This problem involves calculating percentage change. We are given the percentage increase in weight when block B is placed on block A. We need to find the percentage decrease in weight when block B is removed, relative to the combined weight of A and B.
Let the weight of block A be \(W_A\).
Let the weight of block B be \(W_B\).
When B is placed on A, the weight increases by 60% with respect to the weight of A. This means the total weight \(W_A + W_B\) is 60% more than \(W_A\).
\[ W_A + W_B = W_A + 0.60 \times W_A \]\[ W_A + W_B = 1.60 \times W_A \]From the equation above, subtract \(W_A\) from both sides:
\[ W_B = 1.60 \times W_A - W_A \]\[ W_B = 0.60 \times W_A \]The total weight when B is on A is \(W_{\text{total}} = W_A + W_B\).
Substitute \(W_B = 0.60 \times W_A\):
\[ W_{\text{total}} = W_A + 0.60 \times W_A \]\[ W_{\text{total}} = 1.60 \times W_A \]When B is removed, the weight reduces by \(W_B\).
We need to find this reduction as a percentage of the total weight \(W_{\text{total}}\).
\[ \text{Percentage reduction} = \frac{\text{Weight of B}}{\text{Total weight of A and B}} \times 100\% \]\[ \text{Percentage reduction} = \frac{W_B}{W_A + W_B} \times 100\% \]Substitute \(W_B = 0.60 \times W_A\) and \(W_A + W_B = 1.60 \times W_A\) into the percentage reduction formula:
\[ \text{Percentage reduction} = \frac{0.60 \times W_A}{1.60 \times W_A} \times 100\% \]\[ \text{Percentage reduction} = \frac{0.60}{1.60} \times 100\% \]\[ \text{Percentage reduction} = \frac{6}{16} \times 100\% \]\[ \text{Percentage reduction} = \frac{3}{8} \times 100\% \]\[ \text{Percentage reduction} = 0.375 \times 100\% \]\[ \text{Percentage reduction} = 37.5\% \]Correct Option: D) 37.50%
Consider the following Statements and Conclusions:
Statements:
1. Some rats are cats.
2. Some cats are dogs.
3. No dog is a cow.
Conclusions:
I. No cow is a cat.
II. No dog is a rat.
III. Some cats are rats.
Which of the above conclusions is/are drawn from the statements?
I, II and III
Only I and II
Only III
Only II and III
This is a syllogism problem. We will use Venn diagrams to represent the given statements and then check the validity of each conclusion based on these diagrams. For a conclusion to be valid, it must hold true in all possible Venn diagram representations of the statements.
This means there is an overlap between the set of rats and the set of cats. We can draw two intersecting circles, one for "Rats" and one for "Cats", with the overlapping region representing "Some rats are cats" (and "Some cats are rats").
This means there is an overlap between the set of cats and the set of dogs. We draw a third circle for "Dogs" intersecting the "Cats" circle. It may or may not intersect the "Rats" circle, depending on the specific arrangement, but it must intersect "Cats".
This means the set of dogs and the set of cows are entirely separate. We draw a fourth circle for "Cows" that does not overlap with the "Dogs" circle. The "Cows" circle may or may not overlap with "Rats" or "Cats".
From statement 3, we know "No dog is a cow." From statement 2, "Some cats are dogs." This means some cats are inside the 'Dogs' circle. Since the 'Dogs' circle has no overlap with 'Cows', those specific cats that are dogs cannot be cows. However, there might be other cats that are not dogs, and these cats could potentially be cows. We can draw a scenario where the 'Cows' circle intersects the 'Cats' circle (but not the 'Dogs' circle). Since this conclusion is not necessarily true in all possible diagrams, it is not a valid conclusion.
From statement 1, "Some rats are cats." From statement 2, "Some cats are dogs." This means there is an overlap between rats and cats, and an overlap between cats and dogs. However, there is no direct statement linking rats and dogs, nor is there a statement that prevents rats and dogs from overlapping. We can draw a Venn diagram where the 'Dogs' circle intersects the 'Rats' circle (via the 'Cats' circle or directly), and another where it does not. Since this conclusion is not necessarily true in all possible diagrams, it is not a valid conclusion.
This conclusion is a direct restatement of Statement 1, just rephrased. Statement 1 says, "Some rats are cats." If some rats are cats, it logically follows that some cats are rats. This is always true based on Statement 1.
C) Only III
Conclusion III is a direct inference from Statement 1. If "Some rats are cats", then it must also be true that "Some cats are rats". Conclusions I and II are not necessarily true and can be disproven with alternative valid Venn diagrams.
To form a parallelogram, we need to select two parallel lines from one set and two parallel lines from the other set. This is a problem of combinations, where the order of selection does not matter.
D) 36 is the correct answer because there are 6 ways to choose two lines from the first set and 6 ways to choose two lines from the second set, resulting in \(6 \times 6 = 36\) parallelograms.
To find the number of students who play both cricket and football, we need to identify the identification numbers that are divisible by both 4 and 6. This means finding the numbers that are divisible by the Least Common Multiple (LCM) of 4 and 6. We will then count how many such numbers exist between 1 and 100.
Identify the conditions:
Find the LCM of 4 and 6: To be divisible by both 4 and 6, a number must be a multiple of their Least Common Multiple (LCM). Factors of 4 are \(2 \times 2\). Factors of 6 are \(2 \times 3\). \[ \text{LCM}(4, 6) = 2^2 \times 3 = 4 \times 3 = 12 \] So, students who play both sports have ID numbers that are multiples of 12.
Count the multiples of 12 between 1 and 100: We need to find how many numbers from 1 to 100 are divisible by 12. We can list them or use division. The multiples of 12 are: 12, 24, 36, 48, 60, 72, 84, 96. Alternatively, divide 100 by 12: \[ \frac{100}{12} = 8 \text{ with a remainder of } 4 \] The quotient, 8, indicates that there are 8 multiples of 12 up to 100.
Conclusion: There are 8 students whose identification numbers are divisible by both 4 and 6, meaning 8 students play both cricket and football.
Correct Option: B) 8
The problem states that a runner has completed 12 km, which represents 80% of the total race distance. We need to find the total distance of the race. This can be solved by setting up a proportion or by using the concept of percentages.
Correct Option: B) 15
The problem involves calculating the total amount Raju has and comparing it with the required amount to buy a mobile handset. We need to determine the full cost of the handset first, then calculate Raju's total money after borrowing, and finally compare these two values.
Raju has ₹9000, which is 75% of the amount required to buy the handset.
Let \(C\) be the full cost of the handset.
\[ 0.75 \times C = 9000 \]To find \(C\), divide 9000 by 0.75:
\[ C = \frac{9000}{0.75} \] \[ C = \frac{9000}{\frac{3}{4}} \] \[ C = 9000 \times \frac{4}{3} \] \[ C = 3000 \times 4 \] \[ C = 12000 \]So, the full cost of the mobile handset is ₹12000.
Raju initially has ₹9000.
He borrows ₹2000 from a friend.
Total amount Raju has now \( = 9000 + 2000 = 11000 \).
Raju has ₹11000.
The handset costs ₹12000.
Since \(11000 < 12000\), Raju still does not have enough money to buy the handset.
The deficit is \(12000 - 11000 = 1000\).
Correct Option: A) Raju still does not have enough amount to buy the handset.
This problem involves setting up and solving a system of linear equations based on the given age relationships at two different points in time. We will define variables for Meenu's and Meera's ages in a specific year and then use the given ratios to form equations. Solving these equations will give us their ages, from which we can deduce Meenu's birth year.
Define variables for ages in 2002:
Let Meenu's age in 2002 be \(M\).
Let Meera's age in 2002 be \(R\).
Formulate the first equation based on the year 2002:
In 2002, Meenu's age was one-third of Meera's age.
\[ M = \frac{1}{3}R \quad \text{(Equation 1)} \]Define variables for ages in 2010:
The time difference between 2002 and 2010 is \(2010 - 2002 = 8\) years.
So, Meenu's age in 2010 will be \(M + 8\).
And Meera's age in 2010 will be \(R + 8\).
Formulate the second equation based on the year 2010:
In 2010, Meenu's age was half the age of Meera.
\[ M + 8 = \frac{1}{2}(R + 8) \quad \text{(Equation 2)} \]Solve the system of equations:
Substitute Equation 1 into Equation 2:
\[ \frac{1}{3}R + 8 = \frac{1}{2}(R + 8) \]Multiply both sides by 6 (the least common multiple of 2 and 3) to eliminate fractions:
\[ 6 \left( \frac{1}{3}R + 8 \right) = 6 \left( \frac{1}{2}(R + 8) \right) \] \[ 2R + 48 = 3(R + 8) \] \[ 2R + 48 = 3R + 24 \]Subtract \(2R\) from both sides:
\[ 48 = R + 24 \]Subtract 24 from both sides:
\[ R = 24 \]So, Meera's age in 2002 was 24 years.
Now, find Meenu's age in 2002 using Equation 1:
\[ M = \frac{1}{3}R = \frac{1}{3}(24) = 8 \]So, Meenu's age in 2002 was 8 years.
Calculate Meenu's year of birth:
Meenu's age in 2002 was 8 years. To find her birth year, subtract her age from the year:
\[ \text{Birth Year} = 2002 - 8 = 1994 \]Correct Option: B) 1994
In a conference, out of a total 100 participants, 70 are Indians. If 60 of the total participants are vegetarian, then which of the following statements is/are correct?
1. At least 30 Indian participants are vegetarian.
2. At least 10 Indian participants are non-vegetarian.
Select the correct answer using the codes given below:
This problem involves set theory concepts, specifically finding the minimum and maximum possible overlaps between different groups of participants. We will use the principle of inclusion-exclusion or logical deduction to determine the range of Indian vegetarians and Indian non-vegetarians.
Identify the given information:
Calculate the number of non-Indian participants:
\[ \text{Non-Indian participants} = \text{Total participants} - \text{Indian participants} \\ = 100 - 70 = 30 \]Calculate the number of non-vegetarian participants:
\[ \text{Non-vegetarian participants} = \text{Total participants} - \text{Vegetarian participants} \\ = 100 - 60 = 40 \]Evaluate Statement 1: At least 30 Indian participants are vegetarian.
To find the minimum number of Indian vegetarians, we consider the maximum possible number of non-Indian vegetarians. If all 30 non-Indian participants are vegetarian, then the remaining vegetarians must be Indian.
Alternatively, consider the maximum number of non-Indian vegetarians. The total number of non-Indians is 30. The total number of vegetarians is 60. The maximum number of non-Indian vegetarians is 30 (all non-Indians are vegetarian).
The number of Indian vegetarians can be calculated using the formula for the minimum overlap of two sets:
\[ \text{Minimum (Indian} \cap \text{Vegetarian)} = \text{Indian participants} + \text{Vegetarian participants} - \text{Total participants} \] \[ = 70 + 60 - 100 = 130 - 100 = 30 \]This means at least 30 Indian participants are vegetarian. So, Statement 1 is correct.
Evaluate Statement 2: At least 10 Indian participants are non-vegetarian.
To find the minimum number of Indian non-vegetarians, we consider the maximum possible number of non-Indian non-vegetarians. The total number of non-vegetarians is 40. The total number of non-Indians is 30. It is possible that all 30 non-Indians are non-vegetarian.
Using the formula for minimum overlap:
\[ \text{Minimum (Indian} \cap \text{Non-Vegetarian)} = \text{Indian participants} + \text{Non-vegetarian participants} - \text{Total participants} \] \[ = 70 + 40 - 100 = 110 - 100 = 10 \]This means at least 10 Indian participants are non-vegetarian. So, Statement 2 is correct.
C) Both 1 and 2
Both statements are found to be correct based on the calculations. Statement 1 indicates a minimum of 30 Indian vegetarians, and Statement 2 indicates a minimum of 10 Indian non-vegetarians.