We need to find two consecutive even natural numbers whose squares sum to 1924. We can represent these numbers algebraically, form an equation, and solve for the numbers. Once the numbers are found, we can calculate their sum.
Correct Option: C) 62
To find the measure of each interior angle of a regular polygon, we can use the formula that relates the number of sides to the sum of interior angles, and then divide by the number of sides since all angles in a regular polygon are equal.
Identify the number of sides: The problem states that the regular polygon has 15 sides. Let \(n\) be the number of sides.
\[ n = 15 \]Calculate the sum of interior angles: The formula for the sum of the interior angles of any polygon with \(n\) sides is given by:
\[ \text{Sum of interior angles} = (n - 2) \times 180^{\circ} \]Substitute the number of sides into the formula:
\[ \text{Sum of interior angles} = (15 - 2) \times 180^{\circ} \]\[ \text{Sum of interior angles} = 13 \times 180^{\circ} \]\[ \text{Sum of interior angles} = 2340^{\circ} \]Calculate the measure of each interior angle: For a regular polygon, all interior angles are equal. Therefore, to find the measure of each interior angle, divide the sum of interior angles by the number of sides:
\[ \text{Each interior angle} = \frac{\text{Sum of interior angles}}{n} \]\[ \text{Each interior angle} = \frac{2340^{\circ}}{15} \]\[ \text{Each interior angle} = 156^{\circ} \]B) 156° is the correct measure for each interior angle of a regular polygon with 15 sides.
This problem involves calculating the original sales amount given a series of percentage increases over two years. We can work backward from the current sales amount or set up an equation to find the initial sales.
Correct Option: C) ₹1,25,000
To evaluate the given expression, we must follow the order of operations, commonly known as BODMAS/PEMDAS. This acronym stands for Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).
The given expression is:
\[ (-9) - (-60) \div (-12) + (-2) \times 8 \]First, perform the Division operation:
\[ (-60) \div (-12) \]When dividing two negative numbers, the result is positive.
\[ (-60) \div (-12) = 5 \]Substitute this result back into the expression:
\[ (-9) - 5 + (-2) \times 8 \]Next, perform the Multiplication operation:
\[ (-2) \times 8 \]When multiplying a negative number by a positive number, the result is negative.
\[ (-2) \times 8 = -16 \]Substitute this result back into the expression:
\[ (-9) - 5 + (-16) \]Now, perform the Addition and Subtraction operations from left to right. First, subtraction:
\[ (-9) - 5 \]Subtracting a positive number is the same as adding a negative number.
\[ -9 - 5 = -14 \]Finally, perform the addition:
\[ -14 + (-16) \]Adding a negative number is the same as subtracting a positive number.
\[ -14 - 16 = -30 \]Thus, the evaluated expression is \(-30\).
A) \(-30\) is the correct answer, as calculated by following the BODMAS/PEMDAS rule.
Anjani can do a certain piece of work in 30 days. Anjani and Khushbu together can do the same work in 16 days, and Anjani, Khushbu and Sushmita can do the same work together in 15 days. In how many days can Anjani and Sushmita do the same work?
\( \frac{85}{4} \)
\( \frac{80}{3} \)
\( \frac{73}{5} \)
\( \frac{81}{3} \)
This problem involves work and time. We will calculate the individual work rates of Anjani, Khushbu, and Sushmita. Then, we will combine the work rates of Anjani and Sushmita to find the time they take to complete the work together.
Anjani can do the work in 30 days.
\[ \text{Anjani's 1-day work} = \frac{1}{30} \]Anjani and Khushbu together can do the work in 16 days.
\[ \text{(Anjani + Khushbu)'s 1-day work} = \frac{1}{16} \]Subtract Anjani's one-day work from their combined one-day work.
\[ \text{Khushbu's 1-day work} = \frac{1}{16} - \frac{1}{30} = \frac{15 - 8}{240} = \frac{7}{240} \]Anjani, Khushbu, and Sushmita together can do the work in 15 days.
\[ \text{(Anjani + Khushbu + Sushmita)'s 1-day work} = \frac{1}{15} \]Subtract the combined one-day work of Anjani and Khushbu from the combined one-day work of all three.
\[ \text{Sushmita's 1-day work} = \frac{1}{15} - \frac{1}{16} = \frac{16 - 15}{240} = \frac{1}{240} \]Add Anjani's one-day work and Sushmita's one-day work.
\[ \text{(Anjani + Sushmita)'s 1-day work} = \frac{1}{30} + \frac{1}{240} = \frac{8 + 1}{240} = \frac{9}{240} = \frac{3}{80} \]The number of days is the reciprocal of their combined one-day work.
\[ \text{Days} = \frac{1}{\frac{3}{80}} = \frac{80}{3} \text{ days} \]Correct Option: B) \( \frac{80}{3} \)
To solve this problem, we need to calculate the distance covered and time taken in the initial two phases of the journey. Then, we can find the remaining distance and remaining time. Finally, we can determine the required speed for the last part of the journey to reach the destination on time.
Correct Option: A) 52 km/hr
This problem involves compound interest. We need to find the time it takes for a sum of money to become 9 times itself, given that it triples in 5 years. The key property of compound interest is that if a sum becomes \(x\) times in \(T\) years, it will become \(x^n\) times in \(n \times T\) years.
Let the principal amount be \(P\). After 5 years, the amount becomes \(3P\).
Using the compound interest formula, \(A = P(1 + r/100)^t\), where \(A\) is the amount, \(P\) is the principal, \(r\) is the rate of interest, and \(t\) is the time in years.
So, we have:
\[ 3P = P \left(1 + \frac{r}{100}\right)^5 \]Dividing by \(P\):
\[ 3 = \left(1 + \frac{r}{100}\right)^5 \quad \text{(Equation 1)} \]We want to find the time \(T\) when the amount becomes \(9P\).
\[ 9P = P \left(1 + \frac{r}{100}\right)^T \]Dividing by \(P\):
\[ 9 = \left(1 + \frac{r}{100}\right)^T \quad \text{(Equation 2)} \]We know that \(9 = 3^2\). Substitute this into Equation 2:
\[ 3^2 = \left(1 + \frac{r}{100}\right)^T \]From Equation 1, we know that \(3 = \left(1 + \frac{r}{100}\right)^5\). Substitute this into the previous equation:
\[ \left(\left(1 + \frac{r}{100}\right)^5\right)^2 = \left(1 + \frac{r}{100}\right)^T \]Using the power rule \((a^m)^n = a^{mn}\):
\[ \left(1 + \frac{r}{100}\right)^{5 \times 2} = \left(1 + \frac{r}{100}\right)^T \] \[ \left(1 + \frac{r}{100}\right)^{10} = \left(1 + \frac{r}{100}\right)^T \]Comparing the exponents, we get:
\[ T = 10 \text{ years} \]C) 10 years
We need to find the list price of the fan given the profit, the percentage by which the list price is above the cost price, and the discount percentage. We can set up equations based on the given percentages and profit to find the cost price, selling price, and then the list price.
Let the Cost Price (CP) of the fan be \(P\).
The shopkeeper lists the price of the fan at 36% above its cost price. So, the List Price (LP) is:
\[ \text{LP} = \text{CP} + 0.36 \times \text{CP} = 1.36 \times \text{CP} \]\[ \text{LP} = 1.36P \]A 25% discount is offered on the list price. So, the Selling Price (SP) is:
\[ \text{SP} = \text{LP} - 0.25 \times \text{LP} = (1 - 0.25) \times \text{LP} = 0.75 \times \text{LP} \]Substitute the expression for LP from step 2 into the equation for SP:
\[ \text{SP} = 0.75 \times (1.36P) \]\[ \text{SP} = 1.02P \]The profit is given as ₹171. Profit is the difference between the Selling Price and the Cost Price:
\[ \text{Profit} = \text{SP} - \text{CP} \]\[ 171 = 1.02P - P \]\[ 171 = 0.02P \]Now, calculate the Cost Price (P):
\[ P = \frac{171}{0.02} \]\[ P = \frac{171 \times 100}{2} \]\[ P = 171 \times 50 \]\[ P = 8550 \]So, the Cost Price is ₹8550.Finally, calculate the List Price (LP) using the relationship from step 2:
\[ \text{LP} = 1.36 \times P \]\[ \text{LP} = 1.36 \times 8550 \]\[ \text{LP} = 11628 \]The list price of the fan is ₹11,628.
C) 11,628
To determine which of the given numbers divides 78,89,73,289, we can use divisibility rules or perform actual division. For larger numbers, divisibility rules are efficient. If a rule doesn't apply directly, or if the number is prime, direct division might be necessary.
Correct Option: D) 13
The problem involves dividing a total sum of money among three individuals (A, B, and C) based on a given ratio. The relationship between their shares is provided as an equality: 7 times A's share = 4 times B's share = 2 times C's share. We need to find the individual share of A.
Set up the given relationship:
Let the shares of A, B, and C be \(S_A\), \(S_B\), and \(S_C\) respectively. The problem states:
Introduce a common constant:
To find the ratio of their shares, we can set this common value equal to a constant, say \(k\):
Express individual shares in terms of \(k\):
From the above equation, we can write the shares as:
Find the ratio of their shares:
The ratio \(S_A : S_B : S_C\) is:
Calculate the total parts in the ratio:
The sum of the ratio parts is \(4 + 7 + 14 = 25\).
Determine A's share:
The total amount of money is ₹15,500. A's share is 4 parts out of a total of 25 parts.
Thus, the share of A is ₹2,480.
C) ₹2,480 is the correct share for A, calculated based on the given ratios.