When partners invest different amounts in a business, their share of the profit is proportional to their individual investments. First, we find the ratio of their investments. Then, we calculate each person's share of the total profit based on this ratio.
Identify individual investments:
Calculate the ratio of investments:
The ratio of investments for Nikhil : Nidhi : Tejas is:
\[ 1200 : 1070 : 1310 \]We can simplify this by dividing each number by 10:
\[ 120 : 107 : 131 \]Calculate the sum of the ratio parts:
Sum of ratio parts \( = 120 + 107 + 131 = 358 \)
Determine Tejas's share of the profit:
Tejas's share of the profit is his ratio part divided by the total sum of ratio parts, multiplied by the total profit.
Total profit = ₹1,790
\[ \text{Tejas's share} = \frac{\text{Tejas's ratio part}}{\text{Sum of ratio parts}} \times \text{Total profit} \] \[ \text{Tejas's share} = \frac{131}{358} \times 1790 \]Notice that \( 1790 = 5 \times 358 \). So, we can simplify the calculation:
\[ \text{Tejas's share} = 131 \times \frac{1790}{358} \] \[ \text{Tejas's share} = 131 \times 5 \] \[ \text{Tejas's share} = 655 \]Therefore, Tejas's share of the profit is ₹655.
Correct Option: D) ₹655
If X = \(\frac{(1+\cos\theta)(2-2\cos\theta)}{(2+2\sin\theta)(1-\sin\theta)}\), then the value of X + \( \frac{1}{X} \) + 2 is:
The problem asks us to simplify the given expression for \(X\) and then find the value of \(X + \frac{1}{X} + 2\). We will use trigonometric identities to simplify \(X\) first, and then substitute the simplified form into the final expression.
Simplify the expression for \(X\):
Given:
\[ X = \frac{(1+\cos\theta)(2-2\cos\theta)}{(2+2\sin\theta)(1-\sin\theta)} \]Factor out common terms from the numerator and denominator:
\[ X = \frac{(1+\cos\theta) \cdot 2(1-\cos\theta)}{2(1+\sin\theta)(1-\sin\theta)} \]Cancel out the common factor of 2:
\[ X = \frac{(1+\cos\theta)(1-\cos\theta)}{(1+\sin\theta)(1-\sin\theta)} \]Use the identity \((a+b)(a-b) = a^2 - b^2\):
\[ X = \frac{1^2 - \cos^2\theta}{1^2 - \sin^2\theta} \]Apply the Pythagorean identities \(\sin^2\theta + \cos^2\theta = 1\), which implies \(1 - \cos^2\theta = \sin^2\theta\) and \(1 - \sin^2\theta = \cos^2\theta\):
\[ X = \frac{\sin^2\theta}{\cos^2\theta} \]Recall that \(\frac{\sin\theta}{\cos\theta} = \tan\theta\):
\[ X = \tan^2\theta \]Calculate \(X + \frac{1}{X} + 2\):
Substitute \(X = \tan^2\theta\) into the expression:
\[ X + \frac{1}{X} + 2 = \tan^2\theta + \frac{1}{\tan^2\theta} + 2 \]Recall that \(\frac{1}{\tan\theta} = \cot\theta\):
\[ X + \frac{1}{X} + 2 = \tan^2\theta + \cot^2\theta + 2 \]This expression can be recognized as the expansion of \( (a+b)^2 = a^2 + b^2 + 2ab \). Here, \(a = \tan\theta\) and \(b = \cot\theta\). Since \(\tan\theta \cdot \cot\theta = 1\), we have \(2ab = 2 \tan\theta \cot\theta = 2(1) = 2\).
So, we can write:
\[ \tan^2\theta + \cot^2\theta + 2 = (\tan\theta + \cot\theta)^2 \]Further simplify the expression:
Convert \(\tan\theta\) and \(\cot\theta\) back to \(\sin\theta\) and \(\cos\theta\):
\[ (\tan\theta + \cot\theta)^2 = \left(\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta}\right)^2 \]Find a common denominator inside the parenthesis:
\[ = \left(\frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta}\right)^2 \]Use the identity \(\sin^2\theta + \cos^2\theta = 1\):
\[ = \left(\frac{1}{\sin\theta\cos\theta}\right)^2 \]Separate the terms:
\[ = \frac{1}{\sin^2\theta\cos^2\theta} \]Recall that \(\frac{1}{\sin\theta} = \csc\theta\) and \(\frac{1}{\cos\theta} = \sec\theta\):
\[ = \csc^2\theta \sec^2\theta \]Correct Option: D) \(\csc^2\theta\sec^2\theta\)
This problem involves compound interest. We are given the time it takes for an amount to triple and need to find the time it takes for it to become 81 times itself. The key is to recognize the exponential growth pattern of compound interest and express 81 as a power of 3.
Let the principal amount be \(P\) and the rate of compound interest be \(r\) per annum. Let \(n\) be the number of years. The formula for the amount \(A\) after \(n\) years at compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]According to the problem, the sum triples itself in 17 years. So, when \(A = 3P\), \(n = 17\).
\[ 3P = P \left(1 + \frac{r}{100}\right)^{17} \]Dividing both sides by \(P\), we get:
\[ 3 = \left(1 + \frac{r}{100}\right)^{17} \quad \text{(Equation 1)} \]Now, we need to find the number of years, let's call it \(N\), in which the amount will be 81 times the principal. So, when \(A = 81P\), \(n = N\).
\[ 81P = P \left(1 + \frac{r}{100}\right)^{N} \]Dividing both sides by \(P\), we get:
\[ 81 = \left(1 + \frac{r}{100}\right)^{N} \quad \text{(Equation 2)} \]We know that \(81 = 3^4\). Substitute this into Equation 2:
\[ 3^4 = \left(1 + \frac{r}{100}\right)^{N} \]From Equation 1, we know that \(3 = \left(1 + \frac{r}{100}\right)^{17}\). Substitute this expression for 3 into the equation from Step 6:
\[ \left(\left(1 + \frac{r}{100}\right)^{17}\right)^4 = \left(1 + \frac{r}{100}\right)^{N} \]Using the exponent rule \((a^b)^c = a^{bc}\), we simplify the left side:
\[ \left(1 + \frac{r}{100}\right)^{17 \times 4} = \left(1 + \frac{r}{100}\right)^{N} \]\[ \left(1 + \frac{r}{100}\right)^{68} = \left(1 + \frac{r}{100}\right)^{N} \]Since the bases are the same, the exponents must be equal:
\[ N = 68 \]Thus, it will take 68 years for the sum to amount to 81 times itself.
A) 68 years is the correct answer, as derived from the compound interest formula and the relationship between the given multiples.
The value of 8² + \(\sqrt{3^2}\) − 2\(\sqrt{36}\) − 11 is
To solve this problem, we need to evaluate the given expression by performing the operations in the correct order, following the order of operations (PEMDAS/BODMAS). This involves calculating squares, square roots, multiplication, addition, and subtraction.
The first term is \(8^2\).
\[ 8^2 = 8 \times 8 = 64 \]The second term is \(\sqrt{3^2}\). We know that \(\sqrt{a^2} = a\) for non-negative \(a\).
\[ \sqrt{3^2} = \sqrt{9} = 3 \]The third term is \(2\sqrt{36}\). First, find the square root of 36.
\[ \sqrt{36} = 6 \]Now, multiply by 2:
\[ 2\sqrt{36} = 2 \times 6 = 12 \]The original expression is \(8^2 + \sqrt{3^2} - 2\sqrt{36} - 11\).
Substitute the values we found:
\[ 64 + 3 - 12 - 11 \]D) 44 — The step-by-step calculation shows that the value of the expression is 44.
The problem describes a discount scheme where a customer buys a certain number of articles and gets some articles free. To find the discount percentage, we need to determine the total number of articles the customer receives and the number of articles they actually pay for. The discount is calculated based on the free articles as a percentage of the total articles received.
Identify the number of articles bought and free:
The shopkeeper offers 'Buy 20 and get 5 free'.
Number of articles bought = 20
Number of free articles = 5
Calculate the total number of articles received by the customer:
Total articles received = Articles bought + Free articles
Total articles received = \(20 + 5 = 25\) articles
Determine the number of articles the customer pays for:
The customer pays for the 20 articles they buy. The 5 articles are given as a discount.
Calculate the discount percentage:
The discount percentage is calculated as the number of free articles divided by the total number of articles received, multiplied by 100.
Correct Option: C) 20%
Find the value of x satisfying: 5(5x² − 5) − 5(5x² + 7x − 2) = 14
−\( \frac{29}{35} \)
−\( \frac{38}{35} \)
−\( \frac{39}{35} \)
−\( \frac{24}{35} \)
The problem requires solving a linear equation for the variable \(x\). The first step is to simplify the given equation by distributing the constants into the parentheses and combining like terms. After simplification, the equation will be in the form \(ax = b\), which can then be solved for \(x\).
Write down the given equation:
\[ 5(5x^2 - 5) - 5(5x^2 + 7x - 2) = 14 \]Distribute the \(5\) into the first set of parentheses and \(-5\) into the second set of parentheses:
\[ (5 \times 5x^2) - (5 \times 5) - (5 \times 5x^2) - (5 \times 7x) + (5 \times 2) = 14 \] \[ 25x^2 - 25 - 25x^2 - 35x + 10 = 14 \]Combine the like terms. Notice that the \(25x^2\) terms cancel each other out:
\[ (25x^2 - 25x^2) - 35x - 25 + 10 = 14 \] \[ 0 - 35x - 15 = 14 \] \[ -35x - 15 = 14 \]Isolate the term with \(x\) by adding \(15\) to both sides of the equation:
\[ -35x = 14 + 15 \] \[ -35x = 29 \]Solve for \(x\) by dividing both sides by \(-35\):
\[ x = \frac{29}{-35} \] \[ x = -\frac{29}{35} \]Thus, the value of \(x\) that satisfies the given equation is \(-\frac{29}{35}\).
To find the LCM of decimal numbers, we first convert them into fractions. Then, we use the formula for the LCM of fractions: \[\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}\]. Finally, we convert the resulting fraction back to a decimal.
The given numbers are \(2.8\) and \(0.217\).
\[ 2.8 = \frac{28}{10} = \frac{14}{5} \]\[ 0.217 = \frac{217}{1000} \]For the fractions \(\frac{14}{5}\) and \(\frac{217}{1000}\):
We need to find \(\text{LCM}(14, 217)\).
First, find the prime factorization of each number:
\[ 14 = 2 \times 7 \]\[ 217 = 7 \times 31 \]\[ \text{LCM}(14, 217) = 2 \times 7 \times 31 = 14 \times 31 = 434 \]We need to find \(\text{HCF}(5, 1000)\).
Prime factorization:
\[ 5 = 5 \]\[ 1000 = 2^3 \times 5^3 \]\[ \text{HCF}(5, 1000) = 5 \]D) 86.8 is the correct LCM of 2.8 and 0.217.
This problem can be solved by setting up a system of two linear equations with two variables. We will assign variables to the father's age and the son's age, translate the given conditions into equations, and then solve the system of equations simultaneously.
Define variables:
Let \(F\) be the father's age in years.
Let \(S\) be the son's age in years.
Formulate the first equation:
The first condition states: "If 4 times a son's age is added to his father's age, the sum is 100 years."
This translates to:
Formulate the second equation:
The second condition states: "If 2.5 times the father's age is added to the son's age, the sum is 97 years."
This translates to:
Solve the system of equations:
We have the system:
\[
\begin{cases}
F + 4S = 100 \quad (1)\\
2.5F + S = 97 \quad (2)
\end{cases}
\]
From Equation 2, we can express \(S\) in terms of \(F\):
\[ S = 97 - 2.5F \quad \text{(Equation 3)} \]
Substitute Equation 3 into Equation 1:
\[ F + 4(97 - 2.5F) = 100 \]
\[ F + 388 - 10F = 100 \]
\[ 388 - 9F = 100 \]
\[ 9F = 388 - 100 \]
\[ 9F = 288 \]
\[ F = \frac{288}{9} \]
\[ F = 32 \]
Find the son's age (optional, but good for verification):
Substitute \(F = 32\) into Equation 3:
\[ S = 97 - 2.5(32) \]
\[ S = 97 - 80 \]
\[ S = 17 \]
Verify the solution:
Using \(F=32\) and \(S=17\):
Equation 1: \(32 + 4(17) = 32 + 68 = 100\) (Correct)
Equation 2: \(2.5(32) + 17 = 80 + 17 = 97\) (Correct)
The father's age is 32 years.
Correct Option: B) 32
This problem involves work and time, where the efficiency of men and women needs to be related first. We will calculate the total work in terms of "man-days" or "woman-days" and then find the combined efficiency of 3 women and 2 men to determine the time taken to complete the work.
Calculate the total work from men's data:
4 men can complete the work in 12 days.
Total work = \(\text{Number of men} \times \text{Number of days}\)
\[ \text{Total Work} = 4 \text{ men} \times 12 \text{ days} = 48 \text{ man-days} \]Calculate the total work from women's data:
8 women can complete the work in 15 days.
\[ \text{Total Work} = 8 \text{ women} \times 15 \text{ days} = 120 \text{ woman-days} \]Relate the efficiency of men and women:
Since the total work is the same in both cases:
\[ 48 \text{ man-days} = 120 \text{ woman-days} \]To find the relationship between 1 man's work and 1 woman's work:
\[ 1 \text{ man-day} = \frac{120}{48} \text{ woman-days} \] \[ 1 \text{ man-day} = \frac{5}{2} \text{ woman-days} \]This means 1 man's work is equivalent to the work of 2.5 women.
Alternatively, we can express woman's work in terms of man's work:
\[ 1 \text{ woman-day} = \frac{48}{120} \text{ man-days} \] \[ 1 \text{ woman-day} = \frac{2}{5} \text{ man-days} \]Calculate the combined work rate of 3 women and 2 men:
Let's convert everything to "woman-days".
2 men's work = \[2 \times \frac{5}{2} \text{ woman-days} = 5 \text{ woman-days}\]
So, 3 women and 2 men are equivalent to:
\[ 3 \text{ women} + 5 \text{ women} = 8 \text{ women} \]Alternatively, convert everything to "man-days".
3 women's work = \[3 \times \frac{2}{5} \text{ man-days} = \frac{6}{5} \text{ man-days}\]
So, 3 women and 2 men are equivalent to:
\[ \frac{6}{5} \text{ men} + 2 \text{ men} = \frac{6+10}{5} \text{ men} = \frac{16}{5} \text{ men} \]Calculate the number of days required:
Using the "woman-days" equivalent:
Total work = 120 woman-days.
Combined team = 8 women.
\[ \text{Days} = \frac{\text{Total Work}}{\text{Combined team efficiency}} = \frac{120 \text{ woman-days}}{8 \text{ women}} = 15 \text{ days} \]Using the "man-days" equivalent:
Total work = 48 man-days.
Combined team = \(\frac{16}{5}\) men.
\[ \text{Days} = \frac{\text{Total Work}}{\text{Combined team efficiency}} = \frac{48 \text{ man-days}}{\frac{16}{5} \text{ men}} = 48 \times \frac{5}{16} \text{ days} = 3 \times 5 \text{ days} = 15 \text{ days} \]Correct Option: B) 15 days
To find the number of sides of a regular polygon given its interior angle, we can use the formula relating the interior angle to the number of sides, or first find the exterior angle and then use the exterior angle formula.
Let \(n\) be the number of sides of the regular polygon.
The measure of each interior angle of a regular polygon with \(n\) sides is given by the formula:
\[ \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} \]We are given that each interior angle measures \(162^\circ\). So, we can set up the equation:
\[ 162^\circ = \frac{(n-2) \times 180^\circ}{n} \]Now, we solve for \(n\):
\[ 162n = (n-2) \times 180 \]\[ 162n = 180n - 360 \]Rearrange the terms to solve for \(n\):
\[ 360 = 180n - 162n \]\[ 360 = 18n \]Divide by 18:
\[ n = \frac{360}{18} \]\[ n = 20 \]Alternatively, we can use the exterior angle. The sum of an interior angle and its corresponding exterior angle is \(180^\circ\).
\[ \text{Exterior Angle} = 180^\circ - \text{Interior Angle} \]\[ \text{Exterior Angle} = 180^\circ - 162^\circ \]\[ \text{Exterior Angle} = 18^\circ \]For any regular polygon, the sum of its exterior angles is \(360^\circ\). If there are \(n\) sides, then each exterior angle is:
\[ \text{Exterior Angle} = \frac{360^\circ}{n} \]Using the calculated exterior angle:
\[ 18^\circ = \frac{360^\circ}{n} \]Solve for \(n\):
\[ n = \frac{360}{18} \]\[ n = 20 \]C) The number of sides of the regular polygon is 20.