SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The total percentage difference between a 6% loss and a 10% profit is 10% - (-6%) = 16%. This 16% change in the value of the cost price (CP) represents the Rs. 64 increase in the selling price. Using the formula for the difference in prices:

Difference in Percentage × CP = Difference in Selling Price

16% of CP = 64

¹⁶⁄₁₀₀ × CP = 64

CP = (64 × 100)/(16)

CP = 4 × 100 = 400

The cost price of the article is Rs. 400.

Explanation

The total number of students wearing spectacles is calculated as 25% of the total student population: S = 1400 × 0.25 = 350

Out of the students wearing spectacles, ²⁄₇ are boys. Since the group consists of boys and girls, the fraction of girls wearing spectacles is 1 - ²⁄₇ = ⁵⁄₇.

To find the number of girls wearing spectacles, we multiply this fraction by the total number of spectacle-wearers: Number of girls = ⁵⁄₇ × 350 Number of girls = 5 × 50 = 250

Explanation

The speed of a person in still water is the average of their downstream speed and upstream speed. Using the standard formula: Speed in still water v = (d + u)/(2) where d is the speed downstream and u is the speed upstream. Substituting the given values: v = (18 km/hr + 12 km/hr)/(2) v = ³⁰⁄₂ v = 15 km/hr

Explanation

From the given ratio ((a+b)²)/((a-b)²) = ²⁵⁄₄, we can express (a+b)² in terms of (a-b)² as (a+b)² = ²⁵⁄₄(a-b)². Using the standard algebraic identity (a+b)² = (a-b)² + 4ab, we substitute the known value ab = 21 into the equation:

²⁵⁄₄(a-b)² = (a-b)² + 4(21)

By rearranging the terms to solve for (a-b)²:

²⁵⁄₄(a-b)² - (a-b)² = 84 ²¹⁄₄(a-b)² = 84 (a-b)² = (84 × 4)/(21) = 16

Now, find the value of (a+b)² using the initial ratio:

(a+b)² = ²⁵⁄₄ × 16 = 100

The expression to be evaluated is a² + b² + 3ab. This can be simplified by relating it to the expansion of (a+b)²:

a² + b² + 3ab = (a² + b² + 2ab) + ab = (a+b)² + ab

Substituting the calculated values (a+b)² = 100 and ab = 21:

100 + 21 = 121

Explanation

The expression is simplified using the quotient rule for exponents, which states that dᵐ ÷ dⁿ = dᵐ⁻ⁿ. Applying this rule first to the expression within the parentheses:

d⁵⁺ᵗ ÷ d⁵ = d⁽⁵⁺ᵗ⁾⁻⁵ = dᵗ

Now, substituting this result back into the remaining part of the original problem:

dᵗ ÷ dᵗ = dᵗ⁻ᵗ = d⁰

According to the laws of exponents, any non-zero base raised to the power of 0 is equal to 1. Thus, the value of the given expression is 1.

Explanation

According to the Angle Sum Property of a triangle, the sum of all three interior angles must be exactly 180°. For the measures provided in this option, the calculation is 50° + 60° + 70° = 110° + 70° = 180°. Since the sum equals exactly 180°, these three angles can successfully form a triangle.

Explanation

The expression sec⁴θ - tan⁴θ can be simplified using the algebraic identity for the difference of squares, a² - b² = (a + b)(a - b). Treating sec²θ as a and tan²θ as b, the expression becomes:

sec⁴θ - tan⁴θ = (sec²θ + tan²θ)(sec²θ - tan²θ)

From the fundamental trigonometric identity, we know that 1 + tan²θ = sec²θ, which implies that sec²θ - tan²θ = 1. The problem provides the value for the other factor as sec²θ + tan²θ = √3. Substituting these two values into the equation:

sec⁴θ - tan⁴θ = (√3)(1) = √3

Explanation

The largest 6-digit number is 999999. To find the greatest perfect square within this range, we look for the largest integer whose square is less than 1,000,000. Since 1,000,000 is the smallest 7-digit number and is equal to 1000², the largest 6-digit perfect square must be 999². Using the algebraic identity (a - b)² = a² - 2ab + b² where a = 1000 and b = 1, we calculate 999² = (1000 - 1)² = 1000² - 2(1000)(1) + 1² = 1000000 - 2000 + 1 = 998001.

Explanation

The average height of the entire class is calculated by finding the total height of all students and dividing it by the total number of students. The class consists of 50 boys in total. The first group of 30 boys has an average height of 160 cm, and the remaining 20 boys, calculated as 50 - 30 = 20, have an average height of 165 cm.

The total height of the class is the sum of the heights of both groups: Total Height = (30 × 160) + (20 × 165) = 4800 + 3300 = 8100 cm

The average height for the whole class of 50 boys is: Average = ⁸¹⁰⁰⁄₅₀ = 162 cm

This can be calculated more efficiently using the weighted average formula A = (n₁ A₁ + n₂ A₂)/(n₁ + n₂). By simplifying the ratio of the two groups from 30:20 to 3:2, the computation becomes: A = (3 × 160 + 2 × 165)/(3 + 2) = (480 + 330)/(5) = ⁸¹⁰⁄₅ = 162 cm

Explanation

The expression for the difference of two cubes is given by the identity a³ - b³ = (a - b)(a² + ab + b²). Substituting the given values a³ - b³ = 26 and a - b = 2 into this formula, we get 26 = 2(a² + ab + b²), which simplifies to a² + ab + b² = 13.

We also know that (a - b)² = a² - 2ab + b². Since a - b = 2, we have 2² = a² - 2ab + b², which means a² - 2ab + b² = 4.

To find the value of ab, we subtract the second equation from the first: (a² + ab + b²) - (a² - 2ab + b²) = 13 - 4. This yields 3ab = 9, so ab = 3.

The expression (a + b)² can be expanded as a² + 2ab + b². Using the previous result a² + ab + b² = 13, we can add ab to both sides to get a² + 2ab + b² = 13 + ab. Substituting ab = 3, we find (a + b)² = 13 + 3 = 16. Alternatively, using the identity (a + b)² = (a - b)² + 4ab, we get (a + b)² = 2² + 4(3) = 4 + 12 = 16.

Explanation

The expression follows the standard algebraic identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca). Let a = (X - 4), b = (Y - 2), and c = (Z - 3).

First, calculate the sum of these three terms: a + b + c = (X - 4) + (Y - 2) + (Z - 3) a + b + c = X + Y + Z - (4 + 2 + 3) a + b + c = X + Y + Z - 9

Substituting the given condition X + Y + Z = 9 into the equation: a + b + c = 9 - 9 = 0

Whenever the sum of three terms is zero (a + b + c = 0), the identity a³ + b³ + c³ - 3abc evaluates to zero because the entire right-hand side of the identity (a + b + c)(a² + b² + c² - ab - bc - ca) becomes 0 × (a² + b² + c² - ab - bc - ca) = 0. Consequently, the expression (X - 4)³ + (Y - 2)³ + (Z - 3)³ - 3(X - 4)(Y - 2)(Z - 3) equals 0.

Explanation

The medians of a triangle intersect at the centroid G and divide the triangle into six smaller triangles of equal area. These six triangles are Δ AGE, Δ CGE, Δ CGD, Δ BGD, Δ BGF, and Δ AGF. Given that the area of the entire triangle ABC is 36 sq cm, the area of each of these six constituent triangles is exactly one-sixth of the total area.

Area of Δ CGE = ⅙ × Area of Δ ABC

Substituting the given value:

Area of Δ CGE = ⅙ × 36 = 6 sq cm

Explanation

Let the center of the circle be O and the chord be AB. Since the length of the chord AB is equal to the radius r of the circle, we have OA = OB = AB = r. This makes △ OAB an equilateral triangle, where each internal angle is 60°. Therefore, ∠ OAB = 60°.

The radius OA is perpendicular to the tangent drawn at the extremity A. Thus, the angle between the radius OA and the tangent is 90°. The angle between the tangent and the chord AB is the difference between the angle formed by the radius and the tangent and the interior angle of the equilateral triangle. By calculation, this angle is 90° - 60° = 30°.

Alternatively, using the Alternate Segment Theorem, the angle subtended by chord AB at the center is 60°. The angle subtended by the same chord at any point on the major arc of the circumference is half the central angle, which is (60°)/(2) = 30°. The theorem states that the angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment, which confirms the value is 30°.

Explanation

From the given equations π sin θ = 1 and π cos θ = 1, we can divide the first by the second to obtain (π sin θ)/(π cos θ) = ¹⁄₁, which simplifies to tan θ = 1. Since both sin θ and cos θ are positive, θ lies in the first quadrant, giving θ = (π)/(4) (or 45°).

Now, substitute the value of θ into the required expression: √3 tan(⅔θ) + 1 = √3 tan(⅔ · (π)/(4)) + 1 = √3 tan((π)/(6)) + 1

Using the standard trigonometric value tan((π)/(6)) = (1)/(√(3)), we get: √3 ((1)/(√(3))) + 1 = 1 + 1 = 2

Explanation

The difference between compound interest and simple interest for a period of 3 years is given by the formula D = P ( ʳ⁄₁₀₀ )² ( (300 + r)/(100) ), where P is the principal, r is the annual rate of interest, and D is the difference.

Given D = 93 and r = 10, substituting these values into the formula:

93 = P ( ¹⁰⁄₁₀₀ )² ( (300 + 10)/(100) )

93 = P ( ⅒ )² ( ³¹⁰⁄₁₀₀ )

93 = P ( ¹⁄₁₀₀ ) ( ³¹⁄₁₀ )

93 = (31P)/(1000)

P = (93 × 1000)/(31)

P = 3 × 1000 = 3000

Alternatively, the effective rate of simple interest for 3 years at 10% is 10% × 3 = 30%. The effective rate of compound interest for 3 years at 10% is calculated using the successive percentage formula as 33.1%. The difference in rates is 33.1% - 30% = 3.1%.

3.1% of P = 93

(3.1)/(100) × P = 93

P = (93 × 100)/(3.1) = 3000

Explanation

In this problem, let the distance from the observer to the base of the flagpost be d = 30 m. Let hₚₒₛₜ be the height of the flagpost up to the bottom of the flag, and H be the total height from the ground to the top of the flag. The height of the flag itself is h = H - hₚₒₛₜ.

Using the trigonometric relation tan θ = (height)/(distance), we find: For the bottom of the flag: tan 30° = hₚₒₛₜ/30 ⟹ hₚₒₛₜ = 30 tan 30° = (30)/(√(3)) = 10√3 m. For the top of the flag: tan 45° = (H)/(30) ⟹ H = 30 tan 45° = 30(1) = 30 m.

The height of the flag is h = H - hₚₒₛₜ = 30 - 10√3. Substituting √3 = 1.732: h = 30 - 10(1.732) = 30 - 17.32 = 12.68 m.

Explanation

The rate of work of the first son is 1/3 units per hour, and the rate of work of the second son is 1/6 units per hour. When working together, their combined rate of work is the sum of their individual rates: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2 units per hour. Since the father can do the job as fast as both sons working together, his work rate is also 1/2 units per hour. The time taken by the father to complete the job alone is the reciprocal of his work rate, which is 1 / (1/2) = 2 hours.

Explanation

A rhombus is a special type of parallelogram where all sides are of equal length. The perimeter P of a rhombus is given by the formula P = 4s, where s is the length of one side. Given a perimeter of 240 m, the side length is calculated as:

s = (P)/(4) = ²⁴⁰⁄₄ = 60 m

The area of any parallelogram, including a rhombus, can be calculated using the formula Area = base × height. In this case, the side length s acts as the base, and the distance between the parallel sides acts as the height h. Given h = 20 m:

Area = s × h = 60 × 20 = 1200 sq. m.

Explanation

The discount of 16 ⅔% is equivalent to the fraction ⅙. The relationship between the selling price SP and the printed price MP is defined by the formula SP = MP × (1 - d), where d is the discount fraction. Substituting the given values, we get 450 = MP × (1 - ⅙). This simplifies to 450 = MP × ⅚. Solving for MP yields MP = (450 × 6)/(5) = 90 × 6 = 540.

Explanation

The problem states that A receives ²⁄₉ of the combined share of B and C. This establishes a ratio between A and the sum of B and C such that A : (B + C) = 2 : 9. The total sum of Rs. 770 represents the sum of all parts in this ratio. The total number of parts is 2 + 9 = 11. To find the value of A's share, use the proportional division formula: A = ²⁄₁₁ × 770. Dividing 770 by 11 gives 70, and multiplying 70 by 2 results in 140. Thus, A's share is Rs. 140. This can also be verified by letting B + C = x, so A = ²⁄₉x. The equation x + ²⁄₉x = 770 leads to ¹¹⁄₉x = 770, which gives x = 630. Substituting x back into the expression for A yields A = ²⁄₉ × 630 = 140.

Explanation

The total cost price (CP) of the eggs is calculated by summing the costs of the two separate purchases. The first purchase of 4 dozen at ₹ 24 per dozen equals ₹ 96 (4 × 24), and the second purchase of 2 dozen at ₹ 32 per dozen equals ₹ 64 (2 × 32). Thus, the total cost price for 6 dozen eggs is ₹ 160 (96 + 64).

To calculate the required selling price (SP) per dozen for a 20% gain on the whole transaction, we use the formula: SP = (Total CP × (100 + Gain%))/(100 × Total Dozens)

Substituting the known values: SP = (160 × 120)/(100 × 6) SP = (160 × 1.2)/(6) SP = ¹⁹²⁄₆ = ₹ 32 per dozen.

Explanation

When one value is r% higher than another, the percentage by which the second value is lower than the first is calculated using the formula [ (r)/(100 + r) × 100 ]%. Given that P's salary is r = 25% higher than Q's, substituting the value into the formula gives:

Percentage lower = (25)/(100 + 25) × 100

Percentage lower = ²⁵⁄₁₂₅ × 100

Percentage lower = ⅕ × 100 = 20%

This can also be understood by assuming Q's salary is 100. If P's salary is 25% higher, P's salary is 125. The difference is 25. To find the percentage lower than P, we calculate the ratio of the difference to P's salary: ²⁵⁄₁₂₅ × 100, which results in 20%.

Explanation

To find the time when two individuals starting from the same point will meet again at the starting point while moving in a circular path, we calculate the Least Common Multiple (LCM) of the time each takes to complete one full round. In this scenario, A takes 40 seconds and B takes 50 seconds to complete one revolution.

The prime factorization of the given time intervals is: 40 = 2³ × 5¹ 50 = 2¹ × 5²

The LCM is the product of the highest powers of all prime factors involved: LCM(40, 50) = 2³ × 5² LCM(40, 50) = 8 × 25 LCM(40, 50) = 200

Alternatively, using a common divisor approach, we can see that both 40 and 50 are divisible by 10. Dividing them gives 4 and 5, which are co-prime. Therefore, the LCM = 10 × 4 × 5 = 200. This means after 200 seconds, A will have completed 5 rounds and B will have completed 4 rounds, bringing both back to the starting point simultaneously for the first time.

Explanation

The given equation is: x - ¹⁄₃ₓ = ⅓

To find the value, multiply both sides of the equation by 3x: 3x × (x - ¹⁄₃ₓ) = 3x × ⅓

Simplify both sides: 3x² - ³ˣ⁄₃ₓ = ³ˣ⁄₃ 3x² - 1 = x

By rearranging the terms of the equation (subtracting x from both sides and adding 1 to both sides), we obtain the identity: 3x² - x = 1

The initial manipulation 3x(x - ¹⁄₃ₓ) simplifies to 3x² - 1, which is equal to x. In algebraic problems of this type, the constant value is found by identifying the relationship where the variable terms equate to 1.

Explanation

The given equation p = 3 + ¹⁄ₚ can be rearranged into the form p - ¹⁄ₚ = 3. To find the value of p⁴ + (1)/(p⁴), we apply the algebraic identity (x - ¹⁄ₓ)² = x² + (1)/(x²) - 2.

Squaring both sides of p - ¹⁄ₚ = 3 gives: (p - ¹⁄ₚ)² = 3² p² + (1)/(p²) - 2 = 9 p² + (1)/(p²) = 11

Now, square this result again using the identity (x + ¹⁄ₓ)² = x² + (1)/(x²) + 2: (p² + (1)/(p²))² = 11² p⁴ + (1)/(p⁴) + 2 = 121 p⁴ + (1)/(p⁴) = 121 - 2 = 119

Explanation

In an isosceles triangle where AB = AC, the altitude from vertex A bisects the base BC at a point D, creating two congruent right-angled triangles ABD and ACD. By the Pythagorean theorem in △ ABD, we have AB² = AD² + BD². Substituting the given values AB = 15 cm and AD = 12 cm:

15² = 12² + BD² 225 = 144 + BD² BD² = 81 BD = 9 cm

Since the altitude bisects the base, BC = 2 × BD = 2 × 9 = 18 cm. This can also be determined by recognizing the Pythagorean triplet (3, 4, 5) scaled by a factor of 3, resulting in (9, 12, 15). Given the hypotenuse is 15 and the altitude is 12, the half-base must be 9 cm.

Explanation

According to the Midpoint Theorem, the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is exactly half the length of that third side. In triangle ABC, since X and Y are the midpoints of AB and AC, the relationship between XY and BC is given by the formula XY = ½BC, which can also be written as BC = 2XY.

Given the equation BC + XY = 12 units, we substitute BC with 2XY: 2XY + XY = 12 3XY = 12 XY = 4 units

Substituting the value of XY back into the relation for BC: BC = 2 × 4 = 8 units

To find the value of BC - XY: BC - XY = 8 - 4 = 4 units

Explanation

For any angle θ in the range 0 < θ < 90°, the value of sinθ lies in the open interval (0, 1). A fundamental algebraic property states that for any real number x such that 0 < x < 1, the square of that number is strictly less than the number itself, expressed as x² < x. Since sinθ represents such a value in the first quadrant, substituting x = sinθ into the inequality yields sin²θ < sinθ. This relationship can be verified using a standard angle such as θ = 30°, where sin 30° = 1/2 = 0.5 and sin² 30° = 1/4 = 0.25; clearly, 0.5 > 0.25. Thus, within the given boundaries, the sine of the angle is always greater than its square.

Explanation

Let the two numbers be x and y. Based on the given conditions, x - y = 9 and x² - y² = 207. Using the algebraic identity x² - y² = (x + y)(x - y), we can substitute the known values to find the sum of the numbers:

207 = (x + y)(9) x + y = ²⁰⁷⁄₉ = 23

To find the individual numbers, we solve the system of equations: x + y = 23 x - y = 9

Adding the two equations: (x + y) + (x - y) = 23 + 9 2x = 32 x = 16

Subtracting the equations: (x + y) - (x - y) = 23 - 9 2y = 14 y = 7

The numbers are 16 and 7. These satisfy both conditions as 16 - 7 = 9 and 16² - 7² = 256 - 49 = 207.

Explanation

The correct average can be determined by calculating the net adjustment required for the total sum and then distributing that adjustment across all observations. The error occurred because the number 85 was misread as 45, which means the calculated sum was lower than the actual sum.

The net increase required in the total sum is: Correct Value - Incorrect Value = 85 - 45 = 40

This increase of 40 must be distributed over the 20 numbers to find the correction for the average: Increase in Average = (Net Increase in Sum)/(Total Numbers) = ⁴⁰⁄₂₀ = 2

The correct average is the sum of the initial average and this increase: Correct Average = 35 + 2 = 37

Alternatively, the correct sum is calculated as (35 × 20) + (85 - 45) = 700 + 40 = 740. Dividing the correct total sum by the count of numbers results in 740 / 20 = 37.

Explanation

The expression x⁴ + x²y² + y⁴ can be factored using the standard algebraic identity: x⁴ + x²y² + y⁴ = (x² + xy + y²)(x² - xy + y²)

This identity is derived by completing the square for the first and last terms. By adding and subtracting x²y², the expression becomes (x⁴ + 2x²y² + y⁴) - x²y². This is equivalent to (x² + y²)² - (xy)². Applying the difference of squares formula a² - b² = (a + b)(a - b) yields the two quadratic factors.

Substituting the values given in the question: 6 = (x² + xy + y²) × 2

Dividing both sides by 2: x² + xy + y² = 6 / 2 = 3

Explanation

The given equation a² + 13b² + c² - 4ab - 6bc = 0 can be rearranged by regrouping the terms to form perfect squares. We split 13b² into 4b² + 9b² to match the cross-product terms 4ab and 6bc:

a² - 4ab + 4b² + 9b² - 6bc + c² = 0

By applying the identity (x - y)² = x² - 2xy + y², the expression becomes:

(a - 2b)² + (3b - c)² = 0

For the sum of two squares of real numbers to equal zero, each squared term must independently equal zero. Solving these two equations simultaneously:

a - 2b = 0 ⟹ a = 2b ⟹ a:b = 2:1

3b - c = 0 ⟹ c = 3b ⟹ b:c = 1:3

Combining the ratios a:b = 2:1 and b:c = 1:3 yields the ratio a:b:c = 2:1:3.

Explanation

In triangle ABC, the sum of all interior angles is 180°. Given ∠ BAC = 85° and ∠ BCA = 75°, the third angle ∠ ABC is calculated as 180° - (85° + 75°) = 20°. According to the properties of a circumcircle, the angle subtended by a chord AC at the center O is twice the angle subtended by it at the circumference at vertex B. Thus, ∠ AOC = 2 × ∠ ABC = 2 × 20° = 40°.

In triangle OAC, the segments OA and OC are radii of the same circumcircle, making OA = OC. This implies triangle OAC is an isosceles triangle where the angles opposite to the equal sides are equal, so ∠ OAC = ∠ OCA. Using the angle sum property for triangle OAC, we have ∠ OAC + ∠ OCA + ∠ AOC = 180°. Substituting the known values, 2 × ∠ OAC + 40° = 180°, which simplifies to 2 × ∠ OAC = 140°. Dividing by two, we find ∠ OAC = 70°. Alternatively, for an acute angle B, the angle ∠ OAC can be directly determined using the formula 90° - ∠ B, which yields 90° - 20° = 70°.

Explanation

The radius of the incircle r of an equilateral triangle with side length a is given by the formula r = (a)/(2√3). This relation is derived from the property that the incenter of an equilateral triangle coincides with its centroid, which divides the median (or height h = (√3/2)a) in the ratio 2:1. The inradius is the length of the shorter segment, r = ⅓h.

Given the side length a = 2√3 units, we substitute this value into the formula:

x = (2√3)/(2√3)

The terms in the numerator and denominator cancel out completely, resulting in x = 1.

Explanation

Using the trigonometric identity tan A · tan B = 1 ⟺ A + B = 90°, we can equate the sum of the angles to 90°. Given the equation tan 3θ · tan 7θ = 1, we have:

3θ + 7θ = 90° 10θ = 90° θ = 9°

Substituting the value of θ into the expression tan(θ + 36°):

tan(9° + 36°) = tan 45°

The value of tan 45° is 1.

Explanation

Let H be the height of the cloud above the lake surface and h be the height of the observation point above the lake, which is 200 m. The reflection of the cloud in the lake is at a depth H below the surface.

Let x be the horizontal distance from the observation point to the vertical line passing through the cloud. The height of the cloud relative to the observation point is H - h, and the depth of its reflection relative to the observation point is H + h.

From the angle of elevation α = 30°: tan 30° = (H - h)/(x) ⟹ x = (H - 200)/(tan 30°)

From the angle of depression β = 60°: tan 60° = (H + h)/(x) ⟹ x = (H + 200)/(tan 60°)

Equating the two expressions for x: (H - 200)/(tan 30°) = (H + 200)/(tan 60°) (H - 200) √3 = (H + 200)/(√3) 3(H - 200) = H + 200 3H - 600 = H + 200 2H = 800 H = 400 m

Alternatively, using the direct formula for height of a cloud H = h (tan β + tan α)/(tan β - tan α): H = 200 (tan 60° + tan 30°)/(tan 60° - tan 30°) = 200 (√3 + 1/√3)/(√3 - 1/√3) = 200 (4/√3)/(2/√3) = 200 × 2 = 400 m

Explanation

The difference between compound interest and simple interest for a period of 3 years can be calculated using the formula D = PR²(300 + R){100³}, where P is the principal, R is the annual interest rate, and D is the difference.

Given P = 40,000 and R = 8, substituting these values into the formula gives:

D = 40,000 × 8² × (300 + 8){100³}

D = (40,000 × 64 × 308)/(1,000,000)

By simplifying the zeros, the expression becomes:

D = (4 × 64 × 308)/(100)

D = (256 × 308)/(100)

To calculate 256 × 308, split the multiplication: 256 × 300 = 76,800 and 256 × 8 = 2,048. Adding these results gives 78,848.

D = (78,848)/(100) = 788.48

Explanation

The stratosphere is the second major layer of the Earth's atmosphere, extending from the top of the troposphere to approximately 50 kilometers above the surface. It contains the ozone layer, a region where ozone (O₃) molecules are concentrated, primarily between altitudes of 15 and 35 kilometers. This layer is formed when high-energy ultraviolet radiation from the sun strikes oxygen molecules (O₂), splitting them into individual oxygen atoms that then combine with other O₂ molecules. The ozone layer is essential for life on Earth as it absorbs the vast majority of harmful solar ultraviolet radiation, specifically UV-B, preventing it from reaching the surface.

Explanation

The Great Barrier Reef is the world's largest coral reef system, stretching over 2,300 kilometers along the northeast coast of Australia. It is situated in the Coral Sea, which is a marginal sea of the South Pacific Ocean. This vast ecosystem is composed of over 2,900 individual reefs and 900 islands, making it the largest structure on Earth built by living organisms. It is located off the coast of Queensland and is recognized as a major geographical feature of the Pacific region.

Explanation

Let the three consecutive natural numbers divisible by 5 be x-5, x, and x+5. Since they are consecutive multiples of 5, the common difference between them is 5.

According to the given condition: (x - 5) + x + (x + 5) = 225 3x = 225 x = ²²⁵⁄₃ = 75

The three numbers are 75 - 5 = 70, 75, and 75 + 5 = 80. The largest number among them is 80.

Alternatively, for any three numbers in an arithmetic progression, the middle term is the average of the sum. Middle term = ²²⁵⁄₃ = 75 The numbers are consecutive multiples of 5, so the largest number is the middle term plus 5: Largest number = 75 + 5 = 80

Explanation

A can complete the entire work in 18 days. In 12 days, the fraction of work completed by A is 12/18 = 2/3. The remaining work to be done is 1 - 2/3 = 1/3 of the total task. Given that B finishes this remaining 1/3 of the work in 8 days, the time B would take to complete the full unit of work is calculated by dividing the time taken by the fraction of work completed. Thus, B alone can do the whole work in 8 / (1/3) = 8 × 3 = 24 days.

Explanation

The selling price of an article after successive discounts is calculated by applying each discount percentage to the remaining value. For a list price L and successive discounts d₁ and d₂, the selling price SP is given by the formula SP = L × (1 - (d₁)/(100)) × (1 - (d₂)/(100)).

Given the list price L = 900, the first discount d₁ = 20%, and the second discount d₂ = 10%: SP = 900 × (1 - ²⁰⁄₁₀₀) × (1 - ¹⁰⁄₁₀₀) SP = 900 × 0.8 × 0.9 SP = 900 × 0.72 SP = 648

Alternatively, the first discount of 20% on Rs. 900 reduces the price by 0.20 × 900 = 180, resulting in an intermediate price of 900 - 180 = 720. The second discount of 10% is then applied to this reduced price, which is 0.10 × 720 = 72. Subtracting the second discount from the intermediate price yields 720 - 72 = 648.

Explanation

Monthly income is the sum of expenditure and savings. Given the ratio of expenditure to savings is 61 : 6, the total income can be represented as 61 + 6 = 67 units.

The total monthly income is given as Rs. 8710. To find the value of one unit in the ratio, divide the total income by the total units: 1 unit = ⁸⁷¹⁰⁄₆₇ = 130

The monthly savings correspond to 6 units in the ratio. Therefore, the total monthly savings are: Savings = 6 × 130 = 780

Explanation

The relationship between the Marked Price (MP), Selling Price (SP), and Discount percentage (d%) is given by the formula MP = (SP × 100)/(100 - d%). Substituting the given values where SP = 950 and d = 5:

MP = (950 × 100)/(100 - 5)

MP = (950 × 100)/(95)

MP = 10 × 100 = 1000

Explanation

Let the initial amount of money the person had be x. In each bet, the person loses 75% of the money they have at that moment, which means they are left with 100% - 75% = 25%, or ¼ of the amount.

After the first bet, the remaining money is x × ¼. After the second bet, the remaining money is (x × ¼) × ¼ = ˣ⁄₁₆. After the third bet, the remaining money is (ˣ⁄₁₆) × ¼ = ˣ⁄₆₄.

The problem states that the final amount remaining is Rs. 2. Therefore, we can set up the equation:

ˣ⁄₆₄ = 2

Solving for x:

x = 2 × 64 x = 128

The initial amount of money was Rs. 128.

Explanation

When a train crosses a stationary point object such as an electric pole, the total distance covered by the train is equal to its own length. Given the length d = 160 m and the speed v = 72 km/hr, the speed must first be converted into meters per second using the conversion factor ⁵⁄₁₈:

v = 72 × ⁵⁄₁₈ = 4 × 5 = 20 m/s

The time t taken to cross the pole is given by the formula:

t = (Distance)/(Speed)

Substituting the values into the formula:

t = ¹⁶⁰⁄₂₀ = 8 sec

Explanation

To find the value of 9x² + (1)/(9x²), first transform the given equation x + ¹⁄₉ₓ = 4 by multiplying both sides by 3. This results in the expression 3x + ¹⁄₃ₓ = 12. Next, apply the algebraic identity (a + b)² = a² + 2ab + b² by squaring both sides of this equation: (3x + ¹⁄₃ₓ)² = 12². Expanding the left side gives (3x)² + 2(3x)(¹⁄₃ₓ) + (¹⁄₃ₓ)² = 144, which simplifies to 9x² + 2 + (1)/(9x²) = 144. By subtracting 2 from both sides of the equation, the result is 9x² + (1)/(9x²) = 142.

Explanation

The given expression 27a³ - 54a²b + 36ab² - 8b³ follows the algebraic identity (x - y)³ = x³ - 3x²y + 3xy² - y³. By comparing the terms, we can identify x = 3a and y = 2b, since (3a)³ = 27a³ and (2b)³ = 8b³.

The intermediate terms also match the identity: -3(3a)²(2b) = -3(9a²)(2b) = -54a²b 3(3a)(2b)² = 3(3a)(4b²) = 36ab²

Therefore, the expression simplifies to (3a - 2b)³. Substituting the given values a = 2 and b = -3: (3(2) - 2(-3))³ = (6 + 6)³ = 12³ = 1728

Explanation

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. Mathematically, for sides a, b, and c, the conditions a + b > c, a + c > b, and b + c > a must all be true. For the lengths 3 cm, 4 cm, and 5 cm, the sum of the two smallest sides is 3 + 4 = 7 cm. Since 7 > 5, the fundamental property of a triangle is satisfied. Furthermore, these lengths satisfy the Pythagorean theorem 3² + 4² = 5², forming a valid right-angled triangle.

Explanation

The centroid of a triangle is the point of intersection of its medians and possesses the property of dividing each median in the ratio 2 : 1, measured from the vertex to the midpoint of the opposite side. In Δ ABC, for the median AD with centroid O, the segments AO and OD are in the ratio AO : OD = 2 : 1. Since the segment from the vertex A to the centroid O is given as AO = 10 cm, the length of the segment from the centroid to the midpoint D is calculated as OD = ½ × AO = ½ × 10 = 5 cm.