This problem involves calculating the combined work rate of two individuals to determine the time taken to complete a task together. We will first find the individual work rates (boxes packed per hour) and then sum them to get the combined work rate. Finally, we will use the combined work rate to find the total time required.
Calculate Rama's work rate:
Rama packs 36 boxes in 1 hour.
\[ \text{Rama's work rate} = \frac{\text{Number of boxes}}{\text{Time taken}} = \frac{36 \text{ boxes}}{1 \text{ hour}} = 36 \text{ boxes/hour} \]Calculate her sister's work rate:
Her sister packs 36 boxes in 3 hours.
\[ \text{Sister's work rate} = \frac{\text{Number of boxes}}{\text{Time taken}} = \frac{36 \text{ boxes}}{3 \text{ hours}} = 12 \text{ boxes/hour} \]Calculate their combined work rate:
When they work together, their work rates add up.
\[ \text{Combined work rate} = \text{Rama's work rate} + \text{Sister's work rate} \] \[ \text{Combined work rate} = 36 \text{ boxes/hour} + 12 \text{ boxes/hour} = 48 \text{ boxes/hour} \]Calculate the time taken to pack 36 boxes together:
To find the time taken, we divide the total number of boxes by their combined work rate.
\[ \text{Time taken} = \frac{\text{Total number of boxes}}{\text{Combined work rate}} = \frac{36 \text{ boxes}}{48 \text{ boxes/hour}} \] \[ \text{Time taken} = \frac{36}{48} \text{ hours} = \frac{3}{4} \text{ hours} \]Convert the time to minutes:
Since 1 hour = 60 minutes, we convert \(\frac{3}{4}\) hours to minutes.
\[ \text{Time in minutes} = \frac{3}{4} \times 60 \text{ minutes} = 3 \times 15 \text{ minutes} = 45 \text{ minutes} \]Correct Option: D) 45 min
The problem asks us to identify the option where the numbers share the same relationship as the given group of numbers (15, 7, 110). We need to find a mathematical operation or pattern connecting the first two numbers to the third number in the given group and then apply the same pattern to the options.
Let the three numbers be \(A\), \(B\), and \(C\). So, \(A = 15\), \(B = 7\), and \(C = 110\).
We need to find a relationship between \(A\), \(B\), and \(C\). Let's try common arithmetic operations:
Let's explore the relationship \(A \times B + X = C\). If \(X = 5\), then \(15 \times 7 + 5 = 105 + 5 = 110\). This seems like a plausible pattern.
Another possible pattern could be related to the first number itself. What if \(X\) is related to \(A\) or \(B\)?
Consider the pattern: \((A \times B) + (A - B)\).
For \((15, 7, 110)\):
\[ (15 \times 7) + (15 - 7) = 105 + 8 = 113 \neq 110 \]This is not the pattern.
Consider the pattern: \((A \times B) + (A - \text{some constant})\) or \((A \times B) + (B - \text{some constant})\).
Let's re-examine \(15 \times 7 = 105\). We need to add 5 to get 110. The number 5 is \(15 - 10\) or \(7 - 2\). It's not immediately obvious how 5 is derived from 15 or 7.
Let's consider another common pattern in such problems: \((A^2 - B^2)\), \((A+B)^2\), or combinations of \(A\) and \(B\).
What if the pattern is \((A \times B) + \text{some number}\)? We found \(15 \times 7 + 5 = 110\). The number 5 could be a constant, or it could be derived from \(A\) or \(B\).
Let's try \((A \times B) + \text{a constant}\). If the constant is 5, then the pattern is \(A \times B + 5 = C\).
D) (18, 5, 95)
When dealing with similar polygons (which regular polygons with the same number of sides are), the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. We will use this property to find the area of the smaller polygon.
A) 575 cm² is the correct answer, as calculated.
The value of \(\left(\frac{\frac{8}{3}\div\frac{3}{5}\times\frac{7}{5}}{\frac{5}{3}\div\frac{5}{7}\times\frac{8}{9}}\right)\div9\)
\( \frac{1}{3} \)
\( \frac{3}{5} \)
\( \frac{2}{3} \)
\( \frac{1}{5} \)
To solve this problem, we need to simplify the given complex fraction step-by-step. We will first evaluate the numerator, then the denominator, then divide the numerator by the denominator, and finally divide the result by 9. Remember the order of operations: division before multiplication when they appear from left to right.
Correct Option: A) \(\frac{1}{3}\)
The question asks to identify the specific article of the Indian Constitution that designates Hindi as the official language of the Union. This requires knowledge of the constitutional provisions related to official languages.
D) Article 343(1) — This article unequivocally declares that the official language of the Union shall be Hindi in Devanagari script. It also specifies that the form of numerals to be used for official purposes of the Union shall be the international form of Indian numerals.
To find the greatest four-digit number divisible by 18, 25, 45, and 55, we first need to find the Least Common Multiple (LCM) of these numbers. Any number divisible by all these numbers must be a multiple of their LCM. After finding the LCM, we will find the greatest four-digit multiple of the LCM.
Find the prime factorization of each number:
Calculate the LCM: The LCM is found by taking the highest power of all prime factors present in the numbers.
\[\text{LCM}(18, 25, 45, 55) = 2^1 \times 3^2 \times 5^2 \times 11^1\] \[\text{LCM} = 2 \times 9 \times 25 \times 11\] \[\text{LCM} = 18 \times 25 \times 11\] \[\text{LCM} = 450 \times 11\] \[\text{LCM} = 4950\]Find the greatest four-digit number: The greatest four-digit number is 9999.
Divide the greatest four-digit number by the LCM:
\[\frac{9999}{4950} \approx 2.02\]This means that \(4950 \times 2\) will be the greatest multiple of 4950 that is a four-digit number.
Multiply the LCM by the integer part of the quotient:
\[4950 \times 2 = 9900\]Verify the result: 9900 is a four-digit number and is divisible by 4950, and therefore by 18, 25, 45, and 55. Any multiple greater than 9900 (e.g., \(4950 \times 3 = 14850\)) would be a five-digit number.
A) 9900 is the greatest four-digit number divisible by 18, 25, 45, and 55, as calculated.
The sum of the salaries of A and B together is ₹4,300. A spends 95% of his salary and B, 80% of his salary. If their savings are the same, what is A's salary?
Let A's salary be \(S_A\) and B's salary be \(S_B\). We are given the sum of their salaries and their spending percentages. We are also told that their savings are equal. We can set up equations based on this information and solve for A's salary.
Define variables and set up the first equation:
Let A's salary be \(S_A\) and B's salary be \(S_B\).
Given that the sum of their salaries is ₹4,300:
\[ S_A + S_B = 4300 \quad \text{(Equation 1)} \]Calculate their savings:
A spends 95% of his salary, so A saves \(100\% - 95\% = 5\%\) of his salary.
A's savings \( = 0.05 \times S_A \)
B spends 80% of his salary, so B saves \(100\% - 80\% = 20\%\) of his salary.
B's savings \( = 0.20 \times S_B \)
Set up the second equation based on equal savings:
Given that their savings are the same:
\[ 0.05 \times S_A = 0.20 \times S_B \]Simplify the second equation:
Divide both sides by 0.05:
\[ S_A = \frac{0.20}{0.05} \times S_B \]\[ S_A = 4 \times S_B \quad \text{(Equation 2)} \]Substitute Equation 2 into Equation 1:
Substitute \(S_A = 4 S_B\) into \(S_A + S_B = 4300\):
\[ 4 S_B + S_B = 4300 \]\[ 5 S_B = 4300 \]Solve for B's salary (\(S_B\)):
\[ S_B = \frac{4300}{5} \]\[ S_B = 860 \]Solve for A's salary (\(S_A\)):
Using Equation 2, \(S_A = 4 S_B\):
\[ S_A = 4 \times 860 \]\[ S_A = 3440 \]Correct Option: A) ₹3,440
A's salary is ₹3,440.
To arrange the events chronologically, we need to recall the years in which each event occurred during India's freedom struggle. Then, we can order them from earliest to latest.
Based on these dates, the chronological order is:
This corresponds to the sequence: II, IV, I, III.
B) II, IV, I, III is the correct sequence, as it accurately reflects the chronological order of the events: Jallianwala Bagh Massacre (1919), Non-Cooperation Movement (1920), The Salt March (1930), and Quit India Movement (1942).
This is a blood relation problem. We need to deduce the relationship between the girl and the mother of the person in the photo by breaking down the statement given by the girl.
The girl says, "His daughter's paternal uncle's wife is my mother." Let's analyze this statement from the perspective of the person in the photo.
"His daughter": This refers to the daughter of the person in the photo.
"His daughter's paternal uncle": A paternal uncle is the father's brother. So, the daughter's paternal uncle is the brother of the person in the photo.
"His daughter's paternal uncle's wife": This means the wife of the person's brother. This person is the sister-in-law of the person in the photo.
The girl states that "His daughter's paternal uncle's wife is my mother." So, the girl's mother is the sister-in-law of the person in the photo.
This implies that the girl's mother is married to the brother of the person in the photo. Therefore, the girl is the daughter of the brother of the person in the photo. In other words, the girl is the niece of the person in the photo.
Now, we need to find the relation of that girl to the mother of the person in the photo.
The mother of the person in the photo is also the mother of the person's brother (who is the girl's father).
So, the girl's father's mother is the mother of the person in the photo. This means the girl is the daughter of the person's son (her father) to the person's mother.
Therefore, the girl is the son's daughter of the mother of the person in the photo.
A) Son's Daughter: As deduced, the girl is the daughter of the person's brother. The person's mother is also the brother's mother. So, the girl is the daughter of the mother's son (the girl's father).
Calculate the standard deviation for the following data.
3, 4, 5, 6, 7
To calculate the standard deviation, we first need to find the mean of the given data. Then, we calculate the variance by finding the average of the squared differences from the mean. Finally, the standard deviation is the square root of the variance.
List the data:
The given data points are \(x = \{3, 4, 5, 6, 7\}\).
Calculate the mean (\(\bar{x}\)):
The mean is the sum of all data points divided by the number of data points (n).
\[ \bar{x} = \frac{\sum x}{n} = \frac{3 + 4 + 5 + 6 + 7}{5} = \frac{25}{5} = 5 \]Calculate the deviations from the mean (\(x - \bar{x}\)):
Square the deviations (\((x - \bar{x})^2\)):
Calculate the sum of the squared deviations (\(\sum (x - \bar{x})^2\)):
\[ \sum (x - \bar{x})^2 = 4 + 1 + 0 + 1 + 4 = 10 \]Calculate the variance (\(\sigma^2\)):
The variance is the sum of the squared deviations divided by the number of data points (n).
\[ \sigma^2 = \frac{\sum (x - \bar{x})^2}{n} = \frac{10}{5} = 2 \]Calculate the standard deviation (\(\sigma\)):
The standard deviation is the square root of the variance.
\[ \sigma = \sqrt{\sigma^2} = \sqrt{2} \]Correct Option: C) \(\sqrt{2}\)