This problem involves calculating the total number of matches played based on changes in a player's batting average after a series. The batting average is defined as the total runs scored divided by the number of matches played.

Understanding the Cricket Average Problem

We are given information about Mithali's batting performance:

• Her average score before a specific series.
• Her performance (total runs and matches) during that series.
• Her new average score after the series.

Our goal is to find the total number of matches she had played up to the end of that series.

Setting Up the Initial Scenario (Before the Series)

Let's denote the number of matches played by Mithali before the start of the women's national series as \(N\).

Let the total runs scored by Mithali before the start of the women's national series be \(T\).

The average score before the series is given as 45 runs/match.

The formula for average is:

\(\text{Average} = \frac{\text{Total Runs}}{\text{Number of Matches}}\)

So, before the series, we have:

\(45 = \frac{T}{N}\)

This gives us a relationship between the total runs and the number of matches before the series:

\(T = 45N \quad \ldots (1)\)

Analyzing Performance During the Series

The women's national series consisted of 10 matches.

Mithali's total score in this series was 500 runs.

Calculating the Scenario After the Series

After the series, the total number of matches played by Mithali will be the sum of matches played before the series and matches played in the series.

\(\text{Total Matches After Series} = \text{Matches Before Series} + \text{Matches in Series}\)

\(\text{Total Matches After Series} = N + 10\)

Similarly, the total runs scored by Mithali after the series will be the sum of runs scored before the series and runs scored in the series.

\(\text{Total Runs After Series} = \text{Runs Before Series} + \text{Runs in Series}\)

\(\text{Total Runs After Series} = T + 500\)

The new average after the series is given as 47.5 runs/match.

Using the average formula again for the scenario after the series:

\(\text{New Average} = \frac{\text{Total Runs After Series}}{\text{Total Matches After Series}}\)

\(47.5 = \frac{T + 500}{N + 10} \quad \ldots (2)\)

Solving the Equations

Now we have a system of two equations with two variables (\(N\) and \(T\)):

1. \(T = 45N\)
2. \(47.5 = \frac{T + 500}{N + 10}\)

We can substitute the expression for \(T\) from equation (1) into equation (2):

\(47.5 = \frac{45N + 500}{N + 10}\)

Now, we need to solve this equation for \(N\). Multiply both sides by \((N + 10)\):

\(47.5 \times (N + 10) = 45N + 500\)

Distribute 47.5 on the left side:

\(47.5N + 47.5 \times 10 = 45N + 500\)

\(47.5N + 475 = 45N + 500\)

Now, collect terms involving \(N\) on one side and constant terms on the other side. Subtract \(45N\) from both sides:

\(47.5N - 45N + 475 = 500\)

\(2.5N + 475 = 500\)

Subtract 475 from both sides:

\(2.5N = 500 - 475\)

\(2.5N = 25\)

Finally, divide by 2.5 to find the value of \(N\):

\(N = \frac{25}{2.5}\)

To simplify the division, multiply the numerator and denominator by 10:

\(N = \frac{25 \times 10}{2.5 \times 10} = \frac{250}{25}\)

\(N = 10\)

So, Mithali played 10 matches before the start of the women's national series.

Finding the Total Matches Played Till Date

The question asks for the total number of matches played by her till date, which means the total matches including the series.

\(\text{Total Matches Till Date} = \text{Matches Before Series} + \text{Matches in Series}\)

\(\text{Total Matches Till Date} = N + 10\)

Substitute the value of \(N = 10\):

\(\text{Total Matches Till Date} = 10 + 10 = 20\)

Therefore, Mithali played a total of 20 matches till date.

Verification

Let's check if the new average is indeed 47.5 with 20 matches played in total.

• Matches before: \(N = 10\)
• Runs before: \(T = 45N = 45 \times 10 = 450\)
• Matches in series: 10
• Runs in series: 500
• Total matches till date: \(10 + 10 = 20\)
• Total runs till date: \(450 + 500 = 950\)
• New average: \(\frac{\text{Total Runs Till Date}}{\text{Total Matches Till Date}} = \frac{950}{20}\)

Let's perform the division:

\(\frac{950}{20} = \frac{95}{2} = 47.5\)

The calculated new average matches the given new average (47.5), so our calculation for the total number of matches is correct.

Using the relationships: \(T = 45N\) and \(\frac{T+500}{N+10} = 47.5\), we found \(N=10\).

Total matches till date = \(N + 10 = 10 + 10 = 20\).

Revision Table: Key Concepts

Additional Information: Understanding Averages

An average is a single value that represents a set of values. In cricket, the batting average tells us how many runs a batsman scores, on average, per innings or per match.

When new data (like the runs and matches in a series) is added, the total sum and the total count change, which results in a new average.

Problems involving averages often require setting up equations based on the total sum and total count before and after the new data is included. The key is to remember that Total Sum = Average \(\times\) Number of Items.

In this problem, we used this fundamental relationship to express the total runs before the series (\(T\)) in terms of the number of matches before the series (\(N\)) and the initial average. Then, we used the same relationship for the scenario after the series, leading to an equation that we could solve to find the unknown number of matches.

Understanding how totals and counts change when new items are added is crucial for solving average-related word problems effectively.