To find the HCF (Highest Common Factor) of numbers that include decimals, it's often easiest to convert all numbers into integers by multiplying them by a suitable power of 10. Then, find the HCF of these integers and finally divide the result by the same power of 10.
Convert decimals to integers: The number 17.5 has one decimal place. To convert it into an integer, multiply it by 10. We must do the same for all other numbers to maintain their relative values.
Now, we need to find the HCF of 175, 420, and 140.
Find the prime factorization of each integer:
Identify common prime factors and their lowest powers:
Calculate the HCF of the integers:
\[ \text{HCF}(175, 420, 140) = 5^1 \times 7^1 = 5 \times 7 = 35 \]
Adjust for the initial multiplication: Since we multiplied all original numbers by 10, we must now divide the HCF by 10 to get the HCF of the original numbers.
\[ \text{HCF}(17.5, 42, 14) = \frac{35}{10} = 3.5 \]
D) 3.5 is the HCF of 17.5, 42, and 14, as calculated by converting to integers, finding their HCF, and then adjusting for the decimal.
To solve this problem, we first need to calculate the total work done by P and Q individually in terms of 'man-hours' or 'work-hours'. Then, we determine their individual work rates per hour. After finding their combined work rate per hour, we can calculate the total time required to complete the task when they work together for a specified number of hours per day.
Calculate total work for P:
P completes the task in 10 days working 6 hours a day.
\[ \text{Total work by P} = 10 \text{ days} \times 6 \text{ hours/day} = 60 \text{ work-hours} \]Calculate total work for Q:
Q completes the same task in 9 days working 10 hours a day.
\[ \text{Total work by Q} = 9 \text{ days} \times 10 \text{ hours/day} = 90 \text{ work-hours} \]Since it's the same task, the total work required is constant. We can take the LCM of 60 and 90 to represent the total units of work, or simply use 1 unit of work for the entire task.
Let's assume the total work is 1 unit.
Calculate P's work rate per hour:
P completes 1 unit of work in 60 work-hours.
\[ \text{P's work rate} = \frac{1 \text{ unit}}{60 \text{ hours}} = \frac{1}{60} \text{ units/hour} \]Calculate Q's work rate per hour:
Q completes 1 unit of work in 90 work-hours.
\[ \text{Q's work rate} = \frac{1 \text{ unit}}{90 \text{ hours}} = \frac{1}{90} \text{ units/hour} \]Calculate their combined work rate per hour:
\[ \text{Combined work rate} = \text{P's rate} + \text{Q's rate} \] \[ = \frac{1}{60} + \frac{1}{90} \]To add these fractions, find a common denominator, which is 180.
\[ = \frac{3}{180} + \frac{2}{180} = \frac{5}{180} = \frac{1}{36} \text{ units/hour} \]Calculate the total hours required to complete the task together:
\[ \text{Total hours} = \frac{\text{Total work}}{\text{Combined work rate}} = \frac{1 \text{ unit}}{\frac{1}{36} \text{ units/hour}} = 36 \text{ hours} \]Calculate the number of days if they work 6 hours a day:
If they work 6 hours a day, the number of days needed is:
\[ \text{Number of days} = \frac{\text{Total hours}}{\text{Hours per day}} = \frac{36 \text{ hours}}{6 \text{ hours/day}} = 6 \text{ days} \]D) 6 days
Based on our calculations, P and Q working together for 6 hours a day will complete the task in 6 days.
The problem involves calculating compound interest (CI) and simple interest (SI). We are given the CI for a certain principal, rate, and time. We need to find the simple interest for the same principal and rate but for a different time period. The approach will be to first use the given compound interest information to find the principal amount. Once the principal is known, we can calculate the simple interest.
The question asks to identify the founder of the Young Bengal Movement. This requires knowledge of prominent figures and movements during the Indian Renaissance period.
Correct Option: D) Henry Vivian Derozio initiated the Young Bengal Movement. He was a radical thinker and teacher at Hindu College, Calcutta, who inspired his students to question authority, embrace rationalism, and advocate for social reforms like women's rights and the abolition of Sati.
Which of the following amendment granted financial powers to the panchayats and helped to strengthen the local government?
74th Amendment
64th Amendment
63rd Amendment
73rd Amendment
The question asks about the constitutional amendment that granted financial powers to Panchayats and strengthened local self-governance. We need to identify the specific amendment related to Panchayati Raj Institutions.
Correct Option: D) 73rd Amendment — This amendment, enacted in 1992, gave constitutional status to Panchayati Raj Institutions. It added Part IX to the Constitution, titled "The Panchayats," and the Eleventh Schedule. It mandated the establishment of Panchayats at the village, intermediate, and district levels, and included provisions for their financial powers, such as the power to levy taxes and the constitution of State Finance Commissions to review their financial position.
Find the value of \(\frac{(759+253)^2+(759-253)^2}{759\times759+253\times253}\):
The problem asks us to evaluate a given algebraic expression. We can simplify this expression by recognizing a common algebraic identity. Let the two numbers be \(a\) and \(b\). The expression is in the form of \(\frac{(a+b)^2+(a-b)^2}{a^2+b^2}\).
Identify the two numbers in the expression. Let \(a = 759\) and \(b = 253\).
Substitute these values into the given expression:
\[ \frac{(759+253)^2+(759-253)^2}{759\times759+253\times253} = \frac{(a+b)^2+(a-b)^2}{a^2+b^2} \]Recall the algebraic identities for \((a+b)^2\) and \((a-b)^2\):
Substitute these identities into the numerator of the expression:
\[ (a+b)^2 + (a-b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) \]Simplify the numerator by combining like terms:
\[ a^2 + 2ab + b^2 + a^2 - 2ab + b^2 = 2a^2 + 2b^2 \]Factor out 2 from the simplified numerator:
\[ 2a^2 + 2b^2 = 2(a^2 + b^2) \]Now, substitute this back into the full expression:
\[ \frac{2(a^2 + b^2)}{a^2 + b^2} \]Cancel out the common term \((a^2 + b^2)\) from the numerator and the denominator:
\[ \frac{2\cancel{(a^2 + b^2)}}{\cancel{(a^2 + b^2)}} = 2 \]The value of the expression is 2, regardless of the specific values of \(a\) and \(b\) (as long as \(a^2+b^2 \neq 0\), which is true here).
C) The simplified value of the expression is 2.
The question asks to identify the phylum whose members are exclusively free-living marine animals. We need to examine the characteristics of each given phylum regarding their habitat and lifestyle.
Correct Option: C) Echinodermata. Members of the phylum Echinodermata (e.g., starfish, sea urchins, sea cucumbers) are characterized by being exclusively marine and free-living. They are never found in freshwater or terrestrial environments, nor are they parasitic.
The question asks to identify the two main components of the Central Processing Unit (CPU) of a computer. Understanding the basic architecture of a CPU is key to answering this question.
Correct Option: B) Control unit and ALU
The Control Unit (CU) and the Arithmetic Logic Unit (ALU) are the two fundamental and primary components that constitute the CPU. The CU manages and coordinates operations, while the ALU performs computations and logical operations.
We will denote the rate of pipe A as \(R_A\) and the rate of pipe B as \(R_B\). The total work (filling the tank) can be considered as 1 unit. We will set up equations based on the given conditions for filling and emptying, and then solve for the ratio \(R_A : R_B\).
Define rates: Let the rate of pipe A be \(R_A\) (fraction of tank filled per unit time). Let the rate of pipe B be \(R_B\) (fraction of tank filled per unit time).
Case 1: Both A and B are filling pipes. When B is used for filling, the combined rate of A and B is \(R_A + R_B\). They fill the tank in time 't'. So, the total work done is 1 (full tank). \[ (R_A + R_B) \times t = 1 \quad \text{(Equation 1)} \]
Case 2: A is filling, B is emptying. When B is used for emptying, its rate is \(-R_B\). The net filling rate is \(R_A - R_B\). They fill the tank in time '5t'. So, the total work done is 1 (full tank). \[ (R_A - R_B) \times 5t = 1 \quad \text{(Equation 2)} \]
Equate the work done: From Equation 1 and Equation 2, since both equal 1: \[ (R_A + R_B) \times t = (R_A - R_B) \times 5t \]
Solve for the ratio: Divide both sides by 't' (assuming \(t \neq 0\)): \[ R_A + R_B = 5(R_A - R_B) \] \[ R_A + R_B = 5R_A - 5R_B \] Rearrange the terms to group \(R_A\) and \(R_B\): \[ R_B + 5R_B = 5R_A - R_A \] \[ 6R_B = 4R_A \] Now, find the ratio \(R_A : R_B\): \[ \frac{R_A}{R_B} = \frac{6}{4} \] \[ \frac{R_A}{R_B} = \frac{3}{2} \] So, the ratio of the rates of A and B is \(3 : 2\).
Correct Option: C) 3 : 2