SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

To find the required percentage, first calculate the total number of candidates selected from Delhi and Maharashtra over the five-year period from 2012 to 2016.

Total candidates selected from Delhi: Total_Dₑₗₕᵢ = 94 + 48 + 90 + 70 + 82 = 384

Total candidates selected from Maharashtra: Total_Mₐₕₐᵣₐₛₕₜᵣₐ = 85 + 70 + 84 + 60 + 65 = 364

The percentage of candidates selected from Maharashtra relative to those selected from Delhi is calculated using the formula: Percentage = ( Total_Mₐₕₐᵣₐₛₕₜᵣₐ{Total_Dₑₗₕᵢ} ) × 100

Substituting the values into the formula: Percentage = ( ³⁶⁴⁄₃₈₄ ) × 100 Percentage = ( ⁹¹⁄₉₆ ) × 100 ≈ 0.947916... × 100 = 94.7916...%

Rounding to two decimal places, the percentage is approximately 94.79%.

Explanation

To find the value of x, we substitute the standard trigonometric values for the angle (π)/(6) radians (or 30°) into the given expression.

The value of tan (π)/(6) is (1)/(√(3)). The value of sec (π)/(6) is the reciprocal of cos (π)/(6), which is (1)/(√(3)){2} = (2)/(√(3)).

Substituting these values into the equation: x = tan (π)/(6) + sec (π)/(6) x = (1)/(√(3)) + (2)/(√(3))

Since the denominators are the same, we add the numerators: x = (1 + 2)/(√(3)) x = (3)/(√(3))

Simplifying the fraction by dividing 3 by √3 (or rationalizing): x = √3 · √3{√3} x = √3

Explanation

The whole surface area (Total Surface Area) of a cone is calculated using the formula A = π r(l + r), where r is the base radius and l is the slant height. The slant height l is determined by the Pythagorean relationship with the radius r and vertical height h as l = √(r² + h²).

Given r = 6 cm and h = 8 cm, the slant height is l = √(6² + 8²) = √(36 + 64) = √(100) = 10 cm.

Substituting these values into the total surface area formula: A = π × 6 × (10 + 6) A = π × 6 × 16 A = 96π

Using the approximation π ≈ 22/7: A = 96 × ²²⁄₇ = ²¹¹²⁄₇ ≈ 301.7142 cm²

Explanation

The time taken to complete a task is inversely proportional to the work rate. Since 15 men complete the work in 25 days, their combined rate is 1/25 of the work per day. Similarly, 25 women complete the same work in 40 days, so their combined rate is 1/40 of the work per day.

When the 15 men and 25 women work together, their combined daily work rate is the sum of these rates: Combined Rate = ¹⁄₂₅ + ¹⁄₄₀

Using the least common multiple (LCM) of 25 and 40, which is 200, the calculation is: Combined Rate = ⁸⁄₂₀₀ + ⁵⁄₂₀₀ = ¹³⁄₂₀₀

The total number of days required to complete the work is the reciprocal of the combined daily rate: Total Days = ²⁰⁰⁄₁₃ days

To convert this to a mixed fraction, we divide 200 by 13: 200 = (13 × 15) + 5 Total Days = 15 ⁵⁄₁₃ days

Explanation

The value of y can be determined using the weighted average formula for the combined tax rate:

Combined Rate = (R₁ I₁ + R₂ I₂)/(I₁ + I₂)

Where I₁ = 90,000 and R₁ = 30% represent Suhani's income and tax rate, and I₂ = y and R₂ = 40% represent Ritika's income and tax rate. Substituting the given combined rate of 37%:

37 = (30 × 90,000 + 40 × y)/(90,000 + y)

37(90,000 + y) = 30(90,000) + 40y

37 × 90,000 + 37y = 30 × 90,000 + 40y

Rearranging the terms to solve for y:

37 × 90,000 - 30 × 90,000 = 40y - 37y

(37 - 30) × 90,000 = 3y

7 × 90,000 = 3y

y = (6,30,000)/(3)

y = 2,10,000

Explanation

The numerator x⁴ - 2x² + 1 follows the algebraic identity a² - 2ab + b² = (a - b)² where a = x² and b = 1. Therefore, the numerator can be written as (x² - 1)². Applying the difference of squares identity a² - b² = (a - b)(a + b) to the inner term, the numerator becomes ((x - 1)(x + 1))², which simplifies to (x - 1)²(x + 1)².

The denominator x² - 2x + 1 is a perfect square trinomial equivalent to (x - 1)².

By substituting these forms into the original expression: ((x - 1)²(x + 1)²)/((x - 1)²) = (x + 1)²

Expanding (x + 1)² using the identity (a + b)² = a² + 2ab + b² results in x² + 2x + 1.

Explanation

Given x - ¹⁄ₓ = √6. To find the value of x⁸ - (1)/(x⁸), we use the algebraic identity for the difference of squares: a² - b² = (a+b)(a-b). Expanding the target expression gives: x⁸ - (1)/(x⁸) = (x⁴ + (1)/(x⁴))(x⁴ - (1)/(x⁴)) = (x⁴ + (1)/(x⁴))(x² + (1)/(x²))(x² - (1)/(x²)) x⁸ - (1)/(x⁸) = (x⁴ + (1)/(x⁴))(x² + (1)/(x²))(x + ¹⁄ₓ)(x - ¹⁄ₓ)

First, find x² + (1)/(x²) by squaring the given equation: (x - ¹⁄ₓ)² = (√6)² ⟹ x² + (1)/(x²) - 2 = 6 ⟹ x² + (1)/(x²) = 8

Next, find x⁴ + (1)/(x⁴) by squaring the result for x²: (x² + (1)/(x²))² = 8² ⟹ x⁴ + (1)/(x⁴) + 2 = 64 ⟹ x⁴ + (1)/(x⁴) = 62

Now, find x + ¹⁄ₓ using the identity (x + ¹⁄ₓ)² = (x - ¹⁄ₓ)² + 4: (x + ¹⁄ₓ)² = (√6)² + 4 = 10 ⟹ x + ¹⁄ₓ = √(10) (since x > 1, the value is positive)

Substitute all values into the expanded expression: x⁸ - (1)/(x⁸) = 62 × 8 × √(10) × √6 x⁸ - (1)/(x⁸) = 496 × √(60) = 496 × √(4 × 15) = 496 × 2√(15) x⁸ - (1)/(x⁸) = 992√(15)

Explanation

The equivalent single discount for three successive discounts of d₁%, d₂%, and d₃% can be calculated by finding the final multiplier of the marked price. For an initial price of 100, the value after a 12% discount is 100 × (1 - 0.12) = 88. Applying a 20% discount to this result gives 88 × (1 - 0.20) = 88 × 0.8 = 70.4. Finally, applying the 25% discount yields 70.4 × (1 - 0.25) = 70.4 × 0.75 = 52.8. The total discount is the difference between the initial price and the final price, which is 100 - 52.8 = 47.2%.

Alternatively, using the formula for successive discounts a + b - ᵃᵇ⁄₁₀₀, the combined effect of 12% and 20% is 12 + 20 - (12 × 20)/(100) = 32 - 2.4 = 29.6%. Combining this result with the third discount of 25% gives 29.6 + 25 - (29.6 × 25)/(100) = 54.6 - 7.4 = 47.2%.

Explanation

Given sin θ = ⅘, we identify the perpendicular p = 4 and the hypotenuse h = 5. Using the Pythagorean theorem h² = p² + b², the base b is calculated as b = √(5²⁾⁻⁴^^^² = √(25 - 16) = √9 = 3.

The trigonometric ratios are defined as tan θ = (p)/(b) and cot θ = ᵇ⁄ₚ. Substituting the values of p and b: tan θ = ⁴⁄₃ cot θ = ¾

The expression tan θ - cot θ is solved by finding a common denominator: tan θ - cot θ = ⁴⁄₃ - ¾ tan θ - cot θ = (4 × 4 - 3 × 3)/(12) tan θ - cot θ = (16 - 9)/(12) = ⁷⁄₁₂

Explanation

Let the cost price for Ram be x. Ram sells the suitcase to Mohan at a 20% profit, so the selling price for Ram, which becomes the cost price for Mohan, is x × (1 + ²⁰⁄₁₀₀) = 1.2x. Mohan then sells the suitcase to Shyam at a 40% profit. Therefore, the price Shyam pays is 1.2x × (1 + ⁴⁰⁄₁₀₀) = 1.2x × 1.4 = 1.68x.

According to the question, Shyam pays ₹1,430. We can set up the linear equation 1.68x = 1430. To find the initial price x, we divide the total payment by the cumulative profit multiplier: x = (1430)/(1.68). Converting this to a simpler fraction, x = ¹⁴³⁰⁰⁰⁄₁₆₈. Dividing 143000 by 168 results in approximately 851.1904. Rounding to two decimal places, the price at which Ram bought the suitcase is ₹851.19.

Explanation

To find the value of p² - q², we square both given equations: p² = (a cot θ + b csc θ)² = a² cot² θ + b² csc² θ + 2ab cot θ csc θ q² = (b cot θ + a csc θ)² = b² cot² θ + a² csc² θ + 2ab cot θ csc θ

Subtracting q² from p²: p² - q² = (a² cot² θ + b² csc² θ + 2ab cot θ csc θ) - (b² cot² θ + a² csc² θ + 2ab cot θ csc θ) p² - q² = a² cot² θ + b² csc² θ - b² cot² θ - a² csc² θ

Grouping the terms with common coefficients a² and b²: p² - q² = b² (csc² θ - cot² θ) - a² (csc² θ - cot² θ) p² - q² = (b² - a²) (csc² θ - cot² θ)

Using the fundamental trigonometric identity csc² θ - cot² θ = 1: p² - q² = (b² - a²)(1) = b² - a²

Explanation

The standard algebraic identity for the square of a trinomial is (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

By substituting this expansion into the given expression, we get: (a + b + c)² - a² - b² - c² = (a² + b² + c² + 2ab + 2bc + 2ca) - a² - b² - c²

The terms a², b², and c² subtract out, leaving: 2ab + 2bc + 2ca

Factoring out the common coefficient 2 results in the final simplified form: 2(ab + bc + ca)

Explanation

The maximum marks for the examination in Mathematics are 50. To be eligible for higher studies, a student must obtain at least 60% marks.

The qualifying marks are calculated as: Qualifying Marks = 60% of 50 = ⁶⁰⁄₁₀₀ × 50 = 30

The table provides a cumulative classification of students based on their scores. To find the number of students eligible for higher studies, we look at the row for Mathematics and the column representing 30 marks and above. According to the table, the number of students who obtained 30 and above marks in Mathematics is 33. These are the students who met the 60% eligibility threshold.

Explanation

Let the Marked Price be MP and the Cost Price be CP. The selling price after a 15% discount is ₹29,750. Using the formula SP = MP × (1 - d%), we find the Marked Price: MP = 29750 / 0.85 = 35000

If no discount is allowed, the Marked Price becomes the selling price. The problem states that selling at this price results in a 12% gain. Using the relation CP = SP / (1 + g%): CP = 35000 / 1.12 CP = 31250

The calculation 35000 / 1.12 can be simplified by dividing both terms by 7, which gives 5000 / 0.16, and further simplifying results in 31250.

Explanation

Let the total typing work be the Least Common Multiple (LCM) of 10 and 15, which is 30 units. The efficiency of Pratima is calculated as 30 / 10 = 3 units per hour, and the efficiency of Diksha is 30 / 15 = 2 units per hour. Pratima types alone for 4 hours, completing 3 × 4 = 12 units of work. The remaining work to be done is 30 - 12 = 18 units. Diksha completes this remaining work at her own pace, so the time taken by her is the remaining units divided by her efficiency, which is 18 / 2 = 9 hours.

Explanation

A cube consists of six identical square faces. The perimeter P of one square face is determined by the formula P = 4s, where s represents the side length of the cube. Substituting the given value:

4s = 40 s = ⁴⁰⁄₄ = 10 cm

The volume V of a cube is calculated using the formula V = s³. Using the side length derived above:

V = 10³ = 1000 cm³

Explanation

To find the difference between the students placed in all companies in 2014 and 2012, first calculate the total placements for each year from the provided table.

Total placements in the year 2014 (using the first entry for 2014): 19.3 + 13.4 + 13.4 + 11.8 + 7.8 = 65.7 thousand students.

Total placements in the year 2012: 12.6 + 9.8 + 10.4 + 14.9 + 9.8 = 57.5 thousand students.

The difference in thousands is calculated as: 65.7 - 57.5 = 8.2 thousand.

To convert this value into the total number of students, multiply by 1000: 8.2 × 1000 = 8200.

Explanation

The price of a mixture is calculated using the weighted average formula M = P₁ × R₁ + P₂ × R₂{R₁ + R₂}, where M is the mean price, P represents the prices of the components, and R represents their respective ratios. Substituting the given values M = 50, P₁ = 45, R₁ = 3, and R₂ = 2 into the formula:

50 = 45 × 3 + P₂ × 2{3 + 2}

250 = 135 + 2P₂

115 = 2P₂

P₂ = 57.5

Explanation

The side lengths of the triangle are 16, 30, and 34 units. To determine the type of triangle, we check the Pythagorean relation a² + b² = c². Calculating the squares of the sides, we find 16² + 30² = 256 + 900 = 1156, which is equal to 34² = 1156. Since the sum of the squares of the two shorter sides equals the square of the longest side, the triangle is right-angled with a hypotenuse of 34 units.

For any right-angled triangle, the circumcircle's center lies at the midpoint of the hypotenuse, meaning the circumradius R is exactly half the length of the hypotenuse:

R = (Hypotenuse)/(2) = ³⁴⁄₂ = 17 units.

Explanation

To determine if a number is divisible by 132, it must be simultaneously divisible by the factors of 132, which are 3, 4, and 11, since 132 = 3 × 4 × 11.

A number is divisible by 4 if its last two digits form a number divisible by 4. It is divisible by 3 if the sum of its digits is a multiple of 3. It is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.

Applying these rules to the given set: 660: Divisible by 4 (60 is 15 × 4), divisible by 3 (6+6+0=12), and divisible by 11 (6-6+0=0). 924: Divisible by 4 (24 is 6 × 4), divisible by 3 (9+2+4=15), and divisible by 11 (9-2+4=11). 1452: Divisible by 4 (52 is 13 × 4), divisible by 3 (1+4+5+2=12), and divisible by 11 ((1+5)-(4+2)=0). 1980: Divisible by 4 (80 is 20 × 4), divisible by 3 (1+9+8+0=18), and divisible by 11 ((1+8)-(9+0)=0).

The remaining numbers fail the criteria: 754 is not divisible by 4 or 3; 1526 is not divisible by 4 or 3; 2045 is not divisible by 4, 3, or 11; and 2170 is not divisible by 4 or 3. There are 4 numbers in total that meet all conditions.

Explanation

Let the total number of people on the electoral roll be x. The number of votes cast is 90% of the total electoral roll, which is 0.9x. Out of these cast votes, 15% were declared invalid, which means the remaining 85% were valid votes. Thus, the total number of valid votes is 0.85 × 0.9x.

Jeeta secured 60% of these valid votes. Given that Jeeta's total valid votes amounted to 91,800, we can express this relationship with the following equation: 0.60 × 0.85 × 0.90 × x = 91,800

First, calculate the combined product of the percentages: 0.60 × 0.85 × 0.90 = 0.459

Now, substitute this value back into the equation to solve for x: 0.459x = 91,800 x = (91,800)/(0.459) x = 2,00,000

This calculation shows that the total number of people on the electoral roll, considering the rates of voter turnout, vote validity, and the specific portion won by the candidate, is 2,00,000.

Explanation

Let the efficiency of a man be M and a boy be B. Since the total work remains constant, we use the formula W = (n₁M + n₂B) × D₁ = (n₃M + n₄B) × D₂.

(12M + 16B) × 5 = (13M + 24B) × 4 60M + 80B = 52M + 96B 60M - 52M = 96B - 80B 8M = 16B ⟹ M = 2B

This relationship indicates that the work efficiency of one man is equivalent to that of two boys. To find the total work in terms of boy-units, we substitute M = 2B into the first equation: W = (12 × 2B + 16B) × 5 = (24B + 16B) × 5 = 40B × 5 = 200B

The combined efficiency of the target group, which consists of 29 men and 22 boys, is calculated as: Efficiency = 29(2B) + 22B = 58B + 22B = 80B

The time required to complete the work is the total work divided by the total efficiency: Time = (200B)/(80B) = ²⁰⁄₈ = 2.5 days

Explanation

Let the number of 50 p, 25 p, and 20 p coins be 2x, 3x, and 5x respectively, based on the given ratio 2 : 3 : 5. The total value of the coins in rupees is the sum of the product of the number of coins and their respective values in rupees: Total Value = (2x × 0.50) + (3x × 0.25) + (5x × 0.20) 550 = 1x + 0.75x + 1x 550 = 2.75x

Solving for x: x = (550)/(2.75) = 200

The amount contributed by the 50 p coins is 1x = 1 × 200 = 200 rupees. The amount contributed by the 20 p coins is 1x = 1 × 200 = 200 rupees. The difference between the amounts contributed by the 50 p and 20 p coins is 200 - 200 = 0.

Explanation

The length of the direct common tangent (DCT) between two circles is calculated using the formula L = √(d²⁾⁻⁽ʳ^_^¹ ⁻ ʳ^₂⁾^², where d is the distance between the centres and r₁, r₂ are the radii of the circles.

Given the distance between the centres d = 26 cm, the first radius r₁ = 16 cm, and the second radius r₂ = 8 cm, we substitute these values into the formula:

L = √(26²⁾⁻⁽¹⁶⁻⁸⁾^^^² L = √(26²⁾⁻⁸^^^²

Applying the algebraic identity a² - b² = (a - b)(a + b):

L = √((26 - 8)(26 + 8)) L = √(18 × 34) L = √(612)

To express this in the form provided in the options, we factor out a perfect square from 612:

L = √(4 × 153) L = 2√(153)

Explanation

The area of a sector of a circle is calculated using the formula A = (θ)/(360°) × π r², where r is the radius and θ is the central angle in degrees. Given r = 5 cm and θ = 60°, the calculation is:

A = ⁶⁰⁄₃₆₀ × π × 5² A = ⅙ × π × 25 A = (25π)/(6)

Substituting π ≈ 3.14159: A ≈ (25 × 3.14159)/(6) A ≈ (78.53975)/(6) ≈ 13.0899 cm²

Rounding to one decimal place, the area of the sector is 13.1 cm².

Explanation

The expression can be simplified using the distributive property of multiplication over addition, which is defined by the formula a × b + a × c = a × (b + c). In the given expression (- 25) × 8 + (- 25) × 2, let a = -25, b = 8, and c = 2. Applying the formula gives:

(-25) × (8 + 2)

Performing the addition inside the parentheses first:

(-25) × 10 = -250

Explanation

The product can be calculated efficiently using the algebraic identity (a - b)(a + b) = a² - b². By observing that 277 and 323 are equidistant from 300, the expression can be rewritten as:

277 × 323 = (300 - 23)(300 + 23)

Applying the identity:

300² - 23² = 90000 - 529 = 89471

This approach simplifies the calculation by replacing long multiplication with the subtraction of a known square from a power of ten.

Explanation

In the third quadrant, the sine and cosine functions are negative, while the tangent function is positive. Using the fundamental trigonometric identity sin² x + cos² x = 1, we substitute cos x = - ½ to find sin x:

sin² x = 1 - (- ½)² = 1 - ¼ = ¾

Since x lies in the third quadrant, sin x must be negative, so sin x = - √3{2}. The tangent function is defined as the ratio of sine to cosine:

tan x = (sin x)/(cos x) = -{√3{2}}{-½} = √3

Alternatively, considering the reference angle, cos θ = ½ corresponds to an angle of 60°. In the third quadrant, the angle is 180° + 60° = 240°. The value of tan 240° is equal to tan(180° + 60°), which simplifies to tan 60° = √3.

Explanation

The total interest paid on a loan under simple interest where the rate varies annually is calculated by summing the individual interest amounts for each year. Since the principal P remains constant throughout the period, the total interest can be expressed as P × (Σ r)/(100), where Σ r is the sum of the annual interest rates.

The interest rates for the four consecutive years are 5%, 5.5%, 6%, and 6.5%. Adding these rates together gives the cumulative interest percentage for the total duration: 5% + 5.5% + 6% + 6.5% = 23%

Given that the total interest paid after 4 years is ₹4,600, we can set up the equation: (P × 23)/(100) = 4600

Solving for the principal P: P = (4600 × 100)/(23) P = 200 × 100 P = 20,000

Explanation

To find the value of cos 75°, we use the cosine addition formula cos (A + B) = cos A cos B - sin A sin B by choosing standard angles A = 45° and B = 30°. Substituting the known trigonometric values cos 45° = (1)/(√(2)), cos 30° = √3{2}, sin 45° = (1)/(√(2)), and sin 30° = ½ into the identity gives:

cos 75° = cos(45° + 30°) = cos 45° cos 30° - sin 45° sin 30°

cos 75° = ((1)/(√(2))) (√3{2}) - ((1)/(√(2))) (½)

cos 75° = √3{2√2} - (1)/(2√(2)) = √3 - 1{2√2}

To rationalize the denominator and match the given format, multiply both the numerator and the denominator by √2:

cos 75° = (√3 - 1) · √2{2√2 · √2} = √6 - √2{4}

Explanation

To find the time taken by the constable to catch the thief, we use the concept of relative speed since both individuals are moving in the same direction. The relative speed of Ram with respect to Somesh is the difference between their individual speeds: Relative speed = 50 m/min - 30 m/min = 20 m/min

The initial distance separating them is 225 m. The time required to bridge this gap is calculated using the formula: T = (Initial Distance)/(Relative Speed) T = ²²⁵⁄₂₀ minutes T = 11.25 minutes

To convert the decimal portion into seconds, we multiply the fractional part by 60: 0.25 × 60 = 15 seconds

The total time taken is 11 minutes 15 seconds.

Explanation

In this transaction, the Cost Price (CP) of the table is ₹3,000 and the Selling Price (SP) is ₹3,200. Since the Selling Price is greater than the Cost Price (SP > CP), the transaction results in a gain. The amount of gain is calculated using the formula Gain = SP - CP. Substituting the given values into the formula yields Gain = 3200 - 3000 = 200. This indicates a profit of ₹200.

Explanation

The time taken by two individuals working together to complete a task can be calculated using the combined work formula T = (xy)/(x + y), where x and y represent the time taken by each individual to complete the work alone.

Given: Time taken by Sarthak x = 12 days Time taken by Mohan y = 15 days

Substituting these values into the formula: T = (12 × 15)/(12 + 15) T = ¹⁸⁰⁄₂₇

To simplify, divide both the numerator and the denominator by their greatest common divisor, which is 9: T = ²⁰⁄₃ T = 6.666...

Rounding the result to two decimal places, we get 6.67 days. Alternatively, using the efficiency method, if we assume the total work is the LCM of 12 and 15 (60 units), Sarthak's efficiency is 5 units/day and Mohan's is 4 units/day. Together, they complete 9 units/day, so the time taken is 60 / 9 = 6.67 days.

Explanation

To find the value of x⁵ + (1)/(x⁵), we use the standard algebraic identity: x⁵ + (1)/(x⁵) = (x² + (1)/(x²))(x³ + (1)/(x³)) - (x + ¹⁄ₓ)

Given x + ¹⁄ₓ = -6, we first find the squares and cubes. For any value k = x + ¹⁄ₓ, the sum of squares is given by k² - 2: x² + (1)/(x²) = (-6)² - 2 = 36 - 2 = 34

The sum of cubes is given by the formula k³ - 3k: x³ + (1)/(x³) = (-6)³ - 3(-6) = -216 + 18 = -198

Now, substitute these calculated values into the main identity: x⁵ + (1)/(x⁵) = (34)(-198) - (-6)

Performing the multiplication: 34 × (-198) = 34 × (-200 + 2) = -6800 + 68 = -6732

Finally, complete the expression: -6732 - (-6) = -6732 + 6 = -6726

Explanation

The relationship between the efficiency of men (M) and women (W) is found by equating total work: (15M + 21W) × 56 = (12M + 24W) × 64 Dividing by 8 and expanding the terms: 7(15M + 21W) = 8(12M + 24W) 105M + 147W = 96M + 192W 9M = 45W ⟹ M = 5W By taking the efficiency of a woman as 1 unit/day and a man as 5 units/day, the total work is: (15 × 5 + 21 × 1) × 56 = 96 × 56 = 5376 units The combined efficiency of 18 men and 24 women is: 18(5) + 24(1) = 90 + 24 = 114 units/day The time required to complete the job is: ⁵³⁷⁶⁄₁₁₄ = ⁸⁹⁶⁄₁₉ = 47 ³⁄₁₉ days

Explanation

Let the original dividend be D and the original divisor be d. Since the original quotient is 25 and the remainder is 0, the relationship is D = 25d.

Pranjal uses a new dividend D' which is 10% more than D, so D' = 1.1D. He uses a new divisor d' which is 25% more than d, so d' = 1.25d. The quotient Q' obtained by Pranjal is:

Q' = (D')/(d')

Substituting the values of D' and d':

Q' = (1.1D)/(1.25d)

Since (D)/(d) = 25, substitute this into the equation:

Q' = (1.1)/(1.25) × 25

Q' = (1.1 × 25)/(1.25)

Q' = 1.1 × 20

Q' = 22

Explanation

Let the cost price be CP and the initial selling price be SP. The original profit is defined as P = SP - CP. The problem states that when the selling price is doubled to 2SP, the new profit becomes 4P. This relationship is expressed by the following equation:

2SP - CP = 4(SP - CP)

Solving for the relationship between SP and CP: 2SP - CP = 4SP - 4CP 4CP - CP = 4SP - 2SP 3CP = 2SP (SP)/(CP) = ³⁄₂ = 1.5

The original profit percentage is calculated using the formula: Profit % = ((SP)/(CP) - 1) × 100 Profit % = (1.5 - 1) × 100 Profit % = 0.5 × 100 = 50%

Explanation

In the right-angled triangle formed by the building, the ground, and the line of sight, the height of the building serves as the perpendicular side (P) and the distance of the person from the base of the building serves as the base (B). The angle between the ground and the line of sight to the top of the building is the angle of elevation, θ = 30°.

The trigonometric relationship between the perpendicular, the base, and the angle of elevation is defined by the tangent function:

tan(θ) = (P)/(B)

Substituting the given height P = 1000 m and θ = 30° into the formula:

tan(30°) = (1000)/(B)

The value of tan(30°) is known to be (1)/(√(3)). Substituting this value into the equation:

(1)/(√(3)) = (1000)/(B)

Rearranging the equation to solve for the distance B:

B = 1000 × √3

Using the standard mathematical approximation for the square root of 3, where √3 ≈ 1.732:

B = 1000 × 1.732 = 1732

Explanation

The total work done by two individuals working together in a specific time period is the sum of the work each individual completes independently during that same time period. A can complete 20% of the job in 7 days, and B can complete 25% of the job in 7 days. When they work together for 7 days, their individual contributions are additive:

Total percentage completed = Percentage by A + Percentage by B

Substituting the given values into the equation:

Total percentage = 20% + 25% = 45%

Because the time frame for their individual work matches the time frame for their combined work (7 days), we simply sum the percentages to find the total progress made on the job.

Explanation

The numerator sin⁴θ + cos⁴θ is expanded using the algebraic identity a² + b² = (a + b)² - 2ab. By substituting a = sin²θ and b = cos²θ, the expression becomes (sin²θ + cos²θ)² - 2sin²²θ. Using the standard trigonometric identity sin²θ + cos²θ = 1, the numerator simplifies to 1² - 2sin²²θ, which is exactly equal to the denominator 1 - 2sin²²θ. Dividing these identical terms results in 1. This can also be quickly verified by substituting a specific value such as θ = 0°, which yields (0⁴ + 1⁴)/(1 - 0) = 1.

Explanation

The given equation is (X - 5Y)/(X + 5Y) = ⁷⁄₁₃. By applying the principle of cross-multiplication, we can solve for the ratio of X to Y:

13(X - 5Y) = 7(X + 5Y)

Expanding both sides of the equation:

13X - 65Y = 7X + 35Y

To isolate the variables, group the X terms on one side and the Y terms on the other:

13X - 7X = 35Y + 65Y

6X = 100Y

To find the value of (X)/(Y), divide both sides by 6Y:

(X)/(Y) = ¹⁰⁰⁄₆

Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, 2:

(X)/(Y) = ⁵⁰⁄₃

Alternatively, using the rule of Componendo and Dividendo, if (a-b)/(a+b) = (c)/(d), then (a)/(b) = (d+c)/(d-c). Applying this to (X)/(5Y) = ¹³⁺⁷⁄₁₃₋₇ gives (X)/(5Y) = ²⁰⁄₆ = ¹⁰⁄₃. Multiplying by 5 yields (X)/(Y) = ⁵⁰⁄₃.

Explanation

The average score for each player is determined by the formula Average = (Score₁ + Score₂ + Score₃)/(3). Since the number of attempts is constant for all participants, the highest average corresponds to the highest total sum of scores. Preeti's scores are 74.30, 75.00, and 77.50. Her total score is 74.30 + 75.00 + 77.50 = 226.80, and her average score is (226.80)/(3) = 75.60. This average is the highest among all players because her individual scores are consistently greater than the scores achieved by any other competitor in the tournament.

Explanation

Given the algebraic identity x² + (1)/(x²) = (x + ¹⁄ₓ)² - 2, we substitute the given value x + ¹⁄ₓ = 10: x² + (1)/(x²) = 10² - 2 = 100 - 2 = 98

To find the value of x⁴ + (1)/(x⁴), we treat x² as the base variable and apply the identity again: x⁴ + (1)/(x⁴) = (x² + (1)/(x²))² - 2 x⁴ + (1)/(x⁴) = 98² - 2

The square of 98 can be calculated quickly using the identity (a - b)² = a² - 2ab + b²: 98² = (100 - 2)² = 10000 - 400 + 4 = 9604

Substituting this value back into the equation: x⁴ + (1)/(x⁴) = 9604 - 2 = 9602

Explanation

The net gain or loss percentage can be calculated using the successive percentage change formula: a + b + ᵃᵇ⁄₁₀₀, where a represents the percentage markup and b represents the percentage discount.

Given a markup of 20% (a = 20) and a discount of 15% (b = -15):

Net percentage = 20 - 15 + (20 × (-15))/(100) Net percentage = 5 - ³⁰⁰⁄₁₀₀ Net percentage = 5 - 3 = 2%

Since the result is positive, it signifies a 2% gain. Alternatively, if we assume the cost price is 100, the marked price is 120 (20% above cost). A discount of 15% on the marked price equals 0.15 × 120 = 18. Subtracting this discount from the marked price gives a selling price of 120 - 18 = 102. Comparing the selling price of 102 to the cost price of 100 results in a 2% gain.

Explanation

Let the six-digit number be N. Since N is divisible by 33, it can be expressed as N = 33k for some integer k. The new number formed by adding 54 is N + 54. Substituting the value of N, we get:

M = 33k + 54

We can factor out 3 from both terms in the expression because 33 is 3 × 11 and 54 is 3 × 18:

M = 3(11k + 18)

Since 11k + 18 is an integer, the expression 3(11k + 18) represents a multiple of 3. This follows the mathematical principle that if two numbers are divisible by the same divisor, their sum is also divisible by that divisor. Because the original number N is a multiple of 33 (and thus a multiple of 3) and 54 is also a multiple of 3, their sum must be divisible by 3.

Explanation

The area of a sector of a circle is calculated using the formula A = (θ)/(360°) × π r², where θ is the central angle and r is the radius. Given the radius r = 10 cm, the angle θ = 60°, and π = 3.14, the calculation is:

A = ⁶⁰⁄₃₆₀ × 3.14 × (10)²

A = ⅙ × 3.14 × 100

A = ³¹⁴⁄₆

A = 52.333... cm²

Rounding to two decimal places, the area of the sector is 52.33 cm².

Explanation

The volume of a sphere is given by the formula V = ⁴⁄₃π r³. Since the radius r is half of the diameter d, the formula can be expressed in terms of diameter as V = (π)/(6) d³. When a larger sphere is melted and recast into smaller spheres, the total volume remains conserved. Thus, the cube of the diameter of the original ball is equal to the sum of the cubes of the diameters of the three smaller balls.

Let d be the diameter of the original ball, and d₁, d₂, and d₃ be the diameters of the three smaller balls. The relationship is given by:

d³ = d₁³ + d₂³ + d₃³

Substituting the given values into the equation:

3³ = (1.5)³ + 2³ + d₃³ 27 = 3.375 + 8 + d₃³ 27 = 11.375 + d₃³

Rearranging the equation to solve for d₃³:

d₃³ = 27 - 11.375 d₃³ = 15.625

Taking the cube root of both sides to find the diameter of the third ball:

d₃ = ³√(15.625) = 2.5 cm

Explanation

The total distance is the sum of the distances covered in the two parts of the journey. Using the formula Distance = Speed × Time, and converting time from minutes to hours by dividing by 60:

For the first part: D₁ = 90 × (37.5)/(60) = 1.5 × 37.5 = 56.25 km In fractional form, 56.25 = 56 ¼ = ²²⁵⁄₄ km.

For the second part: D₂ = 80 × ¹⁴⁄₆₀ = ⁴⁄₃ × 14 = ⁵⁶⁄₃ = 18 ⅔ km

Total distance D = D₁ + D₂: D = ²²⁵⁄₄ + ⁵⁶⁄₃ = (225 × 3 + 56 × 4)/(12) = (675 + 224)/(12) = ⁸⁹⁹⁄₁₂

Converting the improper fraction to a mixed number: 899 ÷ 12 = 74 with a remainder of 11. Therefore, the total distance is 74 ¹¹⁄₁₂ km.

Explanation

Let A be the average expenditure of all 20 students. According to the problem, the 20th student spent A + 19. The average expenditure is calculated by distributing the excess amount spent by the 20th student equally among the remaining 19 students: A = 175 + (19 / 19) = 175 + 1 = 176 The total money spent by all 20 students is the product of the total number of students and their average expenditure: Total spent = 20 × 176 = 3520

Explanation

The expression is in the form of the algebraic identity a² - b²{a + b}. Applying the difference of squares formula, a² - b² = (a + b)(a - b), the expression simplifies as follows:

a² - b²{a + b} = ((a + b)(a - b))/(a + b) = a - b

By substituting a = 3.35 and b = 1.25:

3.35 - 1.25 = 2.1