A and B are emptying pipes and C is a filling pipe. Pipes A and B can empty a full tank in 15 hours and 18 hours, respectively. When all three pipes are opened together, it takes 1\( \frac{1}{4} \) hours to empty \( \frac{1}{9} \) part of the tank. In how many hours can pipe C alone fill \( \frac{2}{3} \) part of the tank?
We need to determine the time taken by pipe C alone to fill a certain part of the tank. We are given the emptying rates of pipes A and B, and the combined emptying/filling rate of A, B, and C. We will calculate the individual rates, then the combined rate, and finally deduce the rate of pipe C to find the required time.
Calculate the emptying rate of pipe A:
Pipe A can empty the full tank in 15 hours.
\[ \text{Rate of A} = \frac{1}{15} \text{ tank/hour} \]Calculate the emptying rate of pipe B:
Pipe B can empty the full tank in 18 hours.
\[ \text{Rate of B} = \frac{1}{18} \text{ tank/hour} \]Calculate the combined emptying rate of pipes A and B:
Since both A and B are emptying pipes, their rates add up.
\[ \text{Combined rate of A and B} = \frac{1}{15} + \frac{1}{18} = \frac{6+5}{90} = \frac{11}{90} \text{ tank/hour} \]Calculate the combined rate of pipes A, B, and C:
When all three pipes are opened together, \( \frac{1}{9} \) part of the tank is emptied in \( 1\frac{1}{4} \) hours.
First, convert the mixed fraction to an improper fraction:
\[ 1\frac{1}{4} \text{ hours} = \frac{5}{4} \text{ hours} \]The rate of emptying when all three pipes are open is:
\[ \text{Combined rate of A, B, C} = \frac{\text{Part emptied}}{\text{Time taken}} = \frac{1/9}{5/4} = \frac{1}{9} \times \frac{4}{5} = \frac{4}{45} \text{ tank/hour} \]Since the tank is being emptied, this combined rate is an emptying rate.
Calculate the filling rate of pipe C:
Let the filling rate of pipe C be \( R_C \). The combined rate of A, B, and C can be expressed as:
\[ (\text{Rate of A} + \text{Rate of B}) - \text{Rate of C} = \text{Combined rate of A, B, C} \]Substituting the values:
\[ \frac{11}{90} - R_C = \frac{4}{45} \] \[ R_C = \frac{11}{90} - \frac{4}{45} \] \[ R_C = \frac{11}{90} - \frac{8}{90} \] \[ R_C = \frac{3}{90} = \frac{1}{30} \text{ tank/hour} \]This is the filling rate of pipe C. So, pipe C can fill the entire tank in 30 hours.
Calculate the time taken by pipe C to fill \( \frac{2}{3} \) part of the tank:
Time taken = \[\frac{\text{Part to be filled}}{\text{Filling rate}}\]
\[ \text{Time} = \frac{2/3}{1/30} = \frac{2}{3} \times 30 = 2 \times 10 = 20 \text{ hours} \]A) 20 hours is the correct answer, as calculated in the step-by-step solution.
Aruna bought 8 bananas for Rs 25 and sold 6 bananas for Rs 20. What is her profit or loss percentage?
3 \( \frac{2}{3} \) % Profit
6 \( \frac{2}{3} \) % Profit
5 \( \frac{2}{3} \) % Loss
6 \( \frac{1}{3} \) % Loss
To find the profit or loss percentage, we need to calculate the cost price (CP) and selling price (SP) for the same number of bananas. We will first find the CP and SP of one banana, then calculate the total CP and SP for a common multiple of bananas (e.g., 24, which is the LCM of 8 and 6), or directly calculate the CP and SP for a quantity that makes calculation easier.
Aruna bought 8 bananas for Rs 25.
Cost of 1 banana \[= \frac{\text{Total Cost}}{\text{Number of Bananas}}\]
\[ \text{CP of 1 banana} = \frac{25}{8} \text{ Rs} \]
She sold 6 bananas for Rs 20.
Selling Price of 1 banana \[= \frac{\text{Total Selling Price}}{\text{Number of Bananas Sold}}\]
\[ \text{SP of 1 banana} = \frac{20}{6} \text{ Rs} = \frac{10}{3} \text{ Rs} \]
To find profit or loss, we must compare the CP and SP for the same quantity. The least common multiple (LCM) of 8 and 6 is 24. Let's calculate the CP and SP for 24 bananas.
\[ \text{CP of 24 bananas} = 24 \times \frac{25}{8} = 3 \times 25 = 75 \text{ Rs} \]
\[ \text{SP of 24 bananas} = 24 \times \frac{10}{3} = 8 \times 10 = 80 \text{ Rs} \]
Since the Selling Price (Rs 80) is greater than the Cost Price (Rs 75), there is a profit.
\[ \text{Profit} = \text{SP} - \text{CP} = 80 - 75 = 5 \text{ Rs} \]
\[ \text{Profit Percentage} = \frac{\text{Profit}}{\text{CP}} \times 100 \% \]
\[ \text{Profit Percentage} = \frac{5}{75} \times 100 \% \]
\[ \text{Profit Percentage} = \frac{1}{15} \times 100 \% \]
\[ \text{Profit Percentage} = \frac{100}{15} \% = \frac{20}{3} \% \]
Converting the improper fraction to a mixed fraction:
\[ \frac{20}{3} = 6 \frac{2}{3} \% \]
Correct Option: B) 6 \( \frac{2}{3} \) % Profit
This problem involves arranging people in a row based on given positional clues. We need to deduce the relative positions of all individuals and then count the total number of people.
This can be represented as: K L
Combining with the previous point, we place D four positions to the left of K.
D _ _ _ K L
Now, place A four positions to the left of D.
A _ _ _ D _ _ _ K L
There are two possible scenarios for P:
Let's consider Scenario 2 first, as it's usually easier to extend to the right.
A _ _ _ D _ _ _ K L _ _ P
This statement confirms that P must be to the right of K. Our Scenario 2 (A _ _ _ D _ _ _ K L _ _ P) satisfies this condition. If P were to the left of L with two people between them, P would be to the left of K, which contradicts this statement.
So the arrangement is confirmed as: A _ _ _ D _ _ _ K L _ _ P
Let's count the positions:
A (1) _ (2) _ (3) _ (4) D (5) _ (6) _ (7) _ (8) K (9) L (10) _ (11) _ (12) P (13)
The total number of people seated is 13.
A) 13 is the correct total number of people seated in the row.
The question asks to identify the Indian institution that pioneered the first alternative to corneal transplantation. This requires knowledge of recent advancements in medical technology and research in India, specifically related to ophthalmology.
C) IIT Hyderabad
Researchers at the Indian Institute of Technology Hyderabad (IIT Hyderabad) developed a novel hydrogel-based artificial cornea. This innovation is a significant step towards providing an alternative to human donor corneas for transplantation, addressing the shortage of donor tissues and reducing the risk of immune rejection. This development was a collaborative effort with the Centre for Cellular and Molecular Biology (CCMB) and LV Prasad Eye Institute.
Study the given arrangement of letters and symbols and answer the question that follows:
(LEFT) W & T O ! A L % V E N # J * U D $ I X @ C ^ (RIGHT)
How many such consonants are there in the arrangement each of which is both immediately preceded by a vowel and immediately followed by a symbol?
The problem asks us to identify consonants in a given arrangement that are immediately preceded by a vowel and immediately followed by a symbol. We need to scan the arrangement, identify vowels and symbols, and then check for consonants that fit the specified pattern.
B) Four
This problem involves substituting given symbols with mathematical operations and then evaluating the expression using the order of operations (BODMAS/PEMDAS).
The expression becomes: \(15 \div 5 \times 10 - 4 + 2\).
First, perform division:
\[ 15 \div 5 = 3 \]A) 28 is the correct answer, as calculated by substituting the operations and following the BODMAS rule.
This is a circular arrangement puzzle. We need to deduce the seating positions of six cousins (R, S, T, U, V, W) around a circular table based on the given clues. We will place individuals one by one based on the definite clues and then fill in the remaining positions.
Correct Option: A) V is seated to the immediate right of S.
Select the option that is related to the fifth letter-cluster in the same way as the second letter-cluster is related to the first letter-cluster and the fourth letter-cluster is related to the third letter-cluster.
PAIN : APNI :: MODE : OMED :: JUST :?
The problem asks us to find the relationship between letter-clusters. We need to identify the pattern in the given pairs (PAIN : APNI) and (MODE : OMED), and then apply the same pattern to the third pair (JUST : ?).
Therefore, JUST : UJTS.
Correct Option: A) UJTS
The question asks about Gregor Mendel's motivation for studying seven different characters in pea plants. Understanding Mendel's scientific goals and experimental design is key to answering this.
Correct Option: B) To uncover the universal principles of inheritance. Mendel's meticulous work with multiple distinct characters in pea plants was designed to identify the fundamental rules by which traits are passed from one generation to the next, which are applicable across many species.
We need to identify the type of quadrilateral MNOP based on the given properties: all angles are right angles, opposite sides are equal, and diagonals do not intersect at 90 degrees. We will analyze each property and compare it with the definitions of the given options.
A quadrilateral with all four angles equal to 90 degrees is either a rectangle or a square. This property eliminates kites and rhombuses, as they do not necessarily have right angles.
This means that opposite sides are equal in length. This is a characteristic of both rectangles and squares. For a rectangle, opposite sides are equal. For a square, all sides are equal, which implies opposite sides are also equal.
In a square, the diagonals bisect each other at 90 degrees. In a rectangle, the diagonals bisect each other but do not necessarily intersect at 90 degrees (they intersect at 90 degrees only if the rectangle is also a square). Since the diagonals do NOT intersect at 90 degrees, this condition rules out a square.
The only quadrilateral that satisfies all these conditions is a rectangle.
Correct Option: B) Rectangle