To find the speed of the wheel, we first need to calculate the total distance covered. The distance covered in one revolution is equal to the circumference of the wheel. We then multiply this by the number of revolutions to get the total distance. Finally, we divide the total distance by the time taken to find the speed, and convert the speed to km/h.
The circumference (\(C\)) of a circle is given by the formula \(C = 2\pi r\).
\[ C = 2 \times \frac{22}{7} \times 70 \text{ cm} \]\[ C = 2 \times 22 \times 10 \text{ cm} \]\[ C = 440 \text{ cm} \]The total distance (\(D\)) covered is the circumference multiplied by the number of revolutions.
\[ D = C \times N \]\[ D = 440 \text{ cm} \times 10 \]\[ D = 4400 \text{ cm} \]Since \(1 \text{ m} = 100 \text{ cm}\),
\[ D = \frac{4400}{100} \text{ m} \]\[ D = 44 \text{ m} \]Speed (\(S\)) is given by the formula \(S = \frac{\text{Distance}}{\text{Time}}\).
\[ S = \frac{44 \text{ m}}{5 \text{ s}} \]\[ S = 8.8 \text{ m/s} \]To convert m/s to km/h, we multiply by \(\frac{18}{5}\) (since \(1 \text{ km} = 1000 \text{ m}\) and \(1 \text{ hour} = 3600 \text{ seconds}\)).
\[ S = 8.8 \times \frac{18}{5} \text{ km/h} \]\[ S = 8.8 \times 3.6 \text{ km/h} \]\[ S = 31.68 \text{ km/h} \]D) 31.68 km/h is the calculated speed of the wheel.
This is a logical reasoning problem involving scheduling. We need to deduce the exam schedule for seven students (P, Q, R, S, T, U, V) over seven days of a week (Monday to Sunday) based on the given clues. We will build a table representing the days and fill in the students' names as we deduce their exam days.
List the days:
| Day | Student |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | |
| Thursday | |
| Friday | |
| Saturday | |
| Sunday |
Clue 1: P's exam is on one of the days after Thursday.
This means P's exam can be on Friday, Saturday, or Sunday.
Clue 2: Q's exam is on the day immediately after P's exam.
If P is on Friday, Q is on Saturday.
If P is on Saturday, Q is on Sunday.
P cannot be on Sunday, because there is no day after Sunday for Q.
Clue 3: Three students have exams between Q and V.
Let's test the two possibilities for P and Q:
Case 1: P is on Friday, Q is on Saturday.
If Q is on Saturday, and there are three students between Q and V, then V must be on Tuesday (Saturday - 3 days - V = Tuesday).
Schedule so far:
| Day | Student |
|---|---|
| Monday | |
| Tuesday | V |
| Wednesday | |
| Thursday | |
| Friday | P |
| Saturday | Q |
| Sunday |
Case 2: P is on Saturday, Q is on Sunday.
If Q is on Sunday, and there are three students between Q and V, then V must be on Wednesday (Sunday - 3 days - V = Wednesday).
Schedule so far:
| Day | Student |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | V |
| Thursday | |
| Friday | |
| Saturday | P |
| Sunday | Q |
Clue 4: R has the exam on one of the days before V, but not on Monday.
For Case 1 (V on Tuesday): R must be before V (Tuesday) but not on Monday. This is impossible as Monday is the only day before Tuesday. So, Case 1 is invalid.
For Case 2 (V on Wednesday): R must be before V (Wednesday) but not on Monday. This means R can be on Tuesday.
Schedule so far:
| Day | Student |
|---|---|
| Monday | |
| Tuesday | R |
| Wednesday | V |
| Thursday | |
| Friday | |
| Saturday | P |
| Sunday | Q |
Clue 5: S has the exam immediately before T.
We need two consecutive empty slots for S and T. Looking at the current schedule, the only consecutive empty slots are Thursday and Friday.
| Day | Student |
|---|---|
| Monday | |
| Tuesday | R |
| Wednesday | V |
| Thursday | S |
| Friday | T |
| Saturday | P |
| Sunday | Q |
Find U's exam day:
All students except U have been placed. The only remaining empty day is Monday. Therefore, U's exam is on Monday.
Final Schedule:
| Day | Student |
|---|---|
| Monday | U |
| Tuesday | R |
| Wednesday | V |
| Thursday | S |
| Friday | T |
| Saturday | P |
| Sunday | Q |
Correct Option: D) U has the exam on Monday.
The question asks about the specific Indian National Congress session where the Non-cooperation Movement was formally adopted. To answer this, one needs to recall the key Congress sessions and their decisions, particularly those related to major national movements.
Correct Option: B) The Non-cooperation Movement was formally adopted at the Congress session held in Nagpur in December 1920. This session was crucial for solidifying the movement's strategy and gaining widespread support.
We need to find the sum of two numbers given two conditions: their difference and a relationship involving percentages of the numbers. We can set up a system of two linear equations with two variables and solve for the numbers, then find their sum.
Let the larger number be \(x\) and the smaller number be \(y\).
From the first condition, "The difference between two numbers is 45", we can write the equation:
\[ x - y = 45 \quad \text{(Equation 1)} \]From the second condition, "When 20% of the larger number is added to 35% of the smaller number, we get a sum of 31", we can write the equation:
\[ 0.20x + 0.35y = 31 \]To eliminate decimals, multiply the second equation by 100:
\[ 20x + 35y = 3100 \quad \text{(Equation 2)} \]Now we have a system of two linear equations:
Equation 1: \(x - y = 45\)
Equation 2: \(20x + 35y = 3100\)
From Equation 1, express \(x\) in terms of \(y\):
\[ x = 45 + y \]Substitute this expression for \(x\) into Equation 2:
\[ 20(45 + y) + 35y = 3100 \]Distribute and solve for \(y\):
\[ 900 + 20y + 35y = 3100 \] \[ 900 + 55y = 3100 \] \[ 55y = 3100 - 900 \] \[ 55y = 2200 \] \[ y = \frac{2200}{55} \] \[ y = 40 \]Now substitute the value of \(y\) back into the expression for \(x\):
\[ x = 45 + y \] \[ x = 45 + 40 \] \[ x = 85 \]The two numbers are 85 and 40. We need to find their sum:
\[ \text{Sum} = x + y = 85 + 40 = 125 \]Correct Option: A) 125
A alone can complete a task in 30 days, while B alone can complete it in 40 days. A and B, working together, can complete the same task in:
17 \( \frac{1}{7} \) days
11 \( \frac{2}{7} \) days
18 \( \frac{3}{7} \) days
15 \( \frac{2}{7} \) days
To find the time taken by A and B working together, we first calculate the individual work rates of A and B. The work rate is the reciprocal of the time taken to complete the task. Then, we add their individual work rates to find their combined work rate. Finally, the reciprocal of the combined work rate gives the total time taken to complete the task when working together.
Calculate A's work rate:
A alone can complete the task in 30 days. So, in one day, A completes:
\[ \text{A's work rate} = \frac{1}{30} \text{ of the task per day} \]Calculate B's work rate:
B alone can complete the task in 40 days. So, in one day, B completes:
\[ \text{B's work rate} = \frac{1}{40} \text{ of the task per day} \]Calculate their combined work rate:
When A and B work together, their work rates add up:
\[ \text{Combined work rate} = \text{A's work rate} + \text{B's work rate} \] \[ \text{Combined work rate} = \frac{1}{30} + \frac{1}{40} \]To add these fractions, find a common denominator, which is 120 (LCM of 30 and 40):
\[ \text{Combined work rate} = \frac{4}{120} + \frac{3}{120} \] \[ \text{Combined work rate} = \frac{4 + 3}{120} = \frac{7}{120} \text{ of the task per day} \]Calculate the time taken to complete the task together:
The time taken to complete the task together is the reciprocal of their combined work rate:
\[ \text{Time taken} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{7}{120}} \] \[ \text{Time taken} = \frac{120}{7} \text{ days} \]Convert the improper fraction to a mixed number:
Divide 120 by 7:
\[ 120 \div 7 = 17 \text{ with a remainder of } 1 \]So, the time taken is \(17 \frac{1}{7}\) days.
Correct Option: A) 17 \( \frac{1}{7} \) days
The question asks about the number of Zonal Cultural Centres (ZCCs) established by the Government of India as of June 2019. This is a factual question related to cultural institutions in India. To answer this, one needs to recall the number of such centers.
Correct Option: D) Seven
The question asks to identify which site was submitted to UNESCO for World Heritage Site nomination in the 2021-22 cycle. This requires knowledge of recent nominations from India to UNESCO's World Heritage list.
Correct Option: B) Sacred ensemble of Hoysalas
The "Sacred Ensembles of Hoysalas" comprising the Chennakeshava Temple at Belur, the Hoysaleswara Temple at Halebid, and the Keshava Temple at Somanathpur, was India's nomination for the UNESCO World Heritage List for the 2022-2023 cycle. This nomination was submitted in 2021, falling within the 2021-22 period referenced in the question. It was subsequently inscribed on the World Heritage List in September 2023.
The value of x in the equation x + \( \frac{5}{27} \) = \( \frac{12}{27} \) is:
\( \frac{7}{27} \)
\( \frac{10}{27} \)
\( \frac{9}{27} \)
\( \frac{1}{3} \)
To find the value of \(x\) in the given equation, we need to isolate \(x\) on one side of the equation. This can be done by subtracting the fraction \( \frac{5}{27} \) from both sides of the equation.
Start with the given equation:
\[ x + \frac{5}{27} = \frac{12}{27} \]To isolate \(x\), subtract \( \frac{5}{27} \) from both sides of the equation:
\[ x = \frac{12}{27} - \frac{5}{27} \]Since the fractions on the right-hand side have the same denominator, we can subtract their numerators directly:
\[ x = \frac{12 - 5}{27} \]Perform the subtraction in the numerator:
\[ x = \frac{7}{27} \]Thus, the value of \(x\) is \( \frac{7}{27} \).
Correct Option: A) \( \frac{7}{27} \)
To find the smallest number that can be divided exactly by a set of given numbers, we need to find the Least Common Multiple (LCM) of those numbers. The given numbers are 16, 18, 27, and 45.
B) 2160 is the smallest number that is exactly divisible by 16, 18, 27, and 45.
To find Rita's share of the profit, we first need to determine the ratio of the investments made by Sita, Gita, and Rita. The total profit is then distributed among them in this same ratio. We will calculate Rita's share by multiplying her proportion of the total investment by the net profit.
Identify the investments:
Determine the ratio of their investments:
To find the ratio, we can simplify the amounts by dividing by the greatest common divisor, which is 10,000 in this case.
Calculate the sum of the ratios:
\[ \text{Sum of ratios} = 2 + 5 + 4 = 11 \]Determine Rita's share of the profit:
Rita's share of the profit will be her ratio divided by the sum of the ratios, multiplied by the total net profit.
A) ₹4,400 is the correct share of profit earned by Rita, calculated based on her investment proportion.