To find the profit percentage, we first need to calculate the cost price (CP) and selling price (SP) of a single pen. Then, we can use the formula for profit percentage: \[\text{Profit Percentage} = \frac{\text{Profit}}{\text{CP}} \times 100\].
Rishi bought pens at the rate of ₹180 per dozen. A dozen means 12 pens.
\[ \text{CP of 12 pens} = \text{₹180} \]\[ \text{CP of 1 pen} = \frac{\text{₹180}}{12} = \text{₹15} \]He sold them for ₹21 each.
\[ \text{SP of 1 pen} = \text{₹21} \]Profit is the difference between Selling Price and Cost Price.
\[ \text{Profit per pen} = \text{SP} - \text{CP} = \text{₹21} - \text{₹15} = \text{₹6} \]The formula for profit percentage is:
\[ \text{Profit Percentage} = \frac{\text{Profit}}{\text{CP}} \times 100 \]\[ \text{Profit Percentage} = \frac{6}{15} \times 100 \]\[ \text{Profit Percentage} = \frac{2}{5} \times 100 \]\[ \text{Profit Percentage} = 0.4 \times 100 = 40\% \]A) 40% is the correct profit percentage.
To evaluate the given expression, we must follow the order of operations, commonly known as BODMAS/PEMDAS. This acronym stands for Brackets (or Parentheses), Orders (or Exponents), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).
The given expression is:
\[ (-9) - (-60) \div (-10) + (-4) \times 7 \]First, perform Division and Multiplication from left to right.
Division: Calculate \( (-60) \div (-10) \).
\[ (-60) \div (-10) = 6 \]Multiplication: Calculate \( (-4) \times 7 \).
\[ (-4) \times 7 = -28 \]Substitute these results back into the expression:
\[ (-9) - (6) + (-28) \]Now, perform Addition and Subtraction from left to right.
First, calculate \( (-9) - 6 \).
\[ (-9) - 6 = -15 \]Next, calculate \( -15 + (-28) \).
\[ -15 + (-28) = -15 - 28 = -43 \]Thus, the final value of the expression is \(-43\).
B) -43 is the correct answer, as derived by following the order of operations.
This problem involves compound interest. We are given the principal amount, the final amount, and the time period. We need to find the annual rate of interest. The formula for compound interest will be used to solve this problem.
The formula for the amount (\(A\)) after \(n\) years with compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]To find the value of \(\left(1 + \frac{r}{100}\right)\), we need to take the cube root of 1.092727.
We know that \( (1.03)^3 = 1.03 \times 1.03 \times 1.03 = 1.0609 \times 1.03 = 1.092727 \).
So,
\[ \sqrt[3]{1.092727} = 1.03 \]\[ 1.03 = 1 + \frac{r}{100} \]Correct Option: D) The rate of interest per annum is 3%. This is derived by using the compound interest formula and solving for the rate \(r\).
To find the average of the remaining two numbers, we first need to determine the sum of these two numbers. We can do this by subtracting the sum of the first three numbers from the total sum of all five numbers. Once we have the sum of the remaining two numbers, we can divide it by 2 to find their average.
Calculate the sum of the first two numbers:
Given that the average of the first two numbers is 79.
Identify the third number:
Given that the third number is 135.
Calculate the sum of the first three numbers:
\[ \text{Sum of first three numbers} = \text{Sum of first two numbers} + \text{Third number} = 158 + 135 = 293 \]Calculate the sum of the remaining two numbers:
Given that the sum of all five numbers is 655.
Calculate the average of the remaining two numbers:
\[ \text{Average of remaining two numbers} = \frac{\text{Sum of remaining two numbers}}{\text{Number of remaining terms}} = \frac{362}{2} = 181 \]Correct Option: B) 181
This problem involves calculating the total number of registered voters based on percentages of votes polled, invalid votes, and the winning margin. We will work backward from the winning margin to find the total valid votes, then total polled votes, and finally the total registered voters.
Let the total number of registered voters be \(V\).
Votes Polled: 80% of the registered voters cast their vote.
\[ \text{Votes Polled} = 0.80 \times V \]Invalid Votes: 2% of the votes polled were found invalid.
\[ \text{Invalid Votes} = 0.02 \times (0.80 \times V) \]Valid Votes: The number of valid votes is the votes polled minus the invalid votes.
\[ \text{Valid Votes} = (0.80 \times V) - (0.02 \times 0.80 \times V) \]\[ \text{Valid Votes} = 0.80 \times V \times (1 - 0.02) \]\[ \text{Valid Votes} = 0.80 \times V \times 0.98 \]\[ \text{Valid Votes} = 0.784 \times V \]Winning Candidate's Votes: The winning candidate got 52% of the valid votes.
\[ \text{Winning Candidate's Votes} = 0.52 \times (\text{Valid Votes}) \]Losing Candidate's Votes: The losing candidate got the remaining percentage of valid votes.
\[ \text{Losing Candidate's Votes} = (100\% - 52\%) \text{ of Valid Votes} \]\[ \text{Losing Candidate's Votes} = 0.48 \times (\text{Valid Votes}) \]Winning Margin: The winning candidate won by a margin of 1568 votes.
\[ \text{Winning Margin} = \text{Winning Candidate's Votes} - \text{Losing Candidate's Votes} \]\[ 1568 = 0.52 \times (\text{Valid Votes}) - 0.48 \times (\text{Valid Votes}) \]\[ 1568 = (0.52 - 0.48) \times (\text{Valid Votes}) \]\[ 1568 = 0.04 \times (\text{Valid Votes}) \]Calculate Valid Votes:
\[ \text{Valid Votes} = \frac{1568}{0.04} \]\[ \text{Valid Votes} = 39200 \]Calculate Total Registered Voters (\(V\)): We know that \(\text{Valid Votes} = 0.784 \times V\).
\[ 39200 = 0.784 \times V \]\[ V = \frac{39200}{0.784} \]\[ V = 50000 \]Thus, the total number of registered voters was 50,000.
C) 50000
To determine the correct relationship between M and N, we use the divisibility rule for 11. A number is divisible by 11 if the alternating sum of its digits is divisible by 11. We will apply this rule to the given 6-digit number N37M01.
C) M - N = 3 — This matches our derived relationship between M and N.
If secθ + tanθ = \( \frac{3}{2} \), then the value of cosθ is:
\( \frac{12}{13} \)
\( \frac{1}{2} \)
\( \frac{8}{17} \)
\( \frac{4}{5} \)
The problem provides an equation involving \(\sec\theta\) and \(\tan\theta\) and asks for the value of \(\cos\theta\). We can use the fundamental trigonometric identity \(\sec^2\theta - \tan^2\theta = 1\) to find another relationship between \(\sec\theta\) and \(\tan\theta\). By solving the system of two equations, we can find the values of \(\sec\theta\) and \(\tan\theta\), and then determine \(\cos\theta\).
We are given the equation:
\[ \sec\theta + \tan\theta = \frac{3}{2} \quad \text{(Equation 1)} \]Recall the trigonometric identity:
\[ \sec^2\theta - \tan^2\theta = 1 \]Factor the left side using the difference of squares formula \((a^2 - b^2 = (a-b)(a+b))\):
\[ (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1 \]Substitute Equation 1 into this factored identity:
\[ (\sec\theta - \tan\theta)\left(\frac{3}{2}\right) = 1 \]Solve for \(\sec\theta - \tan\theta\):
\[ \sec\theta - \tan\theta = \frac{1}{3/2} = \frac{2}{3} \quad \text{(Equation 2)} \]Now we have a system of two linear equations with \(\sec\theta\) and \(\tan\theta\):
\[ \sec\theta + \tan\theta = \frac{3}{2} \quad \text{(Equation 1)} \]\[ \sec\theta - \tan\theta = \frac{2}{3} \quad \text{(Equation 2)} \]Add Equation 1 and Equation 2:
\[ (\sec\theta + \tan\theta) + (\sec\theta - \tan\theta) = \frac{3}{2} + \frac{2}{3} \]\[ 2\sec\theta = \frac{9 + 4}{6} \]\[ 2\sec\theta = \frac{13}{6} \]Solve for \(\sec\theta\):
\[ \sec\theta = \frac{13}{12} \]Since \(\cos\theta = \frac{1}{\sec\theta}\), we can find \(\cos\theta\):
\[ \cos\theta = \frac{1}{13/12} = \frac{12}{13} \]A) The value of \(\cos\theta\) is \(\frac{12}{13}\), which matches our calculated result.
This problem involves finding two numbers given their difference and the product of their LCM and HCF. We will use the fundamental property that for any two positive integers, the product of the numbers is equal to the product of their LCM and HCF. Then, we can either solve a system of equations or test the given options.
Let the two numbers be \(a\) and \(b\).
We are given that the product of their LCM and HCF is 150. According to the property of numbers:
\[ a \times b = \text{LCM}(a, b) \times \text{HCF}(a, b) \]So, we have:
\[ a \times b = 150 \]We are also given that the difference between the two numbers is 5. Without loss of generality, let's assume \(a > b\).
\[ a - b = 5 \]Now we have a system of two equations:
From equation (ii), we can express \(a\) in terms of \(b\):
\[ a = b + 5 \]Substitute this expression for \(a\) into equation (i):
\[ (b + 5)b = 150 \] \[ b^2 + 5b = 150 \] \[ b^2 + 5b - 150 = 0 \]Solve this quadratic equation for \(b\). We can factor the quadratic equation:
\[ b^2 + 15b - 10b - 150 = 0 \] \[ b(b + 15) - 10(b + 15) = 0 \] \[ (b - 10)(b + 15) = 0 \]This gives two possible values for \(b\): \(b = 10\) or \(b = -15\).
Since the numbers are typically positive in such problems (and LCM/HCF are defined for positive integers), we take \(b = 10\).
Now, find \(a\) using \(a = b + 5\):
\[ a = 10 + 5 \] \[ a = 15 \]So, the two numbers are 15 and 10.
A) 15, 10
Let's verify: Product \(15 \times 10 = 150\). Difference \(15 - 10 = 5\). Both conditions are satisfied.
To find the original number, we can set up an algebraic equation. If a number is increased by 70%, it means the new number is the original number plus 70% of the original number. This can be expressed as 100% of the number plus 70% of the number, which equals 170% of the original number.
Let the unknown number be \(x\).
When the number is increased by 70%, it means we are adding \(70\%\) of \(x\) to \(x\). Mathematically, this can be written as:
\[ x + 70\% \text{ of } x \]Convert the percentage to a decimal or fraction:
\[ 70\% = \frac{70}{100} = 0.70 \]Substitute this into the expression:
\[ x + 0.70x \]Combine the terms:
\[ 1x + 0.70x = 1.70x \]According to the problem, this increased number is equal to 3570. So, we set up the equation:
\[ 1.70x = 3570 \]To find \(x\), divide both sides of the equation by 1.70:
\[ x = \frac{3570}{1.70} \]To simplify the division, we can remove the decimal by multiplying the numerator and denominator by 100:
\[ x = \frac{3570 \times 100}{1.70 \times 100} = \frac{357000}{170} \]Now, perform the division:
\[ x = \frac{35700}{17} \]\[ x = 2100 \]A) 2100 is the correct number. When 2100 is increased by 70%, it becomes \(2100 + (0.70 \times 2100) = 2100 + 1470 = 3570\).
This problem can be solved by setting up a system of two linear equations with two variables. We will represent the son's age and the father's age with variables and then solve for the father's age.
Let \(S\) be the son's age and \(F\) be the father's age.
From the first statement, "2 times a son's age is added to his father's age, the sum is 42 years", we can form the equation:
\[ 2S + F = 42 \quad \text{(Equation 1)} \]From the second statement, "2.2 times the father's age is added to the son's age, the sum is 89 years", we can form the equation:
\[ S + 2.2F = 89 \quad \text{(Equation 2)} \]Now we have a system of two linear equations:
Equation 1: \(2S + F = 42\)
Equation 2: \(S + 2.2F = 89\)
We want to find the father's age, \(F\). We can use the substitution method or elimination method. Let's use substitution. From Equation 1, express \(F\) in terms of \(S\):
\[ F = 42 - 2S \quad \text{(Equation 3)} \]Substitute Equation 3 into Equation 2:
\[ S + 2.2(42 - 2S) = 89 \]Distribute 2.2:
\[ S + 92.4 - 4.4S = 89 \]Combine like terms:
\[ -3.4S = 89 - 92.4 \]\[ -3.4S = -3.4 \]Solve for \(S\):
\[ S = \frac{-3.4}{-3.4} \]\[ S = 1 \]Now that we have the son's age, \(S = 1\) year, substitute it back into Equation 3 to find the father's age, \(F\):
\[ F = 42 - 2(1) \]\[ F = 42 - 2 \]\[ F = 40 \]So, the father's age is 40 years.
Correct Option: B) 40