CAT 2024 Slot 2 — DILR Question 10

Mixed PracticeEasy

Passage / Data

Answer the following questions based on the information given below.

An online e-commerce firm receives daily integer product ratings from 1 through 5 given by buyers. The daily average is the average of the ratings given on that day. The cumulative average is the average of all ratings given on or before that day. The rating system began on Day 1, and the cumulative averages were 3 and 3.1 at the end of Day 1 and Day 2, respectively. The distribution of ratings on Day 2 is given in the figure below.

Distribution of ratings on Day 2

Rating (Day 2)	1	2	3	4	5
Number of buyers	5	10	5	20	10

The following information is known about ratings on Day 3.
1. 100 buyers gave product ratings on Day 3.
2. The modes of the product ratings were 4 and 5.
3. The numbers of buyers giving each product rating are non-zero multiples of 10.
4. The same number of buyers gave product ratings of 1 and 2, and that number is half the number of buyers who gave a rating of 3.

How many buyers gave ratings on Day 1?

Answer & solution

Answer: 150

Solution

Easy

Use the cumulative-average jump from Day 1 (3.0) to Day 2 (3.1). Compute the Day-2 total from the bar chart, then solve for the Day-1 count.

Day 2 distribution (from chart): 1→5, 2→10, 3→5, 4→20, 5→10 buyers (50 buyers, rating-sum 170). Cumulative average = 3.0 after Day 1, 3.1 after Day 2.

Day-2 rating total. Multiply each rating by its frequency.

\begin{aligned} &1(5)+2(10)+3(5)+4(20)+5(10)\\ &\Rightarrow\ 5+20+15+80+50=170 \quad\text{(50 buyers)} \end{aligned}

Solve for Day-1 buyers. Let $n_1$ be the Day-1 count; its rating-sum is $3n_1$ .

\begin{aligned} &\frac{3n_1+170}{n_1+50}=3.1\\ &\Rightarrow\ 3n_1+170=3.1n_1+155 \quad\text{(cross-multiply)}\\ &\Rightarrow\ 15=0.1\,n_1\\ &\Rightarrow\ n_1=150 \end{aligned}

150