The average of eight numbers is 80. Find the least number if it is \( \frac{1}{16} \) of the sum of the numbers.
We are given the average of eight numbers and a relationship between the least number and the sum of all numbers. We will first use the average formula to find the sum of the numbers. Then, we will use the given relationship to find the value of the least number.
Given that the average of eight numbers is 80, we have:
\[ 80 = \frac{\text{Sum of numbers}}{8} \]Multiplying both sides by 8, we get the sum of the numbers:
\[ \text{Sum of numbers} = 80 \times 8 = 640 \]Substitute the sum of numbers we calculated in Step 1:
\[ L = \frac{1}{16} \times 640 \]Perform the division:
\[ L = 40 \]Thus, the least number is 40.
Correct Option: A) 40 is the correct answer as calculated.
This is a syllogism problem. We will use Venn diagrams to represent the relationships given in the statements and then evaluate the conclusions based on these diagrams. We need to assume the statements are true, even if they contradict common knowledge.
D) Only Conclusion (I) is true
By how much is 60% of 40 greater than \( \frac{2}{5} \) of 55?
To solve this problem, we need to calculate two values: first, 60% of 40, and second, \( \frac{2}{5} \) of 55. After finding these two values, we will subtract the second value from the first to determine how much greater the first value is than the second.
To find a percentage of a number, convert the percentage to a decimal or a fraction and multiply it by the number.
\[ 60\% \text{ of } 40 = \frac{60}{100} \times 40 \]\[ = \frac{6}{10} \times 40 \]\[ = 6 \times 4 \]\[ = 24 \]To find a fraction of a number, multiply the fraction by the number.
\[ \frac{2}{5} \text{ of } 55 = \frac{2}{5} \times 55 \]\[ = 2 \times \frac{55}{5} \]\[ = 2 \times 11 \]\[ = 22 \]Now, subtract the second calculated value from the first calculated value to find out by how much the first is greater than the second.
\[ \text{Difference} = (\text{60% of 40}) - (\frac{2}{5} \text{ of 55}) \]\[ = 24 - 22 \]\[ = 2 \]Thus, 60% of 40 is greater than \( \frac{2}{5} \) of 55 by 2.
B) 2 is the correct answer, as calculated in the step-by-step explanation.
The problem presents two triads of words, where each word in a triad is derived from the previous one following a specific logical rule. We need to identify this rule and then apply it to the given options to find the one that follows the same logic.
B) WIND - IWND - DIWN
Therefore, option B follows the same logic.
This problem involves substituting given symbols with mathematical operators and then evaluating the expression using the order of operations (BODMAS/PEMDAS).
Correct Option: B) 55
The problem asks us to identify a set of numbers that follows the same logical relationship as the given example sets. We need to find a mathematical operation or pattern connecting the three numbers in each set.
Let the numbers be \(A=23\), \(B=253\), \(C=11\).
We need to find a relationship between \(A\), \(B\), and \(C\). Let's try multiplication or division.
If we multiply \(A\) and \(C\): \(23 \times 11 = 253\). This matches \(B\).
So, the relationship is \(A \times C = B\).
Let the numbers be \(A=46\), \(B=1104\), \(C=24\).
Using the derived relationship \(A \times C = B\):
\(46 \times 24 = 1104\). This also matches \(B\).
The established pattern is that the product of the first and third numbers equals the second number.
D) (25, 525, 21)
Let \(A=25\), \(B=525\), \(C=21\).
Check if \(A \times C = B\):
\(25 \times 21 = 525\).
This matches the pattern. Therefore, option D is the correct answer.
Here, \(A=4\), \(B=11\), \(C=2\).
\(A \times C = 4 \times 2 = 8\). This is not equal to \(B=11\).
Here, \(A=1\), \(B=12\), \(C=9\).
\(A \times C = 1 \times 9 = 9\). This is not equal to \(B=12\).
Here, \(A=25\), \(B=253\), \(C=10\).
\(A \times C = 25 \times 10 = 250\). This is not equal to \(B=253\).
The problem presents a series of letter groups. We need to identify the pattern in the alphabetical positions of the letters within each group and between consecutive groups to find the missing group.
Correct Option: C) LOQ
The problem presents a series of letter-number clusters. We need to identify the pattern in both the letters and the numbers to find the missing cluster. We will analyze the progression of each letter individually and the sequence of numbers.
Correct Option: D) DGD 125
Simplify: \(\frac{\cot(\theta)}{\mathrm{cosec}(\theta)+1}+\frac{\mathrm{cosec}(\theta)+1}{\cot(\theta)}\)
The question asks us to simplify a trigonometric expression involving cotangent and cosecant functions. We need to apply the standard trigonometric identity relating cotangent and cosecant, combine the fractions and simplify the result.
Identify the given expression:
\[ \frac{\cot\theta}{\operatorname{cosec}\theta+1} + \frac{\operatorname{cosec}\theta+1}{\cot\theta} \]We need to simplify this expression.
Take the LCM of the denominators:
\[ =\frac{\cot^2\theta+(\operatorname{cosec}\theta+1)^2} {\cot\theta(\operatorname{cosec}\theta+1)} \]Expand the square in the numerator:
\[ =\frac{\cot^2\theta+\operatorname{cosec}^2\theta+2\operatorname{cosec}\theta+1} {\cot\theta(\operatorname{cosec}\theta+1)} \]Apply the trigonometric identity:
We know that:
\[ 1+\cot^2\theta=\operatorname{cosec}^2\theta \]Substituting \(\cot^2\theta+1=\operatorname{cosec}^2\theta\) in the numerator:
\[ =\frac{2\operatorname{cosec}^2\theta+2\operatorname{cosec}\theta} {\cot\theta(\operatorname{cosec}\theta+1)} \]Factorise and simplify:
Taking \(2\operatorname{cosec}\theta\) as the common factor:
\[ =\frac{2\operatorname{cosec}\theta(\operatorname{cosec}\theta+1)} {\cot\theta(\operatorname{cosec}\theta+1)} \]Cancel the common factor \((\operatorname{cosec}\theta+1)\):
\[ =\frac{2\operatorname{cosec}\theta}{\cot\theta} \]Convert into sine and cosine:
Using the identities:
\[ \operatorname{cosec}\theta=\frac{1}{\sin\theta} \] \[ \cot\theta=\frac{\cos\theta}{\sin\theta} \]Substitute these values:
\[ =2\times\frac{1}{\sin\theta} \times\frac{\sin\theta}{\cos\theta} \] \[ =\frac{2}{\cos\theta} \]Since \(\sec\theta=\frac{1}{\cos\theta}\), we get:
\[ =2\sec\theta \]Therefore, the simplified expression is \(2\sec\theta\).
Correct Option: C) \(2\sec\theta\) — By taking the LCM, applying the identity \(1+\cot^2\theta=\operatorname{cosec}^2\theta\), and cancelling the common factors, the expression simplifies to \(\frac{2\operatorname{cosec}\theta}{\cot\theta}=\frac{2}{\cos\theta}=2\sec\theta\). Hence, Option C is the correct answer.
A) \(2\operatorname{cosec}\theta\): This is incorrect because the simplified expression is \(\frac{2}{\cos\theta}\), not \(\frac{2}{\sin\theta}\).
B) \(2\sin\theta\): This is incorrect because the result involves the reciprocal of cosine, not sine.
D) \(2\cos\theta\): This is incorrect because the simplified expression is \(2\sec\theta=\frac{2}{\cos\theta}\), not \(2\cos\theta\).