12 men finished \( \frac{1}{4} \) part of whole work in 6 days. Find the number of additional men required to complete the job in next 6 days.
This problem involves the concept of work and time, specifically using the formula \(M_1 D_1 W_2 = M_2 D_2 W_1\), where \(M\) is the number of men, \(D\) is the number of days, and \(W\) is the amount of work. We will first calculate the total work units and then determine how many additional men are needed to complete the remaining work in the given time.
Calculate the work done by the initial group:
Given that 12 men finished \(\frac{1}{4}\) part of the whole work in 6 days.
Let the total work be \(W_{\text{total}}\). The work done in the first phase is \(W_1 = \frac{1}{4} W_{\text{total}}\).
The remaining work is \[W_{\text{remaining}} = W_{\text{total}} - \frac{1}{4} W_{\text{total}} = \frac{3}{4} W_{\text{total}}\].
Apply the work-time formula:
The fundamental principle for work-time problems is that the total work units are constant. We can use the formula: \[ M_1 D_1 W_2 = M_2 D_2 W_1 \] Where: \(M_1 = 12\) men \(D_1 = 6\) days \(W_1 = \frac{1}{4}\) (part of work done) We need to find the total number of men (\(M_2\)) required to complete the remaining work (\(W_2 = \frac{3}{4}\)) in \(D_2 = 6\) days.
Substitute the values into the formula:
\[ 12 \times 6 \times \frac{3}{4} = M_2 \times 6 \times \frac{1}{4} \]Solve for \(M_2\):
Simplify the equation:
\[ 12 \times 6 \times \frac{3}{4} = M_2 \times 6 \times \frac{1}{4} \]We can cancel out common terms. Cancel \(6\) from both sides:
\[ 12 \times \frac{3}{4} = M_2 \times \frac{1}{4} \]Cancel \(\frac{1}{4}\) from both sides (or multiply both sides by 4):
\[ 12 \times 3 = M_2 \] \[ M_2 = 36 \]So, 36 men are required to complete the remaining \(\frac{3}{4}\) of the work in 6 days.
Calculate the additional men required:
The number of additional men required is the total men needed minus the men already working.
Additional men = \(M_2 - M_1 = 36 - 12 = 24\).
Correct Option: D) 24
To solve this blood relation problem, we will construct a family tree based on the given statements. We will denote males with a '+' sign and females with a '-' sign, and use lines to indicate relationships (e.g., horizontal for siblings, vertical for parent-child, double horizontal for spouse).
Correct Option: A) Maternal Aunt. D is the sister of C, and C is the mother of F. Thus, D is F's maternal aunt.
The question asks to identify which of the given options is NOT one of the twelve traditional zodiac signs. This requires knowledge of the standard astrological zodiac.
A) Ophiuchus — Ophiuchus is a constellation that lies along the ecliptic, the apparent path of the Sun across the sky. While it is sometimes referred to as the "13th zodiac sign" in some modern astronomical contexts due to the precession of the equinoxes, it is not part of the traditional twelve astrological zodiac signs that have been used for millennia.
The problem requires us to first unscramble the jumbled letters in each option to form meaningful words. After forming the words, we need to identify the word that is different from the others based on a common category or characteristic shared by the rest.
Correct Option: C) TORRAC
Rearranging TORRAC gives CARROT, which is a vegetable. The other options form names of flowers.
The question asks to identify the field to which taxonomy is fundamentally related. Taxonomy is a core concept in biology, dealing with the classification of living organisms. Understanding its definition will directly lead to the correct answer.
A) Biodiversity. Taxonomy is the science of classifying and naming organisms, which is fundamental to understanding and managing the Earth's biodiversity.
The problem involves calculating the original cost price of a memory card given its selling price (including sales tax) and the seller's profit percentage. We will first find the price before sales tax, then use the profit percentage to determine the cost price.
Calculate the price before sales tax:
Let the price of the memory card before sales tax be \(P\). The sales tax is 25%.
The price including sales tax is Rs. 625.
\[ P + 0.25P = 625 \] \[ 1.25P = 625 \] \[ P = \frac{625}{1.25} \] \[ P = \frac{625}{\frac{5}{4}} \] \[ P = 625 \times \frac{4}{5} \] \[ P = 125 \times 4 \] \[ P = 500 \]So, the selling price of the memory card before sales tax is Rs. 500.
Calculate the cost price:
The seller made a gain (profit) of 50% on the cost price. Let the cost price be \(C\).
The selling price (before tax, which is the price the seller received) is Rs. 500.
\[ \text{Selling Price} = \text{Cost Price} + \text{Profit} \] \[ 500 = C + 0.50C \] \[ 500 = 1.50C \] \[ C = \frac{500}{1.50} \] \[ C = \frac{500}{\frac{3}{2}} \] \[ C = 500 \times \frac{2}{3} \] \[ C = \frac{1000}{3} \] \[ C = 333.333... \]Rounding to the nearest whole number, the cost price is Rs. 333.
A) Rs. 333 is the calculated cost price of the memory card.
The shadow of a standing tower of height 25\(\sqrt{3}\) m is found to be 50 m longer when the sun's elevation changes from 60° to X°. Find the measure of X?
This problem involves trigonometry, specifically the tangent function, to relate the height of the tower to the length of its shadow at different angles of elevation of the sun. We will set up two right-angled triangles and use the given information to find the unknown angle X.
Let the height of the tower be \(h\). Given \(h = 25\sqrt{3}\) m.
Let the initial length of the shadow be \(L_1\) when the sun's elevation is \(60^\circ\).
In the right-angled triangle formed by the tower, its shadow, and the line of sight to the sun:
\[ \tan(60^\circ) = \frac{\text{Height}}{\text{Shadow Length}} = \frac{h}{L_1} \]We know that \(\tan(60^\circ) = \sqrt{3}\). So,
\[ \sqrt{3} = \frac{25\sqrt{3}}{L_1} \]Solving for \(L_1\):
\[ L_1 = \frac{25\sqrt{3}}{\sqrt{3}} = 25 \text{ m} \]When the sun's elevation changes to \(X^\circ\), the shadow is 50 m longer. So, the new length of the shadow, \(L_2\), is:
\[ L_2 = L_1 + 50 = 25 + 50 = 75 \text{ m} \]Now, consider the new right-angled triangle with the same tower height \(h\) and the new shadow length \(L_2\), with the angle of elevation \(X^\circ\):
\[ \tan(X^\circ) = \frac{\text{Height}}{\text{New Shadow Length}} = \frac{h}{L_2} \]Substitute the values of \(h\) and \(L_2\):
\[ \tan(X^\circ) = \frac{25\sqrt{3}}{75} \] \[ \tan(X^\circ) = \frac{\sqrt{3}}{3} \]We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), which is equivalent to \(\frac{\sqrt{3}}{3}\) (by multiplying numerator and denominator by \(\sqrt{3}\)).
Therefore,
\[ X^\circ = 30^\circ \]Correct Option: B) 30°
This is a problem involving work and time, where the total work is constant. We will calculate the total work units, the work done in the initial days, and then determine how many people are needed to complete the remaining work in the new timeframe. Finally, we'll find out how many people should leave.
Calculate Total Work:
If 30 persons can finish a job in 20 days, the total work can be expressed in "person-days".
\[ \text{Total Work} = \text{Number of Persons} \times \text{Number of Days} \]\[ \text{Total Work} = 30 \text{ persons} \times 20 \text{ days} = 600 \text{ person-days} \]Calculate Work Done in the First 6 Days:
For the first 6 days, all 30 persons worked.
\[ \text{Work Done} = 30 \text{ persons} \times 6 \text{ days} = 180 \text{ person-days} \]Calculate Remaining Work:
Subtract the work done from the total work.
\[ \text{Remaining Work} = \text{Total Work} - \text{Work Done} \]\[ \text{Remaining Work} = 600 \text{ person-days} - 180 \text{ person-days} = 420 \text{ person-days} \]Calculate Remaining Days:
The work is to be completed in a total of 26 days. 6 days have already passed.
\[ \text{Remaining Days} = \text{Total Days} - \text{Days Passed} \]\[ \text{Remaining Days} = 26 \text{ days} - 6 \text{ days} = 20 \text{ days} \]Calculate Number of Persons Required for Remaining Work:
To complete 420 person-days of work in 20 remaining days, we need a certain number of persons.
\[ \text{Persons Required} = \frac{\text{Remaining Work}}{\text{Remaining Days}} \]\[ \text{Persons Required} = \frac{420 \text{ person-days}}{20 \text{ days}} = 21 \text{ persons} \]Calculate Number of Persons Who Should Leave:
Initially, there were 30 persons. Now, only 21 persons are required.
\[ \text{Persons to Leave} = \text{Initial Persons} - \text{Persons Required} \]\[ \text{Persons to Leave} = 30 - 21 = 9 \text{ persons} \]Correct Option: A) 9
The calculation shows that 9 persons should leave the job for the work to be completed in a total of 26 days.
An investor invested \( \frac{1}{2} \) of his capital at 5%, \( \frac{1}{4} \) at 10% and rest at 8%. If his income after 2 years is Rs. 2,800, find the capital amount.
To find the total capital, we will first calculate the simple interest earned from each part of the investment. Since the total income for 2 years is given, we can set up an equation where the sum of interests from all parts equals the total income. We will then solve for the total capital.
Define the total capital:
Let the total capital be \(P\).
Calculate the amount invested in each part:
Calculate the simple interest for each part for 2 years:
The formula for simple interest is \( \text{SI} = \frac{P \times R \times T}{100} \), where \(P\) is principal, \(R\) is rate, and \(T\) is time.
Calculate the total interest:
The total income after 2 years is Rs. 2,800. So, the sum of interests from all parts must equal this amount.
Solve for \(P\):
Find a common denominator for 10 and 25, which is 50.
Thus, the capital amount is Rs. 20,000.
Correct Option: C) Rs. 20,000
We are given two poles of different heights and the distance between them is unknown. The angles of elevation of the top of each pole from the base of the other are complementary. We need to find the distance between the poles. This problem can be solved using trigonometry, specifically the tangent function, and the property of complementary angles.
Let the heights of the two poles be \(h_1 = 16 \text{ m}\) and \(h_2 = 9 \text{ m}\).
Let the distance between the two poles be \(X \text{ m}\).
Let the angle of elevation of the top of the first pole (height \(h_1\)) from the bottom of the second pole be \(\theta\).
Let the angle of elevation of the top of the second pole (height \(h_2\)) from the bottom of the first pole be \(\phi\).
According to the problem, these two angles are complementary, which means \(\theta + \phi = 90^\circ\). This implies \(\phi = 90^\circ - \theta\).
From the first right-angled triangle formed by the first pole, the ground, and the line of sight to its top:
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h_1}{X} = \frac{16}{X} \quad \text{(Equation 1)} \]From the second right-angled triangle formed by the second pole, the ground, and the line of sight to its top:
\[ \tan \phi = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h_2}{X} = \frac{9}{X} \quad \text{(Equation 2)} \]Substitute \(\phi = 90^\circ - \theta\) into Equation 2:
\[ \tan (90^\circ - \theta) = \frac{9}{X} \]Using the trigonometric identity \(\tan (90^\circ - \theta) = \cot \theta\):
\[ \cot \theta = \frac{9}{X} \]We know that \(\cot \theta = \frac{1}{\tan \theta}\). So,
\[ \frac{1}{\tan \theta} = \frac{9}{X} \]\[ \tan \theta = \frac{X}{9} \quad \text{(Equation 3)} \]Now, equate Equation 1 and Equation 3:
\[ \frac{16}{X} = \frac{X}{9} \]Cross-multiply to solve for \(X\):
\[ X^2 = 16 \times 9 \]\[ X^2 = 144 \]Take the square root of both sides:
\[ X = \sqrt{144} \]\[ X = 12 \text{ m} \]D) The calculated distance between the poles is \(12 \text{ m}\).