To find the minimum number of working days, we need to consider a month with the maximum possible number of holidays. The holidays are defined as all Sundays and the second and fourth Saturdays. We will analyze months with different lengths (28, 29, 30, 31 days) and determine the maximum number of holidays possible in each case.
Working days = Total days - Total holidays
Working days = \(28 - 6 = 22\) days.
If February has 29 days, it will have 4 full weeks and 1 extra day. To maximize holidays, this extra day should be a Sunday. If it's a Sunday, there will be 5 Sundays and 4 Saturdays (2 of which are holidays). Total holidays = \(5 \text{ Sundays} + 2 \text{ Saturdays} = 7\). Working days = \(29 - 7 = 22\).
A 30-day month can have a maximum of 5 Sundays and 5 Saturdays. To maximize holidays, let's assume the month starts on a Saturday. Then it will have 5 Saturdays and 5 Sundays. The 5 Saturdays will include the 1st, 8th, 15th, 22nd, 29th. The 2nd and 4th Saturdays are the 8th and 22nd. So, 2 Saturdays are holidays. Total holidays = \(5 \text{ Sundays} + 2 \text{ Saturdays} = 7\). Working days = \(30 - 7 = 23\).
A 31-day month can have a maximum of 5 Sundays and 5 Saturdays. To maximize holidays, let's assume the month starts on a Saturday. Then it will have 5 Saturdays and 5 Sundays. The 2nd and 4th Saturdays are holidays. Total holidays = \(5 \text{ Sundays} + 2 \text{ Saturdays} = 7\). Working days = \(31 - 7 = 24\).
The minimum number of working days is 22.
Correct Option: B) 22
This problem involves calculating the remaining population after a certain fraction migrates each year for a specified number of years. We can solve this by determining the fraction of the population remaining each year and then multiplying these fractions over the given period.
Understand the migration rate: Each year, \(1/3\)rd of the population migrates. This means that if the initial population is \(P\), then \(P/3\) migrates.
Calculate the remaining population fraction each year: If \(1/3\)rd migrates, then the fraction of the population that remains is \(1 - 1/3 = 2/3\).
Calculate the population after the first year: If the initial population is \(P_0\), after the first year, the population \(P_1\) will be:
\[ P_1 = P_0 \times \left(1 - \frac{1}{3}\right) = P_0 \times \frac{2}{3} \]Calculate the population after the second year: The migration happens from the *current* population. So, after the second year, the population \(P_2\) will be:
\[ P_2 = P_1 \times \frac{2}{3} = \left(P_0 \times \frac{2}{3}\right) \times \frac{2}{3} = P_0 \times \left(\frac{2}{3}\right)^2 \]Generalize for \(n\) years: Following this pattern, after \(n\) years, the remaining population \(P_n\) will be:
\[ P_n = P_0 \times \left(\frac{2}{3}\right)^n \]Apply for the sixth year: In this problem, \(n = 6\). So, the leftover population after the sixth year will be:
\[ P_6 = P_0 \times \left(\frac{2}{3}\right)^6 \]Calculate the final fraction:
\[ \left(\frac{2}{3}\right)^6 = \frac{2^6}{3^6} = \frac{2 \times 2 \times 2 \times 2 \times 2 \times 2}{3 \times 3 \times 3 \times 3 \times 3 \times 3} = \frac{64}{729} \]Therefore, the leftover population is \(64/729\)th part of the original population.
Correct Option: D) 64/729th part of the population
This is a logical reasoning problem involving the sequencing of events. We need to deduce the order of four tests (Physics, Chemistry, Mathematics, Biology) conducted on four consecutive days based on the given clues. We will use a step-by-step approach to place each test in its correct position.
Let the four days be Day 1, Day 2, Day 3, and Day 4.
Clue 1: Chemistry is conducted exactly after two tests are held.
This means Chemistry is the third test. So, Day 3 is Chemistry.
Clue 2: The Physics test is held before the test which is conducted after Biology.
Let's break this down:
From Clue 1, we know Chemistry is on Day 3. This leaves Day 1, Day 2, and Day 4 for Physics, Biology, and Mathematics.
Since Biology cannot be on Day 4 (as it's followed by another test), Biology must be on Day 1 or Day 2.
Consider the possibilities for Biology:
If Biology is on Day 1:
According to Clue 2, Physics is held before the test on Day 2. This is impossible because Physics would have to be before Day 1, which is not possible.
If Biology is on Day 2:
Now, apply Clue 2: "Physics test is held before the test which is conducted after Biology." The test after Biology is on Day 4. So, Physics must be held before Day 4. The only remaining day before Day 4 is Day 1. Thus, Physics is on Day 1.
So the order is:
The only test remaining is Mathematics, and the only day remaining is Day 4.
Therefore, the final order is:
The question asks for the last test held, which is on Day 4.
Correct Option: C) Mathematics
The last test held is Mathematics.
This problem involves setting up a system of inequalities and equations based on the given statements about the incomes of A, B, C, and D. By manipulating these expressions, we can compare the incomes and determine who earns the highest.
Let the incomes of A, B, C, and D be represented by \(A\), \(B\), \(C\), and \(D\) respectively.
Translate the given statements into mathematical expressions:
"The sum of income of A and B is more than that of C and D taken together."
\[ A + B > C + D \quad \text{(1)} \]"The sum of income of A and C is the same as that of B and D taken together."
\[ A + C = B + D \quad \text{(2)} \]"A earns half as much as the sum of the income of B and D."
\[ A = \frac{1}{2} (B + D) \quad \text{(3)} \]Substitute equation (3) into equation (2):
From (3), we have \(B + D = 2A\). Substitute this into (2):
\[ A + C = 2A \]\[ C = 2A - A \]\[ C = A \quad \text{(4)} \]Now, substitute \(C = A\) (from equation 4) and \(B + D = 2A\) (from equation 3) into inequality (1):
\[ A + B > C + D \]\[ A + B > A + 2A \]\[ A + B > 3A \]\[ B > 3A - A \]\[ B > 2A \quad \text{(5)} \]From equation (4), we know \(C = A\).
From inequality (5), we know \(B > 2A\).
To compare D, use \(B + D = 2A\):
\[ D = 2A - B \]Since \(B > 2A\), let's substitute this into the expression for D. If \(B > 2A\), then \(2A - B\) must be negative. However, income cannot be negative. This indicates a contradiction or an issue in assuming D can be negative. Let's re-evaluate. The incomes must be positive. So \(D > 0\).
From \(B + D = 2A\), since \(D > 0\), it implies \(B < 2A\). This contradicts our finding \(B > 2A\).
Let's recheck the steps carefully.
From (3): \(B + D = 2A\).
From (2): \(A + C = B + D\).
Substitute \(B + D = 2A\) into (2):
\[ A + C = 2A \]\[ C = A \]Now, substitute \(C = A\) and \(B + D = 2A\) into inequality (1):
\[ A + B > C + D \]\[ A + B > A + (2A - B) \]\[ A + B > 3A - B \]\[ 2B > 2A \]\[ B > A \quad \text{(6)} \]Let's re-evaluate D using \(B + D = 2A\):
\[ D = 2A - B \]We know \(B > A\). Let's compare D with A and C.
Since \(B > A\), then \(2A - B < 2A - A\), which means \(D < A\).
So, we have the following relationships:
Combining these, we get \(D < A = C < B\).
Therefore, B has the highest income.
B) B is the correct option because, based on the derived inequalities, \(B\) has the highest income (\(B > A\), \(A = C\), and \(D < A\)).
A) A — A's income is equal to C's income, but less than B's income (\(A < B\)). So A is not the highest.
C) C — C's income is equal to A's income, but less than B's income (\(C < B\)). So C is not the highest.
D) D — D's income is less than A's income (\(D < A\)), and therefore also less than B's income. So D is not the highest.
Consider the following :
Statements: Good voice is a natural gift but one has to keep practicing to improve and excel well in the field of music.
Conclusions:
I. Natural gits need nurturing and care.
II. Even though one's voice is not good, one can keep practicing.
Which one of the following is correct in respect of the above statement and conclusions?
Only conclusion I follows from the statement.
Only conclusion II follows from the statement.
Either conclusion I or conclusion II follows from the statement.
Neither conclusion I nor conclusion II follows from the statement.
This is a statement and conclusion type of question. We need to analyze the given statement carefully and determine which of the conclusions logically follows from it. The statement discusses the importance of practice even for a natural gift.
Correct Option: A) Only conclusion I follows from the statement.
The problem describes three spiders climbing pillars with different climbing and slipping rates. They all reach the top in 40 chances. We need to find the height of the shortest pillar. The key is to calculate the effective height climbed in each chance and then consider the last chance where the spider might not slip.
B) The height of pillar Y is 163 cm, which is the shortest among the three pillars.
The question asks for the correct understanding of rights based on the provided statement: "Rights are certain advantageous conditions of social well-being indispensable to the true development of the citizen." We need to analyze the key components of this statement to determine whether rights focus on individual good, social good, or both.
C) Rights aim at both individual and social good. The statement clearly links "social well-being" to the "true development of the citizen." This implies a symbiotic relationship where the conditions of society enable individual growth, and the development of individuals contributes to a better society. Therefore, rights serve both purposes simultaneously.
This problem involves determining a student's rank from the bottom of a list, given their rank from the top and the total number of students in that list. We need to first calculate the number of students who passed, as the merit list only includes passed students.
The total number of students in the class is 52.
The number of students who failed is 15.
So, the number of students who passed and are included in the merit list is:
\[ \text{Number of passed students} = \text{Total students} - \text{Failed students} \]\[ \text{Number of passed students} = 52 - 15 = 37 \]Ramesh's position from the top in the merit list is 22nd.
The total number of students in the merit list is 37.
To find his position from the bottom, we use the formula:
\[ \text{Position from bottom} = (\text{Total number of students in list} - \text{Position from top}) + 1 \]\[ \text{Position from bottom} = (37 - 22) + 1 \]\[ \text{Position from bottom} = 15 + 1 \]\[ \text{Position from bottom} = 16 \]C) 16th is the correct position of Ramesh from the bottom of the merit list.
Consider the following :
A + B means A is the son of B.
A - B means A is the wife of B.
What does the expression P + R - Q mean?
Q is the son of P.
Q is the wife of P.
Q is the father of P.
None of the above.
This is a blood relations problem. We need to decode the given symbols and then apply them to the expression to determine the relationship between P and Q.
According to the first rule, "A + B means A is the son of B", so "P + R" means P is the son of R.
According to the second rule, "A - B means A is the wife of B", so "R - Q" means R is the wife of Q.
From step 3, P is the son of R.
From step 4, R is the wife of Q.
If R is the wife of Q, then Q is the husband of R.
Since P is the son of R, and R is married to Q, it implies that Q is the father of P.
Correct Option: C) Q is the father of P.