The problem asks for the greatest number that divides 128, 303, and 247, leaving specific remainders. This means that if we subtract the respective remainders from each number, the resulting numbers will be perfectly divisible by the greatest number we are looking for. Therefore, the problem reduces to finding the HCF (Highest Common Factor) of these new numbers.
When a number \(N\) is divided by \(D\) leaving a remainder \(R\), it means \(N - R\) is perfectly divisible by \(D\).
We need to find the HCF of 124, 296, and 228.
The HCF is the product of the lowest powers of common prime factors. The only common prime factor among 124, 296, and 228 is 2. The lowest power of 2 present in all factorizations is \(2^2\).
Therefore, the HCF is \(2^2 = 4\).
A) 4 is the greatest number that divides 128, 303, and 247 leaving remainders 4, 7, and 19 respectively. This is because 4 is the HCF of \(128-4=124\), \(303-7=296\), and \(247-19=228\).
To find Ravi's present age, we need to work backward from the given information. First, determine the daughter's present age, then the wife's present age, and finally Ravi's present age.
It is given that three years ago, her daughter's age was 4 years.
So, Daughter's present age \( = 4 + 3 = 7 \) years.
The wife is 5 times as old as her daughter.
Wife's present age \( = 5 \times (\text{Daughter's present age}) \)
Wife's present age \( = 5 \times 7 = 35 \) years.
Ravi is 5 years older than his wife.
Ravi's present age \( = (\text{Wife's present age}) + 5 \)
Ravi's present age \( = 35 + 5 = 40 \) years.
Correct Option: C) 40 years
To find the arithmetic mean of the first 15 natural numbers, we need to sum these numbers and then divide by the count of the numbers (which is 15). We can use the formula for the sum of the first \(n\) natural numbers or directly sum them if the count is small.
A) 8 is the correct arithmetic mean of the first 15 natural numbers.
To convert a decimal number of hours into hours, minutes, and seconds, we need to separate the integer part (hours) from the decimal part. The decimal part is then converted into minutes, and any remaining decimal part of the minutes is converted into seconds.
Identify the whole hours: The given time is 4.17 hours. The whole number part is 4, so we have 4 hours.
Convert the decimal part of hours to minutes: The decimal part is 0.17 hours. Since 1 hour = 60 minutes, we multiply the decimal part by 60:
\[ \text{Minutes} = 0.17 \times 60 \text{ minutes} \]\[ \text{Minutes} = 10.20 \text{ minutes} \]Identify the whole minutes: From 10.20 minutes, the whole number part is 10, so we have 10 minutes.
Convert the decimal part of minutes to seconds: The remaining decimal part is 0.20 minutes. Since 1 minute = 60 seconds, we multiply this decimal part by 60:
\[ \text{Seconds} = 0.20 \times 60 \text{ seconds} \]\[ \text{Seconds} = 12 \text{ seconds} \]Combine the results: So, 4.17 hours is equal to 4 hours, 10 minutes, and 12 seconds.
Correct Option: A) 4 hours 10 minutes and 12 seconds
To find the average speed for the whole journey, we need to calculate the total distance traveled and the total time taken. The journey involves traveling a certain distance by car and returning the same distance by scooter. We can assume the distance to be 'd' and then use the formula for average speed: \[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\].
Define variables:
Calculate total distance:
Sachin covers distance \(d\) while going and distance \(d\) while returning. So, the total distance traveled is:
\[ \text{Total Distance} = d + d = 2d \text{ km} \]Calculate time taken for each part of the journey:
Time taken to cover distance \(d\) by car (\(T_1\)) is:
\[ T_1 = \frac{\text{Distance}}{\text{Speed}_1} = \frac{d}{80} \text{ hours} \]Time taken to cover distance \(d\) by scooter (\(T_2\)) is:
\[ T_2 = \frac{\text{Distance}}{\text{Speed}_2} = \frac{d}{50} \text{ hours} \]Calculate total time:
The total time taken for the whole journey is:
\[ \text{Total Time} = T_1 + T_2 = \frac{d}{80} + \frac{d}{50} \]To add these fractions, find a common denominator, which is 400:
\[ \text{Total Time} = \frac{5d}{400} + \frac{8d}{400} = \frac{13d}{400} \text{ hours} \]Calculate average speed:
Now, use the formula for average speed:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{13d}{400}} \]Simplify the expression:
\[ \text{Average Speed} = 2d \times \frac{400}{13d} = \frac{800d}{13d} = \frac{800}{13} \]Perform the division:
\[ \text{Average Speed} \approx 61.53846 \text{ km/h} \]Round to two decimal places:
Rounding to two decimal places, the average speed is 61.54 km/h.
D) 61.54 km/h — This is the correct average speed calculated by dividing the total distance by the total time taken for the entire journey.
The foundation day of the UN Charter was on:
29th October 1946
21st October 1950
20th October 1932
24th October 1945
The question asks for the foundation day of the UN Charter. This is a factual question related to the history of international organizations. To answer correctly, one needs to recall the specific date when the United Nations officially came into existence with the ratification of its Charter.
Correct Option: D) 24 th October 1945 — This is the date when the UN Charter officially came into force, marking the establishment of the United Nations.
If the price of tea increases by 20%, by what percentage should a household reduce its consumption of tea so that the budget remains the same?
16 \( \frac{2}{3} \) %
14 \( \frac{2}{3} \) %
15 \( \frac{2}{3} \) %
13 \( \frac{2}{3} \) %
This problem involves calculating the percentage reduction in consumption required to maintain a constant budget when the price of a commodity increases. We can solve this by assuming an initial price and consumption, calculating the new price, and then finding the new consumption that keeps the total expenditure the same.
Let the original price of tea be \(P = 100\) units (e.g., ₹100 per kg).
Let the original consumption of tea be \(C = 100\) units (e.g., 100 kg).
The original budget (expenditure) will be: \[\text{Budget} = P \times C = 100 \times 100 = 10000\]
The price of tea increases by 20%. So, the new price \(P'\) will be:
\[P' = P + (20\% \text{ of } P) = 100 + \left(\frac{20}{100} \times 100\right) = 100 + 20 = 120\]The budget remains the same, i.e., \(10000\). Let the new consumption be \(C'\).
So, \(P' \times C' = \text{Budget}\)
\(120 \times C' = 10000\)
\[C' = \frac{10000}{120} = \frac{1000}{12} = \frac{250}{3}\] units.
Reduction in consumption \( = C - C' = 100 - \frac{250}{3}\)
\[= \frac{300 - 250}{3} = \frac{50}{3}\] units.
Percentage reduction \[= \frac{\text{Reduction in consumption}}{\text{Original consumption}} \times 100\%\]
\[= \frac{\frac{50}{3}}{100} \times 100\% = \frac{50}{3}\%\]
Converting this to a mixed fraction:
\[\frac{50}{3}\% = 16 \frac{2}{3}\%\]
If the price of an item increases by \(x\%\), the percentage reduction in consumption required to keep the expenditure constant is given by the formula:
\[\frac{x}{100 + x} \times 100\%\]
In this case, \(x = 20\%\).
Percentage reduction \( = \frac{20}{100 + 20} \times 100\%\)
\[= \frac{20}{120} \times 100\% = \frac{1}{6} \times 100\%\]
\[= \frac{100}{6}\% = \frac{50}{3}\% = 16 \frac{2}{3}\%\]
This matches the previous calculation.
Correct Option: A) \(16 \frac{2}{3}\) %
The value of 3.14 (with 4 recurring) is:
3 \( \frac{12}{90} \)
3 \( \frac{13}{90} \)
3 \( \frac{14}{90} \)
3 \( \frac{11}{90} \)
The problem asks us to convert a recurring decimal into a mixed fraction. A recurring decimal like \(X.YZ\overline{A}\) can be converted to a fraction by setting it equal to \(x\), multiplying by powers of 10 to align the recurring part, and then subtracting the equations to eliminate the recurring part.
B) 3 \( \frac{13}{90} \) is the correct representation of the given recurring decimal.
The question asks to identify which option does NOT fall under the category of "Infrastructure". Infrastructure refers to the fundamental facilities and systems serving a country, city, or area, such as transportation and communication systems, power plants, and public institutions. These are essential for the functioning of a society and economy.
A) Entertainment — Entertainment is generally considered a service or an industry, but not a basic physical or organizational structure that forms the foundation of a society's functioning, unlike utilities or transport networks.
The question asks about the location of cytoplasm within a cell. To answer this, one needs to recall the basic structure of an animal or plant cell.
B) nucleus and cell membrane
The cytoplasm is the entire content within the cell membrane, excluding the nucleus. It is a jelly-like substance that fills the cell and is present between the cell membrane and the nucleus. It contains various organelles suspended in a fluid called cytosol.