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class 9 ganita manjari chapter 10
📖 Textbook Reference: 10.1.1 Average of Averages
When you mix two groups to get a combined average, the final average will always be closer to the group that has more members (the larger weight).
You can visually imagine this as a seesaw. If the balancing point (the overall average) is not exactly in the middle, the seesaw must have more weight (more items/people) on the side it leans closer to.
Where $x$ and $y$ are the number of members in each respective group.
7. The following table shows the weight data of langurs in an animal facility. Average weight: Male = 16.5 kg, Female = 13.8 kg, All 60 langurs = 14.925 kg.
Without doing computations, can you tell whether there are more male or female langurs? Also, identify the correct algebraic expression and find the exact number of male langurs.
Without doing computations, can you tell whether there are more male or female langurs? Also, identify the correct algebraic expression and find the exact number of male langurs.
Select the correct expression(s) for this scenario:
(a) $$\frac{16.5x + 13.8y}{x + y} = 14.925$$
(b)$$\frac{16.5x + 13.8y}{60} = 14.925$$
(c)$$\frac{16.5x + 13.8y}{2} = 14.925$$
(d)$$\frac{16.5x + 13.8y}{16.5 + 13.8} = 14.925$$
(b)
(c)
(d)
1
A weighted average always lies between the two individual averages. It is pulled magnetically closer to the group with the larger "weight" (higher number of members).
2
The general formula is $\frac{\text{Total Sum of Males} + \text{Total Sum of Females}}{\text{Total Number of Langurs}} = \text{Overall Average}$.
1
Logical Deduction: Calculate the distance between the Overall Average ($14.925$) and the Male Average ($16.5$). Then calculate the distance to the Female Average ($13.8$). Which distance is smaller?
2
Expressions: In the formula $\frac{16.5x + 13.8y}{x + y}$, the denominator $(x + y)$ is the total number of langurs. Does the problem state the total number of langurs? Yes, $60$.
3
Solving: If total langurs = $60$, then Females $y = 60 - x$. Substitute this into the correct expression and solve the linear equation for $x$ (Males).
!
Expression (c) Trap: Many students mistakenly choose expression (c) because they think they are averaging two things (Males and Females) so they divide by $2$. This is completely wrong because there are $60$ individual langurs being averaged, not $2$!
Part 1: Logical Deduction
Distance from Overall Avg to Male Avg: $16.5 - 14.925 = 1.575$
Distance from Overall Avg to Female Avg: $14.925 - 13.8 = 1.125$
Because the overall average is closer to the female average ($1.125 < 1.575$), there must be a greater "weight" pulling it there.
Ans: There are more female langurs.
Part 2: Evaluating Expressions
Expression (a) is correct. It is the standard weighted average formula where $(x+y)$ is the total population.
Expression (b) is also correct because the total population $(x+y)$ is explicitly given as $60$.
Ans: Expressions (a) and (b) are correct.
Part 3: Solving for Males ($x$)
Since $x + y = 60$, we can substitute $y = 60 - x$ into expression (b):
$\frac{16.5x + 13.8(60 - x)}{60} = 14.925$
$16.5x + 828 - 13.8x = 14.925 \times 60$
$2.7x + 828 = 895.5$
$2.7x = 67.5 \implies x = \frac{67.5}{2.7} = 25$
Ans: There are 25 male langurs.
(iii) A female langur weighing 15.2 kg is admitted to the facility. What is the average weight of the female langurs after this?
1
To find a new average after adding a data point, you must find the Total Sum first. Total Sum = Average × Count.
1
From the main problem, we found there are $25$ males. Since total is $60$, there must be $35$ females.
2
Step 1: Find the total weight of the original $35$ females ($35 \times 13.8$).
3
Step 2: Add the new langur's weight ($15.2\text{ kg}$) to that total.
4
Step 3: Divide the new grand total by the new count ($36$).
!
Do not simply average $13.8$ and $15.2$! The $13.8$ represents $35$ langurs, while $15.2$ represents just $1$ langur. You must find the total mass first.
Step-by-Step Calculation
1. Find Old Total Weight:
$\text{Total}_{35} = 35 \times 13.8 = 483\text{ kg}$
2. Find New Total Weight:
$\text{Total}_{36} = 483 + 15.2 = 498.2\text{ kg}$
3. Calculate New Average:
$\text{New Average} = \frac{498.2}{36} \approx 13.84\text{ kg}$
Ans: The new average weight is 13.84 kg.
(iv) Two male langurs weighing 16.9 kg and 16.1 kg are released from the facility. What is the average weight of the male langurs after this?
1
When removing data points, you subtract their values from the Total Sum, and you decrease the count in the denominator.
1
We know there are $25$ males averaging $16.5\text{ kg}$. Find their total weight ($25 \times 16.5$).
2
Subtract the weight of the two released langurs ($16.9 + 16.1 = 33.0\text{ kg}$).
3
Divide the new total by the new count of males ($25 - 2 = 23$).
!
Look closely at the numbers before doing complex math. What is the average weight of the two langurs leaving? $(16.9 + 16.1) / 2 = 16.5\text{ kg}$. If you remove items that perfectly match the existing average, the average will not change!
Step-by-Step Calculation
1. Find Old Total Weight:
$\text{Total}_{25} = 25 \times 16.5 = 412.5\text{ kg}$
2. Find New Total Weight:
Total removed = $16.9 + 16.1 = 33\text{ kg}$
$\text{New Total}_{23} = 412.5 - 33 = 379.5\text{ kg}$
3. Calculate New Average:
$\text{New Average} = \frac{379.5}{23} = 16.5\text{ kg}$
Ans: The average weight remains 16.5 kg.
(v) Now, suppose one of the $25$ male langurs lost $1\text{ kg}$ of weight due to illness. What is the new average weight of all the male langurs after this?
A. 15.5 kg
B. 16.0 kg
C. 16.46 kg
D. 16.5 kg
1
When the total sum of a group changes by a specific amount, that change is distributed equally across the entire group when calculating the new average.
Logical Breakdown
The total weight of the male group decreases by $1\text{ kg}$.
Because the average represents weight "per langur", this $1\text{ kg}$ loss is effectively shared among all $25$ male langurs.
Therefore, the average decreases by $\frac{1}{25}\text{ kg}$.
$\frac{1}{25} = 0.04\text{ kg}$.
New Average = $16.5 - 0.04 = 16.46\text{ kg}$.
Ans: Option (C)
In the main langur problem, why is expression (c) mathematically incorrect?
$$\frac{16.5x + 13.8y}{2} = 14.925$$
A. Because the numerator should just be $x + y$.
B. Because the denominator must be the sum of the "weights" (the total number of langurs, 60), not the number of subgroups (2).
C. Because 16.5 + 13.8 is not equal to 60.
D. Because the right side of the equation should be 60.
1
In the weighted average formula, the denominator is always the total *count* of individual items (the weights), never the number of categories.
Logical Explanation
The formula is $\frac{\text{Total Weight of all Langurs}}{\text{Total Number of Langurs}}$.
The number of langurs is $(x + y)$, which equals $60$.
Expression (c) uses $2$ as the denominator simply because there are two groups (Male and Female). This is a fatal error, as it attempts to average the *groups* rather than the *individuals*.
Ans: Option (B)
