SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

The ratio of the flight fare increase is 11:13, which implies that the original fare is represented by 11 parts and the new fare by 13 parts. The increase in the fare is the difference between these parts, which is 13 - 11 = 2 units.

According to the problem, the original fare of 11 units is equal to Rs 12100. We can find the value of one unit by dividing the original fare by 11: 1 unit = ¹²¹⁰⁰⁄₁₁ = 1100

To find the actual increase in the fare, we multiply the value of one unit by the number of units representing the increase: Increase = 2 × 1100 = 2200

Explanation

Let the first number be x and the sum of the other two numbers be S. According to the problem, the average of the three numbers is 112, so their total sum is 112 × 3 = 336. This gives us the equation x + S = 336.

The problem also states that the first number is 1/6th the sum of the other two, which can be expressed as x = ⅙S, or S = 6x. Substituting S = 6x into the sum equation:

x + 6x = 336 7x = 336 x = ³³⁶⁄₇ = 48

Explanation

The total cost price (CP) for 800 kg of tomatoes is Rs 7200. To achieve a profit of 30%, the total selling price (SP) is calculated using the formula: Total SP = CP × (1 + (Profit%)/(100)) Total SP = 7200 × (1 + 0.30) = 7200 × 1.3 = 9360

The quantity of tomatoes available for sale decreases because 10% are damaged. The usable quantity is: Remaining weight = 800 × (1 - 0.10) = 800 × 0.9 = 720 kg

To find the selling rate per kg, divide the total required selling price by the remaining weight: Rate = (Total SP)/(Remaining weight) = ⁹³⁶⁰⁄₇₂₀ = ⁹³⁶⁄₇₂ = 13 Rs/kg

Explanation

Let the maximum marks of the subject be M. The passing marks are defined as the sum of the marks obtained by the student and the marks by which the student failed. Since the student secured 82 marks and fell short by 23 marks, the passing marks are 82 + 23 = 105.

The problem states that the student needs to secure 35% of the maximum marks to pass. This provides the equation: 35% of M = 105

To find the maximum marks M, we set up the calculation as follows: ³⁵⁄₁₀₀ × M = 105 M = (105 × 100)/(35)

Dividing 105 by 35 gives 3. Therefore: M = 3 × 100 = 300

The maximum marks for the subject are 300.

Explanation

To find the distance in metres, we first convert the speed from kilometres per hour (km/hr) to metres per second (m/s) using the conversion factor 1 km/hr = ⁵⁄₁₈ m/s.

The speed of the spaceship in m/s is calculated as: Speed = 810 × ⁵⁄₁₈ = 45 × 5 = 225 m/s

Now, using the formula Distance = Speed × Time, we find the distance travelled in ⅕ of a second: Distance = 225 × ⅕ = 45 metres

Explanation

In compound interest, the amount at the end of any year serves as the principal for the following year. Therefore, the difference between the amounts at the end of two consecutive years represents the interest earned on the amount from the earlier year.

The interest earned during the third year is: Interest = 2268 - 2100 = 168

The rate of interest r is calculated by expressing this interest as a percentage of the amount at the end of the second year: r = (A₃ - A₂)/(A₂) × 100 r = ¹⁶⁸⁄₂₁₀₀ × 100 r = ¹⁶⁸⁄₂₁ r = 8

The annual rate of interest is 8%.

Explanation

The given equation is 5x - ½(2x - 7) = 5.5. Distributing the -½ across the terms inside the parentheses simplifies the expression to 5x - x + 3.5 = 5.5. Combining the like terms yields 4x + 3.5 = 5.5. Subtracting 3.5 from both sides of the equation gives 4x = 5.5 - 3.5, which simplifies to 4x = 2. Dividing both sides by 4 results in x = ²⁄₄, which reduces to the final value x = ½.

Explanation

The value of a³ + b³ can be calculated using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Given the values a + b = 4 and ab = -5, we substitute them directly into the formula.

Substituting the values, we get: a³ + b³ = (4)³ - 3(-5)(4) a³ + b³ = 64 - (-60) a³ + b³ = 64 + 60 a³ + b³ = 124

Alternatively, using the identity a³ + b³ = (a + b)(a² - ab + b²), we first find a² + b² using (a + b)² = a² + b² + 2ab. Substituting the knowns gives 16 = a² + b² - 10, so a² + b² = 26. Then, a³ + b³ = 4(26 - (-5)) = 4(31) = 124.

Explanation

Let the number be x. According to the problem, the sum of the number and 4 times its reciprocal is 5. This relationship can be expressed by the equation x + ⁴⁄ₓ = 5. Multiplying the entire equation by x to remove the denominator results in the quadratic equation x² + 4 = 5x, which can be rearranged to x² - 5x + 4 = 0. Factoring this quadratic equation gives (x - 4)(x - 1) = 0, leading to the possible values x = 4 or x = 1. Among the choices provided, the value 4 satisfies the condition. Testing the value directly, 4 + ⁴⁄₄ = 4 + 1 = 5, which confirms the result.

Explanation

In an arithmetic progression, the nᵗʰ term is given by the formula aₙ = a + (n-1)d. The common difference d can be determined by the relation d = (aₙ - aₘ)/(n - m).

Given the 3rd term a₃ = -13 and the 8th term a₈ = 2, we find d: d = (2 - (-13))/(8 - 3) = ¹⁵⁄₅ = 3

To find the 14th term a₁₄, we use the property aₙ = aₘ + (n-m)d starting from the 8th term: a₁₄ = a₈ + (14 - 8)d a₁₄ = 2 + 6(3) a₁₄ = 2 + 18 a₁₄ = 20

Alternatively, calculating from the 3rd term yields the same result: a₁₄ = a₃ + (14 - 3)d a₁₄ = -13 + 11(3) a₁₄ = -13 + 33 = 20

Explanation

The reflection of a point P(x, y) in a horizontal line y = k results in an image point P'(x', y') where the x-coordinate remains unchanged because the reflection is purely vertical. The line y = k serves as the midpoint between the y-coordinate of the original point and the y-coordinate of its image. This relationship is expressed by the formula y' = 2k - y.

For the point (4, -3) and the line of reflection y = 1, the x-coordinate stays 4. To find the new y-coordinate, substitute k = 1 and y = -3 into the formula: y' = 2(1) - (-3). This calculation yields y' = 2 + 3 = 5. Therefore, the coordinates of the reflected point are (4, 5).

Explanation

Let the point T(x, 0) divide the line segment joining S(5, 1) and U(-1, -2) in the ratio m : n. According to the section formula, the y-coordinate of a point dividing a line segment joining (x₁, y₁) and (x₂, y₂) is given by y = my₂ + ny₁{m + n}.

Given the y-coordinate of point T is 0, we substitute y = 0, y₁ = 1, and y₂ = -2 into the formula: 0 = (m(-2) + n(1))/(m + n)

Multiplying both sides by (m + n) gives: 0 = -2m + n 2m = n ᵐ⁄ₙ = ½

This result shows that the point T divides the segment SU in the ratio 1 : 2.

Explanation

A line intersects the Y-axis at the specific point where its x-coordinate is equal to zero. To determine the coordinates of this intersection for the linear equation 3x + 2y = 12, substitute x = 0 into the expression:

3(0) + 2y = 12

0 + 2y = 12

2y = 12

Dividing both sides by 2 gives:

y = ¹²⁄₂

y = 6

Since the intersection occurs at x = 0 and the corresponding value of y is 6, the line cuts the Y-axis at the coordinate point (0, 6).

Explanation

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. This relationship is given by the formula:

(Area(Δ ABC))/(Area(Δ PQR)) = ((AB)/(PQ))²

Substituting the given area values and the length of PQ:

¹²¹⁄₆₄ = ((AB)/(12))²

To find the ratio of the sides, take the square root of both sides of the equation:

√(¹²¹⁄₆₄) = (AB)/(12)

¹¹⁄₈ = (AB)/(12)

Now, solve for AB using cross-multiplication:

AB = (11 × 12)/(8)

AB = (11 × 3)/(2)

AB = ³³⁄₂ = 16.5

Explanation

The trigonometric values for an angle of 60° are based on standard geometric ratios. The value of tan 60° is √3, and the value of csc 60° is the reciprocal of sin 60°. Since sin 60° equals √3{2}, it follows that csc 60° is (2)/(√(3)). Substituting these specific values into the given expression yields: tan 60° + csc 60° = √3 + (2)/(√(3))

To simplify this sum, we find a common denominator by expressing √3 as a fraction with √3 in the denominator: √3 = √3 · √3{√3} = (3)/(√(3))

Now, adding the two terms together results in: (3)/(√(3)) + (2)/(√(3)) = (3 + 2)/(√(3)) = (5)/(√(3))

Explanation

In the right-angled triangle Δ XYZ, the sum of angles is 180°. Since m∠ Y = 90° and m∠ Z = 60°, the measure of angle X is calculated as: m∠ X = 180° - (90° + 60°) = 30°

The value of sec 30° is 2/√3. Substituting this into the given expression: (1)/(√(2)) sec X = (1)/(√(2)) × sec 30° = (1)/(√(2)) × (2)/(√(3)) = (2)/(√(6))

Explanation

In trigonometry, csc θ = (Hypotenuse)/(Perpendicular). Given csc θ = ¹⁷⁄₈, the Hypotenuse is 17 units and the Perpendicular is 8 units. Using the Pythagorean theorem, the Base (B) is calculated as follows:

B = √(Hypotenuse²⁾⁻^Pᵉʳᵖᵉⁿᵈⁱᶜᵘˡᵃʳ^^^² B = √(17²⁾⁻⁸^^^² = √(289 - 64) = √(225) = 15

The cosine function is defined as cos θ = (Base)/(Hypotenuse). Substituting the calculated Base and the given Hypotenuse:

cos θ = ¹⁵⁄₁₇

This relation is derived from the standard Pythagorean triplet (8, 15, 17), where 17 represents the longest side (hypotenuse).

Explanation

To find the least number to be added to 4131 so that it becomes perfectly divisible by 19, we first divide 4131 by 19 to find the remainder.

Using the division algorithm: 4131 = (19 × 217) + 8

The remainder is 8. The least number that must be added to a dividend to make it exactly divisible by a divisor is given by the formula: Required Number = Divisor - Remainder

Substituting the given values into the formula: 19 - 8 = 11

When 11 is added to 4131, the result is 4142, which is 19 × 218.

Explanation

The time taken to complete a job working together is determined by the sum of the individual work rates. Using the LCM method, let the total work be the Least Common Multiple of the individual times 12, 8, and 24, which is 24 units. The efficiency of A is 24 / 12 = 2 units per day, the efficiency of B is 24 / 8 = 3 units per day, and the efficiency of C is 24 / 24 = 1 unit per day. Their combined efficiency when working together is 2 + 3 + 1 = 6 units per day. The total time taken to finish the job is the total work divided by the combined efficiency, calculated as 24 / 6 = 4 days.

Explanation

The area A of an equilateral triangle with side length s is determined by the formula A = √3{4} s². Given that the side length s = 14 cm, we substitute this value into the equation to find Area = √3{4} × (14)². Calculating the square of the side gives 14 × 14 = 196. By dividing 196 by 4, we obtain 49. Thus, the area of the triangle is 49√3 sq cm.

Explanation

The net discount for two successive discounts of x% and y% is calculated using the formula x + y - ˣʸ⁄₁₀₀. Substituting the given values x = 20 and y = 10 into this formula, the calculation becomes 20 + 10 - (20 × 10)/(100). This simplifies to 30 - ²⁰⁰⁄₁₀₀, which results in 30 - 2 = 28%. This formula accounts for the second discount being applied to the price remaining after the first discount has been deducted.

Explanation

The total profit of Rs 1,03,500 is divided among A, B, and C in the ratio 11:7:5. To find the share of B, first calculate the sum of the ratio parts: 11 + 7 + 5 = 23.

The value of one part is determined by dividing the total profit by the sum of the ratios: (1,03,500)/(23) = 4,500

B's share corresponds to 7 parts of the total ratio. Therefore, the amount B receives is calculated by multiplying the value of one part by B's ratio: 7 × 4,500 = 31,500

Explanation

The total weight of P, Q, and R is calculated by multiplying their average weight by the number of people: 45 × 3 = 135 kg. The sum of the weights of P and Q is 36.5 × 2 = 73 kg, while the sum of the weights of Q and R is 52 × 2 = 104 kg.

By adding the sums of P+Q and Q+R, we obtain the value for P + 2Q + R, which is 73 + 104 = 177 kg. To isolate the weight of Q, subtract the total weight of the three individuals from this combined sum: Q = (P + 2Q + R) - (P + Q + R) = 177 - 135 = 42 kg.

Explanation

Let the selling price (SP) of one item be x. The total selling price for 21 items is 21x. The problem states that the profit earned is equal to the selling price of 1 item, so Profit = x. The cost price (CP) of the 21 items is determined by the relation CP = SP - Profit. Substituting the values, we get CP = 21x - x = 20x. The profit percentage is calculated using the formula Profit% = (Profit)/(CP) × 100. Plugging in the values results in ˣ⁄₂₀ₓ × 100, which simplifies to ¹⁄₂₀ × 100 = 5%.

Explanation

The value is determined by the product of the base number and the decimal equivalents of the percentages using the formula Value = Number × Rate₁ × Rate₂. Substituting the given figures, the expression becomes 400 × ¹⁵⁰⁄₁₀₀ × ¹⁰⁄₁₀₀. Simplifying the terms leads to 400 × 1.5 × 0.1, which equals 600 × 0.1 = 60. Alternatively, one can calculate the effective percentage first by multiplying 10% and 150% to get 15%. Calculating 15% of 400 (0.15 × 400) yields the same result of 60.

Explanation

The average speed is calculated using the formula Speed = (Distance)/(Time). Given the distance is 144 km and the time is 3.2 hours, the speed in km/h is determined first:

Speed = (144)/(3.2) = ¹⁴⁴⁰⁄₃₂ = 45 km/h

To convert the speed from kilometers per hour to meters per second, the value is multiplied by the conversion factor ⁵⁄₁₈:

Speed in m/s = 45 × ⁵⁄₁₈

By simplifying the calculation, dividing both 45 and 18 by 9 yields:

Speed = 5 × ⁵⁄₂ = ²⁵⁄₂ = 12.5 m/s

Explanation

The rate of simple interest is calculated using the relation R = (Increase Percentage)/(Time). Since the principal increases by 18% in 2 years, the annual rate of interest R is 18% / 2 = 9% per annum.

For the compound interest calculation on Rs 7000 for 3 years at 9% per annum, the effective rate of interest over 3 years can be found using the successive percentage formula or the standard compound interest formula CI = P[(1 + (R)/(100))^T - 1].

Substituting the values: CI = 7000 × [(1 + ⁹⁄₁₀₀)³ - 1] CI = 7000 × [(1.09)³ - 1] CI = 7000 × [1.295029 - 1] CI = 7000 × 0.295029 = 2065.203

Rounding to the nearest decimal as per the options, the interest earned is Rs 2065.2.

Explanation

To solve for x, first simplify the expression within the brackets and distribute the external fraction: ˣ⁄₃ - ⁵⁄₂(⁷ˣ⁄₅ - ⁴⁄₃) = -ˣ⁄₆

Distributing ⁵⁄₂ into the parenthesis: ˣ⁄₃ - (⁵⁄₂ · ⁷ˣ⁄₅ - ⁵⁄₂ · ⁴⁄₃) = -ˣ⁄₆ ˣ⁄₃ - (⁷ˣ⁄₂ - ¹⁰⁄₃) = -ˣ⁄₆

Removing the parentheses by distributing the negative sign: ˣ⁄₃ - ⁷ˣ⁄₂ + ¹⁰⁄₃ = -ˣ⁄₆

To clear the denominators, multiply the entire equation by the least common multiple of 2, 3, and 6, which is 6: 6(ˣ⁄₃) - 6(⁷ˣ⁄₂) + 6(¹⁰⁄₃) = 6(-ˣ⁄₆) 2x - 21x + 20 = -x

Combine the x terms on one side: -19x + 20 = -x 20 = 19x - x 20 = 18x

Solving for x: x = ²⁰⁄₁₈ x = ¹⁰⁄₉

Explanation

The algebraic identity for the sum of two cubes is a³ + b³ = (a + b)³ - 3ab(a + b). Substituting the given values a³ + b³ = 19 and a + b = 1 into this formula, we get 19 = (1)³ - 3ab(1). This simplifies the equation to 19 = 1 - 3ab. By rearranging the terms, we find 3ab = 1 - 19, which gives 3ab = -18. Dividing both sides by 3 results in ab = -6.

Explanation

Let the required fraction be x. According to the given condition, the fraction is greater than its reciprocal by ⁷²⁄₇₇, which can be expressed as the equation x - ¹⁄ₓ = ⁷²⁄₇₇.

By substituting the value x = ¹¹⁄₇, the reciprocal is ⁷⁄₁₁. Calculating the difference between the two:

¹¹⁄₇ - ⁷⁄₁₁ = 11² - 7²{77} = (121 - 49)/(77) = ⁷²⁄₇₇

Alternatively, the equation x - ¹⁄ₓ = ⁷²⁄₇₇ can be rearranged into the quadratic form x² - 1{x} = ⁷²⁄₇₇, leading to 77x² - 72x - 77 = 0. Solving for x using the quadratic formula x = -b ± √(b²⁾⁻⁴ᵃᶜ{2a}:

x = 72 ± √((-72)²⁾⁻⁴⁽⁷⁷⁾⁽⁻⁷⁷⁾{2(77)} = 72 ± √(5184 + 23716){154} = 72 ± √(28900){154}

x = (72 ± 170)/(154)

Taking the positive root, x = (72 + 170)/(154) = ²⁴²⁄₁₅₄. Dividing both terms by 22 simplifies the fraction to ¹¹⁄₇.

Explanation

The sum of the first n terms of an arithmetic progression is determined by the formula Sₙ = ⁿ⁄₂(a + l), where a is the first term, l is the last term, and n is the number of terms. Using the given values a = -19, l = 36, and n = 12, the sum is calculated as:

S₁₂ = ¹²⁄₂(-19 + 36) S₁₂ = 6 × 17 S₁₂ = 102

Explanation

When a point (x, y) is reflected in a vertical line defined by x = k, the y-coordinate of the point remains unchanged while the x-coordinate is transformed. The new x-coordinate x' is calculated using the formula x' = 2k - x, which ensures that the line x = k is the midpoint between the original x and the reflected x'.

Given the point (-2, 5) and the line of reflection x = -1, we have x = -2, y = 5, and k = -1. Applying the formula, the new x-coordinate is x' = 2(-1) - (-2) = -2 + 2 = 0. Since the y-coordinate remains 5, the coordinates of the reflected point are (0, 5).

Explanation

The midpoint formula states that the coordinates of the midpoint P(xₚ, yₚ) of a line segment with endpoints A(x₁, y₁) and B(x₂, y₂) are given by xₚ = (x₁ + x₂)/(2) and yₚ = (y₁ + y₂)/(2). Given the midpoint P(3, 1) and point B(5, -4), the coordinates of A(x₁, y₁) are found by substituting these values into the formula. For the horizontal coordinate, 3 = (x₁ + 5)/(2), which implies 6 = x₁ + 5 and x₁ = 1. For the vertical coordinate, 1 = (y₁ - 4)/(2), which implies 2 = y₁ - 4 and y₁ = 6.

Explanation

The slope of the line passing through (-5, 1) and (-2, 0) is calculated using the formula (y2 - y1) / (x2 - x1), which gives (0 - 1) / (-2 - (-5)) = -1/3. The slope of a perpendicular line is the negative reciprocal of this value, which is -1 / (-1/3) = 3.

Explanation

To find the value of the expression (1)/(√(2)) cot 45° + (1)/(√(3)) csc 60°, substitute the standard trigonometric values cot 45° = 1 and csc 60° = (2)/(√(3)).

The expression becomes: ((1)/(√(2)) × 1) + ((1)/(√(3)) × (2)/(√(3))) = (1)/(√(2)) + ⅔

To combine these terms, find the common denominator, which is 3√2. Adjusting the numerators accordingly: = 3 + 2√2{3√2}

Explanation

In △ ABC right-angled at B, the sum of the angles is 180°. Given m∠B = 90° and m∠A = 60°, the measure of angle C is calculated as 180° - (90° + 60°) = 30°.

To find the value of sec C · sin A, substitute the values of the angles: sec 30° · sin 60°

Using standard trigonometric values: sec 30° = 1/cos 30° = 2/√3 sin 60° = √3/2

The product is: (2/√3) · (√3/2) = 1

Explanation

Given tan θ = 7/24, we can use the fundamental trigonometric identity sec² θ = 1 + tan² θ. Substituting the given value:

sec² θ = 1 + (7/24)² sec² θ = 1 + 49/576 sec² θ = (576 + 49)/576 sec² θ = 625/576

Taking the square root of both sides: sec θ = √(625/576) = 25/24

Alternatively, in a right-angled triangle, tan θ = Perpendicular/Base = 7/24. Using the Pythagorean theorem, the Hypotenuse is √(7² + 24²) = √(49 + 576) = √625 = 25. Since sec θ = Hypotenuse/Base, we get 25/24. The values (7, 24, 25) form a standard Pythagorean triplet, allowing for a direct calculation of the ratio.

Explanation

To find the quotient, we determine the largest integer q such that 66 × q ≤ 7251. A quick estimation using a multiple of 10 shows that 66 × 110 = 7260. Since 7260 is slightly higher than the dividend 7251, we subtract one unit of the divisor to check the value for 109: 66 × 109 = 66 × (110 - 1) = 7260 - 66 = 7194. Subtracting 7194 from 7251 leaves a remainder of 57. Since this remainder is non-negative and less than the divisor 66, the integer quotient is 109.

Explanation

Let the efficiency of Bashir be x units of work per day. Since Asif is twice as good a workman as Bashir, Asif efficiency is 2x units of work per day. When working together, their combined efficiency is the sum of their individual efficiencies: Combined Efficiency = x + 2x = 3x units per day

The total work is the product of their combined efficiency and the time taken to complete the piece of work together: Total Work = 3x × 30 = 90x units

To find the time taken by Asif to finish the work alone, we divide the total work by Asif individual efficiency: Time taken by Asif = (Total Work)/(Asif Efficiency) = ⁹⁰ˣ⁄₂ₓ = 45 days

This relationship follows the principle that efficiency is inversely proportional to the time taken. Because Asif is twice as efficient as Bashir, the combined group performs at three times the rate of Bashir alone. Since they finish in 30 days, Bashir would take 90 days, and Asif, being twice as fast as Bashir, takes half that time.

Explanation

To find the time taken by A, B, and C to finish the job together, we first determine the total work by taking the Least Common Multiple (LCM) of their individual completion times. The LCM of 72, 24, and 36 is 72 units.

We then calculate the efficiency (work done per day) for each individual: Efficiency of A = 72 / 72 = 1 unit per day Efficiency of B = 72 / 24 = 3 units per day Efficiency of C = 72 / 36 = 2 units per day

The combined efficiency of A, B, and C working together is the sum of their individual efficiencies: Total Efficiency = 1 + 3 + 2 = 6 units per day

The number of days required to complete the work together is given by the formula: Time = Total Work / Total Efficiency Time = 72 / 6 = 12 days

Alternatively, using the reciprocal method: 1/T = 1/72 + 1/24 + 1/36 1/T = (1 + 3 + 2) / 72 1/T = 6 / 72 T = 72 / 6 = 12 days

Explanation

The area of the four walls of a cuboid is given by its lateral surface area formula, A = 2h(l + b), where h is the height, l is the length, and b is the breadth.

Given: Area A = 448 sq cm Length l = 18 cm Height h = 8 cm

Substituting the given values into the formula: 448 = 2 × 8 × (18 + b) 448 = 16 × (18 + b)

To find the value of b, first divide the total area by 16: 448 / 16 = 18 + b 28 = 18 + b

Subtracting 18 from both sides gives the breadth: b = 28 - 18 b = 10 cm

Explanation

The Selling Price (SP) is related to the Marked Price (MP) and the Discount (D%) by the formula SP = MP × (100 - D)/(100). Since the Marked Price remains constant for both cases, we can use the ratio of selling prices:

SP₁{100 - D₁} = SP₂{100 - D₂}

Substituting the given values:

(4500)/(100 - 10) = SP₂{100 - 27.5}

⁴⁵⁰⁰⁄₉₀ = SP₂{72.5}

50 = SP₂{72.5}

SP₂ = 50 × 72.5

SP₂ = 3625

Explanation

If four numbers a, b, c, and x are in proportion such that a:b = c:x, then x is the fourth proportional. The formula to calculate the fourth proportional is x = (b × c) / a.

Substituting the given values a = 72, b = 168, and c = 150 into the formula: x = (168 × 150) / 72

To simplify the calculation, divide both 168 and 72 by their common factor 24: 168 / 24 = 7 72 / 24 = 3

The expression becomes: x = (7 × 150) / 3 x = 7 × 50 x = 350

Explanation

The numbers between 11 and 80 that are divisible by 6 form an arithmetic progression. The first number in this range divisible by 6 is 12, and the last number is 78 (as 6 × 13 = 78).

For any series in an arithmetic progression, the average is given by the formula:

Average = (a + l)/(2)

where a is the first term and l is the last term of the series.

Substituting the values:

Average = (12 + 78)/(2)

Average = ⁹⁰⁄₂

Average = 45

Explanation

Using the rule of alligation, the ratio of the quantities sold at 20% profit and 8% profit is calculated by the difference between the mean profit and the individual profit rates.

Ratio of quantity at 20% profit to quantity at 8% profit: (12 - 8) : (20 - 12) 4 : 8 1 : 2

The total quantity of rice is 960 kg. To find the amount sold at 8% profit, we divide the total quantity in the ratio 1 : 2. The part representing the 8% profit is the second term of the ratio.

Quantity sold at 8% profit = (2)/(1 + 2) × 960 ⅔ × 960 2 × 320 = 640 kg

Explanation

In a rectangle, the diagonal and the two adjacent sides form a right-angled triangle. According to the Pythagoras theorem, the square of the diagonal is equal to the sum of the squares of the two sides: d² = a² + b². Given the diagonal d = 25 cm and one side a = 24 cm, the other side b is calculated as:

b = √(d² - a²) b = √(25² - 24²) b = √(625 - 576) b = √(49) b = 7 cm

The area of a rectangle is the product of its two adjacent sides. Using the side lengths 24 cm and 7 cm:

Area = a × b Area = 24 × 7 Area = 168 sq cm

Explanation

The given condition is 40% of x = y, which can be written as y = ⁴⁰⁄₁₀₀x = 0.4x. The expression to be evaluated is y% of 40, which is ʸ⁄₁₀₀ × 40 = 0.4y. By substituting the value of y into this expression, we get 0.4 × (0.4x) = 0.16x. To represent 0.16x as a percentage of x, we multiply by 100, which gives 16%. Therefore, y% of 40 is equal to 16% of x.

Explanation

The duration for which a bus stops per hour can be determined using the formula: Stop time per hour = (Excluding speed - Including speed)/(Excluding speed) × 60 minutes. In one hour, the bus should ideally cover 72 km, but due to stoppages, it only covers 60 km. The loss in distance is 12 km. The time taken to cover this 12 km at the speed of 72 kmph is ¹²⁄₇₂ hours. Converting this into minutes, we get ¹²⁄₇₂ × 60 = ⅙ × 60 = 10 minutes.

Explanation

Simple interest is earned uniformly each year. Since the simple interest for two years is Rs 1500, the interest earned in the first year is SI₁ = ¹⁵⁰⁰⁄₂ = 750 Rs. For the first year, the simple interest and compound interest are exactly the same.

The difference between compound interest and simple interest over a two-year period is the interest earned on the first year's interest during the second year. This difference D is calculated as 1575 - 1500 = 75 Rs.

To find the rate of interest R, we calculate the percentage that this difference D represents relative to the first year's interest SI₁:

R = (D)/(SI₁₎ × 100

Substituting the values into the formula:

R = ⁷⁵⁄₇₅₀ × 100 = 10

The rate of interest is 10% per annum.

Explanation

To find the value of x, we solve the equation (7(5x)/(3) - ³⁄₂){2} + ³⁄₂ = ¼ by isolating the variable step by step. First, subtract ³⁄₂ from both sides to get (7(5x)/(3) - ³⁄₂){2} = ¼ - ⁶⁄₄ = -⁵⁄₄. Multiplying both sides by 2 yields 7(⁵ˣ⁄₃ - ³⁄₂) = -⁵⁄₂. Dividing both sides by 7 gives ⁵ˣ⁄₃ - ³⁄₂ = -⁵⁄₁₄. Next, add ³⁄₂ to both sides, which is ²¹⁄₁₄ in common denominator form: ⁵ˣ⁄₃ = ²¹⁄₁₄ - ⁵⁄₁₄ = ¹⁶⁄₁₄. Simplifying the fraction results in ⁵ˣ⁄₃ = ⁸⁄₇. Finally, multiply both sides by ⅗ to solve for x: x = ⁸⁄₇ × ⅗ = ²⁴⁄₃₅.