A alone can complete \( \frac{2}{5} \)of a task in 12 days, while B alone can complete \( \frac{3}{4} \) of the same task in 25 days. In how many days can they complete the task if they work together?
\( \frac{75}{19} \)
\( \frac{150}{19} \)
\( \frac{1}{19} \)
\( \frac{300}{19} \)
To solve this problem, we need to determine the individual rates at which A and B complete the task. Once we have their individual rates (fraction of work per day), we can add them to find their combined rate. Finally, the total time to complete the task together will be the reciprocal of their combined rate.
Calculate A's rate:
A completes \( \frac{2}{5} \) of the task in 12 days.
So, A completes the full task in: \[ 12 \text{ days} \times \frac{5}{2} = 30 \text{ days} \]
A's one day's work (rate) is: \[ \frac{1}{30} \text{ of the task per day} \]
Calculate B's rate:
B completes \( \frac{3}{4} \) of the task in 25 days.
So, B completes the full task in: \[ 25 \text{ days} \times \frac{4}{3} = \frac{100}{3} \text{ days} \]
B's one day's work (rate) is: \[ \frac{1}{\frac{100}{3}} = \frac{3}{100} \text{ of the task per day} \]
Calculate their combined rate:
When A and B work together, their combined one day's work is the sum of their individual rates:
\[ \text{Combined rate} = \text{A's rate} + \text{B's rate} \]
\[ \text{Combined rate} = \frac{1}{30} + \frac{3}{100} \]
To add these fractions, find a common denominator, which is 300:
\[ \text{Combined rate} = \frac{1 \times 10}{30 \times 10} + \frac{3 \times 3}{100 \times 3} \]
\[ \text{Combined rate} = \frac{10}{300} + \frac{9}{300} \]
\[ \text{Combined rate} = \frac{10 + 9}{300} = \frac{19}{300} \text{ of the task per day} \]
Calculate the total time to complete the task together:
The total time taken to complete the task together is the reciprocal of their combined rate:
\[ \text{Time} = \frac{1}{\text{Combined rate}} = \frac{1}{\frac{19}{300}} = \frac{300}{19} \text{ days} \]
D) \( \frac{300}{19} \) is the correct answer, as calculated above.
The question asks about the year when the British Parliament transferred the powers of the East India Company to the British Crown. This event was a direct consequence of a major historical uprising in India.
A) 1858 — The Government of India Act 1858, passed by the British Parliament, transferred the powers of the East India Company to the British Crown. This act was a direct response to the Indian Rebellion of 1857.
This problem involves counting combinations, specifically the number of ways to choose 2 people out of a group of 40 for a handshake. Since a handshake involves two people and the order in which they shake hands does not matter (person A shaking person B's hand is the same as person B shaking person A's hand), this is a combination problem.
Correct Option: A) 780
The question asks for the specific term describing the process of depositing a layer of one metal onto another material using electricity. We need to identify the correct scientific term from the given options.
Correct Option: A) Electroplating is the process of depositing a layer of any desired metal on another material by means of electricity. This technique is widely used for decorative purposes, corrosion prevention, and to improve surface properties like hardness or wear resistance.
To solve this problem, we will use the relationship between two numbers, their ratio, their Least Common Multiple (LCM), and their Highest Common Factor (HCF). If two numbers are in a given ratio, we can express them as multiples of their HCF. We know that for any two numbers, the product of the numbers is equal to the product of their LCM and HCF.
Let the two numbers be \(a\) and \(b\). The problem states that the numbers are in the ratio of \(1:7\).
So, we can write the numbers as \(a = 1x\) and \(b = 7x\), where \(x\) is their Highest Common Factor (HCF).
The LCM of the two numbers is given as 721.
We know the property that for any two numbers \(a\) and \(b\):
\[ a \times b = \text{LCM}(a, b) \times \text{HCF}(a, b) \]In our case, \(a = x\), \(b = 7x\), \(\text{LCM} = 721\), and \(\text{HCF} = x\).
Substitute these values into the formula:
\[ (x) \times (7x) = 721 \times x \]Simplify the equation:
\[ 7x^2 = 721x \]Since \(x\) cannot be zero (as it's an HCF), we can divide both sides by \(x\):
\[ 7x = 721 \]Solve for \(x\):
\[ x = \frac{721}{7} \] \[ x = 103 \]So, the HCF of the two numbers is 103.
Now, find the two numbers:
First number \(a = 1x = 1 \times 103 = 103\)
Second number \(b = 7x = 7 \times 103 = 721\)
The question asks for the sum of the numbers:
Sum \( = a + b = 103 + 721 = 824\)
A) The sum of the numbers is 824, which matches our calculated value.
The question asks to identify the specific book for which Rabindranath Tagore was awarded the Nobel Prize in Literature. This requires knowledge of Rabindranath Tagore's literary works and the historical context of his Nobel Prize.
Correct Option: D) Gitanjali
The question asks to identify the individual who has never served as a Governor of the Reserve Bank of India (RBI) from the given options. This requires knowledge of the past Governors of the RBI.
Correct Option: B) K Kasturi Rangan
K. Kasturi Rangan is a distinguished Indian space scientist who served as the Chairman of the Indian Space Research Organisation (ISRO) and was a Member of the Rajya Sabha. He has never held the position of Governor of the Reserve Bank of India.
To find the ratio of the total surface areas of two cubes given the ratio of their volumes, we first need to determine the ratio of their side lengths. Once we have the side length ratio, we can use it to find the ratio of their total surface areas.
Correct Option: A) 16 : 121
This problem requires us to substitute the given new meanings for mathematical operators into the expression and then evaluate the expression following the order of operations (BODMAS/PEMDAS).
\(15 \div (5 + 8)\)
Replace '÷' with 'addition' and '+' with 'subtraction'.
The expression becomes:
\[ 15 + (5 - 8) \]First, solve the operation inside the parentheses:
\[ 5 - 8 = -3 \]The value of the expression is 12.
B) 12 is the correct answer, as derived from substituting the new operator meanings and following the order of operations.
The question asks to identify the Viceroy of India associated with the Ilbert Bill. This requires knowledge of key historical events and the Viceroys during the British Raj in India.
D) Lord Ripon — Lord Ripon served as the Viceroy of India from 1880 to 1884. He was a liberal Viceroy who introduced several reforms, including the repeal of the Vernacular Press Act and the introduction of local self-government. The Ilbert Bill, introduced in 1883, was a significant legislative measure during his tenure. It sought to remove racial discrimination in the judicial system by allowing Indian judges and magistrates to try European subjects in criminal cases. This bill faced strong opposition from the European community in India, leading to widespread protests and eventually its modification, which diluted its original intent. This controversy is a hallmark of Lord Ripon's viceroyalty.