The question asks us to find a relationship between the first two terms and apply the same relationship to the third term to find the fourth. This is an analogy-based question. We need to identify the core component from which the beverage is made.
Coffee is made from coffee beans, which are the roasted seeds of the coffee plant. Just as tea is derived from leaves, coffee is derived from beans.
To find a number that is 30% less than 70, we first need to calculate 30% of 70. Then, we subtract this calculated value from 70.
To find a percentage of a number, we convert the percentage to a decimal or a fraction and multiply it by the number.
\[ 30\% \text{ of } 70 = \frac{30}{100} \times 70 \]\[ = 0.30 \times 70 \]\[ = 21 \]The problem asks for a number that is 30% less than 70. So, we subtract the value calculated in step 1 from 70.
\[ \text{Required Number} = 70 - 21 \]\[ = 49 \]Thus, 49 is 30% less than 70.
Correct Option: C) 49
To find the gain percentage, we need to establish a relationship between the Cost Price (CP) and Selling Price (SP) based on the given information. The problem states that the gain is equal to the selling price of a certain number of pens. We will use the fundamental formula for profit and profit percentage.
A) 25% is the correct gain percentage calculated from the given conditions.
The question asks to identify the statement that is NOT correct about mangrove forests. We need to evaluate each option based on the known characteristics of mangroves.
A) Mangrove trees grow in areas with oxygen-rich soil. This statement is incorrect. Mangroves thrive in waterlogged, anaerobic (oxygen-poor) soils, which is why they have specialized root structures like pneumatophores to absorb oxygen from the atmosphere.
If x + y = 75 and sin x : sin y = \(\frac{1}{\sqrt{2}}\) :\( \frac{1}{2} \), then x : y is:
The problem provides two equations: one relating the sum of two angles \(x\) and \(y\), and another relating the ratio of their sines. We need to find the ratio \(x:y\). We will use the given ratio of sines to find the values of \(x\) and \(y\), and then calculate their ratio.
We are given the equation:
\[ x + y = 75^\circ \quad \text{(1)} \]We are also given the ratio of sines:
\[ \sin x : \sin y = \frac{1}{\sqrt{2}} : \frac{1}{2} \]This can be written as:
\[ \frac{\sin x}{\sin y} = \frac{1/\sqrt{2}}{1/2} \]Simplify the right side:
\[ \frac{\sin x}{\sin y} = \frac{1}{\sqrt{2}} \times 2 = \frac{2}{\sqrt{2}} = \sqrt{2} \]So, we have:
\[ \sin x = \sqrt{2} \sin y \quad \text{(2)} \]From equation (1), we can express \(x\) in terms of \(y\):
\[ x = 75^\circ - y \]Substitute this into equation (2):
\[ \sin(75^\circ - y) = \sqrt{2} \sin y \]Use the sine subtraction formula, \(\sin(A-B) = \sin A \cos B - \cos A \sin B\):
\[ \sin 75^\circ \cos y - \cos 75^\circ \sin y = \sqrt{2} \sin y \]Recall the values for \(\sin 75^\circ\) and \(\cos 75^\circ\):
\[ \sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \]\[ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]\[ \cos 75^\circ = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \]\[ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]Substitute these values back into the equation:
\[ \frac{\sqrt{3} + 1}{2\sqrt{2}} \cos y - \frac{\sqrt{3} - 1}{2\sqrt{2}} \sin y = \sqrt{2} \sin y \]Multiply the entire equation by \(2\sqrt{2}\):
\[ (\sqrt{3} + 1) \cos y - (\sqrt{3} - 1) \sin y = 4 \sin y \]Rearrange the terms to group \(\sin y\):
\[ (\sqrt{3} + 1) \cos y = (4 + \sqrt{3} - 1) \sin y \]\[ (\sqrt{3} + 1) \cos y = (3 + \sqrt{3}) \sin y \]Factor out \(\sqrt{3}\) from the right side:
\[ (\sqrt{3} + 1) \cos y = \sqrt{3}(\sqrt{3} + 1) \sin y \]Since \(\sqrt{3} + 1 \neq 0\), we can divide both sides by \((\sqrt{3} + 1)\):
\[ \cos y = \sqrt{3} \sin y \]Divide by \(\cos y\) (assuming \(\cos y \neq 0\)):
\[ 1 = \sqrt{3} \frac{\sin y}{\cos y} \]\[ 1 = \sqrt{3} \tan y \]Solve for \(\tan y\):
\[ \tan y = \frac{1}{\sqrt{3}} \]This implies \(y = 30^\circ\).
Substitute \(y = 30^\circ\) back into equation (1) to find \(x\):
\[ x + 30^\circ = 75^\circ \]\[ x = 75^\circ - 30^\circ \]\[ x = 45^\circ \]Now, find the ratio \(x:y\):
\[ x : y = 45^\circ : 30^\circ \]Divide both sides by the greatest common divisor, which is 15:
\[ x : y = \frac{45}{15} : \frac{30}{15} \]\[ x : y = 3 : 2 \]Correct Option: D) 3 : 2
The LCM of \( \frac{3}{2} \),\( \frac{4}{6} \), \( \frac{5}{8} \) and \( \frac{6}{10} \) is:
To find the LCM (Least Common Multiple) of fractions, we use the formula: LCM of fractions = (LCM of numerators) / (HCF of denominators). First, we need to simplify the given fractions if possible before applying the formula.
The given fractions are \( \frac{3}{2} \), \( \frac{4}{6} \), \( \frac{5}{8} \), and \( \frac{6}{10} \).
Simplify \( \frac{4}{6} \) by dividing the numerator and denominator by 2: \[\frac{4 \div 2}{6 \div 2} = \frac{2}{3}\].
Simplify \( \frac{6}{10} \) by dividing the numerator and denominator by 2: \[\frac{6 \div 2}{10 \div 2} = \frac{3}{5}\].
The fractions become: \( \frac{3}{2} \), \( \frac{2}{3} \), \( \frac{5}{8} \), and \( \frac{3}{5} \).
The numerators are 3, 2, 5, and 3.
LCM(3, 2, 5, 3):
LCM is the product of the highest powers of all prime factors involved.
\[ \text{LCM}(3, 2, 5, 3) = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30 \]The denominators are 2, 3, 8, and 5.
HCF(2, 3, 8, 5):
HCF is the product of the lowest powers of common prime factors. In this case, there are no common prime factors other than 1.
\[ \text{HCF}(2, 3, 8, 5) = 1 \]Using the formula:
\[ \text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} = \frac{30}{1} = 30 \]B) 30 is the correct LCM of the given fractions.
The question asks for the reason behind the red color of soils commonly observed in many regions of the countryside. This is a general knowledge question related to soil science and geology.
Correct Option: A) Abundance of ferric oxides is the primary reason for the red color of soils. The oxidation of iron-bearing minerals in the parent rock leads to the formation of ferric oxides like hematite, which imparts the distinctive red hue.
To solve this problem, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This means we perform operations in the following order: Brackets (Parentheses), Orders (Exponents), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). We will work from the innermost brackets outwards.
Thus, the value of the given expression is 7.
A) 7 — The step-by-step calculation following the BODMAS/PEMDAS rule correctly leads to the result 7.
This problem can be solved using the concept of work and time, where the total work done is constant. The total work is the product of the number of men and the number of days taken to complete the task. We will set up an equation based on the two given scenarios and solve for the original number of men.
The total work (\(W\)) is constant. We know that Work = Number of Men \(\times\) Number of Days.
So, for the first scenario:
\[ W = M \times D_1 = M \times 70 \]For the second scenario:
\[ W = (M + 10) \times D_2 = (M + 10) \times 60 \]Since the total work is the same in both cases:
\[ M \times 70 = (M + 10) \times 60 \]Expand the equation:
\[ 70M = 60M + 600 \]Subtract \(60M\) from both sides:
\[ 70M - 60M = 600 \] \[ 10M = 600 \]Divide by 10:
\[ M = \frac{600}{10} \] \[ M = 60 \]So, there were originally 60 men.
Correct Option: C) 60
The problem asks for the least number which, when increased by 7, becomes divisible by 16, 18, 24, and 28. This means that if we add 7 to our required number, the result must be a common multiple of 16, 18, 24, and 28. To find the least such number, we first need to find the Least Common Multiple (LCM) of 16, 18, 24, and 28. Once we have the LCM, we subtract 7 from it to get our answer.
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.
So, the LCM is:
\[ \text{LCM}(16, 18, 24, 28) = 2^4 \times 3^2 \times 7^1 \] \[ = 16 \times 9 \times 7 \] \[ = 144 \times 7 \] \[ = 1008 \]Determine the required number:
The problem states that when the required number is increased by 7, it becomes divisible by 16, 18, 24, and 28. This means the required number plus 7 is equal to the LCM.
Let the required number be \(X\).
\[ X + 7 = 1008 \] \[ X = 1008 - 7 \] \[ X = 1001 \]Thus, the least number is 1001.
C) 1001 — When 1001 is increased by 7, it becomes 1008. 1008 is the LCM of 16, 18, 24, and 28, meaning it is divisible by all these numbers. Therefore, 1001 is the correct answer.