SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Let r denote the radius of the circle. The geometric properties of the circle are defined by the following formulas: Area x = π r² Circumference y = 2π r Diameter z = 2r

The expression to be evaluated is x/yz. Substituting the expressions for x, y, and z into this formula gives: x/yz = (π r²)/((2π r)(2r)) x/yz = (π r²)/(4π r²)

By canceling the common terms π and r² from the numerator and the denominator, we obtain: x/yz = ¼

The value 1/4 is expressed as the ratio 1 : 4.

Explanation

To evaluate the expression √(9 + 216) + ³√(512), we simplify each radical term sequentially. First, the innermost square root is √(16) = 4. Multiplying this by the coefficient 2 results in 2 × 4 = 8. Next, the term ³√(512) is evaluated. Since 8 × 8 × 8 = 512, the cube root of 512 is 8. Substituting these simplified values into the main expression yields √(9 + 8 + 8). Adding the integers under the radical gives √(25). The square root of 25 is 5.

Explanation

The diagonals of a rhombus bisect each other at right angles, forming four congruent right-angled triangles. The side s of the rhombus acts as the hypotenuse of these triangles, with legs equal to half the lengths of the diagonals (d₁/2 and d₂/2).

Using the Pythagorean theorem: s = √(((d₁)/(2))² + ((d₂)/(2))²)

Given d₁ = 24 cm and d₂ = 10 cm: s = √((²⁴⁄₂)² + (¹⁰⁄₂)²) s = √(12² + 5²) s = √(144 + 25) = √(169) = 13 cm

The perimeter of a rhombus is 4s. Perimeter = 4 × 13 = 52 cm.

Explanation

The gain percentage for a dishonest shopkeeper who sells goods at cost price but uses a false weight is calculated using the formula:

Gain % = (Error)/(False Weight) × 100

In this problem, the shopkeeper uses a weight of 950 g instead of 1 kg (1000 g). The error in weight is the difference between the true weight and the weight actually given:

Error = 1000 g - 950 g = 50 g

Substituting the values into the formula:

Gain % = ⁵⁰⁄₉₅₀ × 100

Simplifying the fraction by dividing both the numerator and denominator by 50:

Gain % = ¹⁄₁₉ × 100 Gain % = ¹⁰⁰⁄₁₉

When 100 is divided by 19, the quotient is 5 and the remainder is 5. Therefore, the gain percentage expressed as a mixed fraction is 5 ⁵⁄₁₉% .

Explanation

A's current age is calculated by subtracting the time interval from his future age: A = 26 - 6 = 20 years. Given that the ratio of A's age to B's age is 4:3, we can express their ages in terms of a common variable x such that A = 4x and B = 3x. Substituting the known value of A into the first equation, we get 4x = 20, which implies x = 5. Using this value for B's age, we calculate B = 3 × 5 = 15. Thus, B's current age is 15 years.

Explanation

The arithmetic mean is defined by the formula x = (Σ xᵢ)/(n), where Σ xᵢ represents the sum of all observations and n represents the total number of observations in the set. For the given data set {7, 5, 13, x, 9}, there are n = 5 observations and the mean x is 10.

Substituting the given values into the mean formula results in the following linear equation: 10 = (7 + 5 + 13 + x + 9)/(5)

To solve for x, first multiply both sides of the equation by 5 to eliminate the denominator: 50 = 7 + 5 + 13 + x + 9

Next, simplify the right side of the equation by adding the known numerical values: 50 = 34 + x

Finally, isolate x by subtracting 34 from both sides: x = 50 - 34 x = 16

The value of x that satisfies the condition for the arithmetic mean to be 10 is 16.

Explanation

The difference between the two profit percentages represents the additional amount for which the article was sold. The initial profit was 15% and the new profit is 18%. The difference in the percentage of the cost price is 18% - 15% = 3%.

According to the problem, this 3% difference is equal to the additional ₹ 6 in the selling price. Let the cost price of the article be x.

3% of x = 6

³⁄₁₀₀ × x = 6

x = (6 × 100)/(3)

x = 2 × 100 = 200

The man bought the article for ₹ 200.

Explanation

When a value is subjected to two successive percentage changes of x% and y%, the net percentage change is calculated using the formula x + y + ˣʸ⁄₁₀₀. In this case, the salary is first decreased by 50%, so x = -50, and then increased by 50%, so y = +50. Substituting these values into the formula gives:

Net change = -50 + 50 + ((-50 × 50))/(100) = 0 - ²⁵⁰⁰⁄₁₀₀ = -25%

The negative result indicates a net decrease or loss in the salary. Alternatively, if the initial salary is assumed to be 100 units, a 50% decrease reduces it to 50 units. A subsequent 50% increase on this new amount adds 25 units (50% of 50), resulting in a final salary of 75 units. The total loss is 100 - 75 = 25 units, which is a 25% reduction from the original 100 units.

Explanation

Relative speed is the speed of one object as observed from another. Since the car and the man are moving in the same direction, the relative speed Vᵣₑₗ is the difference between the speed of the car V_c and the speed of the man Vₘ.

Visibility distance is the relative distance covered by the car ahead of the man during the time it remains visible. Given: Visibility distance d = 130 m = ¹³⁰⁄₁₀₀₀ km = 0.13 km Time t = 3 minutes = ³⁄₆₀ hours = ¹⁄₂₀ hours Speed of man Vₘ = 4 km/h

Using the formula Relative Speed = (Relative Distance)/(Time): V_c - Vₘ = ᵈ⁄ₜ V_c - 4 = (0.13)/(1/20) V_c - 4 = 0.13 × 20 V_c - 4 = 2.6 V_c = 4 + 2.6 = 6.6 km/h

Converting the decimal to a mixed fraction: 6.6 = 6 ⁶⁄₁₀ = 6 ⅗ km/h

Explanation

The total simple interest received is the sum of the interest from both loans. Using the standard formula for simple interest SI = (P × R × T)/(100), where P is the principal, R is the annual rate of interest, and T is the time in years, the equation for the total interest can be established as:

Total Interest = P₁ × R × T₁{100} + P₂ × R × T₂{100}

Substituting the given values into the equation:

160 = (400 × R × 3)/(100) + (500 × R × 4)/(100)

Simplifying the terms:

160 = 12R + 20R

160 = 32R

Solving for R:

R = ¹⁶⁰⁄₃₂

R = 5

The rate of interest per annum is 5%.

Explanation

Rationalizing the denominator of x by multiplying both the numerator and denominator by the conjugate (√2 - 1) gives: x = 1(√2 - 1) / ((√2 + 1)(√2 - 1)) = (√2 - 1) / (2 - 1) = √2 - 1

Rearranging the equation to isolate the radical term yields: x + 1 = √2

Squaring both sides of the equation leads to: (x + 1)² = (√2)² x² + 2x + 1 = 2

Subtracting 2 from both sides results in the value of the required expression: x² + 2x + 1 - 2 = 0 x² + 2x - 1 = 0

Explanation

To find the value of the expression (3x)/(x²⁾⁻¹, divide both the numerator and the denominator by x to simplify it to (3)/(x - 1){x}.

Using the standard algebraic identity (x - ¹⁄ₓ)² = (x + ¹⁄ₓ)² - 4, substitute the given value x + ¹⁄ₓ = √(13):

(x - ¹⁄ₓ)² = (√(13))² - 4 = 13 - 4 = 9

Taking the square root of both sides, we get x - ¹⁄ₓ = 3. Substituting this value back into the simplified expression:

(3)/(x - 1){x} = ³⁄₃ = 1

Explanation

The radius of a circle is always perpendicular to the tangent at the point of contact. Let O be the center of the circle, P be the external point, and T be the point of contact of the tangent. In the right-angled triangle OTP, the angle at T is 90 degrees, making the distance OP the hypotenuse.

Using the Pythagoras theorem: OT² + PT² = OP²

Substituting the given values where the radius OT = 5 cm and the distance OP = 13 cm: 5² + PT² = 13² 25 + PT² = 169 PT² = 169 - 25 PT² = 144 PT = √(144) = 12 cm

The length of the tangent from point P to the point of contact is 12 cm. This follows the standard Pythagorean triplet (5, 12, 13).

Explanation

Given the trigonometric equation sin(θ + 18°) = 1/2, we identify the angle whose sine value is 1/2. For an acute angle θ, the value θ + 18° falls within the first or second quadrant. Since sin 30° = 1/2, we equate the arguments: θ + 18° = 30°. Solving for θ yields θ = 30° - 18° = 12°. To express this value in circular measure, we use the conversion formula where 180° = π radians. Therefore, θ = 12 × (π / 180) = π / 15 radians.

Explanation

From the given information, 18 men are equivalent to 36 boys in terms of work capacity. This establishes the efficiency ratio where 1 man (M) equals 2 boys (B), or 1M = 2B. The total work can be calculated using the formula W = M × D × H, where M is the number of workers, D is the number of days, and H is the number of hours per day.

For the first case, using men as the base unit: M₁ = 18 D₁ = 24 H₁ = 6

For the second case, we convert 24 men and 24 boys into a single unit of men. Since 2B = 1M, then 24B = 12M. Thus, the total power of the second group is 24M + 12M = 36M. M₂ = 36 H₂ = 9 D₂ = ?

Using the work equivalence formula M₁ × D₁ × H₁ = M₂ × D₂ × H₂: 18 × 24 × 6 = 36 × D₂ × 9

Solving for D₂: 2592 = 324 × D₂ D₂ = ²⁵⁹²⁄₃₂₄ D₂ = 8

Explanation

From the given equations, we have a(x + y) = 2ab and b(x - y) = 2ab. Dividing the first equation by a gives x + y = 2b, and dividing the second equation by b gives x - y = 2a.

Using the algebraic identity (x + y)² + (x - y)² = 2(x² + y²), we can substitute the values derived above. This gives 2(x² + y²) = (2b)² + (2a)². Simplifying the right side of the equation yields 4b² + 4a². Factoring out the constant, we get 4(a² + b²).

Explanation

To find the value of x² + (1)/(x²) given x + ¹⁄ₓ = 6, we use the algebraic identity (a + b)² = a² + b² + 2ab.

By squaring both sides of the given equation x + ¹⁄ₓ = 6, we get: (x + ¹⁄ₓ)² = 6² x² + (¹⁄ₓ)² + 2 · x · ¹⁄ₓ = 36 x² + (1)/(x²) + 2 = 36

To isolate the required expression, subtract 2 from both sides: x² + (1)/(x²) = 36 - 2 x² + (1)/(x²) = 34

As a standard mathematical shortcut, if x + ¹⁄ₓ = k, then x² + (1)/(x²) = k² - 2. Substituting k = 6 into this formula results in 6² - 2 = 36 - 2 = 34.

Explanation

In an equilateral triangle, the distance from a vertex to the centroid is equal to the circumradius R of the triangle. For an equilateral triangle with side length a, the height h is given by the formula h = √3{2}a. The centroid divides the median (which is also the altitude in an equilateral triangle) in the ratio 2:1 from the vertex. Therefore, the distance AG is calculated as:

AG = ⅔ × h = ⅔ × √3{2}a = (a)/(√(3))

Given the side length AB = 9 cm, we substitute a = 9 into the formula:

AG = (9)/(√(3))

To simplify the expression, we multiply the numerator and the denominator by √3:

AG = 9√3{3} = 3√3 cm

This relationship between the side and the distance to the centroid is a standard property of equilateral triangles where the circumradius R = (a)/(√(3)) and the inradius r = (a)/(2√(3)).

Explanation

Given the equation 4sin²θ = 3, we solve for sinθ as follows: sin²θ = ¾ sinθ = √¾ = √3{2}

Since θ is a positive acute angle (0° < θ < 90°), the value of θ is 60°.

Substituting θ = 60° into the expression tanθ - cot(θ)/(2): tan 60° - cot((60°)/(2)) = tan 60° - cot 30°

Substituting the standard trigonometric values tan 60° = √3 and cot 30° = √3: √3 - √3 = 0

Explanation

In a circle with center O, let the tangent at point A be represented by a line passing through A. According to the fundamental property of circles, the radius OA is perpendicular to the tangent at the point of contact A. Thus, the total angle between the radius OA and the tangent line is 90°. The problem states that the chord AB makes an angle θ with the tangent at A. By observing the geometry at point A, the angle ∠OAB is the difference between the angle formed by the radius and the tangent (90°) and the angle formed by the chord and the tangent (θ). Mathematically, this relation is expressed as ∠OAB = 90° - θ.

Now consider the triangle OAB. Since OA and OB are both radii of the same circle, their lengths are equal. In any triangle where two sides are of equal length, the angles opposite those sides are also equal. This makes △OAB an isosceles triangle where the base angles ∠OAB and ∠OBA are equal. Since the question asks for the measure of ∠ABO, which is the same as ∠OBA, we can directly use the value derived for ∠OAB. Consequently, the angle measure is ∠ABO = 90° - θ.

Explanation

Let P be the point on the bridge and A, B be the points on the opposite banks of the river. The perpendicular height of the bridge from the river bed is PD = 2.5 m. The angles of depression to the banks are 30° and 45°, which are equal to the angles of elevation ∠ PAD and ∠ PBD respectively due to the property of alternate interior angles.

The width of the river is AB = AD + DB. In the right-angled triangle PAD, the base AD is calculated as PD cot 30°, which gives AD = 2.5 × √3. In the right-angled triangle PBD, the base DB is calculated as PD cot 45°, which gives DB = 2.5 × 1 = 2.5.

The total width of the river is W = 2.5√3 + 2.5 = 2.5(√3 + 1). Substituting the given value √3 = 1.732: W = 2.5(1.732 + 1) W = 2.5 × 2.732 W = 6.83 m

Explanation

Let the total work be the Least Common Multiple (LCM) of 24 and 8, which is 24 units. The combined efficiency of A and B is 24 / 24 = 1 unit per day. The combined efficiency of A, B, and C together is 24 / 8 = 3 units per day. The individual efficiency of C is the difference between these two rates: 3 - 1 = 2 units per day. The time taken by C alone to finish the job is the total work divided by C's efficiency, which results in 24 / 2 = 12 days.

Explanation

The diagonal d of a square is related to its side s by the formula d = s√2. Given the diagonal is 6√2 cm, we substitute this into the formula to find 6√2 = s√2, which results in s = 6 cm.

For a circle inscribed in a square, the diameter of the circle is equal to the side length of the square. Therefore, the diameter of the circle is 6 cm, making its radius r = 3 cm. The area of a circle is given by the formula A = π r². Substituting the radius, we get A = π × 3² = 9π sq cm.

Explanation

The final price is determined by applying successive percentage changes to the original price. The formula for the final amount after a discount and a subsequent increase is:

Final Price = P × (1 - ᵈ⁄₁₀₀) × (1 + ⁱ⁄₁₀₀)

Where P is the original price, d is the discount percentage, and i is the increase percentage for the service contract.

Substituting the given values into the formula: Final Price = 6000 × (1 - ²⁰⁄₁₀₀) × (1 + ¹⁰⁄₁₀₀) Final Price = 6000 × 0.80 × 1.10

First, calculate the price after the 20% discount: 6000 × 0.80 = 4800

Next, apply the 10% increase for the service contract on this discounted price: 4800 × 1.10 = 4800 + 480 = 5280

Alternatively, the effective multiplier for the two changes is 0.80 × 1.10 = 0.88. Multiplying the original price by this factor yields 6000 × 0.88 = 5280.

Explanation

The total sum is divided in the ratio 5:6:9, making the total number of parts 5 + 6 + 9 = 20. A's share corresponds to 5 parts of this total. Given that 5 parts equals Rs. 450, the value of one part is calculated as 450 / 5 = 90. To find the total sum divided, multiply the value of one part by the total number of parts: 20 × 90 = 1800.

Explanation

The Selling Price (SP) and the Profit Percentage (P%) are related to the Cost Price (CP) by the formula SP = CP × (100 + P%)/(100). Since the cost price of the bag remains constant, the selling price is directly proportional to (100 + P%).

Initially, a profit of 15% is made on a selling price of Rs. 230. This can be expressed as 115% of CP = 230. To find the selling price required for a 20% profit, we need to determine the value of 120% of CP.

Using the unitary method: New Selling Price = ²³⁰⁄₁₁₅ × 120

Dividing 230 by 115 gives 2. Therefore: New Selling Price = 2 × 120 = 240

Explanation

To determine the percentage weight of the first ball with respect to the second ball, use the following formula:

Percentage = (Weight of 1st ball)/(Weight of 2nd ball) × 100

Substitute the given values into the formula:

Percentage = (3.5)/(7.5) × 100

To simplify the calculation, multiply both the numerator and the denominator by 10 to remove the decimals:

Percentage = ³⁵⁄₇₅ × 100

Reduce the fraction by dividing both 35 and 75 by their greatest common divisor, which is 5:

Percentage = ⁷⁄₁₅ × 100

Percentage = ⁷⁰⁰⁄₁₅

Dividing 700 by 15:

Percentage = 46.66...%

The integer value for this calculation is 46%.

Explanation

To determine the distance covered by the bus in one second, the speed must be converted from kilometers per hour (km/hr) to meters per second (m/s). The conversion factor is derived from the fact that 1 kilometer equals 1000 meters and 1 hour equals 3600 seconds.

1 km/hr = (1000 meters)/(3600 seconds) = ⁵⁄₁₈ m/s

Given the speed of the bus is 36 km/hr, the conversion to meters per second is:

Speed in m/s = 36 × ⁵⁄₁₈

Speed in m/s = 2 × 5 = 10 m/s

By definition, speed in meters per second represents the distance in meters covered in one second. Therefore, at a speed of 10 m/s, the bus covers a distance of 10 meters in one second.

Explanation

To simplify the expression (a)/(a-b) + (b)/(b-a) , we first observe that the denominator of the second term, b-a , can be rewritten as -(a-b) . By substituting this into the expression, we get (a)/(a-b) + (b)/(-(a-b)) , which is equivalent to (a)/(a-b) - (b)/(a-b) . Since the terms now share a common denominator, they can be combined into a single fraction (a-b)/(a-b) . Dividing the numerator by the identical denominator yields a value of 1.

Explanation

The given expression is a telescoping series of the form ₙ₌₁¹⁵ (√n - √(n+1)). In such a series, when the terms are expanded, each intermediate term is cancelled by a subsequent term. Expanding the sum gives:

(1 - √2) + (√2 - √3) + (√3 - √4) + … + (√(14) - √(15)) + (√(15) - √(16))

Observing the pattern of addition, the -√2 from the first bracket is cancelled by the +√2 from the second bracket. Similarly, the -√3 is cancelled by the +√3, and this process continues until the final +√(15) is cancelled by the -√(15) in the preceding term. After all possible cancellations, only the first part of the first term and the second part of the last term remain.

The expression simplifies to 1 - √(16). Since the square root of 16 is 4, the calculation becomes 1 - 4. This results in a final value of -3.

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides. Given that Δ ABC ∼ Δ DEF, the relationship between their perimeters and the corresponding sides AB and DE can be expressed as:

(Perimeter(Δ ABC))/(Perimeter(Δ DEF)) = (AB)/(DE)

Substituting the given values into the formula:

³⁰⁄₁₈ = (AB)/(36)

To find AB, we can simplify the ratio of the perimeters by dividing both 30 and 18 by their greatest common divisor, 6:

⁵⁄₃ = (AB)/(36)

Now, solving for AB by rearranging the equation:

AB = ⁵⁄₃ × 36

AB = 5 × 12

AB = 60 cm

Explanation

Let r be the radius of the circle and L be the length of the chord. Since L = r, the triangle formed by the chord and the two radii connecting the center to the endpoints of the chord is an equilateral triangle. In an equilateral triangle, all internal angles are 60°.

To convert the angle from degrees to radians, the following formula is used: ᵣₐ_d = _dₑ_g × (π)/(180°)

Substituting the value of the central angle: ᵣₐ_d = 60° × (π)/(180°) = (π)/(3) radians

The magnitude of the angle subtended at the centre is (π)/(3) radians, where the numerical coefficient is ⅓.

Explanation

The expression can be evaluated by substituting the standard values of trigonometric ratios. We know that sec 45° = √2, cot 45° = 1, sin 30° = 1/2, and sin 60° = √3/2. Substituting these values into the given expression:

(sec² 45° - cot² 45°) - (sin² 30° + sin² 60°) = ((√2)² - (1)²) - ((1/2)² + (√3/2)²) = (2 - 1) - (1/4 + 3/4) = 1 - (4/4) = 1 - 1 = 0

Alternatively, using trigonometric identities, sec² θ - tan² θ = 1 and sin² θ + cos² θ = 1. Since cot 45° = tan 45° = 1, the first term (sec² 45° - cot² 45°) equals 1. Since sin 60° = cos 30° by the complementary angle identity sin(90° - θ) = cos θ, the second term (sin² 30° + sin² 60°) becomes (sin² 30° + cos² 30°), which equals 1. The final calculation is 1 - 1 = 0.

Explanation

The ratio of the number of male employees to female employees can be determined using the rule of alligation. Let M be the average salary of males, F be the average salary of females, and A be the mean salary of all employees.

The ratio of Males (nₘ) to Females (n_f) is given by: nₘ / n_f = (A - F) / (M - A)

Substituting the given values: nₘ / n_f = (5000 - 4200) / (5200 - 5000) = 800 / 200 = 4 / 1

This means for every 4 male employees, there is 1 female employee. The total parts in the ratio are 4 + 1 = 5.

The percentage of female employees is calculated as: Percentage of Females = (n_f / (nₘ + n_f)) × 100 Percentage of Females = (1 / 5) × 100 = 20%

Explanation

Given the algebraic condition 4x = √5 + 2, we find the value of the reciprocal term ¹⁄₄ₓ through rationalization: ¹⁄₄ₓ = (1)/(√(5) + 2) × √5 - 2{√5 - 2} = √5 - 2{5 - 4} = √5 - 2

To solve for x - ¹⁄₁₆ₓ most efficiently, we factor out ¼ from the expression to make use of the terms 4x and ¹⁄₄ₓ directly: x - ¹⁄₁₆ₓ = ¼ ( 4x - ¹⁄₄ₓ )

By substituting the established values into this transformed expression, we obtain: ¼ [(√5 + 2) - (√5 - 2)]

Expanding the terms within the brackets results in the cancellation of the irrational part: √5 + 2 - √5 + 2 = 4

The final calculation yields the value: ¼ × 4 = 1

Explanation

To find the cube of 105, we use the algebraic identity (a + b)³ = a³ + 3a²b + 3ab² + b³. By setting a = 100 and b = 5, the expansion is calculated as follows:

105³ = (100 + 5)³ = 100³ + 3(100²)(5) + 3(100)(5²) + 5³ = 1,000,000 + 150,000 + 7,500 + 125 = 1,157,625

Alternatively, using the Vedic math shortcut for numbers near the base 100 with a deviation d = 5: The result is partitioned into three parts: (100 + 3d) | (3d²) | (d³). The first part is 100 + 3(5) = 115. The second part is 3(5²) = 75. The third part is 5³ = 125. Since the base is 100, we retain only two digits in the second and third sections and carry over any excess. Carrying the 1 from 125 to the middle section gives 75 + 1 = 76. Combining the parts yields 1157625.

Explanation

In a right-angled triangle ABC where ∠ B = 90°, the length of the hypotenuse AC is determined by the Pythagorean theorem: AC = √(AB² + BC²). Using the given values AB = 6 cm and BC = 8 cm, we find AC = √(6² + 8²) = √(36 + 64) = √(100) = 10 cm.

A key geometric property of right-angled triangles is that the median drawn from the right-angle vertex to the hypotenuse is equal to half the length of the hypotenuse. This is because the midpoint of the hypotenuse, D, acts as the circumcenter of the triangle, making the distances DA, DC, and DB equal to the circumradius R. Since D is the midpoint of AC, BD is the median to the hypotenuse, and its length is BD = (AC)/(2) = ¹⁰⁄₂ = 5 cm.

Explanation

The area of a square A can be calculated using its diagonal d with the formula A = d²{2}. Since the area is directly proportional to the square of the diagonal, the ratio of the areas of two squares is equal to the ratio of the squares of their diagonals.

Given the ratio of the diagonals is d₁ : d₂ = 5 : 2, the ratio of their areas A₁ : A₂ is calculated as:

A₁{A₂} = (1)/(2) d₁²{½ d₂²} = (d₁{d₂})²

Substituting the given ratio:

(⁵⁄₂)² = ²⁵⁄₄

The ratio of the areas is therefore 25:4.

Explanation

In the right-angled triangle formed by the ladder, the wall, and the ground, the ladder acts as the hypotenuse while the distance from the foot of the ladder to the wall represents the base (adjacent side). The relationship between the base and the hypotenuse for a given angle of elevation is provided by the cosine trigonometric ratio.

The formula is defined as: cos(θ) = (Base)/(Hypotenuse)

Given that the angle of elevation θ is 60° and the distance from the wall (base) is 4.6 m, the equation is: cos(60°) = (4.6)/(Length of the ladder)

Using the standard trigonometric value cos(60°) = 0.5 or ½, we can substitute it into the equation: ½ = (4.6)/(Length of the ladder)

By cross-multiplying to solve for the length of the ladder: Length of the ladder = 4.6 × 2 = 9.2 m

Explanation

Let the two numbers be 12x and 12y, where x and y are co-prime integers since their H.C.F. is 12. Given that the product of these two 2-digit numbers is 2160, we have the equation (12x) × (12y) = 2160. Simplifying this, 144xy = 2160, which leads to xy = ²¹⁶⁰⁄₁₄₄ = 15.

The pairs of co-prime factors of 15 are (1, 15) and (3, 5). If we take the pair (1, 15), the numbers would be 12 × 1 = 12 and 12 × 15 = 180. However, the problem states that both are 2-digit numbers, so 180 is excluded. Taking the pair (3, 5), the numbers are 12 × 3 = 36 and 12 × 5 = 60. Both 36 and 60 are 2-digit numbers that satisfy the given conditions of having an H.C.F. of 12 and a product of 2160.

Explanation

The difference between compound interest (CI) and simple interest (SI) for a period of 2 years is given by the formula D = P((R)/(100))², where D is the difference, P is the principal sum, and R is the annual rate of interest. This formula arises because the difference after two years is simply the interest earned on the first year's interest.

Given that the difference D = 1 and the rate R = 4, we can substitute these values into the formula to find the principal:

1 = P × (⁴⁄₁₀₀)²

1 = P × (¹⁄₂₅)²

1 = P × ¹⁄₆₂₅

P = 1 × 625 = 625

Alternatively, using the effective rate method, the simple interest for 2 years at 4% is 4 + 4 = 8%. The compound interest for 2 years at 4% is 4 + 4 + (4 × 4)/(100) = 8.16%. The difference between them is 0.16%. According to the question, 0.16% of the sum P is Rs. 1. Therefore, P × (0.16)/(100) = 1, which leads to P = (100)/(0.16) = ¹⁰⁰⁰⁰⁄₁₆ = 625.

Explanation

The initial mixture of 25 litres has a milk to water ratio of 4:1, making the total parts 4 + 1 = 5. The volume of each part is 25 / 5 = 5 litres. Thus, the quantity of milk is 4 × 5 = 20 litres and the quantity of water is 1 × 5 = 5 litres. When 3 litres of water are added, the new quantity of water becomes 5 + 3 = 8 litres. The quantity of milk remains unchanged at 20 litres. The new ratio of milk to water is 20 : 8, which simplifies to 5 : 2 after dividing both sides by 4.

Explanation

Time taken by A to complete the whole work is 5 × 3 = 15 days. Time taken by B to complete the whole work is 10 × ⁵⁄₂ = 25 days. Using the combined work formula T = (xy)/(x+y) where x and y are the individual times, we calculate:

T = (15 × 25)/(15 + 25) = ³⁷⁵⁄₄₀ days

Dividing both the numerator and denominator by their greatest common divisor, 5, results in ⁷⁵⁄₈ days. To convert this into a mixed fraction, divide 75 by 8, which gives a quotient of 9 and a remainder of 3, resulting in 9 ⅜ days.

Explanation

The volume V of a right pyramid is determined by the formula V = ⅓ × B × h, where B represents the area of the base and h represents the vertical height. Given the base area B = 57 sq. units and the height h = 10 units, the volume is calculated as V = ⅓ × 57 × 10. Dividing 57 by 3 results in 19, and multiplying 19 by 10 yields a total volume of 190 cubic units.

Explanation

The selling price (SP) is determined by deducting the discount amount from the marked price (MP). Using the standard relation SP = MP × (1 - (Discount %)/(100)), we substitute the given values MP = 1200 and Discount % = 5. The calculation is SP = 1200 × (1 - ⁵⁄₁₀₀) = 1200 × ⁹⁵⁄₁₀₀ = 12 × 95 = 1140. Alternatively, finding 5% of 1200 yields 60, and subtracting this from the marked price results in 1200 - 60 = 1140.

Explanation

Let A and B represent the respective incomes. Based on the given condition: 35% of A = 25% of B ³⁵⁄₁₀₀ × A = ²⁵⁄₁₀₀ × B 35 × A = 25 × B

To find the ratio of A's income to B's income, we rearrange the equation: (A)/(B) = ²⁵⁄₃₅

Simplifying the fraction by dividing both the numerator and the denominator by their common factor 5: (A)/(B) = ⁵⁄₇

The ratio A:B is 5:7.

Explanation

In a business partnership where the time period of investment is the same for all partners, the profit or loss is shared in the direct ratio of their investments. The ratio of the investments of A and B is 3000 : 2400. By dividing both terms by their greatest common divisor, 600, the simplified ratio of their investments is 5 : 4.

The total loss incurred is Rs. 720. To find B's share, the total loss is divided by the sum of the ratio parts, 5 + 4 = 9. The loss per unit of the ratio is 720 / 9 = 80. Since B's investment represents 4 parts of the total ratio, B's share of the loss is calculated as 4 × 80 = 320. Using the direct proportion formula, B's share = (2400)/(3000 + 2400) × 720 = ²⁴⁄₅₄ × 720 = ⁴⁄₉ × 720 = 320.

Explanation

The expression 6 ¼ % of 1600 + 12 ½ % of 800 can be solved efficiently by converting the percentages into their fractional equivalents.

The first percentage 6 ¼ % is equivalent to ²⁵⁄₄ %. To convert this to a fraction, we divide by 100: (25)/(4 × 100) = ²⁵⁄₄₀₀ = ¹⁄₁₆

The second percentage 12 ½ % is equivalent to ²⁵⁄₂ %. Converting this to a fraction: (25)/(2 × 100) = ²⁵⁄₂₀₀ = ⅛

Now, we apply these fractions to the respective values provided in the question: First part: ¹⁄₁₆ × 1600 = 100 Second part: ⅛ × 800 = 100

Summing the two results yields the final value: 100 + 100 = 200

Using these fractional shortcuts for common percentages like 6.25% and 12.5% allows for rapid calculation by reducing complex multiplication to simple division.

Explanation

The distance between two locations is determined by the formula Distance = Speed × Time. In this case, the Shatabdi Express maintains an average speed of 80 km/hr and completes the journey from Kolkata to Ranchi in 7 hours. Substituting these values into the formula:

Distance = 80 km/hr × 7 hrs = 560 km

The calculation yields a total distance of 560 kilometers, matching the units of the average speed provided.

Explanation

The value of a² + b² can be determined using standard algebraic identities. We are given a³ - b³ = 56 and a - b = 2.

First, we use the expansion of the cube of a binomial: a³ - b³ = (a - b)³ + 3ab(a - b)

Substituting the given values into this equation: 56 = (2)³ + 3ab(2) 56 = 8 + 6ab 48 = 6ab ab = 8

Next, we relate a² + b² to the given a - b and the calculated ab using the square of a binomial: (a - b)² = a² + b² - 2ab a² + b² = (a - b)² + 2ab

Substituting a - b = 2 and ab = 8: a² + b² = (2)² + 2(8) a² + b² = 4 + 16 a² + b² = 20