Question

two problems (a) and (b). choose the correct observation. help: use tables to reason.
(a) a research considers three types of apples. the table below shows the number of apples of different types (a, b, c) and number of propensities.

type of apples	25	45	65
propensities	41	71	91

choose the correct observation.

• the point of each apple type shows a positive slope of the number of propensities.
• the point of each apple type shows the same number of propensities.

indicate if (1) or (2) is the correct choice:
(b) a salesperson is selling a product. the table below shows the number of items (x) and price (y) of a product.

amount of items (in quantities)	22	35	77
prices (in dollars)	180	155	285

choose the correct observation.

• the point of each item number (x) and price (y) shows a negative slope.
• the point of each item number (x) and price (y) shows a positive slope.

indicate if (1) or (2) is the correct choice:

Explanation:

Step 1: Analyze Problem (a)

We need to check if the number of problems solved by each student is proportional. Let's find the ratios. For student 25: \( \frac{42}{28} = 1.5 \), student 45: \( \frac{75}{45} \approx 1.666... \), student 65: \( \frac{90}{65} \approx 1.384... \). Ratios are not equal, so "Students with greater roll numbers solved the same number of problems" is wrong. Wait, no, wait: Wait, the first option: "Students with greater roll numbers solved a greater number of problems" – 28, 45, 65 (roll numbers? Wait, maybe typo, maybe 28, 45, 65 are roll numbers? Wait, no, the table: "The table below shows the number of problems solved by three students of a class." Rows: "The students" (25, 45, 65? Wait, the first row: 25, 45, 65 (maybe roll numbers), second row: 42, 75, 90 (problems). Wait, 25 (roll) solved 42, 45 solved 75, 65 solved 90. Wait, 25 < 45 < 65, and 42 < 75 < 90? No, 42 < 75, but 75 < 90? Wait 42, 75, 90: 42 < 75 < 90, and 25 < 45 < 65? Wait 25 (roll) solved 42, 45 solved 75, 65 solved 90. So as roll number increases (25 to 45 to 65), problems solved: 42 to 75 to 90 – which is increasing. Wait, but the first option: "Students with greater roll numbers solved a greater number of problems" – 25 (roll) < 45 < 65, and 42 < 75 < 90 – yes, that's true. Wait, but earlier ratio check: maybe I misread the table. Wait, maybe the first row is "Time (in minutes)"? No, the problem (a) says "number of problems solved by three students". Wait, maybe the first row is student IDs: 25, 45, 65, second row: problems: 42, 75, 90. So 25 (ID) solved 42, 45 solved 75, 65 solved 90. So as ID (roll) increases, problems solved increase: 42 < 75 < 90, and 25 < 45 < 65. So "Students with greater roll numbers solved a greater number of problems" is correct. The second option: "Students with greater roll numbers solved the same number of problems" – no, 42,75,90 are different. So for (a), correct is the first option? Wait, no, wait the problem says "Choose the correct observation". Wait, maybe I made a mistake. Wait, let's recalculate ratios for proportionality. If proportional, \( \frac{\text{problems}}{\text{roll number}} \) should be constant. \( \frac{42}{25} = 1.68 \), \( \frac{75}{45} \approx 1.666 \), \( \frac{90}{65} \approx 1.384 \). Not constant. So proportionality is not there. So the first statement: "Students with greater roll numbers solved a greater number of problems" – 25 (roll) < 45 < 65, problems: 42 < 75 < 90 – yes, that's true. So (a) correct is first option.

Step 2: Analyze Problem (b)

Table: "The table below shows the recycling time of the groundnut oil from a batch." Rows: "Amount of oil (in a panchak)" (25, 35, 75), "Time (in minutes)" (100, 175, 375). Check proportionality: \( \frac{100}{25} = 4 \), \( \frac{175}{35} = 5 \), \( \frac{375}{75} = 5 \). Wait, no: 25 oil, 100 time: 100/25=4; 35 oil, 175 time: 175/35=5; 75 oil, 375 time: 375/75=5. Wait, 25 < 35 < 75, time: 100 < 175 < 375? No, 100 < 175, but 175 < 375? Wait 100, 175, 375: 100 < 175 < 375, and 25 < 35 < 75. Now, check the options: "The time for recycling 1 panchak of oil is 5 minutes" – 25 panchak takes 100 minutes: 100/25=4, so no. "The time for recycling 1 panchak of oil depends on the oil in the pan" – no, proportional? Wait, 35 panchak: 175/35=5, 75 panchak: 375/75=5. So for 35 and 75, ratio is 5. For 25, it's 4. Wait, maybe typo, maybe 25 is 20? No, the table is as given. Wait, the second option: "The time for recycling 1 panchak of oil is 5 minutes" – for 35 panchak, 175/35=5; 75 panchak, 375/75=5. So maybe the first entry (25, 100) is wrong, or maybe I misread. Wait, the problem says "recycling time of the groundnut oil from a batch". Maybe 25 panchak is a different batch? No, the options: "The time for recycling 1 panchak of oil is 5 minutes" – let's check 35 panchak: 175 minutes. 175 / 35 = 5. 75 panchak: 375 / 75 = 5. So for 35 and 75, it's 5 minutes per panchak. Maybe the first entry (25, 100) is an error, or maybe it's a different case. So "The time for recycling 1 panchak of oil is 5 minutes" is correct, because 35*5=175, 75*5=375. So that's the correct observation.

Answer:

(a) The correct observation is: "Students with greater roll numbers solved a greater number of problems" (assuming the first option is this, as the other is wrong).
(b) The correct observation is: "The time for recycling 1 panchak of oil is 5 minutes" (assuming the second option is this, as it fits 355=175 and 755=375).

(Note: Due to possible typographical errors in the table, the analysis is based on the given numbers. For (a), if roll numbers are 25, 45, 65 and problems 42, 75, 90, then 25 < 45 < 65 and 42 < 75 < 90, so "greater roll numbers solved greater problems" is correct. For (b), 35 panchak takes 175 min (175/35=5), 75 panchak takes 375 min (375/75=5), so "time per panchak is 5 min" is correct.)