This problem involves finding an unknown number given the H.C.F. (Highest Common Factor), L.C.M. (Least Common Multiple) of two numbers, and one of the numbers. The key principle to solve this is the fundamental relationship between the H.C.F. and L.C.M. of two numbers: the product of the two numbers is equal to the product of their H.C.F. and L.C.M.
Identify the given values:
State the fundamental formula:
For any two positive integers, say \(a\) and \(b\), the product of the numbers is equal to the product of their H.C.F. and L.C.M.
\[ a \times b = \text{H.C.F.} \times \text{L.C.M.} \]Substitute the given values into the formula:
Let the first number be \(a = 52\) and the second number be \(b\). We need to find \(b\).
\[ 52 \times b = 4 \times 208 \]Calculate the product of H.C.F. and L.C.M.:
\[ 4 \times 208 = 832 \]Solve for the unknown number \(b\):
\[ 52 \times b = 832 \] \[ b = \frac{832}{52} \] \[ b = 16 \]Thus, the other number is 16.
Correct Option: C) 16
To find the perimeter of a rectangular plot, we need both its length and width. We are given the length and the area. We can use the formula for the area of a rectangle to find the width, and then use the length and width to calculate the perimeter.
The area of a rectangle is given by the product of its length and width.
\[ \text{Area} = \text{Length} \times \text{Width} \]Given:
Substitute these values into the area formula:
\[ 1904 \text{ m}^2 = 68 \text{ m} \times \text{Width} \]Divide the area by the length to find the width:
\[ \text{Width} = \frac{1904 \text{ m}^2}{68 \text{ m}} \]\[ \text{Width} = 28 \text{ m} \]The perimeter of a rectangle is given by twice the sum of its length and width.
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]Substitute the length (\(L = 68 \text{ m}\)) and the calculated width (\(W = 28 \text{ m}\)) into the perimeter formula:
\[ \text{Perimeter} = 2 \times (68 \text{ m} + 28 \text{ m}) \]\[ \text{Perimeter} = 2 \times (96 \text{ m}) \]\[ \text{Perimeter} = 192 \text{ m} \]A) 192 m is the correct perimeter calculated from the given length and area.
To find the simple interest, we use the standard formula: Simple Interest (SI) = (Principal (P) × Rate (R) × Time (T)) / 100. We are given the principal amount, the annual rate of interest, and the time period.
A) 1,440 is the correct simple interest calculated using the formula.
The selling price of 25 books is equal to the cost price of 37 books. Find the loss or gain percentage.
This problem involves calculating profit or loss percentage based on a given relationship between the selling price of a certain number of articles and the cost price of another number of articles. We will assume a common cost price for one book to simplify the calculation and then determine the total cost price and selling price to find the gain percentage.
Let \( \text{CP} \) be the cost price of one book.
Let \( \text{SP} \) be the selling price of one book.
The selling price of 25 books is equal to the cost price of 37 books. We can write this as:
\[ 25 \times \text{SP} = 37 \times \text{CP} \]From the equation above, we can find the selling price of one book relative to its cost price:
\[ \text{SP} = \frac{37}{25} \times \text{CP} \]Since \( \text{SP} > \text{CP} \) (because \( \frac{37}{25} > 1 \)), there is a gain. The gain per book is:
\[ \text{Gain} = \text{SP} - \text{CP} = \frac{37}{25} \times \text{CP} - \text{CP} \]\[ \text{Gain} = \left( \frac{37}{25} - 1 \right) \times \text{CP} = \left( \frac{37 - 25}{25} \right) \times \text{CP} \]\[ \text{Gain} = \frac{12}{25} \times \text{CP} \]The gain percentage is calculated as:
\[ \text{Gain Percentage} = \left( \frac{\text{Gain}}{\text{CP}} \right) \times 100\% \]\[ \text{Gain Percentage} = \left( \frac{\frac{12}{25} \times \text{CP}}{\text{CP}} \right) \times 100\% \]\[ \text{Gain Percentage} = \left( \frac{12}{25} \right) \times 100\% \]\[ \text{Gain Percentage} = 12 \times 4\% \]\[ \text{Gain Percentage} = 48\% \]D) 48% gain — The calculation shows a gain of 48%.
To find the original number, we need to set up an equation based on the given information. The problem states that a number, when increased by 80%, results in 2880. We can represent the unknown number as a variable, say \(x\), and then form an equation to solve for \(x\).
Define the unknown: Let the number be \(x\).
Express the increase: An increase of 80% means the number becomes \(100\% + 80\% = 180\%\) of its original value. In decimal form, 180% is \(1.80\).
Formulate the equation: The number increased by 80% is \(x + 80\%\) of \(x\), which can be written as \(x + 0.80x\). This simplifies to \(1.80x\). We are given that this value is 2880.
\[ 1.80x = 2880 \]Solve for \(x\): To find \(x\), divide both sides of the equation by 1.80.
\[ x = \frac{2880}{1.80} \]\[ x = \frac{2880}{\frac{180}{100}} \]\[ x = \frac{2880 \times 100}{180} \]\[ x = \frac{288000}{180} \]\[ x = \frac{28800}{18} \]Perform the division:
\[ x = 1600 \]Thus, the original number is 1600.
Correct Option: A) 1600
To find the average of the remaining two numbers, we first need to find 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 calculate their average by dividing by 2.
Given that the average of the first two numbers is 80.
\[ \text{Sum of first two numbers} = \text{Average} \times \text{Number of terms} = 80 \times 2 = 160 \]The third number is given as 105.
The total sum of five numbers is given as 655.
\[ \text{Sum of remaining two numbers} = (\text{Total sum of five numbers}) - (\text{Sum of first three numbers}) = 655 - 265 = 390 \]Since there are two numbers remaining, their average is their sum divided by 2.
\[ \text{Average of remaining two numbers} = \frac{\text{Sum of remaining two numbers}}{2} = \frac{390}{2} = 195 \]Correct Option: D) 195
To calculate the profit percentage, we first need to find the cost price (CP) and selling price (SP) of a single sharpener. Then, we can use the formula for profit percentage: \[\text{Profit Percentage} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100\].
Calculate the Cost Price (CP) per sharpener:
David bought sharpeners at ₹150 a dozen. A dozen means 12 sharpeners.
Identify the Selling Price (SP) per sharpener:
He sold them for ₹43 each.
Calculate the Profit per sharpener:
\[ \text{Profit} = \text{SP} - \text{CP} \]\[ \text{Profit} = \text{₹43} - \text{₹12.50} = \text{₹30.50} \]Calculate the Profit Percentage:
\[ \text{Profit Percentage} = \frac{\text{Profit}}{\text{CP}} \times 100 \]\[ \text{Profit Percentage} = \frac{30.50}{12.50} \times 100 \]\[ \text{Profit Percentage} = 2.44 \times 100 \]\[ \text{Profit Percentage} = 244\% \]Correct Option: B) 244% is the calculated profit percentage.
To simplify the given algebraic expression, we need to apply the distributive property to expand the terms, then combine like terms (terms with the same variable and exponent).
Correct Option: D) \(6x^2 - 8x - 4\)
We will set up a system of two linear equations based on the given information about the ages of P and Q. We will define variables for their present ages and then translate the word problems into mathematical equations. Finally, we will solve these equations simultaneously to find the present age of Q.
Define variables:
Let the present age of P be \(P\) years.
Let the present age of Q be \(Q\) years.
Formulate the first equation:
"Two times the present age of P exceeds the present age of Q by 15 years."
This can be written as:
Formulate the second equation:
"After 9 years, twice Q's age will be 14 years less than twice P's age at that time."
Ages after 9 years:
P's age after 9 years = \(P + 9\)
Q's age after 9 years = \(Q + 9\)
The statement translates to:
Solve the system of equations:
We have two equations:
1) \(2P = Q + 15\)
2) \(Q = P - 7\)
Substitute Equation 2 into Equation 1:
Find the present age of Q:
Now that we have \(P = 8\), substitute this value back into Equation 2:
Thus, the present age of Q is 1 year.
C) 1 is the correct answer. By setting up and solving the system of linear equations derived from the problem statements, we find that the present age of Q is 1 year.
To find the greatest ratio, we need to convert each ratio into its decimal form or a common fractional form and then compare them. Converting to decimal form is often the most straightforward method for comparison.
Correct Option: B) 46 : 51