To find the time taken by A and B to complete the work together, we first calculate the individual work rates of A and B. Then, we add their individual work rates to find their combined work rate. Finally, we take the reciprocal of the combined work rate to find the total time taken to complete the work together.
Calculate A's one day work:
A completes the work in 10 days. So, in one day, A completes \(\frac{1}{10}\) of the work.
Calculate B's one day work:
B completes the work in 15 days. So, in one day, B completes \(\frac{1}{15}\) of the work.
Calculate their combined one day work:
When A and B work together, their combined one day work is the sum of their individual one day works.
To add these fractions, find a common denominator, which is 30.
\[ \text{Combined 1 day work} = \frac{3}{30} + \frac{2}{30} \] \[ \text{Combined 1 day work} = \frac{3+2}{30} = \frac{5}{30} \] \[ \text{Combined 1 day work} = \frac{1}{6} \]Calculate the total time taken to complete the work together:
If A and B together complete \(\frac{1}{6}\) of the work in one day, then they will complete the entire work in the reciprocal of this fraction.
Correct Option: B) 6 days
This problem involves calculating the time a bus stops per hour by comparing its speed with and without stoppages. The key idea is to find the distance lost due to stoppages over a certain period and then convert that lost distance back into time.
The reduction in speed is due to the time spent stopping.
\[ \text{Speed difference} = \text{Speed without stoppages} - \text{Speed with stoppages} \]\[ \text{Speed difference} = 54 \text{ kmph} - 45 \text{ kmph} = 9 \text{ kmph} \]A speed difference of 9 kmph means that in one hour, the bus travels 9 km less than it would if it didn't stop. This 9 km represents the distance that would have been covered during the stoppage time.
The time for which the bus stops is equivalent to the time it would take to cover the "lost" distance (9 km) if it were moving at its actual speed (without stoppages).
\[ \text{Time stopped} = \frac{\text{Distance lost}}{\text{Actual speed}} \]\[ \text{Time stopped} = \frac{9 \text{ km}}{54 \text{ kmph}} \]\[ \text{Time stopped} = \frac{1}{6} \text{ hour} \]Since 1 hour = 60 minutes, convert the fraction of an hour into minutes.
\[ \text{Time stopped in minutes} = \frac{1}{6} \times 60 \text{ minutes} \]\[ \text{Time stopped in minutes} = 10 \text{ minutes} \]Thus, the bus stops for 10 minutes per hour.
B) 10 is the correct answer. The bus stops for 10 minutes per hour, as calculated by finding the time equivalent to the distance not covered due to stoppages at the bus's actual running speed.
To find the time taken for a train to pass a tree, we need to consider the distance the train travels as its own length. The speed is given in km/hr, and the length in meters, so we must first convert the speed to meters per second (m/s) to ensure consistent units. Then, we can use the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\) to calculate the time.
To convert km/hr to m/s, multiply by \(\frac{5}{18}\).
\[ S = 63 \times \frac{5}{18}\ \text{m/s} \]\[ S = \frac{7 \times 9 \times 5}{2 \times 9}\ \text{m/s} \]\[ S = \frac{7 \times 5}{2}\ \text{m/s} \]\[ S = \frac{35}{2}\ \text{m/s} \]\[ S = 17.5\ \text{m/s} \]The distance the train needs to cover to pass a tree (a point object) is equal to its own length.
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]\[ \text{Time} = \frac{280\ \text{m}}{17.5\ \text{m/s}} \]\[ \text{Time} = \frac{2800}{175}\ \text{s} \]\[ \text{Time} = \frac{560}{35}\ \text{s} \]\[ \text{Time} = \frac{112}{7}\ \text{s} \]\[ \text{Time} = 16\ \text{s} \]Correct Option: B) 16 sec.
The train will pass the tree in 16 seconds.
This problem can be solved using the rule of alligation, which is a shortcut method to find the ratio in which two ingredients at different prices should be mixed to produce a mixture of a desired price. The rule involves finding the difference between the mean price and the individual prices.
Identify the given prices:
Apply the Rule of Alligation:
The rule of alligation can be visualized as follows:
\[ \begin{array}{ccc} \text{Cheaper Price} & & \text{Dearer Price} \\ 62 & & 72 \\ & \diagdown & \diagup \\ & 64.50 & \\ & \diagup & \diagdown \\ (72 - 64.50) & : & (64.50 - 62) \\ 7.50 & : & 2.50 \\ \end{array} \]
The ratio of the quantities of the cheaper tea to the dearer tea is given by the difference between the dearer price and the mean price, and the difference between the mean price and the cheaper price, respectively.
Calculate the differences:
Determine the ratio:
The required ratio of the quantity of cheaper tea to the quantity of dearer tea is \(D_1 : D_2\).
\[ \text{Ratio} = 7.50 : 2.50 \]
Simplify the ratio:
To simplify, divide both sides by the greatest common divisor, which is \(2.50\).
\[ \text{Ratio} = \frac{7.50}{2.50} : \frac{2.50}{2.50} = 3 : 1 \]
Thus, the tea at Rs 62 per kg must be mixed with tea at Rs 72 per kg in the ratio of \(3:1\).
A) The calculated ratio is \(3:1\), which matches this option.
The problem asks us to identify the "odd man out" from a given series of numbers: 41, 43, 47, 53, 61, 71, 73, 81. This typically involves finding a common property shared by most numbers and identifying the one that does not share that property.
Correct Option: D) 81 is the odd man out because all other numbers in the list (41, 43, 47, 53, 61, 71, 73) are prime numbers, while 81 is a composite number (\(81 = 9 \times 9\)).
To find the wrong number in the sequence, we need to identify the pattern or rule that governs the sequence. We will examine the differences between consecutive numbers and look for a consistent mathematical operation or series.
D) 40 — The established pattern is to alternately add 7 and 5 to the previous number. Following this pattern, the number after 37 should be \(37 + 5 = 42\), not 40. Thus, 40 is the wrong number in the sequence.
The question asks for the Chief Minister of Rajasthan as of 2014. This requires knowledge of political history and the tenure of Chief Ministers in the state of Rajasthan.
Correct Option: B) Vasundhara Raje
The question asks to identify the planet closest to the Sun. This requires knowledge of the order of planets in our solar system starting from the Sun.
Correct Option: C) Mercury
This question tests knowledge about the constitutional provisions related to the oath of office for the Vice-President of India. We need to identify the authority responsible for administering this oath as per the Indian Constitution.
Correct Option: A) President
The question asks for the city where Nahargarh Fort is located. This is a factual question testing knowledge of Indian geography and historical sites, specifically forts in Rajasthan.
A) Jaipur is the correct answer. Nahargarh Fort, along with Amber Fort and Jaigarh Fort, forms a strong defence ring for the city of Jaipur. It was built by Maharaja Sawai Jai Singh II in 1734.