Understanding the Problem of Raghav's Weight Estimation

The question asks for the average of the different possible integer weights of Raghav, given several constraints based on estimations from himself, his sister, mother, and father. All estimates are stated to be correct. We need to find the range of weights that satisfies all these conditions simultaneously and then calculate the average of the integer values within that range.

Analyzing Each Person's Estimation Range

Let \(W\) represent Raghav's weight in kg.

We will break down the constraints provided by each person:

• Raghav's view: His weight is more than 64 kg but less than 74 kg.

  \(64 < W < 74\)

  The possible integer weights according to Raghav are 65, 66, 67, 68, 69, 70, 71, 72, 73.

• Sister's view: Her thinking is that his weight is more than 60 kg but less than 69 kg.

  \(60 < W < 69\)

  The possible integer weights according to his sister are 61, 62, 63, 64, 65, 66, 67, 68.

• Mother's view: Her view is that his weight cannot be more than 68 kg. This means his weight must be less than or equal to 68 kg.

  \(W \leq 68\)

  The possible integer weights according to his mother are ..., 66, 67, 68.

• Father's view: His view is that his weight cannot be more than 67 kg. This means his weight must be less than or equal to 67 kg.

  \(W \leq 67\)

  The possible integer weights according to his father are ..., 65, 66, 67.

Finding the Probable Weight Range based on Combined Estimations

Since all of them are correct in their estimation, Raghav's actual weight must satisfy all the conditions at the same time. We need to find the intersection of all these ranges.

The conditions are:

1. \(W > 64\)
2. \(W < 74\)
3. \(W > 60\)
4. \(W < 69\)
5. \(W \leq 68\)
6. \(W \leq 67\)

Let's combine the lower bounds: \(W\) must be greater than the maximum of the lower bounds from Raghav and his sister. Maximum of (64, 60) is 64. So, \(W > 64\).

Let's combine the upper bounds: \(W\) must be less than the minimum of the upper bounds from Raghav, his sister, mother, and father. Minimum of (74, 69, 68, 67) is 67. So, \(W \leq 67\).

Combining the derived lower and upper bounds, the probable range for Raghav's weight is:

\(64 < W \leq 67\)

Identifying Probable Integer Weights

The question asks for the average of different probable weights, implying integer weights are being considered within the derived range \(64 < W \leq 67\). The integers that satisfy this condition are 65, 66, and 67.

The probable integer weights of Raghav are 65 kg, 66 kg, and 67 kg.

Calculating the Average of Probable Weights

To find the average, we sum the probable weights and divide by the number of probable weights.

Sum of probable weights = \(65 + 66 + 67\)

Number of probable weights = 3

Sum = \(65 + 66 + 67 = 198\)

Average weight = \(\frac{\text{Sum of weights}}{\text{Number of weights}}\)

Average weight = \(\frac{198}{3}\)

Average weight = \(66\)

The average of the different probable weights of Raghav is 66 kg.

Summary of Probable Weight Calculation

Combined Lower Bound: \(W > \max(64, 60) \implies W > 64\)

Combined Upper Bound: \(W \leq \min(74, 69, 68, 67) \implies W \leq 67\)

Combined Range: \(64 < W \leq 67\)

Probable Integer Weights: 65, 66, 67

Average: \(\frac{65 + 66 + 67}{3} = \frac{198}{3} = 66\)

Conclusion on Raghav's Weight Average

Based on the combined estimations, the probable integer weights for Raghav are 65 kg, 66 kg, and 67 kg. The average of these probable weights is 66 kg.

Revision Table: Raghav's Weight Problem

Review the key concepts used to solve this problem:

Additional Information: Range Problems and Averages

Problems involving estimations or measurements often define a range of possible values. When multiple conditions on a variable's value are given, finding the value that satisfies all conditions involves finding the intersection of the ranges. For inequalities, this typically means taking the maximum of all lower bounds and the minimum of all upper bounds to define the overall valid range.

If the question asks for integer values within the range, you list all whole numbers that fall strictly between the lower bound (if exclusive, >) and up to the upper bound (if inclusive, ≤ or ≥). If the range is inclusive at both ends, you include the boundary values.

Calculating the average is a fundamental statistical concept. It represents a central value of a set of numbers. In this case, it gives us the mean of the possible weights Raghav could be, assuming all integer values within the determined range are equally probable.