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class 9 ganita manjari chapter 10
📖 Textbook Reference: 10.1.1 Average of Averages
The weighted mean can also be used to calculate the mean when certain values are added to or removed from data.
To find the new average when a new data point is added, we cannot simply average the old average and the new point. We must first find the Total Sum.
Once you have the total sum of the original data, add the new value to it to get the New Total Sum, and divide by the New Count.
A white stork's migration is tracked. The average daily distance travelled, calculated over 20 days, is 44.5 km. On the 21st day, it flew 55 km. What is the average daily distance travelled over these 21 days? Make a guess before you calculate.
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When modifying data, the most reliable method is to revert the average back into a Total Sum.
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$\text{Total Distance} = \text{Average Daily Distance} \times \text{Number of Days}$.
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The Guess: On the 21st day, the stork flew $55\text{ km}$, which is greater than the current average of $44.5\text{ km}$. Therefore, the new average must be greater than $44.5\text{ km}$.
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Step 1: Calculate the total distance flown in the first $20$ days ($20 \times 44.5$).
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Step 2: Add the distance flown on the 21st day ($55\text{ km}$) to get the total distance for $21$ days.
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Step 3: Divide the new total distance by the new total number of days ($21$).
!
The Simple Average Mistake: A very common error is to just average the old average and the new day: $\frac{44.5 + 55}{2} = 49.75\text{ km}$. This completely ignores that the $44.5\text{ km}$ represents the heavy "weight" of $20$ whole days, while $55\text{ km}$ is just a single day!
Step-by-Step Calculation
1. Calculate Total Distance for 20 days:
$\text{Total Distance}_{20} = 20 \times 44.5$
$= 890\text{ km}$
2. Add the 21st day to find New Total:
$\text{Total Distance}_{21} = 890 + 55$
$= 945\text{ km}$
3. Calculate the New Average:
$\text{New Average} = \frac{\text{New Total Distance}}{\text{Total Days}}$
$\text{New Average} = \frac{945}{21} = 45\text{ km}$
Ans: The new average is 45 km per day.
A batsman's average score over 10 innings is 48 runs. In his 11th inning, he scores 103 runs (a century!). What is his new batting average?
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This is mathematically identical to the Stork migration problem. You must find the total runs first.
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Step 1: Total runs after 10 innings = $10 \times 48$.
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Step 2: Add the runs from the 11th inning ($103$) to the total.
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Step 3: Divide the new grand total by the new number of innings ($11$).
!
Do not average 48 and 103! The 48 represents the weight of 10 entire matches, while 103 is just a single match.
Step-by-Step Calculation
1. Find Old Total Runs:
$\text{Total}_{10} = 10 \times 48 = 480\text{ runs}$
2. Find New Total Runs:
$\text{Total}_{11} = 480 + 103 = 583\text{ runs}$
3. Calculate New Average:
$\text{New Average} = \frac{583}{11} = 53\text{ runs}$
Ans: His new batting average is 53.
Before doing any math, what can you logically conclude about the stork's new 21-day average given that it flew 55 km on the 21st day (previous avg was 44.5 km)?
A. It must be exactly 50 km (the midpoint).
B. It must be greater than 44.5 km.
C. It will decrease because 21 is a larger divisor than 20.
D. It will remain exactly 44.5 km.
1
Averages are responsive. If you add a value that is higher than the current average, it pulls the average up.
Logical Explanation
The stork's established average over $20$ days was $44.5\text{ km}$.
On the 21st day, the stork flew $55\text{ km}$. Since $55$ is significantly larger than $44.5$, this exceptionally long flight day will pull the overall average up.
Therefore, without doing any math, we know the new average must be strictly greater than $44.5\text{ km}$.
Ans: Option (B)
How do you correctly calculate the total distance the stork travelled in the first 20 days?
A. $44.5 \div 20$
B. $20 + 44.5$
C. $20 \times 44.5$
D. $55 \times 20$
1
To find a total sum from an average, you use the fundamental relationship: $\text{Total Sum} = \text{Average} \times \text{Count}$.
Reverse Engineering the Average
The formula for an average is: $\text{Average} = \frac{\text{Total Sum}}{\text{Count}}$.
By multiplying both sides by the Count, we get: $\text{Total Sum} = \text{Average} \times \text{Count}$.
Substituting our values: $\text{Total Distance} = 44.5 \times 20 = 890\text{ km}$.
Ans: Option (C)
