Exercise Set 10.2 Class 9 Maths New NCERT ( Ganita Manjari 2)

 

Ganita Manjari - Class 9 Mathematics Part II

Exercise Set 10.2

Weighted Mean - Q1 and Applications

Important Formula: Weighted Mean

Weighted mean is used when different values have different levels of importance.

\[ \boxed{\text{Weighted Mean}= \frac{\sum(\text{Value}\times\text{Weight})}{\sum\text{Weights}}} \]

When marks are given out of different maximum marks, first convert them into percentages:

\[ \boxed{\text{Percentage}= \frac{\text{Marks Obtained}}{\text{Maximum Marks}}\times100} \]

Question 1

Savitri's marks in Kashmiri are as follows: 35 out of 50 in internal tests, 44 out of 60 in the project, and 80 out of 100 in the final exam. The annual percentage score is calculated by combining the internals, project, and final exam in the ratio \(3:4:5\). Which of the following expression(s) gives her annual score as a percentage in Kashmiri?

(i) \[ \frac{35\times3+44\times4+80\times5}{3+4+5} \]
(ii) \[ \frac{\left(\frac{35}{100}\right)3+\left(\frac{44}{100}\right)4+\left(\frac{80}{100}\right)5}{3+4+5} \]
(iii) \[ \frac{\left(\frac{35}{50}\right)3+\left(\frac{44}{60}\right)4+\left(\frac{80}{100}\right)5}{3+4+5}\times100 \]
(iv) \[ \frac{\left(\frac{35}{50}\times100\right)3+\left(\frac{44}{60}\times100\right)4+\left(\frac{80}{100}\times100\right)5}{3+4+5} \]

Solution

Step 1: Convert each score into a percentage.

\[ \frac{35}{50}\times100=70\% \]

\[ \frac{44}{60}\times100=73.333\ldots\% \]

\[ \frac{80}{100}\times100=80\% \]

Step 2: Apply the weights \(3:4:5\).

\[ 3+4+5=12 \]

\[ \text{Annual Percentage} = \frac{70(3)+73.333\ldots(4)+80(5)}{12} \]

Step 3: Calculate.

\[ 70(3)=210,\qquad 73.333\ldots(4)=293.333\ldots,\qquad 80(5)=400 \]

\[ \frac{210+293.333\ldots+400}{12} = 75.2777\ldots\% \]

Annual score: \[ \boxed{75.28\%\text{ approximately}} \]

Now Evaluate the Correct Options Directly

We will calculate the annual percentage using Option (iii) and Option (iv) exactly as given. This makes it clear why both expressions are correct.

Option (iii)

\[ \frac{ \left(\frac{35}{50}\right)3+ \left(\frac{44}{60}\right)4+ \left(\frac{80}{100}\right)5 }{3+4+5}\times100 \]

Step 1: Substitute the values.

\[ = \frac{ \left(\frac{35}{50}\right)3+ \left(\frac{44}{60}\right)4+ \left(\frac{80}{100}\right)5 }{12}\times100 \]

Step 2: Simplify the fractions.

\[ \frac{35}{50}=\frac{7}{10},\qquad \frac{44}{60}=\frac{11}{15},\qquad \frac{80}{100}=\frac45 \]

\[ = \frac{ \frac{7}{10}\times3+ \frac{11}{15}\times4+ \frac45\times5 }{12}\times100 \]

Step 3: Multiply.

\[ = \frac{ \frac{21}{10}+\frac{44}{15}+4 }{12}\times100 \]

Step 4: Add the terms.

\[ \frac{21}{10}+\frac{44}{15}+4 = \frac{63+88+120}{30} = \frac{271}{30} \]

\[ = \frac{\frac{271}{30}}{12}\times100 = \frac{271}{360}\times100 = 75.2777\ldots\% \]

\[ \boxed{\text{Option (iii)}=75.28\%\text{ approximately}} \]

---

Option (iv)

\[ \frac{ \left(\frac{35}{50}\times100\right)3+ \left(\frac{44}{60}\times100\right)4+ \left(\frac{80}{100}\times100\right)5 }{3+4+5} \]

Step 1: Convert each score into a percentage.

\[ \frac{35}{50}\times100=70 \] \[ \frac{44}{60}\times100=73.333\ldots \] \[ \frac{80}{100}\times100=80 \]

Step 2: Substitute the percentages.

\[ = \frac{70\times3+ 73.333\ldots\times4+ 80\times5}{12} \]

Step 3: Multiply.

\[ 70\times3=210,\qquad 73.333\ldots\times4=293.333\ldots,\qquad 80\times5=400 \]

Step 4: Add and divide by the total weight.

\[ = \frac{210+293.333\ldots+400}{12} = \frac{903.333\ldots}{12} = 75.2777\ldots\% \]

\[ \boxed{\text{Option (iv)}=75.28\%\text{ approximately}} \]

Final Comparison

\[ \boxed{\text{Option (iii)}=75.28\%} \] \[ \boxed{\text{Option (iv)}=75.28\%} \]

Both expressions give the same result. Option (iii) keeps the scores as fractions and multiplies by \(100\) at the end. Option (iv) converts each score into a percentage first and then calculates the weighted mean.

Final Answer: \[ \boxed{\text{Correct options are (iii) and (iv).}} \] \[ \boxed{\text{Annual score}=75.28\%\text{ approximately}} \]

Application 1: Ratings of a Hotel/Eatery

The experience of eating at a restaurant is influenced by the taste or quality of the food, the ambience, and the service provided. Suppose a customer gives ratings of 5 for food, 3 for service, and 4 for ambience. The weights assigned to food, service, and ambience are \(4:3:2\), respectively. Find the customer's combined rating for the restaurant.

Solution

Aspect	Rating	Weight
Food	5	4
Service	3	3
Ambience	4	2

Step 1: Multiply each rating by its weight.

\[ 5\times4=20,\qquad3\times3=9,\qquad4\times2=8 \]

Step 2: Add the weighted ratings.

\[ 20+9+8=37 \]

Step 3: Add the weights.

\[ 4+3+2=9 \]

Step 4: Find the weighted mean.

\[ \text{Combined Rating} = \frac{37}{9} = 4.111\ldots \]

Answer: \[ \boxed{4.11\text{ approximately}} \]

Application 2: Ratings of a Product

A store makes and sells portable furniture. The store owners make changes to their products regularly. To calculate the store's average rating, it makes sense to give current products more weight than older products. For example, they may choose to assign twice the weight to customer reviews in the last 3 months compared to older reviews.

Solution and Explanation

The main idea is that recent reviews are more important for a product that changes regularly.

If an older review has weight \(1\), a review from the last 3 months can have weight \(2\).

\[ \boxed{\text{Older Review Weight}=1} \qquad \boxed{\text{Recent Review Weight}=2} \]

Let:

• \(R_o\) = average rating of older reviews
• \(W_o\) = total weight of older reviews
• \(R_r\) = average rating of recent reviews
• \(W_r\) = total weight of recent reviews

Then the general weighted-mean formula is:

\[ \boxed{ \text{Combined Rating} = \frac{R_oW_o+R_rW_r}{W_o+W_r} } \]

If equal numbers of older and recent reviews are being compared, and each recent review has twice the weight of an older review, we may take the weights as \(1\) and \(2\). Then:

\[ \boxed{ \text{Combined Rating} = \frac{R_o(1)+R_r(2)}{1+2} = \frac{R_o+2R_r}{3} } \]

Note: The textbook paragraph does not give specific numerical ratings for older and recent reviews. Therefore, it does not have a single numerical answer. Its purpose is to show why a weighted mean is useful when recent reviews should have greater importance.

Conclusion: Recent reviews can be given greater weight than older reviews, so the combined rating reflects the current product more strongly.

Quick Revision

\[ \boxed{ \text{Weighted Mean} = \frac{\sum(\text{Value}\times\text{Weight})}{\sum\text{Weights}} } \]

• Exercise 10.2 Q1: Correct options are (iii) and (iv).
• Q1 annual percentage: \(\boxed{75.28\%}\) approximately.
• Hotel/Eatery rating: \(\boxed{4.11}\) approximately.
• Product rating: Recent reviews can receive twice the weight of older reviews; the textbook paragraph provides no numerical data for a fixed answer.