The sum of two numbers is 40 and their product is 60. The sum of their reciprocals is:
\( \frac{1}{2} \)
\( \frac{2}{3} \)
\( \frac{3}{2} \)
\( \frac{3}{4} \)
The problem provides the sum and product of two numbers and asks for the sum of their reciprocals. We can represent the two numbers with variables, form equations based on the given information, and then manipulate the expression for the sum of their reciprocals to use the given sum and product.
The sum of the two numbers is 40:
\[ x + y = 40 \quad \text{(Equation 1)} \]The product of the two numbers is 60:
\[ xy = 60 \quad \text{(Equation 2)} \]We need to find the sum of their reciprocals, which is \[\frac{1}{x} + \frac{1}{y}\].
To add fractions, we find a common denominator, which is \(xy\):
\[ \frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{x + y}{xy} \]Now, substitute the values from Equation 1 and Equation 2 into the simplified expression:
\[ \frac{x + y}{xy} = \frac{40}{60} \]Simplify the fraction:
\[ \frac{40}{60} = \frac{4}{6} = \frac{2}{3} \]Thus, the sum of their reciprocals is \( \frac{2}{3} \).
B) The sum of the reciprocals is \( \frac{2}{3} \), which matches our calculated value.
The question asks about the tenure of members of the Rajya Sabha, which is the upper house of the Indian Parliament. Understanding the structure and functioning of the Indian Parliament, specifically the Rajya Sabha, is key to answering this question.
Correct Option: B) 6 years. Members of the Rajya Sabha are elected for a term of six years. This ensures continuity of the house while allowing for periodic renewal of its membership.
This problem can be solved by using the concept of work rates. If a person completes a task in 'n' days, their one-day work rate is \(1/n\). We are given the relationship between Karan's and Shyam's work rates and the combined time they take to complete the task. We need to find the time Karan takes alone.
Define variables for work rates:
Let Shyam's one-day work rate be \(S\).
Let Karan's one-day work rate be \(K\).
Establish the relationship between their work rates:
The problem states that Karan works twice as fast as Shyam. This means Karan's work rate is twice Shyam's work rate.
Calculate their combined work rate:
When they work together, their work rates add up. Their combined one-day work rate is \(K + S\).
Substitute \(K = 2S\) into the combined work rate:
Use the information about their combined completion time:
They can complete the task together in 20 days. This means their combined one-day work rate is \(1/20\) of the task.
Solve for Shyam's work rate (\(S\)):
Equate the combined work rate expressions:
Solve for Karan's work rate (\(K\)):
Since \(K = 2S\):
Determine the time Karan takes alone:
If Karan's one-day work rate is \(1/30\), it means Karan can complete the entire task in 30 days.
Correct Option: C) 30 days
To find the smallest angle of a quadrilateral when its angles are given in a ratio, we first need to know the sum of the interior angles of a quadrilateral. This sum is always 360 degrees. We can then represent the angles using a common multiple and set up an equation to solve for that multiple.
A) 50° is the smallest angle, calculated as \(5 \times 10^\circ\).
To find the sum of the cubes of natural numbers from 5 to 14, we can use the formula for the sum of the cubes of the first \(n\) natural numbers. The required sum can be calculated by subtracting the sum of the cubes of natural numbers from 1 to 4 from the sum of the cubes of natural numbers from 1 to 14.
Recall the formula for the sum of the cubes of the first \(n\) natural numbers:
\[ \sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2 \]Calculate the sum of the cubes of natural numbers from 1 to 14 (\(S_{14}\)):
Here, \(n = 14\).
\[ S_{14} = \left( \frac{14(14+1)}{2} \right)^2 = \left( \frac{14 \times 15}{2} \right)^2 \]\[ S_{14} = \left( 7 \times 15 \right)^2 = (105)^2 \]\[ S_{14} = 105 \times 105 = 11025 \]Calculate the sum of the cubes of natural numbers from 1 to 4 (\(S_4\)):
Here, \(n = 4\).
\[ S_4 = \left( \frac{4(4+1)}{2} \right)^2 = \left( \frac{4 \times 5}{2} \right)^2 \]\[ S_4 = \left( 2 \times 5 \right)^2 = (10)^2 \]\[ S_4 = 100 \]Subtract \(S_4\) from \(S_{14}\) to find the sum of the cubes of natural numbers from 5 to 14:
Required Sum \( = S_{14} - S_4 \)
Required Sum \( = 11025 - 100 \)
Required Sum \( = 10925 \)
B) 10925 is the correct sum of the cubes of natural numbers from 5 to 14.
We are given four numbers and need to find a value that, when subtracted from each, makes the resulting numbers proportional. We will set up an equation based on the definition of proportion and solve for the unknown value.
Let the unknown number to be subtracted be \(x\).
After subtracting \(x\) from each number, the new numbers will be:
For these numbers to be in proportion, the ratio of the first two must be equal to the ratio of the last two. That is:
\[ \frac{26 - x}{44 - x} = \frac{56 - x}{104 - x} \]Now, we cross-multiply to solve for \(x\):
\[ (26 - x)(104 - x) = (44 - x)(56 - x) \]Expand both sides of the equation:
Set the expanded expressions equal to each other:
\[ 2704 - 130x + x^2 = 2464 - 100x + x^2 \]Subtract \(x^2\) from both sides:
\[ 2704 - 130x = 2464 - 100x \]Rearrange the terms to solve for \(x\). Move \(100x\) to the left and \(2704\) to the right:
\[ -130x + 100x = 2464 - 2704 \]\[ -30x = -240 \]Divide by \(-30\):
\[ x = \frac{-240}{-30} \]\[ x = 8 \]C) 8. When 8 is subtracted from each number, the new numbers are 18, 36, 48, and 96. The ratio \[\frac{18}{36} = \frac{1}{2}\] and \[\frac{48}{96} = \frac{1}{2}\]. Since the ratios are equal, the numbers are in proportion.
To form the smallest five-digit number using a given set of digits, we need to arrange the digits in ascending order. However, there's a special rule when zero is one of the digits: zero cannot be placed in the leftmost (highest place value) position of a number if it's meant to be a number with a specific number of digits, as it would reduce the number of digits. Therefore, the smallest non-zero digit must be placed first, followed by zero, and then the remaining digits in ascending order.
Correct Option: A) 20345
The problem asks us to find the next number in the given series: 4, 13, 27, 82, 165, ?. We need to identify the pattern or rule governing the sequence.
Correct Option: D) 496
We are given the initial ratio of two numbers and the new ratio after a constant value is added to each number. We can set up an algebraic equation based on these ratios to find the numbers and then identify the greater one.
A) The greater number is 22.
The question asks to identify the odd one out from a list of geometric shapes: Rectangle, Triangle, Square, Parallelogram. The approach is to analyze the properties of each shape, specifically focusing on the number of sides or vertices, to find a common characteristic shared by most and one that deviates.
Correct Option: A) Triangle is the odd one out because it is a three-sided polygon, whereas a Rectangle, Square, and Parallelogram are all four-sided polygons (quadrilaterals).