A boy can complete a piece of work alone in 10 days, but with the help of his friend he can complete it in 6 days. In how much time will his friend be able to complete the same piece of work alone?
This problem involves work and time, where we need to find the time taken by an individual to complete a task, given the individual's time and the combined time with another person. We will use the concept that the rate of work is the reciprocal of the time taken to complete the work.
Let the total work be 1 unit.
Time taken by the boy alone to complete the work = 10 days.
Boy's rate of work (work done per day) = \( \frac{1}{10} \) of the work per day.
Time taken by the boy and his friend together to complete the work = 6 days.
Combined rate of work (boy + friend) = \( \frac{1}{6} \) of the work per day.
Let the time taken by the friend alone to complete the work be \( x \) days.
Friend's rate of work (work done per day) = \( \frac{1}{x} \) of the work per day.
The combined rate of work is the sum of individual rates of work.
\[ \text{Boy's rate} + \text{Friend's rate} = \text{Combined rate} \]\[ \frac{1}{10} + \frac{1}{x} = \frac{1}{6} \]To find \( x \), we need to isolate \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{6} - \frac{1}{10} \]The least common multiple (LCM) of 6 and 10 is 30.
\[ \frac{1}{x} = \frac{5}{30} - \frac{3}{30} \]\[ \frac{1}{x} = \frac{5 - 3}{30} \]\[ \frac{1}{x} = \frac{2}{30} \]\[ \frac{1}{x} = \frac{1}{15} \]Since \[\frac{1}{x} = \frac{1}{15}\], then \( x = 15 \).
So, the friend alone can complete the work in 15 days.
Correct Option: A) 15 days
This is a logical reasoning problem involving sequencing. We need to determine the exam schedule for five people (A, B, C, D, E) over five days (Monday to Friday) based on the given clues. We will assign days to people step-by-step using the provided information.
A) A has the exam on Wednesday, as determined by the step-by-step deduction.
The question asks to identify the incorrect statement about effective nuclear charge (\(Z_{\text{eff}}\)). We need to analyze each option based on the definition and trends of effective nuclear charge in the periodic table.
D) It depends only upon the nuclear charge. — This statement is incorrect because effective nuclear charge (\(Z_{\text{eff}}\)) is the net positive charge experienced by an electron, taking into account the shielding effect of inner electrons. It is calculated as \(Z_{\text{eff}} = Z - S\), where \(Z\) is the atomic number (nuclear charge) and \(S\) is the shielding constant. Thus, it depends on both the nuclear charge and the shielding effect.
This question deals with electromagnetic induction, specifically Faraday's Law of Induction and Lenz's Law. Faraday's Law states that the magnitude of the induced electromotive force (potential difference) is proportional to the rate of change of magnetic flux. Lenz's Law determines the direction (sign) of the induced potential difference, stating that the induced current will flow in a direction that opposes the change in magnetic flux that produced it.
Correct Option: C) the same, opposite
The question asks to identify the organization responsible for the 'School to School' programme, which fosters empathy and dignity between urban and rural children. This requires knowledge of the specific initiatives undertaken by prominent NGOs in India.
D) Goonj runs the 'School to School' programme. This initiative focuses on collecting educational material, stationery, and other resources from urban schools and distributing them to rural and underprivileged schools. Beyond material support, it aims to create a sense of connection, empathy, and dignity between children from different backgrounds, aligning perfectly with the description in the question.
Select the number from among the given options that can replace the question mark (?) in the following series.
3, 16, 5, 13, 9, 8, ?, ?, 23, -8
The given series is an alternating series, meaning it consists of two interleaved sub-series. We need to identify the pattern in each sub-series to find the missing terms.
Let's write down the given series: 3, 16, 5, 13, 9, 8, ?, ?, 23, -8.
Separate the series into two alternating sub-series:
Sub-series 1 (Odd positions): 3, 5, 9, ?, 23
Sub-series 2 (Even positions): 16, 13, 8, ?, -8
Analyze Sub-series 1 (Odd positions): 3, 5, 9, ?, 23
Difference between 5 and 3 is \(5 - 3 = 2\).
Difference between 9 and 5 is \(9 - 5 = 4\).
The differences are increasing by 2 each time (2, 4, ...). So, the next difference should be \(4 + 2 = 6\).
The missing term in Sub-series 1 is \(9 + 6 = 15\).
Let's check the next difference: \(6 + 2 = 8\). And \(15 + 8 = 23\). This confirms the pattern for Sub-series 1.
Analyze Sub-series 2 (Even positions): 16, 13, 8, ?, -8
Difference between 13 and 16 is \(13 - 16 = -3\).
Difference between 8 and 13 is \(8 - 13 = -5\).
The differences are decreasing by 2 each time (-3, -5, ...). So, the next difference should be \(-5 - 2 = -7\).
The missing term in Sub-series 2 is \(8 + (-7) = 8 - 7 = 1\).
Let's check the next difference: \(-7 - 2 = -9\). And \(1 + (-9) = 1 - 9 = -8\). This confirms the pattern for Sub-series 2.
The two missing terms are 15 and 1, in that order.
B) 15, 1 — This matches the calculated missing terms for both sub-series.
A) 13, 2 — The first term 13 is incorrect for Sub-series 1 (it should be 15). The second term 2 is incorrect for Sub-series 2 (it should be 1).
C) 12, 3 — Both terms are incorrect based on the identified patterns.
D) 11, 3 — Both terms are incorrect based on the identified patterns.
To solve this problem, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This rule dictates the sequence for evaluating mathematical expressions:
The given expression is: \(6 \times 7 - \{9 \div 3 + [3 \times (5 - 2)]\}\)
Thus, the value of the expression is 30.
B) 30 is the correct answer, as derived by following the BODMAS/PEMDAS rule.
Refer to the following letter, symbol series and answer the question.
L J + K – Y % T # U P
If all symbols are dropped from the series, which letter will be at the middlemost position in the newly formed series?
The problem asks us to identify the middlemost letter in a new series formed by dropping all symbols from the given original series. We need to first filter out the symbols and then count the remaining letters to find the middle position.
Correct Option: B) Y is the letter at the middlemost position in the newly formed series.
The problem involves arranging six persons around a circular table based on given conditions. We need to deduce the seating arrangement step-by-step and then identify the person who is an immediate neighbour of both Madhuram and Prem.
Total persons: There are six persons sitting around a circular table, facing the centre.
Condition 1: Madhuram is an immediate neighbour of both Rahim and Amit.
This means Madhuram is between Rahim and Amit. We can represent this as R - M - A or A - M - R.
Condition 2: Sunit is sitting second to the right of Rahim.
Let's assume Rahim is at a certain position. Counting two positions to the right from Rahim, we place Sunit. Since there are 6 people, "second to the right" means there is one person between Rahim and Sunit.
Condition 3: David is sitting third to the left of Amit.
Counting three positions to the left from Amit, we place David. For 6 people, "third to the left" means there are two people between Amit and David.
Condition 4: Prem is an immediate neighbour of both Sunit and Amit.
This means Prem is between Sunit and Amit. We can represent this as S - P - A or A - P - S.
Combining the conditions:
Notice that Amit is common in both. This implies a sequence: R - M - A - P - S. This accounts for 5 people: Rahim, Madhuram, Amit, Prem, Sunit.
Placing the sixth person and verifying:
We have the sequence R - M - A - P - S. Since it's a circular table with 6 people, the remaining person, David, must fit into the last available spot to complete the circle. The sequence becomes R - M - A - P - S - D (or D - R - M - A - P - S, etc., in a circle).
Let's check the conditions with this arrangement:
This means our initial assumption for R-M-A and S-P-A must be in the opposite direction for one of them, or the circular arrangement needs careful placement.
Let's try placing Amit first, as he is mentioned in multiple conditions.
If Amit is at position 1:
This implies that the sequence A-M-R and A-P-S must be arranged such that M and P are on opposite sides of A.
Let's draw a circle with 6 spots.
1. Place Amit (A) at any position, say position 1.
2. Madhuram (M) is an immediate neighbour of Amit. So, M can be at position 2 or 6.
3. Prem (P) is an immediate neighbour of Amit. So, P can be at position 2 or 6. Since M and P are distinct, one must be at 2 and the other at 6.
Let's fix: A at position 1, M at position 2, P at position 6.
Current arrangement: P - A - M (where P and M are immediate neighbours of A).
4. Madhuram is an immediate neighbour of both Rahim and Amit. Since M is next to A, Rahim (R) must be next to M on the other side. So, R is at position 3 (P - A - M - R).
5. Prem is an immediate neighbour of both Sunit and Amit. Since P is next to A, Sunit (S) must be next to P on the other side. So, S is at position 5 (S - P - A - M - R).
Now we have 5 people: S, P, A, M, R. The remaining person is David (D), who must be at position 4.
The arrangement is: S - P - A - M - R - D (in clockwise or anti-clockwise order).
Let's verify all conditions with this arrangement (clockwise):
The final arrangement (clockwise) is: Amit, Madhuram, Rahim, David, Sunit, Prem.
Or, if we list them starting from Amit: A - M - R - D - S - P (clockwise).
The question asks: Who is an immediate neighbour of both Madhuram and Prem?
Looking at our arrangement (A - M - R - D - S - P):
The common neighbour for both Madhuram and Prem is Amit.
D) Amit is an immediate neighbour of both Madhuram and Prem, as deduced from the final seating arrangement.