To find the selling price after successive discounts, we need to apply each discount one after the other to the marked price. First, calculate the price after the first discount, and then apply the second discount to this new price.
Identify the Marked Price (MP) and Discounts:
Marked Price (MP) = ₹18,000
First Discount (\(D_1\)) = 10%
Second Discount (\(D_2\)) = 5%
Calculate the price after the first discount:
The first discount is 10% of the Marked Price.
\[\text{Discount Amount}_1 = 10\% \times \text{₹}18,000 = \frac{10}{100} \times 18,000 = \text{₹}1,800\]Price after first discount = Marked Price - Discount Amount\(_{1}\)
\[\text{Price}_1 = \text{₹}18,000 - \text{₹}1,800 = \text{₹}16,200\]Calculate the price after the second discount:
The second discount is 5% of the price after the first discount (₹16,200).
\[\text{Discount Amount}_2 = 5\% \times \text{₹}16,200 = \frac{5}{100} \times 16,200 = \text{₹}810\]Selling Price (SP) = Price after first discount - Discount Amount\(_{2}\)
\[\text{SP} = \text{₹}16,200 - \text{₹}810 = \text{₹}15,390\]Alternatively, using successive discount formula:
If there are two successive discounts \(d_1\%\) and \(d_2\%\), the effective discount percentage is given by:
\[\text{Effective Discount} = \left(d_1 + d_2 - \frac{d_1 \times d_2}{100}\right)\%\] \[\text{Effective Discount} = \left(10 + 5 - \frac{10 \times 5}{100}\right)\% = \left(15 - \frac{50}{100}\right)\% = (15 - 0.5)\% = 14.5\%\]Now, calculate the total discount amount:
\[\text{Total Discount Amount} = 14.5\% \times \text{₹}18,000 = \frac{14.5}{100} \times 18,000 = \text{₹}2,610\]Selling Price (SP) = Marked Price - Total Discount Amount
\[\text{SP} = \text{₹}18,000 - \text{₹}2,610 = \text{₹}15,390\]Correct Option: D) ₹15,390
The value of \(\sqrt{256}\) + \(\sqrt{0.0256}\) − \(\sqrt{4.41}\) is _____.
To solve this problem, we need to evaluate each square root term individually and then perform the addition and subtraction operations in the correct order. We will use the knowledge of perfect squares and decimal square roots.
We know that \(16 \times 16 = 256\).
\[ \sqrt{256} = 16 \]First, consider the number without the decimal point, which is \(256\). Its square root is \(16\). Since there are four decimal places in \(0.0256\), its square root will have half that number of decimal places, which is two.
\[ \sqrt{0.0256} = 0.16 \]First, consider the number without the decimal point, which is \(441\). We know that \(21 \times 21 = 441\). Since there are two decimal places in \(4.41\), its square root will have half that number of decimal places, which is one.
\[ \sqrt{4.41} = 2.1 \]The given expression is \(\sqrt{256} + \sqrt{0.0256} - \sqrt{4.41}\).
\[ 16 + 0.16 - 2.1 \]First, add \(16\) and \(0.16\):
\[ 16 + 0.16 = 16.16 \]Next, subtract \(2.1\) from \(16.16\):
\[ 16.16 - 2.1 = 14.06 \]A) 14.06 is the correct answer as calculated above.
To determine if a number is divisible by 14, we need to check if it is divisible by both 2 and 7. A number is divisible by 2 if its last digit is even. A number is divisible by 7 if, when we subtract twice the last digit from the number formed by the remaining digits, the result is divisible by 7. We will apply these divisibility rules to each option.
Let's check option C: 7048048.
Since 7048048 is divisible by both 2 and 7, it is divisible by 14.
In a business partnership, the profit is distributed among partners in the ratio of their investments, assuming the investment period is the same for all. We will first find the ratio of profits, which will give us the ratio of investments. Then, using the total investment, we can find individual investments and their difference.
Calculate Salman's share of profit:
Total Profit = ₹16,000
Vivek's Share = ₹6,400
Salman's Share = Total Profit - Vivek's Share
Salman's Share = \(\text{₹}16,000 - \text{₹}6,400 = \text{₹}9,600\)
Determine the ratio of their profits:
Ratio of Profits (Salman : Vivek) = Salman's Share : Vivek's Share
Ratio of Profits = \(9600 : 6400\)
To simplify the ratio, divide both numbers by their greatest common divisor. We can divide by 100 first, then by 32:
Ratio of Profits = \(96 : 64\)
Dividing by 32:
\[ \frac{96}{32} : \frac{64}{32} = 3 : 2 \]So, the ratio of profits (Salman : Vivek) is \(3:2\).
Relate profit ratio to investment ratio:
Since profit is distributed in the ratio of investments, the ratio of their investments (Salman : Vivek) is also \(3:2\).
Calculate individual investments:
Total Investment = ₹45,200
Sum of the ratio parts = \(3 + 2 = 5\)
Salman's Investment = \[\frac{3}{5} \times \text{Total Investment}\]
\[ \text{Salman's Investment} = \frac{3}{5} \times 45200 = 3 \times 9040 = \text{₹}27,120 \]Vivek's Investment = \[\frac{2}{5} \times \text{Total Investment}\]
\[ \text{Vivek's Investment} = \frac{2}{5} \times 45200 = 2 \times 9040 = \text{₹}18,080 \]Find the difference between their investments:
Difference = Salman's Investment - Vivek's Investment
Difference = \(\text{₹}27,120 - \text{₹}18,080 = \text{₹}9,040\)
Correct Option: C) ₹9,040
We need to find the amount invested at 14% per annum. We can set up equations based on the simple interest formula for each part of the investment and use the condition that the interests earned are equal.
Let the total investment be \(P = \text{₹}20,400\).
Let the sum invested at 14% per annum be \(P_1\). Then the sum invested at 11% per annum will be \(P_2 = P - P_1 = \text{₹}(20,400 - P_1)\).
The rate of interest for the first part is \(R_1 = 14\%\) and for the second part is \(R_2 = 11\%\).
The time period for both investments is \(T = 9\) years.
The formula for simple interest (SI) is: \[\text{SI} = \frac{P \times R \times T}{100}\]
Interest from the first investment (\(\text{SI}_1\)):
\[\text{SI}_1 = \frac{P_1 \times 14 \times 9}{100}\]Interest from the second investment (\(\text{SI}_2\)):
\[\text{SI}_2 = \frac{(20400 - P_1) \times 11 \times 9}{100}\]According to the problem, the interests earned are equal: \(\text{SI}_1 = \text{SI}_2\).
\[\frac{P_1 \times 14 \times 9}{100} = \frac{(20400 - P_1) \times 11 \times 9}{100}\]We can cancel out \(\frac{9}{100}\) from both sides:
\[P_1 \times 14 = (20400 - P_1) \times 11\] \[14P_1 = 20400 \times 11 - 11P_1\] \[14P_1 + 11P_1 = 20400 \times 11\] \[25P_1 = 224400\]Now, solve for \(P_1\):
\[P_1 = \frac{224400}{25}\] \[P_1 = 8976\]So, the sum invested at the rate of 14% per annum is ₹8,976.
A) 8,976
Find the value of: 26-19 ÷ 2618 × 263
The problem involves simplifying an expression with exponents using the laws of exponents. Specifically, we will use the rules for multiplication and division of terms with the same base: \(a^m \times a^n = a^{m+n}\) and \(a^m \div a^n = a^{m-n}\).
The given expression is: \(26^{-19} \div 26^{18} \times 26^3\).
First, let's address the division part: \(26^{-19} \div 26^{18}\). Using the rule \(a^m \div a^n = a^{m-n}\), we get:
\[ 26^{-19} \div 26^{18} = 26^{(-19) - (18)} = 26^{-19 - 18} = 26^{-37} \]Now, substitute this back into the original expression:
\[ 26^{-37} \times 26^3 \]Next, address the multiplication part. Using the rule \(a^m \times a^n = a^{m+n}\), we get:
\[ 26^{-37} \times 26^3 = 26^{(-37) + 3} = 26^{-37 + 3} = 26^{-34} \]Thus, the simplified value of the expression is \(26^{-34}\).
Correct Option: C) \(26^{-34}\)
To find the surface area of a sphere, we need its radius. The problem provides the diameter, so we will first calculate the radius from the given diameter and then use the formula for the surface area of a sphere.
The diameter of the sphere is \(d = 14 \text{ cm}\).
The radius is half of the diameter.
\[ r = \frac{d}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm} \]The surface area (\(A\)) of a sphere with radius \(r\) is given by:
\[ A = 4\pi r^2 \]D) \(196\pi \text{ cm}^2\) is the calculated surface area of the sphere.
By how much is 65% of 75 greater than \( \frac{2}{5} \) of 55?
To solve this problem, we need to calculate two values: first, 65% of 75, and second, \( \frac{2}{5} \) of 55. After calculating both values, we will find the difference between the first value and the second value to determine how much greater the first is than the second.
To find a percentage of a number, convert the percentage to a decimal or a fraction and multiply it by the number.
\[ 65\% \text{ of } 75 = \frac{65}{100} \times 75 \]\[ = \frac{13}{20} \times 75 \]\[ = 13 \times \frac{75}{20} \]\[ = 13 \times \frac{15}{4} \]\[ = \frac{195}{4} \]\[ = 48.75 \]To find a fraction of a number, multiply the fraction by the number.
\[ \frac{2}{5} \text{ of } 55 = \frac{2}{5} \times 55 \]\[ = 2 \times \frac{55}{5} \]\[ = 2 \times 11 \]\[ = 22 \]Now, subtract the second value from the first value to find out how much greater the first value is.
\[ \text{Difference} = 48.75 - 22 \]\[ = 26.75 \]B) 26.75 is the correct answer. The calculations show that 65% of 75 is 48.75, and \( \frac{2}{5} \) of 55 is 22. The difference between these two values is 26.75.
We need to find the measure of \(\angle AEB\) in \(\triangle ABC\). We are given that \(BD \perp AC\), \(\angle DBC = 25°\), and \(\angle CAE = 80°\). We will use the properties of triangles, specifically the sum of angles in a triangle and the exterior angle theorem.
Analyze \(\triangle BDC\):
Given that \(BD \perp AC\), it means \(\angle BDC = 90°\). In \(\triangle BDC\), the sum of angles is \(180°\).
\[ \angle BCD + \angle DBC + \angle BDC = 180° \]\[ \angle BCD + 25° + 90° = 180° \]\[ \angle BCD + 115° = 180° \]\[ \angle BCD = 180° - 115° \]\[ \angle BCD = 65° \]Identify \(\angle ACB\):
From the previous step, we found \(\angle BCD = 65°\). Since \(D\) is on \(AC\), \(\angle BCD\) is the same as \(\angle ACB\).
\[ \angle ACB = 65° \]Analyze \(\triangle AEC\):
We need to find \(\angle AEB\). Notice that \(\angle AEB\) is an exterior angle to \(\triangle AEC\). The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two opposite interior angles.
For \(\triangle AEC\), \(\angle AEB\) is an exterior angle, and its opposite interior angles are \(\angle CAE\) and \(\angle ACE\).
We are given \(\angle CAE = 80°\). We found \(\angle ACE = \angle ACB = 65°\).
\[ \angle AEB = \angle CAE + \angle ACE \]\[ \angle AEB = 80° + 65° \]\[ \angle AEB = 145° \]D) 145° is the correct answer, as calculated by applying the sum of angles in a right-angled triangle and the exterior angle theorem.
The problem asks us to find the greater of two consecutive positive natural numbers whose product is 812. We can solve this by setting up an algebraic equation or by using estimation and checking the given options.
Correct Option: A) 29 is the greater of the two consecutive positive natural numbers whose product is 812.