Simplify:
\(\frac{\left(\frac{1}{4}\div\frac{1}{4}\left(\frac{1}{4}\right)\right)}{\left(\frac{1}{4}\div\frac{1}{4}\times\frac{1}{4}\right)}\)
To simplify the given expression, we need to apply the BODMAS/PEMDAS rule (Brackets, Orders/Exponents, Division, Multiplication, Addition, Subtraction). It's crucial to correctly interpret the order of operations, especially for division and multiplication, and the 'of' operation (which is multiplication but takes precedence over regular multiplication and division).
Let the given expression be \(E\). We need to simplify:
\[ E = \frac{\left(\frac{1}{4}\div\frac{1}{4}\left(\frac{1}{4}\right)\right)}{\left(\frac{1}{4}\div\frac{1}{4}\times\frac{1}{4}\right)} \]First, let's simplify the numerator:
According to BODMAS, the 'of' operation (represented by parentheses here, \[\frac{1}{4}(\frac{1}{4})\]) comes before division. So, we calculate \[\frac{1}{4}(\frac{1}{4})\] first.
\[ \frac{1}{4}\left(\frac{1}{4}\right) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \]Now, substitute this back into the numerator expression:
\[ N = \frac{1}{4}\div\frac{1}{16} \]To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
\[ N = \frac{1}{4} \times \frac{16}{1} = \frac{16}{4} = 4 \]So, the numerator simplifies to 4.
Now, let's simplify the denominator:
\[ D = \frac{1}{4}\div\frac{1}{4}\times\frac{1}{4} \]According to BODMAS, division and multiplication have the same precedence and are performed from left to right.
First, perform the division:
\[ \frac{1}{4}\div\frac{1}{4} = \frac{1}{4} \times \frac{4}{1} = 1 \]Now, substitute this back into the denominator expression:
\[ D = 1 \times \frac{1}{4} = \frac{1}{4} \]So, the denominator simplifies to \(\frac{1}{4}\).
Now, substitute the simplified numerator and denominator back into the original expression:
\[ E = \frac{N}{D} = \frac{4}{\frac{1}{4}} \]To divide by a fraction, we multiply by its reciprocal:
\[ E = 4 \times \frac{4}{1} = 16 \]Thus, the simplified value of the expression is 16.
Correct Option: B) 16
Which of the following fractions is a terminating decimal?
\( \frac{5}{8} \)
\( \frac{1}{3} \)
\( \frac{1}{7} \)
\( \frac{2}{11} \)
A fraction represents a terminating decimal if, when reduced to its simplest form, the prime factors of its denominator are only 2 and/or 5. We will examine each option to determine if its denominator meets this condition.
A) \(\frac{5}{8}\) is a terminating decimal because its denominator, 8, can be expressed as \(2^3\), which only contains the prime factor 2.
This problem involves concepts of boats and streams. We need to find the ratio of the speed of the boat in still water to the speed of the water current. We are given the time taken to travel a certain distance upstream and the time taken to return the same distance downstream.
Let the speed of the boat in still water be \(B\) km/hr and the speed of the water current be \(C\) km/hr.
When the boat moves upstream, its effective speed is \(B - C\) km/hr.
When the boat moves downstream, its effective speed is \(B + C\) km/hr.
Given time taken to cover the distance upstream = 8 hours 48 minutes.
Convert 48 minutes to hours: \[48 \text{ minutes} = \frac{48}{60} \text{ hours} = \frac{4}{5} \text{ hours} = 0.8 \text{ hours}\].
So, upstream time \(T_u = 8 + 0.8 = 8.8\) hours.
Given time taken to return downstream = 4 hours.
So, downstream time \(T_d = 4\) hours.
Let the distance covered be \(D\).
We know that \(\text{Distance} = \text{Speed} \times \text{Time}\).
For upstream journey:
\[ D = (B - C) \times T_u \] \[ D = (B - C) \times 8.8 \quad \text{(Equation 1)} \]For downstream journey:
\[ D = (B + C) \times T_d \] \[ D = (B + C) \times 4 \quad \text{(Equation 2)} \]Since the distance \(D\) is the same for both journeys, we can equate Equation 1 and Equation 2:
\[ (B - C) \times 8.8 = (B + C) \times 4 \]Now, solve for the ratio \(B : C\):
\[ 8.8B - 8.8C = 4B + 4C \] \[ 8.8B - 4B = 4C + 8.8C \] \[ 4.8B = 12.8C \] \[ \frac{B}{C} = \frac{12.8}{4.8} \]To simplify the ratio, multiply the numerator and denominator by 10 to remove decimals:
\[ \frac{B}{C} = \frac{128}{48} \]Divide both by their greatest common divisor. Both are divisible by 16:
\[ 128 \div 16 = 8 \] \[ 48 \div 16 = 3 \] \[ \frac{B}{C} = \frac{8}{3} \]So, the ratio of the speed of the boat in still water to that of the water current is \(8 : 3\).
Correct Option: C) 8 : 3
To find the volume of a cylinder, we need its radius (r) and height (h). We are given the curved surface area and the base circumference. We can use these two pieces of information to find 'r' and 'h' first, and then calculate the volume.
Find the radius (r) from the base circumference:
The formula for the circumference of the base of a cylinder is \(C = 2\pi r\).
Given \(C = 44 \text{ cm}\).
So, \(44 = 2 \times \frac{22}{7} \times r\)
\(44 = \frac{44}{7} \times r\)
\(r = \frac{44 \times 7}{44}\)
\(r = 7 \text{ cm}\)
Find the height (h) from the curved surface area:
The formula for the curved surface area of a cylinder is \(CSA = 2\pi rh\).
Given \(CSA = 440 \text{ cm}^2\).
We know \(r = 7 \text{ cm}\).
So, \(440 = 2 \times \frac{22}{7} \times 7 \times h\)
\(440 = 2 \times 22 \times h\)
\(440 = 44h\)
\(h = \frac{440}{44}\)
\(h = 10 \text{ cm}\)
Calculate the volume (V) of the cylinder:
The formula for the volume of a cylinder is \(V = \pi r^2 h\).
We have \(r = 7 \text{ cm}\) and \(h = 10 \text{ cm}\).
\(V = \frac{22}{7} \times (7)^2 \times 10\)
\(V = \frac{22}{7} \times 49 \times 10\)
\(V = 22 \times 7 \times 10\)
\(V = 154 \times 10\)
\(V = 1540 \text{ cm}^3\)
Correct Option: C) 1540 cm³
This is a coding-decoding problem where we need to find the word corresponding to 'and' by comparing the given coded statements.
Correct Option: A) 'it' is the word that means 'and'.
The question asks to identify the most appropriate description of the Indian economy. We need to understand the characteristics of different economic systems (Mixed, Capitalist, Traditional, Socialist) and compare them with the features of the Indian economy.
Correct Option: A) Mixed — The Indian economy is characterized by the coexistence of both the private and public sectors. While private enterprises operate for profit, the government also plays a significant role in economic planning, regulation, and the provision of public goods and services. This blend of market forces and state intervention is the hallmark of a mixed economy.
The question asks to identify which of the given options is NOT a cereal. To answer this, we need to understand what cereals are and then classify each option accordingly.
Correct Option: D) Mustard
This problem involves understanding the relationship between efficiency and wages. Wages are typically distributed in proportion to the work done or the efficiency of the workers. If one person is less efficient, they do less work in the same amount of time, and thus their share of wages should be proportionally less.
Understand the efficiency relationship: The problem states that P is 25% less efficient than Q. Let's assume Q's efficiency is 100 units. Then P's efficiency will be 25% less than 100.
\[ \text{Q's efficiency} = 100 \% \\ \text{P's efficiency} = 100 \% - 25 \% = 75 \% \]Express efficiencies as a ratio: Now, we can write the ratio of P's efficiency to Q's efficiency.
\[ \text{P's efficiency} : \text{Q's efficiency} = 75 : 100 \]Simplify the ratio: Divide both sides of the ratio by their greatest common divisor, which is 25.
\[ \frac{75}{25} : \frac{100}{25} = 3 : 4 \]Relate efficiency to wages: Wages should be shared in the same ratio as their efficiencies, as more efficient workers contribute more and thus earn more. Therefore, the ratio of their wages will be the same as the ratio of their efficiencies.
\[ \text{P's wages} : \text{Q's wages} = 3 : 4 \]Correct Option: C) 3 : 4
To complete the series, we need to identify the pattern or rule governing the sequence of numbers. We will examine the relationship between consecutive terms to find a consistent mathematical operation.
The first term is 2, and the second term is 4.
\[ 2 \ 4 \]We can observe that \(2 \times 1 + 2 = 4\).The second term is 4, and the third term is 11.
\[ 4 \ 11 \]If we try a similar pattern, \(4 \times 2 + 3 = 11\). This suggests a pattern of multiplying by an increasing integer and adding an increasing integer.The third term is 11, and the fourth term is 37.
\[ 11 \ 37 \]Following the pattern, we should multiply by 3 and add 4: \(11 \times 3 + 4 = 33 + 4 = 37\). This confirms the pattern.The fourth term is 37, and the fifth term is 153.
\[ 37 \ 153 \]Following the pattern, we should multiply by 4 and add 5: \(37 \times 4 + 5 = 148 + 5 = 153\). This further confirms the pattern.The pattern is: \(\text{Next Term} = \text{Current Term} \times n + (n+1)\), where \(n\) increases by 1 for each step, starting from \(n=1\).
For the next term after 153, the multiplier will be 5, and the number to be added will be 6.
\[ \text{Next Term} = 153 \times 5 + 6 \]\[ \text{Next Term} = 765 + 6 \]\[ \text{Next Term} = 771 \]D) 771 — The pattern identified is \(T_{n+1} = T_n \times n + (n+1)\), where \(n\) starts from 1 for the first step. Applying this pattern to the last given term, \(153 \times 5 + 6 = 765 + 6 = 771\).
The question asks to identify the location in India where nuclear explosive devices were tested. This is a factual question related to India's nuclear program and its history.
A) Pokhran is the correct answer. Both India's first nuclear test in 1974 ("Smiling Buddha") and the subsequent series of tests in 1998 ("Operation Shakti") were conducted at the Pokhran Test Range in Rajasthan.