A and B working alone can finish a task in 12 and 16 days respectively. In how many days can the task be finished if they work for one day each alternatively, and A begins the work?
12 \( \frac{1}{3} \)days
12 \( \frac{2}{3} \) days
13 \( \frac{1}{3} \) days
13 \( \frac{2}{3} \) days
This problem involves calculating the time taken to complete a task when two individuals work alternatively. We will first determine the individual work rates, then calculate the work done in one cycle of two days, and finally find the total time.
Calculate individual work rates:
Calculate work done in one cycle:
Since they work alternatively and A begins, the work cycle is A's day + B's day. A cycle consists of 2 days.
Work done in 1st day (by A) \( = \frac{1}{12} \)
Work done in 2nd day (by B) \( = \frac{1}{16} \)
Work done in 2 days (1 cycle) \[= \frac{1}{12} + \frac{1}{16}\]
To add these fractions, find the Least Common Multiple (LCM) of 12 and 16. The LCM of 12 and 16 is 48.
\[ \text{Work in 1 cycle} = \frac{4}{48} + \frac{3}{48} = \frac{7}{48} \]Determine the number of cycles for most of the work:
We need to find how many cycles are needed to complete most of the work without exceeding the total work (1 unit).
Let \(n\) be the number of cycles. We want \(n \times \frac{7}{48}\) to be close to 1, but less than 1.
If \(n = 6\), work done \[= 6 \times \frac{7}{48} = \frac{42}{48}\].
If \(n = 7\), work done \[= 7 \times \frac{7}{48} = \frac{49}{48}\], which is more than 1. So, 6 cycles are completed.
Calculate remaining work and time:
After 6 cycles (which is \(6 \times 2 = 12\) days), the work completed is \(\frac{42}{48}\).
Remaining work \[= 1 - \frac{42}{48} = \frac{48 - 42}{48} = \frac{6}{48} = \frac{1}{8}\].
Now it's A's turn (13th day). A's one day's work is \(\frac{1}{12}\).
Since \[\frac{1}{8} < \frac{1}{12}\] is false (\[\frac{1}{8} > \frac{1}{12}\]), A can do the remaining work in less than a full day.
Time taken by A to complete \(\frac{1}{8}\) of the work \[= \frac{\text{Remaining Work}}{\text{A's 1 day work}} = \frac{\frac{1}{8}}{\frac{1}{12}} = \frac{1}{8} \times 12 = \frac{12}{8} = \frac{3}{2}\] days.
This is incorrect because A's one day's work is \(\frac{1}{12}\). The remaining work is \(\frac{1}{8}\). A will complete this work in:
\[ \text{Time} = \frac{\text{Remaining Work}}{\text{A's 1 day work}} = \frac{1/8}{1/12} = \frac{1}{8} \times 12 = \frac{12}{8} = \frac{3}{2} \text{ days} \]Wait, this calculation is wrong. A's one day's work is \(\frac{1}{12}\). To complete \(\frac{1}{8}\) of the work, A will take:
\[ \text{Days} = \frac{\text{Amount of Work}}{\text{Work Rate}} = \frac{1/8}{1/12} = \frac{1}{8} \times 12 = \frac{12}{8} = \frac{3}{2} \text{ days} \]This means A needs 1.5 days to complete the remaining work. But A works for only one day at a time. This implies that the remaining work is more than what A can do in one day. Let's recheck the calculation.
Remaining work is \[\frac{6}{48} = \frac{1}{8}\].
On the 13th day, A works. A completes \(\frac{1}{12}\) of the work.
Work remaining after A's 13th day \[= \frac{1}{8} - \frac{1}{12}\].
LCM of 8 and 12 is 24.
\[ \text{Work remaining} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24} \]So, after 13 days (12 days for 6 cycles + 1 day for A), the work completed is \[\frac{42}{48} + \frac{1}{12} = \frac{7}{8} + \frac{1}{12} = \frac{21}{24} + \frac{2}{24} = \frac{23}{24}\].
The remaining work is \(\frac{1}{24}\).
Now it's B's turn (14th day). B's one day's work is \(\frac{1}{16}\).
Time taken by B to complete the remaining \(\frac{1}{24}\) work \[= \frac{\text{Remaining Work}}{\text{B's 1 day work}} = \frac{\frac{1}{24}}{\frac{1}{16}} = \frac{1}{24} \times 16 = \frac{16}{24} = \frac{2}{3}\] days.
Total time taken \[= 12 \text{ days (for 6 cycles)} + 1 \text{ day (for A)} + \frac{2}{3} \text{ days (for B)}\].
Total time \( = 13 \frac{2}{3} \) days.
D) \(13 \frac{2}{3}\) days is the correct answer as calculated above.
To find the next number in the series, we need to identify the pattern or relationship between consecutive numbers. We will calculate the difference between adjacent terms to see if there's a consistent arithmetic progression or another mathematical sequence.
Therefore, the next number in the series is 20.
Correct Option: B) 20
When two objects move in the same direction, their relative speed is the difference between their individual speeds. We can use this relative speed to calculate the time it takes for them to be a certain distance apart.
They start in the same direction.
Since they are moving in the same direction, the relative speed (\(S_R\)) is the difference between their speeds. The faster person moves away from the slower person.
\[ S_R = S_M - S_S \] \[ S_R = 4 \text{ km/hr} - 3.5 \text{ km/hr} \] \[ S_R = 0.5 \text{ km/hr} \]The distance they need to be apart (\(D\)) = 1.5 km.
Using the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\), we can find the time (\(T\)) it takes for them to be 1.5 km apart.
\[ T = \frac{D}{S_R} \] \[ T = \frac{1.5 \text{ km}}{0.5 \text{ km/hr}} \] \[ T = 3 \text{ hours} \]Correct Option: B) 3 hours
The question asks to identify a plant hormone from the given options. We need to recall the functions and classifications of the substances listed in each option.
Based on this analysis, Auxin is the only plant hormone among the given options.
Correct Option: D) Auxin is a well-known plant hormone (phytohormone) that regulates plant growth and development.
The question asks to identify the "Saffron town" of Kashmir. This requires knowledge of the geographical and agricultural specializations of different regions within Kashmir.
Pampore, located in the Pulwama district of Jammu and Kashmir, is famously known as the "Saffron Town" or "Saffron Bowl" of Kashmir due to its extensive cultivation of high-quality saffron.
C) Pampore — Pampore is globally recognized for its saffron cultivation, making it the correct answer.
This problem can be solved using the concept of work rate. We can determine the individual work rates of Jaya and the combined work rate of Reena and Jaya. Then, by subtracting Jaya's work rate from the combined work rate, we can find Reena's individual work rate. Finally, the time Reena takes to complete the work alone is the reciprocal of her work rate.
Calculate the combined work rate of Reena and Jaya:
If Reena and Jaya together can complete the work in 6 days, their combined work rate is:
Calculate Jaya's individual work rate:
If Jaya alone can complete the work in 8 days, her individual work rate is:
Calculate Reena's individual work rate:
Reena's work rate can be found by subtracting Jaya's work rate from their combined work rate:
To subtract these fractions, find a common denominator, which is 24:
\[ \text{Work rate (Reena)} = \frac{4}{24} - \frac{3}{24} \] \[ \text{Work rate (Reena)} = \frac{4 - 3}{24} = \frac{1}{24} \text{ work per day} \]Calculate the time Reena takes to complete the work alone:
The time taken is the reciprocal of the work rate:
Correct Option: D) 24 days
A) 48 days — This would imply Reena's work rate is \(1/48\), which is incorrect. This might arise from multiplying the days (6 * 8) instead of using work rates.
B) 12 days — This would imply Reena's work rate is \(1/12\). This could be a result of incorrectly adding the work rates or some other miscalculation.
C) 2 days — This would imply Reena works much faster than the combined rate, which is illogical as Jaya also contributes positively. This might arise from subtracting the days directly (8 - 6).
The question asks to identify which option is NOT a part of the growth areas under the Digital India programme. To answer this, we need to recall the nine pillars or growth areas of the Digital India initiative.
D) Public school education for all is not explicitly listed as one of the nine pillars or growth areas of the Digital India programme. While digital education initiatives exist and are supported by the government, "Public school education for all" as a broad concept is a fundamental right and a long-standing government objective, not a specific growth area defined under Digital India's nine pillars.
The question asks to identify the founder of Visva Bharati, an institution that later became a Central University. This is a factual question related to Indian history and education, specifically concerning prominent personalities and their contributions.
D) Rabindranath Tagore founded Visva Bharati. He envisioned it as a center for learning that would blend the best of Indian and Western traditions, promoting cultural understanding and universal brotherhood. It started as a school in Shantiniketan and was formally inaugurated as Visva Bharati University in 1921.
(\(\sqrt{5}\) + \(\sqrt{7}\))² is a:
whole number
natural number
rational number
irrational number
To determine the nature of the given expression \( (\sqrt{5} + \sqrt{7})^2 \), we need to expand it using the algebraic identity \( (a+b)^2 = a^2 + 2ab + b^2 \) and then classify the resulting number.
Expand the expression: We have the expression \( (\sqrt{5} + \sqrt{7})^2 \). Let \( a = \sqrt{5} \) and \( b = \sqrt{7} \). Using the identity \( (a+b)^2 = a^2 + 2ab + b^2 \):
\[ (\sqrt{5} + \sqrt{7})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{7}) + (\sqrt{7})^2 \]Simplify the terms:
Combine the simplified terms:
\[ (\sqrt{5} + \sqrt{7})^2 = 5 + 2\sqrt{35} + 7 = 12 + 2\sqrt{35} \]Classify the result: We know that \( \sqrt{35} \) is an irrational number because 35 is not a perfect square. The product of a non-zero rational number (2) and an irrational number \( (\sqrt{35}) \) is an irrational number \( (2\sqrt{35}) \). The sum of a rational number (12) and an irrational number \( (2\sqrt{35}) \) is always an irrational number.
Therefore, \( 12 + 2\sqrt{35} \) is an irrational number.
D) irrational number. The simplified expression \( 12 + 2\sqrt{35} \) contains \( \sqrt{35} \), which is an irrational number. The sum of a rational number and an irrational number is always irrational.
In order to celebrate earth's environment and raise public awareness about pollution, which of the following days is observed worldwide as Earth Day each year?
The question asks to identify the specific date on which Earth Day is observed globally to celebrate the environment and raise awareness about pollution. This requires knowledge of important environmental observances.
Earth Day is an annual event celebrated worldwide on April 22nd to demonstrate support for environmental protection. First celebrated in 1970, it now includes a wide range of events coordinated globally by EarthDay.org (formerly Earth Day Network) and involves over 1 billion people in more than 193 countries. The day aims to raise public awareness about pollution and celebrate the Earth's environment.
Correct Option: C) 22nd April is the correct date for the worldwide observance of Earth Day.