Question

The percentage of respondents aged less than 40 years is at least

• A. 10%
• B. 16.67%
• C. 20.0%
• D. 30%
• A. 30%
• B. 73.33%
• C. 76.67%
• D. 90%
• A. 6.67%
• B. 10%
• C. 13.33%
• D. 26.67%

Answer 1

The Correct Option is D

Step 1: Understand what 'at least' asks:
The table gives, for each group, the head count and the youngest and oldest ages. It does not give every person's age.
So the real percentage under 40 can change from one possible set of ages to another. 'At least' asks for the smallest value it can ever take, which means the number of people who are under 40 in every possible case.

Step 2: Rule for each group:
Take any group. The youngest person in it has the minimum age shown. If that minimum is below 40, this one person is certainly under 40.
The other members can be placed anywhere between the minimum and the maximum. If the maximum is 40 or more, they can be pushed up to 40 or more, so we do not have to count them.
If the maximum itself is below 40, every person in the group is under 40, and all of them count.

Step 3: Apply the rule to the male groups:
Male, 0 children: one person, age 38. He is under 40. Count 1.
Male, 1 child: one person, age 32. Count 1.
Male, 2 children: 8 people, youngest 21, oldest 65. Only the youngest is sure to be under 40, the rest can be 65. Count 1.
Male, 3 children: 2 people, ages from 32 to 33. Even the oldest is 33, so both are under 40. Count 2.

Step 4: Apply the rule to the female groups:
Female, 0 children: 5 people, youngest 34, oldest 49. Only the youngest is sure to be under 40. Count 1.
Female, 1 child: 8 people, youngest 35, oldest 57. Count 1.
Female, 2 children: 3 people, youngest 37, oldest 63. Count 1.
Female, 3 children: 2 people, youngest 27, oldest 40. The oldest is exactly 40, which is not less than 40, so only the youngest counts. Count 1.

Step 5: Add and convert to a percentage:
Males: \( 1 + 1 + 1 + 2 = 5 \). Females: \( 1 + 1 + 1 + 1 = 4 \). Total sure-to-be-under-40 people \( = 5 + 4 = 9 \).
\[ \frac{9}{30} \times 100 = 30\% \]
This count can really be reached, because all the other people can be given ages of 40 or more inside their allowed ranges. So 30% is the lowest the percentage can go.

Step 6: Check the other options:
Option (1) 10% is 3 people, option (2) 16.67% is 5 people, option (3) 20% is 6 people. We have shown that at least 9 people are under 40, so the percentage can never be this low. None of these is the minimum possible value.
Option (4) 30% matches exactly 9 people, so it is the right choice.

Final Answer:
At least 9 of the 30 respondents are under 40 years old, which is 30 percent. This is option (4). \[ \boxed{\text{Option (4): } 30\%} \]

Answer 2

Step 1: Understand the question:
'At most' asks for the largest number of respondents who can be older than 35. We have to push as many people as possible above 35, but the youngest and oldest ages of each group are fixed, so some people cannot be moved.

Step 2: Rule for each group:
The youngest person in a group has exactly the minimum age. If this age is 35 or below, he or she is not older than 35, and nothing can change it.
Every other person can be given any age up to the group's maximum. If the maximum is above 35, all of them can be made older than 35.
If the maximum itself is 35 or below, nobody in the group is older than 35.

Step 3: Count group by group:
Male, 0 children: 1 person aged 38, older than 35. Count 1.
Male, 1 child: 1 person aged 32. Count 0.
Male, 2 children: 8 people, youngest 21, oldest 65. The youngest is not older than 35, the other 7 can be. Count 7.
Male, 3 children: ages 32 and 33, both below 35. Count 0.
Female, 0 children: 5 people, youngest 34, oldest 49. Count 4.
Female, 1 child: 8 people, youngest 35. Age 35 is not older than 35, so the youngest is out; the other 7 can be up to 57. Count 7.
Female, 2 children: 3 people, youngest 37, so all three are older than 35. Count 3.
Female, 3 children: 2 people, ages 27 and 40. Only the person aged 40 is older than 35. Count 1.

Step 4: Add them:
\[ 1 + 0 + 7 + 0 + 4 + 7 + 3 + 1 = 23 \]
\[ \frac{23}{30} \times 100 = 76.67\% \]

Step 5: Check the options:
Option (1) 30% is only 9 people. We can do much better than that, so it is far from the largest value.
Option (2) 73.33% is 22 people. It is possible, but we showed that 23 people can be older than 35, so 22 is not the maximum.
Option (4) 90% is 27 people. That needs only 3 people to be 35 or younger, but at least 7 people (the youngest member of five groups and both people of the 32-33 group) cannot be older than 35, so 27 is impossible.
Option (3) 76.67% is 23 people and matches our count.

Final Answer:
At most 23 of the 30 respondents can be older than 35, which is 76.67 percent. This is option (3). \[ \boxed{\text{Option (3): } 76.67\%} \]

Answer 3

Step 1: Understand the question:
We need the least possible number of respondents whose age lies from 35 to 40, both ends included. We only know the youngest and oldest age of each group, so we count the people who must fall in this band in every possible case.

Step 2: Rule for each group:
The youngest and the oldest person of a group have exactly the minimum and maximum ages. For a group of one person, both are the same person. For a group of two or more, they are different people.
The people in between can have any age from the minimum to the maximum. If the group allows an age outside 35 to 40 (below 35 or above 40), we can place all these free people outside the band, so they need not be counted.
So only the youngest and oldest persons can be forced into the band.

Step 3: Check the male groups:
Male, 0 children: one person, age 38. This is inside 35 to 40. Count 1.
Male, 1 child: one person, age 32. Outside the band. Count 0.
Male, 2 children: 8 people from 21 to 65. The youngest is 21 and the oldest 65, both outside. The six in between can be 21 or 65. Count 0.
Male, 3 children: ages 32 and 33, both below 35. Count 0.

Step 4: Check the female groups:
Female, 0 children: 5 people from 34 to 49. The youngest is 34 and the oldest 49, both outside. The three in between can be 34 or 49. Count 0.
Female, 1 child: 8 people from 35 to 57. The youngest is exactly 35, so he or she is inside. The oldest is 57, outside. The six in between can be 57. Count 1.
Female, 2 children: 3 people from 37 to 63. The youngest is 37, inside. The other two can be 63. Count 1.
Female, 3 children: 2 people, 27 and 40. The person aged 40 is inside the band (40 is included). The other is 27, outside. Count 1.

Step 5: Add and convert:
\[ 1 + 0 + 0 + 0 + 0 + 1 + 1 + 1 = 4 \]
\[ \frac{4}{30} \times 100 = 13.33\% \]
We can really reach this value, since all other people can be given ages outside the band within their ranges.

Step 6: Check the options:
6.67% is 2 people and 10% is 3 people. Both are below the 4 people who are forced into the band, so the percentage can never be that low.
26.67% is 8 people. It is possible but not the minimum.
Only 13.33%, which is option (3), equals the minimum.

Final Answer:
At least 4 of the 30 respondents must be aged 35 to 40, which is 13.33 percent. This is option (3). \[ \boxed{\text{Option (3): } 13.33\%} \]