The following pie chart shows the percentage distribution of runs scored by a batsman in a test innings.
Question: 1
If the batsman has scored a total of 306 runs, how many 4s and 6s did he hit?
Show Hint
Always convert percentages of runs into actual runs by multiplying with the total score.
Divide by the shot value (4 or 6) to find the number of boundaries.
Step 1: From the pie chart, percentage of runs scored in 4s = \(43.14%\), in 6s = \(3.92%\). Total runs = 306.
\[
\text{Runs from 4s} = \frac{43.14}{100} \times 306 \approx 132.0, \quad
\text{Runs from 6s} = \frac{3.92}{100} \times 306 \approx 12.0.
\]
Step 2: Number of 4s = \(\frac{132}{4} = 33\).
Number of 6s = \(\frac{12}{6} = 2\).
Step 3: Therefore, the batsman hit 33 fours and 2 sixes.
\[
\boxed{33\ \text{fours and 2 sixes}}
\]
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Question: 2
If 5 of the dot balls had been hit for 4s, and if two of the shots for which the batsman scored 3 runs each had fetched him one run instead, what would have been the central angle of the sector corresponding to the percentage of runs scored in 4s? (Use data from the previous question)
Show Hint
When percentages are given in pie charts, use total runs to convert them into absolute values first.
Adjust totals carefully when modifications are made to the scoring pattern.
Central angle = Percentage \(\times 3.6\).
Step 1: Original runs from 4s = 132. Runs from 3s = \(\frac{3.92}{100}\times306 \approx 12\). This means there were 4 shots of 3 runs each. Step 2: Changes:
- 5 dot balls converted to 4s: add \(5 \times 4 = 20\) runs to 4s. New runs from 4s = \(132+20=152\).
- 2 shots of 3 runs each reduced to 1 run: decrease of 4 runs in total score.
Step 3: New total runs = \(306 + 20 - 4 = 322\).
Percentage of runs from 4s = \(\frac{152}{322}\times100 \approx 47.2%\).
Step 4: Central angle = \(\frac{47.2}{100}\times360 \approx 170^\circ\).
\[
\boxed{170^\circ}
\]
