The problem involves distributing profit among three partners in a given ratio and then using the difference between the smallest and largest shares to find the total profit. We will represent the shares as multiples of a common factor, calculate the difference in terms of this factor, and then solve for the factor to find the total profit.
Understand the ratio: The profit is distributed among three partners in the ratio \(12 : 3 : 14\). Let the shares of the three partners be \(12x\), \(3x\), and \(14x\), respectively, where \(x\) is a common factor.
Identify smallest and largest shares: From the ratio \(12 : 3 : 14\), the smallest share is \(3x\) and the largest share is \(14x\).
Formulate the equation based on the given difference: The difference between the largest and the smallest shares is given as ₹61,699.
So, we can write the equation:
\[ 14x - 3x = 61,699 \]Solve for \(x\):
\[ 11x = 61,699 \] \[ x = \frac{61,699}{11} \] \[ x = 5,609 \]Calculate the total profit: The total profit is the sum of all the shares, which is \(12x + 3x + 14x\).
\[ \text{Total Profit} = (12 + 3 + 14)x \] \[ \text{Total Profit} = 29x \]Substitute the value of \(x\) we found:
\[ \text{Total Profit} = 29 \times 5,609 \] \[ \text{Total Profit} = 162,661 \]A) 1,62,661 is the calculated total profit.
The problem states a relationship between the shares of A, B, and C. We need to find the ratio of their shares first and then distribute the total amount according to this ratio to find A's share.
Set up the given relationship: We are given that 2 times the share of A = 4 times the share of B = 7 times the share of C. Let's denote the shares as A, B, and C respectively.
\[ 2A = 4B = 7C \]Find a common multiple: To find the ratio A : B : C, we can equate each part to a constant, say \(k\), or find the Least Common Multiple (LCM) of the coefficients (2, 4, 7). The LCM of 2, 4, and 7 is 28.
\[ 2A = 4B = 7C = 28k \]Express A, B, and C in terms of \(k\):
So, the ratio of their shares is \(A : B : C = 14k : 7k : 4k = 14 : 7 : 4\).
Calculate the total ratio parts: The sum of the ratio parts is \(14 + 7 + 4 = 25\).
Calculate A's share: The total amount to be divided is ₹21,300. A's share is the fraction of the total amount corresponding to A's ratio part.
\[ \text{Share of A} = \frac{\text{A's ratio part}}{\text{Total ratio parts}} \times \text{Total amount} \] \[ \text{Share of A} = \frac{14}{25} \times 21300 \] \[ \text{Share of A} = 14 \times \frac{21300}{25} \] \[ \text{Share of A} = 14 \times 852 \] \[ \text{Share of A} = 11928 \]Thus, the share of A is ₹11,928.
Correct Option: D) ₹11,928
The problem states that a number, when increased by 60%, results in 3640. We need to find the original number. We can represent the original number as a variable and set up an algebraic equation based on the given percentage increase.
Correct Option: D) 2275
To determine which discount scheme is most beneficial to the customer, we need to calculate the percentage discount offered in each scheme. The scheme with the highest percentage discount will be the most beneficial.
Correct Option: C) (iv) is the most beneficial scheme to the customers as it offers the highest percentage discount of approximately 42.85%.
The problem asks for the sum of two consecutive even natural numbers, given that the sum of their squares is 452. We can represent the numbers algebraically, set up an equation based on the given condition, solve for the numbers, and then find their sum.
Let the first even natural number be \(x\). Since the numbers are consecutive even natural numbers, the next even natural number will be \(x+2\).
According to the problem statement, the sum of the squares of these two numbers is 452. So, we can write the equation:
\[ x^2 + (x+2)^2 = 452 \]Expand the equation:
\[ x^2 + (x^2 + 4x + 4) = 452 \]\[ 2x^2 + 4x + 4 = 452 \]Rearrange the terms to form a quadratic equation:
\[ 2x^2 + 4x + 4 - 452 = 0 \]\[ 2x^2 + 4x - 448 = 0 \]Divide the entire equation by 2 to simplify:
\[ x^2 + 2x - 224 = 0 \]Now, solve this quadratic equation for \(x\). We can use factorization or the quadratic formula. For factorization, we need two numbers that multiply to -224 and add up to 2. These numbers are 16 and -14.
\[ x^2 + 16x - 14x - 224 = 0 \]\[ x(x+16) - 14(x+16) = 0 \]\[ (x-14)(x+16) = 0 \]This gives two possible values for \(x\): \(x = 14\) or \(x = -16\).
Since the problem specifies "natural numbers", \(x\) must be a positive integer. Therefore, we take \(x = 14\).
The first even natural number is \(x = 14\).
The second consecutive even natural number is \(x+2 = 14+2 = 16\).
The numbers are 14 and 16. Let's verify the condition: \(14^2 + 16^2 = 196 + 256 = 452\). This is correct.
The problem asks for the sum of the numbers. Sum \(= 14 + 16 = 30\).
Correct Option: B) 30
To evaluate the given expression, we need to 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).
First, evaluate the division part: \((-60) \div (-12)\).
\[ (-60) \div (-12) = 5 \]The expression now becomes:
\[ (-9) - 5 + (-2) \times 9 \]Next, evaluate the multiplication part: \((-2) \times 9\).
\[ (-2) \times 9 = -18 \]The expression now becomes:
\[ (-9) - 5 + (-18) \]Now, perform the subtraction:
\[ (-9) - 5 = -14 \]Finally, perform the addition:
\[ -14 + (-18) = -14 - 18 = -32 \]Thus, the value of the expression is \(-32\).
Correct Option: B) -32
To find the LCM of decimal numbers, we first convert them into fractions. Then, we find the LCM of the numerators and the HCF of the denominators. The LCM of fractions is given by the formula: \[\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}\].
C) \(25.8\) is the correct LCM of \(8.6\) and \(0.12\).
The problem involves simple interest calculations for two different investments. We are given the total investment, the two different interest rates, the time period, and the condition that the interest earned from both investments is equal. We need to find the amount invested at 12% per annum.
The formula for simple interest (SI) is \(SI = \frac{P \times R \times T}{100}\).
According to the problem, the interests earned from the two investments are equal, so \(SI_1 = SI_2\).
\[ \frac{84 P_1}{100} = \frac{91 (20000 - P_1)}{100} \]Multiply both sides by 100 to clear the denominators:
\[ 84 P_1 = 91 (20000 - P_1) \] \[ 84 P_1 = 91 \times 20000 - 91 P_1 \] \[ 84 P_1 + 91 P_1 = 91 \times 20000 \] \[ 175 P_1 = 1820000 \] \[ P_1 = \frac{1820000}{175} \]To simplify the division:
\[ P_1 = \frac{1820000}{25 \times 7} \]Divide 1820000 by 7:
\[ 1820000 \div 7 = 260000 \] \[ P_1 = \frac{260000}{25} \]Divide 260000 by 25 (which is equivalent to multiplying by 4 and dividing by 100, or just direct division):
\[ P_1 = 10400 \]Thus, the sum invested at 12% per annum is ₹10,400.
Correct Option: C) 10,400
To find the volume of a sphere, we first need to determine its radius from the given diameter. Then, we use the standard formula for the volume of a sphere.
The diameter of the sphere is \(D = 12 \text{ cm}\).
The radius (\(r\)) of a sphere is half of its diameter.
\[ r = \frac{D}{2} = \frac{12 \text{ cm}}{2} = 6 \text{ cm} \]The volume (\(V\)) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]Substitute \(r = 6 \text{ cm}\) into the volume formula:
\[ V = \frac{4}{3} \pi (6 \text{ cm})^3 \]\[ V = \frac{4}{3} \pi (216 \text{ cm}^3) \]\[ V = 4 \pi \times \frac{216}{3} \text{ cm}^3 \]\[ V = 4 \pi \times 72 \text{ cm}^3 \]\[ V = 288 \pi \text{ cm}^3 \]Correct Option: B) 288π cm³
Simplify: \( 2\left(\frac{7}{2}x^{2} - 28x + 17\right) - 7(x^{2} + 6x - 11) \)
To simplify the given algebraic expression, we will first distribute the constants into their respective parentheses. Then, we will combine like terms (terms with the same variable and exponent) to arrive at the simplified expression.
Correct Option: C) −98x + 111