Section - Mathematics
The value of \( \frac{3}{4} \) + \(\left(\frac{1}{1+\frac{5}{6}}\right)\) - \( \frac{3}{5} \) is
\(\frac{145}{229}\)
\(\frac{153}{220}\)
\(\frac{146}{225}\)
\(\frac{162}{227}\)
To solve this problem, we need to evaluate the given expression by following the order of operations (PEMDAS/BODMAS). First, simplify the term inside the parenthesis, then perform the addition and subtraction from left to right.
Simplify the term inside the parenthesis: \[\left( \frac{1}{1+\frac{5}{6}} \right)\]
First, calculate the denominator:
\[ 1 + \frac{5}{6} = \frac{6}{6} + \frac{5}{6} = \frac{6+5}{6} = \frac{11}{6} \]Now, substitute this back into the parenthesis term:
\[ \frac{1}{1+\frac{5}{6}} = \frac{1}{\frac{11}{6}} = 1 \times \frac{6}{11} = \frac{6}{11} \]So, the expression becomes:
\[ \frac{3}{4} + \frac{6}{11} - \frac{3}{5} \]Find a common denominator for the fractions \(\frac{3}{4}\), \(\frac{6}{11}\), and \(\frac{3}{5}\).
The denominators are 4, 11, and 5. Since 4, 11, and 5 are coprime (or their prime factors are distinct), the least common multiple (LCM) is their product:
\[ \text{LCM}(4, 11, 5) = 4 \times 11 \times 5 = 220 \]Convert each fraction to an equivalent fraction with the denominator 220.
\[ \frac{3}{4} = \frac{3 \times 55}{4 \times 55} = \frac{165}{220} \]\[ \frac{6}{11} = \frac{6 \times 20}{11 \times 20} = \frac{120}{220} \]\[ \frac{3}{5} = \frac{3 \times 44}{5 \times 44} = \frac{132}{220} \]Substitute these equivalent fractions back into the expression and perform the addition and subtraction:
\[ \frac{165}{220} + \frac{120}{220} - \frac{132}{220} \]\[ = \frac{165 + 120 - 132}{220} \]\[ = \frac{285 - 132}{220} \]\[ = \frac{153}{220} \]B) The calculated value of the expression is \(\frac{153}{220}\).
The simple interest on a certain sum for 3 \( \frac{5}{9} \) years at the rate of 2.7% per annum is ₹2,808.
What will be the amount payable on the same sum at the rate of 9.2% per annum simple interest in 5 \( \frac{2}{3} \) years?
First, we need to calculate the principal sum using the given simple interest, time, and rate. Once the principal is found, we will use it to calculate the amount payable (Principal + Simple Interest) for the second scenario with the new rate and time.
The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]Given:
Substitute the values into the formula:
\[ 2808 = \frac{P \times 2.7 \times \frac{32}{9}}{100} \] \[ 2808 = \frac{P \times 2.7 \times 32}{9 \times 100} \] \[ 2808 = \frac{P \times 0.3 \times 32}{100} \] \[ 2808 = \frac{P \times 9.6}{100} \] \[ P = \frac{2808 \times 100}{9.6} \] \[ P = \frac{280800}{9.6} \] \[ P = \frac{2808000}{96} \] \[ P = \text{₹}29,250 \]Now, we use the calculated principal \(P = \text{₹}29,250\) with the new rate and time.
Given:
Using the SI formula:
\[ \text{SI}_2 = \frac{P \times R \times T}{100} \] \[ \text{SI}_2 = \frac{29250 \times 9.2 \times \frac{17}{3}}{100} \] \[ \text{SI}_2 = \frac{29250 \times 9.2 \times 17}{3 \times 100} \] \[ \text{SI}_2 = \frac{292.50 \times 9.2 \times 17}{3} \] \[ \text{SI}_2 = 97.5 \times 9.2 \times 17 \] \[ \text{SI}_2 = 897 \times 17 \] \[ \text{SI}_2 = \text{₹}15,249 \]The amount payable is the sum of the Principal and the Simple Interest.
\[ A = P + \text{SI}_2 \] \[ A = 29250 + 15249 \] \[ A = \text{₹}44,499 \]Correct Option: A) ₹44,499
Find the volume (in cm³) of a sphere having a diameter of 16.8 cm, giving your answer correct to two decimal places.
Take π = \( \frac{22}{7} \).
To find the volume of a sphere, we need its radius. The problem provides the diameter, so we first calculate the radius. Then, we use the formula for the volume of a sphere, \(V = \frac{4}{3}\pi r^3\), substituting the given value of \(\pi = \frac{22}{7}\) and the calculated radius. Finally, we round the result to two decimal places.
The diameter of the sphere is given as 16.8 cm.
The radius is half of the diameter:
\[ r = \frac{\text{Diameter}}{2} = \frac{16.8 \text{ cm}}{2} = 8.4 \text{ cm} \]The formula for the volume of a sphere is:
\[ V = \frac{4}{3}\pi r^3 \]Substitute the values \(\pi = \frac{22}{7}\) and \(r = 8.4\text{ cm}\):
\[ V = \frac{4}{3} \times \frac{22}{7} \times (8.4)^3 \]Alternatively, we can simplify before multiplying:
\[ V = \frac{4}{3} \times \frac{22}{7} \times (8.4) \times (8.4) \times (8.4) \]Since \(8.4 = 7 \times 1.2\), we can simplify:
\[ V = \frac{4}{3} \times \frac{22}{7} \times (7 \times 1.2) \times (8.4) \times (8.4) \] \[ V = \frac{4}{3} \times 22 \times 1.2 \times 8.4 \times 8.4 \] \[ V = 4 \times 22 \times \frac{1.2}{3} \times 8.4 \times 8.4 \] \[ V = 88 \times 0.4 \times 8.4 \times 8.4 \] \[ V = 35.2 \times 70.56 \] \[ V = 2483.712 \text{ cm}^3 \]Rounding 2483.712 to two decimal places gives 2483.71.
Correct Option: D) 2483.71
To find the mode of a given set of data, we need to identify the value that appears most frequently in the dataset. The mode is a measure of central tendency.
Correct Option: C) 12 — The number 12 appears 3 times in the given data set, which is the highest frequency among all the numbers. Therefore, 12 is the mode.
The value of \(\sqrt{400}\) + \(\sqrt{0.0289}\)- \(\sqrt{7.84}\) =
To solve this problem, we need to calculate the square root of each term individually and then perform the addition and subtraction operations in the given order. We will convert decimal numbers to fractions to simplify finding their square roots.
We know that \(20 \times 20 = 400\).
\[ \sqrt{400} = 20 \]First, convert the decimal to a fraction:
\[ 0.0289 = \frac{289}{10000} \]\[ \sqrt{0.0289} = \sqrt{\frac{289}{10000}} \]\[ \sqrt{289} = 17 \quad (\text{since } 17 \times 17 = 289) \]\[ \sqrt{10000} = 100 \quad (\text{since } 100 \times 100 = 10000) \]\[ \sqrt{0.0289} = \frac{17}{100} = 0.17 \]First, convert the decimal to a fraction:
\[ 7.84 = \frac{784}{100} \]\[ \sqrt{7.84} = \sqrt{\frac{784}{100}} \]\[ \sqrt{784} = 28 \quad (\text{since } 28 \times 28 = 784) \]\[ \sqrt{100} = 10 \quad (\text{since } 10 \times 10 = 100) \]\[ \sqrt{7.84} = \frac{28}{10} = 2.8 \]The expression is \(\sqrt{400} + \sqrt{0.0289} - \sqrt{7.84}\).
\[ 20 + 0.17 - 2.8 \]Thus, the value of the expression is \(17.37\).
B) 17.37 is the correct answer, as calculated by finding the square roots of each term and then performing the arithmetic operations.
Simplify: 3((\( \frac{6}{3} \))x² - 29x + 14) - 6(x² + 5x - 13)
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 form.
The first part of the expression is \(3\left(\left(\frac{6}{3}\right)x^2 - 29x + 14\right)\).
First, simplify the fraction inside the parenthesis: \(\frac{6}{3} = 2\).
So, the expression becomes \(3(2x^2 - 29x + 14)\).
Now, distribute \(3\) to each term inside the parenthesis:
\[ 3 \cdot 2x^2 - 3 \cdot 29x + 3 \cdot 14 \]\[ 6x^2 - 87x + 42 \]The second part of the expression is \(-6(x^2 + 5x - 13)\).
Distribute \(-6\) to each term inside the parenthesis:
\[ -6 \cdot x^2 - 6 \cdot 5x - 6 \cdot (-13) \]\[ -6x^2 - 30x + 78 \]Now, combine the two simplified parts:
\[ (6x^2 - 87x + 42) + (-6x^2 - 30x + 78) \]\[ 6x^2 - 87x + 42 - 6x^2 - 30x + 78 \]Group the terms with \(x^2\), terms with \(x\), and constant terms:
For \(x^2\) terms:
\[ 6x^2 - 6x^2 = 0x^2 = 0 \]For \(x\) terms:
\[ -87x - 30x = (-87 - 30)x = -117x \]For constant terms:
\[ 42 + 78 = 120 \]Combining all the simplified terms:
\[ 0 - 117x + 120 \]\[ -117x + 120 \]D) -117x + 120 is the correct simplified expression.
To find the number of diagonals in a regular polygon, we first need to determine the number of sides (n) of the polygon. We are given a relationship between the sum of its interior angles and the sum of its exterior angles. We will use the standard formulas for these sums and then the formula for the number of diagonals.
Formulate the sum of interior angles:
The sum of the interior angles of a polygon with \(n\) sides is given by the formula:
Formulate the sum of exterior angles:
The sum of the exterior angles of any convex polygon (including regular polygons) is always \(360^\circ\).
Set up the given equation:
According to the problem statement, the sum of the interior angles is equal to four times the sum of its exterior angles:
Solve for \(n\):
Divide both sides by \(180^\circ\):
Calculate the number of diagonals:
The number of diagonals in a polygon with \(n\) sides is given by the formula:
The selling price of 50 books is equal to the cost price of 15 books. Find the loss or gain percentage on these books.
\( \frac{100}{35} \)% gain
70% loss
\( \frac{100}{35} \)% loss
70% gain
To find the loss or gain percentage, we need to establish a relationship between the selling price (SP) and the cost price (CP) of a single item. We are given a relationship between the selling price of 50 books and the cost price of 15 books. We will use this to find the CP and SP of one book and then calculate the percentage loss or gain.
Let the Cost Price of one book be \(CP\).
Let the Selling Price of one book be \(SP\).
According to the problem statement, the selling price of 50 books is equal to the cost price of 15 books. We can write this as an equation:
\[ 50 \times SP = 15 \times CP \]Now, we can find the ratio of \(SP\) to \(CP\):
\[ \frac{SP}{CP} = \frac{15}{50} \] \[ \frac{SP}{CP} = \frac{3}{10} \]From this ratio, we can infer that if \(CP = 10\) units, then \(SP = 3\) units. Since \(SP < CP\), there is a loss.
Calculate the Loss:
\[ \text{Loss} = CP - SP \] \[ \text{Loss} = 10 - 3 = 7 \text{ units} \]Calculate the Loss Percentage:
\[ \text{Loss Percentage} = \left( \frac{\text{Loss}}{CP} \right) \times 100 \% \] \[ \text{Loss Percentage} = \left( \frac{7}{10} \right) \times 100 \% \] \[ \text{Loss Percentage} = 70 \% \]B) 70% loss
To find the mean proportional of two numbers, say \(a\) and \(b\), we need to find a number \(x\) such that \(a, x, b\) are in continued proportion. This means the ratio of the first to the second is equal to the ratio of the second to the third, i.e., \[\frac{a}{x} = \frac{x}{b}\]. From this, we can derive the formula for the mean proportional.
Let the two given numbers be \(a = 81\) and \(b = 36\).
Let the mean proportional be \(x\). According to the definition of mean proportional, we have:
\[ \frac{a}{x} = \frac{x}{b} \]Substitute the given values of \(a\) and \(b\):
\[ \frac{81}{x} = \frac{x}{36} \]Cross-multiply to solve for \(x\):
\[ x \times x = 81 \times 36 \]\[ x^2 = 81 \times 36 \]To find \(x\), take the square root of both sides:
\[ x = \sqrt{81 \times 36} \]Calculate the square roots of 81 and 36 separately:
\[ \sqrt{81} = 9 \]\[ \sqrt{36} = 6 \]Multiply these values to find \(x\):
\[ x = 9 \times 6 \]\[ x = 54 \]Thus, the mean proportional of 81 and 36 is 54.
Correct Option: C) 54
By how much is 65% of 75 greater than \( \frac{3}{5} \) of 35?
To solve this problem, we need to calculate two separate values: first, 65% of 75, and second, \( \frac{3}{5} \) of 35. After calculating both values, we will find the difference between the first value and the second value to determine how much the first is greater 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.
\[ \frac{3}{5} \text{ of } 35 = \frac{3}{5} \times 35 \]\[ = 3 \times \frac{35}{5} \]\[ = 3 \times 7 \]\[ = 21 \]Subtract the second value from the first value to find out by how much the first value is greater than the second.
Difference \( = \) (65% of 75) \( - \) (\( \frac{3}{5} \) of 35)
\[ \text{Difference} = 48.75 - 21 \]\[ = 27.75 \]A) 27.75 is the correct answer. The difference between 65% of 75 (which is 48.75) and \( \frac{3}{5} \) of 35 (which is 21) is 27.75.