Question

The average weight of 11 players of Indian cricket team is increased by \(1 \mathrm{~kg}\), when one player of the team weighing 55 kg replaced by a new player. The weight of the new player is: (a) \(55 \mathrm{~kg}\) (b) \(64 \mathrm{~kg}\) (c) \(66 \mathrm{~kg}\) (d) none of these

Short answer

Answer: None of the given options.

Step-by-step solution

Define the variables and equation

We will use the following variables: \(n\) represents the number of players in the team (which is 11) \(a\) represents the initial average weight of the 11 players \(w\) represents the weight of the new player We are given that when the player weighing \(55\mathrm{~kg}\) is replaced by the new player, the average weight increases by \(1\mathrm{~kg}\). So, the sum of weights of the remaining 10 players (excluding the replaced player) remains the same. The equation representing the sum of weights is: \(11a = 10a + 55 + w\)

Solve for the average weight

Let's solve the equation for \(a\): \(11a = 10a + 55 + w\) Rearranging the equation, we have: \(a = 55 + w\)

Remember the information about the average weight increase and solve for the new player's weight

We are given that the average weight of the team increases by \(1\mathrm{~kg}\) when the new player replaces the old one. This means: \(a + 1 = \frac{10a+55+w}{11}\) Now, substitute the value of \(a\) from Step 2 into this equation: \((55 + w) + 1 = \frac{10(55+w)+55+w}{11}\) Solve for \(w\): \(56 + w = \frac{605+11w}{11}\) Multiply both sides by 11 to eliminate the fraction: \(11(56 + w) = 605+11w\) \(616 + 11w = 605 + 11w\)

Find the weight of the new player

Now, we can solve for the weight of the new player (\(w\)): \(11w-11w=605-616\) \(0=-11\) This is impossible. So, the answer is (d) none of these. The weight of the new player is not amongst the given options.

Key concepts

Algebraic Equations

Algebraic equations are the cornerstone of solving mathematical problems involving unknown variables. They enable us to establish relationships between different quantities and find values that satisfy these relationships. In the context of an average weight problem, an algebraic equation can express how the total weight of a group changes when an individual's weight is replaced. By setting up an equation, we fix the relationships that are caused by the removal and addition of team members. For example, if the average weight of a team increases when a member is replaced, we derive the equation based on the fact that the sum of individual weights has changed accordingly.

In our specific problem, we have the equation \(11a = 10a + 55 + w\), where \(a\) represents the initial average weight, and \(w\) is the weight of the new player. The equation succinctly captures the entire scenario, enabling us to proceed towards a solution systematically. Understanding how to manipulate this equation to isolate variables is a critical skill in solving such problems.

Average Increase Calculation

The concept of average increase is integral to understanding how the addition or replacement of a single entity affects the overall average of a group. In average increase problems, we calculate the overall impact on group quantities caused by the change. The calculation provides insights into proportional relationships and distributions.

In our problem, the average weight increase by \(1\) kg indicates that the total weight of the cricket team has increased by the number of players times the average increase. Applying this to our equation, \(a + 1 = \frac{10a+55+w}{11}\), allows us to work towards finding the unknown weight of the new player. By understanding the average increase calculation, students can better gauge the effects of such changes in various context—not just in weight, but in any averaged quantity.

Quantitative Aptitude

Quantitative aptitude involves the ability to handle numerical problems swiftly and accurately, applying mathematical concepts from algebra, arithmetic, geometry, and beyond. It is a measure of one's ability to perform tasks ranging from basic arithmetic operations to complex problem-solving involving multiple steps and concepts.

In solving our weight-related problem, quantitative aptitude comes into play as we interpret the mathematical information given, set up equations correctly, and manipulate them to find a solution. Competence in quantitative aptitude means recognizing that when the standard methods seem to lead to an absurd conclusion (such as \(0=-11\)), alternative approaches or solutions outside of the initial options must be considered. This aptitude is essential not only in academic settings but also in various professional and real-life situations where quantitative reasoning is required.