4 questions: Q1, Q2, Q3, Q4. Solve each one here: tap the answer, press Check, then open the solution.
Data Interpretation
The set
A few salesmen are employed to sell a product called ZENTRA among households in various housing complexes. On each day, a salesman is assigned to visit one housing complex. Once a salesman enters a housing complex, he can meet any number of households in the time available. However, if a household makes a complaint against the salesman, then he must leave the housing complex immediately and cannot meet any other household on that day. A household may buy any number of ZENTRA items or may not buy any item. The salesman needs to record the total number of ZENTRA items sold as well as the number of households met in each day. The success rate of a salesman for a day is defined as the ratio of the number of items sold to the number of households met on that day.
Some details about the performances of three salesmen — Vikram, Arjun and Rohit — on two particular days are given below. 1. Over the two days, all three of them met the same total number of households, and each of them sold a total of 100 items. 2. On both days, Rohit met the same number of households and sold the same number of items. 3. Arjun could not sell any item on the second day because the first household he met on that day complained against him. 4. Vikram met 30 more households on the second day than on the first day. 5. Vikram's success rate was twice that of Rohit's on the first day, and it was 75% of Rohit's on the second day.
How many households did Rohit meet on the second day?
Answer and solution
Correct answer: D
Solution
Answer: D — between 21 and 29
1. Rohit sells 100 items over two identical days, hence 50 items per day. If he visits h households each day, every salesman visits 2h households in total.
2. Let Vikram’s visits be a and a+30. Then 2a+30=2h, so a=h−15 and his second-day visits are h+15.
3. Rohit’s rate is 50/h. Vikram sells 100(h−15)/h on Day 1 and 37.5(h+15)/h on Day 2.
4. These sales sum to 100: 100h−1500+37.5h+562.5=100h. Thus 37.5h=937.5 and h=25.
5. Vikram: visits 10,40 and sells 40,60. Rohit: visits 25,25 and sells 50,50. Arjun meets 1 household on Day 2 due to the complaint, so visits 49,1 and sells 100,0.
6. Check every salesman’s totals: 50 households and 100 items. Vikram’s rates 4 and 1.5 are 2 and 0.75 times Rohit’s rate 2.
7. Rohit meets 25 on Day 2, between 21 and 29. Answer D.
Why the rate model is valid: Success rate means items per household, and a household can buy several items. Thus a rate of 4 is permitted; it is not a probability capped at 1. Multiplying the rate by households gives items sold. Both the daily split and the two-day totals must be satisfied together; the completed table checks all these conditions.
G Strategy shortcut
Method: Normalize Rohit’s daily success rate to r. Vikram’s rates are 2r and 0.75r, and his combined average must be r. Alligation gives household weights (1−0.75):(2−1)=1:4. Their 30-household difference is 3 parts, so Vikram visits 10 and 40. Rohit visits 25 each day. Verification: 7. Rohit meets 25 on Day 2, between 21 and 29. Answer D.
G Strategy: Method: Normalize Rohit’s daily success rate to r. Vikram’s rates are 2r and 0.75r, and his combined average must be r. Alligation gives household weights (1−0.75):(2−1)=1:4. Their 30-household difference is 3 parts, so Vikram visits 10 and 40. Rohit visits 25 each day.
Verification: 7. Rohit meets 25 on Day 2, between 21 and 29. Answer D.
How many households did Vikram meet on the first day?
Answer and solution
Correct answer: D
Solution
Answer: D — 10 or less
1. Rohit sells 100 items over two identical days, hence 50 items per day. If he visits h households each day, every salesman visits 2h households in total.
2. Let Vikram’s visits be a and a+30. Then 2a+30=2h, so a=h−15 and his second-day visits are h+15.
3. Rohit’s rate is 50/h. Vikram sells 100(h−15)/h on Day 1 and 37.5(h+15)/h on Day 2.
4. These sales sum to 100: 100h−1500+37.5h+562.5=100h. Thus 37.5h=937.5 and h=25.
5. Vikram: visits 10,40 and sells 40,60. Rohit: visits 25,25 and sells 50,50. Arjun meets 1 household on Day 2 due to the complaint, so visits 49,1 and sells 100,0.
6. Check every salesman’s totals: 50 households and 100 items. Vikram’s rates 4 and 1.5 are 2 and 0.75 times Rohit’s rate 2.
7. Vikram meets 10 on Day 1, so “10 or less”. Answer D.
Why the rate model is valid: Success rate means items per household, and a household can buy several items. Thus a rate of 4 is permitted; it is not a probability capped at 1. Multiplying the rate by households gives items sold. Both the daily split and the two-day totals must be satisfied together; the completed table checks all these conditions.
G Strategy shortcut
Method: Normalize Rohit’s daily success rate to r. Vikram’s rates are 2r and 0.75r, and his combined average must be r. Alligation gives household weights (1−0.75):(2−1)=1:4. Their 30-household difference is 3 parts, so Vikram visits 10 and 40. Rohit visits 25 each day. Verification: 7. Vikram meets 10 on Day 1, so “10 or less”. Answer D.
G Strategy: Method: Normalize Rohit’s daily success rate to r. Vikram’s rates are 2r and 0.75r, and his combined average must be r. Alligation gives household weights (1−0.75):(2−1)=1:4. Their 30-household difference is 3 parts, so Vikram visits 10 and 40. Rohit visits 25 each day.
Verification: 7. Vikram meets 10 on Day 1, so “10 or less”. Answer D.
How many ZENTRA items were sold by Vikram on the first day?
Answer and solution
Correct answer: 40
Solution
Answer: 40
1. Rohit sells 100 items over two identical days, hence 50 items per day. If he visits h households each day, every salesman visits 2h households in total.
2. Let Vikram’s visits be a and a+30. Then 2a+30=2h, so a=h−15 and his second-day visits are h+15.
3. Rohit’s rate is 50/h. Vikram sells 100(h−15)/h on Day 1 and 37.5(h+15)/h on Day 2.
4. These sales sum to 100: 100h−1500+37.5h+562.5=100h. Thus 37.5h=937.5 and h=25.
5. Vikram: visits 10,40 and sells 40,60. Rohit: visits 25,25 and sells 50,50. Arjun meets 1 household on Day 2 due to the complaint, so visits 49,1 and sells 100,0.
6. Check every salesman’s totals: 50 households and 100 items. Vikram’s rates 4 and 1.5 are 2 and 0.75 times Rohit’s rate 2.
7. Vikram sells 10×4=40 items on Day 1. TITA answer 40.
Why the rate model is valid: Success rate means items per household, and a household can buy several items. Thus a rate of 4 is permitted; it is not a probability capped at 1. Multiplying the rate by households gives items sold. Both the daily split and the two-day totals must be satisfied together; the completed table checks all these conditions.
G Strategy shortcut
Method: Normalize Rohit’s daily success rate to r. Vikram’s rates are 2r and 0.75r, and his combined average must be r. Alligation gives household weights (1−0.75):(2−1)=1:4. Their 30-household difference is 3 parts, so Vikram visits 10 and 40. Rohit visits 25 each day. Verification: 7. Vikram sells 10×4=40 items on Day 1. TITA answer 40.
G Strategy: Method: Normalize Rohit’s daily success rate to r. Vikram’s rates are 2r and 0.75r, and his combined average must be r. Alligation gives household weights (1−0.75):(2−1)=1:4. Their 30-household difference is 3 parts, so Vikram visits 10 and 40. Rohit visits 25 each day.
Verification: 7. Vikram sells 10×4=40 items on Day 1. TITA answer 40.
What was the total number of households met by Vikram, Arjun and Rohit on the first day?
Answer and solution
Correct answer: 84
Solution
Answer: 84
1. Rohit sells 100 items over two identical days, hence 50 items per day. If he visits h households each day, every salesman visits 2h households in total.
2. Let Vikram’s visits be a and a+30. Then 2a+30=2h, so a=h−15 and his second-day visits are h+15.
3. Rohit’s rate is 50/h. Vikram sells 100(h−15)/h on Day 1 and 37.5(h+15)/h on Day 2.
4. These sales sum to 100: 100h−1500+37.5h+562.5=100h. Thus 37.5h=937.5 and h=25.
5. Vikram: visits 10,40 and sells 40,60. Rohit: visits 25,25 and sells 50,50. Arjun meets 1 household on Day 2 due to the complaint, so visits 49,1 and sells 100,0.
6. Check every salesman’s totals: 50 households and 100 items. Vikram’s rates 4 and 1.5 are 2 and 0.75 times Rohit’s rate 2.
7. Day 1 households=10+49+25=84. TITA answer 84.
Why the rate model is valid: Success rate means items per household, and a household can buy several items. Thus a rate of 4 is permitted; it is not a probability capped at 1. Multiplying the rate by households gives items sold. Both the daily split and the two-day totals must be satisfied together; the completed table checks all these conditions.
G Strategy shortcut
Method: Normalize Rohit’s daily success rate to r. Vikram’s rates are 2r and 0.75r, and his combined average must be r. Alligation gives household weights (1−0.75):(2−1)=1:4. Their 30-household difference is 3 parts, so Vikram visits 10 and 40. Rohit visits 25 each day. Verification: 7. Day 1 households=10+49+25=84. TITA answer 84.
G Strategy: Method: Normalize Rohit’s daily success rate to r. Vikram’s rates are 2r and 0.75r, and his combined average must be r. Alligation gives household weights (1−0.75):(2−1)=1:4. Their 30-household difference is 3 parts, so Vikram visits 10 and 40. Rohit visits 25 each day.
Verification: 7. Day 1 households=10+49+25=84. TITA answer 84.
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