define the 6th number as \( x \) and the sum of the first 5 numbers as \( y \). 3. Set up the equation based on the relationship between \( y \) and \( x \). 4. Substitute \( y \) in the total sum equation and simplify. 5. Solve for \( x \) to find the 6th number.
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