SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The time taken to complete a task is inversely proportional to the rate of work. Since Koushik can complete the work in X days, his work rate is 1/X of the total work per day. Similarly, Krishna completes the work in Y days, making his work rate 1/Y per day.

When they work together, their combined daily work rate is the sum of their individual rates: Combined Rate = (1)/(X) + (1)/(Y)

By taking the least common multiple of the denominators, the expression simplifies to: Combined Rate = (X + Y)/(XY)

The total time required to finish the work is the reciprocal of the combined daily rate: Total Time = (1)/(Combined Rate) = (XY)/(X + Y) days

Explanation

The volume of a hollow hemisphere is V = ⅔π(R³ - r³). Recasting this volume into a solid cone of base radius R and height h uses the volume conservation principle, where V_cₒₙₑ = ⅓π R² h. Equating the two volumes:

⅔π(R³ - r³) = ⅓π R² h

Substituting the outer radius R = 8 cm and inner radius r = 4 cm:

2(8³ - 4³) = 8² × h 2(512 - 64) = 64h 2 × 448 = 64h 896 = 64h h = ⁸⁹⁶⁄₆₄ = 14 cm

Explanation

The difference in the discount percentages directly results in the difference in profit earned by the shopkeeper, as the cost price remains constant. The initial discount is 7% and the second discount is 9%, making a difference of 9% - 7% = 2% of the market price.

According to the problem, this 2% difference corresponds to a decrease of Rs. 15 in profit. Let the market price be M. We can set up the equation:

2% of M = 15

²⁄₁₀₀ × M = 15

M = (15 × 100)/(2)

M = 15 × 50

M = 750

The market price of the shirt is Rs. 750.

Explanation

The total collection from a railway class is calculated by multiplying the fare per passenger by the total number of passengers in that class. Based on the given ratios, the ratio of total collections for the air conditioned (AC) sleeper and the ordinary sleeper class can be determined by multiplying the corresponding parts of the fare ratio and the passenger ratio.

Ratio of collections (AC : Ordinary) = (4 × 3) : (1 × 25) = 12 : 25

Let the collection from the AC sleeper class be 12x and the collection from the ordinary sleeper class be 25x. The sum of these collections represents the total revenue:

12x + 25x = 37,000 37x = 37,000 x = 1,000

The amount paid by the air conditioned sleeper passengers is represented by 12x. Substituting the value of x:

AC sleeper collection = 12 × 1,000 = 12,000

Explanation

Let the cost price of one book be CP and the selling price of one book be SP. Based on the problem statement, the total cost of 20 books equals the total revenue from 25 books, which gives the equation 20 × CP = 25 × SP. Rearranging this equation to find the ratio of selling price to cost price, we get (SP)/(CP) = ²⁰⁄₂₅, which simplifies to ⅘.

The loss percentage is determined by the formula Loss % = (CP - SP)/(CP) × 100. By substituting the ratio values where CP = 5 and SP = 4, the calculation becomes (5 - 4)/(5) × 100 = ⅕ × 100 = 20.

Explanation

Let the initial number of students be x. An increase of 20% on the initial number means the new number is 120% of x. This relationship can be expressed using the formula for percentage increase: New Value = Original Value × (1 + (Percentage Increase)/(100)) 66 = x × (1 + ²⁰⁄₁₀₀) 66 = x × ¹²⁰⁄₁₀₀ 66 = x × ⁶⁄₅ To find x, rearrange the equation: x = (66 × 5)/(6) x = 11 × 5 = 55 Alternatively, since 20% is equal to the fraction 1/5, an increase of 1/5 means the new value is 6/5 of the original. Dividing 66 by 6 gives 11, and multiplying by 5 results in the initial number of 55 students.

Explanation

When two trains travel in the same direction on parallel tracks, their relative speed is the difference between their individual speeds. Here, the relative speed is 45 km/hr - 40 km/hr = 5 km/hr. The time given is 45 minutes, which is converted to hours by dividing by 60, resulting in 45/60 = 0.75 hours. The distance between the trains is found using the formula Distance = Relative Speed × Time. Substituting the values gives 5 km/hr × 0.75 hr = 3.75 km.

Explanation

The given equation is x + 5 + ¹⁄ₓ₊₁ = 6. To simplify this expression in terms of (x+1), we subtract 4 from both sides or rearrange the terms as (x+1) + 4 + ¹⁄ₓ₊₁ = 6. This simplifies to:

(x+1) + ¹⁄ₓ₊₁ = 6 - 4 (x+1) + ¹⁄ₓ₊₁ = 2

Let a = x+1. The equation becomes a + ¹⁄ₐ = 2. For any real number a, the relation a + ¹⁄ₐ = 2 is satisfied only when a = 1.

We need to find the value of (x+1)³ + (1)/((x+1)³), which is equivalent to a³ + (1)/(a³). Substituting a = 1 into this expression:

1³ + (1)/(1³) = 1 + 1 = 2

Alternatively, using the algebraic identity a³ + (1)/(a³) = (a + ¹⁄ₐ)³ - 3(a + ¹⁄ₐ), we substitute the value 2:

2³ - 3(2) = 8 - 6 = 2

Explanation

Given the quadratic expression a² + 1 = 9a, divide both sides by a to obtain the sum of the variable and its reciprocal: (a² + 1)/(a) = ⁹ᵃ⁄ₐ a + ¹⁄ₐ = 9

To find the value of a² + (1)/(a²), apply the identity x² + y² = (x + y)² - 2xy. Setting x = a and y = ¹⁄ₐ, the term 2xy simplifies to 2(a)(¹⁄ₐ) = 2. Thus the relationship is: a² + (1)/(a²) = (a + ¹⁄ₐ)² - 2

Substitute the calculated value of a + ¹⁄ₐ = 9 into this expression: a² + (1)/(a²) = 9² - 2 a² + (1)/(a²) = 81 - 2 a² + (1)/(a²) = 79

Explanation

In an isosceles triangle Δ ABC with vertical angle A, the base angles B and C are equal, so B = C. According to the angle sum property of a triangle, A + B + C = 180°. Substituting C = B and the given condition A = 3B into the equation, we get 3B + B + B = 180°. This simplifies to 5B = 180°, which gives B = 36°. The measure of the vertical angle A is calculated as 3 × 36° = 108°.

Explanation

When two circles touch each other internally, the distance between their centers is equal to the difference of their radii. Let R represent the radius of the larger circle and r represent the radius of the smaller circle. The distance d between the centers is expressed by the formula d = R - r. Given that the radius of the larger circle R = 6 cm and the distance between the centers d = 2 cm, substituting these values into the formula gives 2 = 6 - r. Solving for r yields r = 6 - 2 = 4 cm.

Explanation

The given equations are y = 2 sec θ and x = 3 tan θ. Dividing the equations by their respective constants, we get ʸ⁄₂ = sec θ and ˣ⁄₃ = tan θ. Squaring both sides of these relations yields (y²)/(4) = sec² θ and (x²)/(9) = tan² θ.

According to the fundamental trigonometric identity, sec² θ - tan² θ = 1. Rearranging this identity to match the required expression, we have tan² θ - sec² θ = -1. Substituting the values of tan² θ and sec² θ into this equation, we obtain (x²)/(9) - (y²)/(4) = -1.

Explanation

Using the properties of remainders, if a number n leaves a remainder r when divided by a divisor d, then the product kn leaves a remainder equal to (k × r) d. In this case, n ≡ 3 4. When the number is doubled to 2n, the remainder is also doubled, resulting in 2 × 3 = 6. Since the remainder cannot be greater than or equal to the divisor, we divide 6 by 4 to find the final remainder: 6 = (1 × 4) + 2.

Alternatively, we can express n as 4q + 3, where q is the quotient. Multiplying this entire expression by 2 gives 2n = 2(4q + 3) = 8q + 6. We can rewrite this as 2n = 8q + 4 + 2, which simplifies to 2n = 4(2q + 1) + 2. Since 4(2q + 1) is a multiple of 4, the remaining constant 2 is the remainder.

Explanation

Let the average donation of all nine members be x. The President's donation is x + 50, so the total donation is 8(100) + (x + 50) = 9x, which simplifies to 8x = 850 or x = 106.25. The President's contribution is 106.25 + 50 = 156.25.

Explanation

The given equation ¹⁄ₐ + (1)/(b) + (1)/(c) = (71)/(abc) can be simplified by taking the least common multiple of the denominators on the left side, which gives (bc + ac + ab)/(abc) = (71)/(abc). By comparing the numerators, we find that ab + bc + ca = 71.

To find the value of a³ + b³ + c³ - 3abc, we use the algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca) This can be rewritten in terms of (a + b + c) and (ab + bc + ca) as: a³ + b³ + c³ - 3abc = (a + b + c)((a + b + c)² - 3(ab + bc + ca))

Substituting the known values a + b + c = 15 and ab + bc + ca = 71 into the formula: 15 × (15² - 3 × 71) 15 × (225 - 213) 15 × 12 = 180

Explanation

The right-hand side of the equation is in the form of the difference of squares identity (A+B)(A-B) = A² - B². By setting A = p²+q² and B = kpq, the expression expands as follows: (p²+q²+kpq)(p²+q²-kpq) = (p²+q²)² - (kpq)² p⁴ + 2p²q² + q⁴ - k²p²q² = p⁴ + (2-k²)p²q² + q⁴ Comparing this result to the left-hand side p⁴ + q⁴, the coefficient of the p²q² term must be zero for the identity to hold for all p and q: 2 - k² = 0 k² = 2 k = ±√2 The question asks for the largest possible real number k. Comparing the two possible values, the largest is √2.

Explanation

According to the theorem on the areas of similar triangles, when two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. This relationship is derived from the fact that the area of a triangle is proportional to the product of its base and height. In similar triangles, both the base and the height scale by the same linear factor.

Given that Δ ABC ∼ Δ DEF and the ratio of their corresponding sides is k:1, we can express the ratio of their areas as:

(Area(Δ ABC))/(Area(Δ DEF)) = (Side_A_B_C{Side_D_E_F})²

Substituting the given side ratio k:1 into the formula:

(Area(Δ ABC))/(Area(Δ DEF)) = (ᵏ⁄₁)² = (k²)/(1²) = (k²)/(1)

Thus, the ratio of the areas of the two triangles is k² : 1.

Explanation

Since BC is the diameter of the circle and CD is the tangent at point C, the radius (or diameter) is perpendicular to the tangent at the point of contact. This established that ∠ BCD = 90°. In the triangle BCD, the vertex D is formed by the intersection of the tangent at C and the extension of the chord BA. The angle ∠ CBD is the same as ∠ ABC, which is given as 36°.

In any triangle, the sum of the interior angles is 180°. Applying this to Δ BCD, we have ∠ BCD + ∠ CBD + ∠ ADC = 180°. Substituting the known values gives 90° + 36° + ∠ ADC = 180°. Solving for the required angle, ∠ ADC = 180° - 126° = 54°.

Explanation

Let the height of the vertical pole be h and the length of its shadow be s. The problem states that the length of the shadow is √3 times the height of the pole, which gives the relation s = √3h. In the right-angled triangle formed by the pole and the shadow, the angle of elevation of the sun θ is the angle between the ground and the line of sight to the sun. Using the trigonometric ratio for tangent, tan θ = (Opposite)/(Adjacent) = ʰ⁄ₛ. Substituting the given value of s, the equation becomes tan θ = (h)/(√(3)h) = (1)/(√(3)). Since tan 30° = (1)/(√(3)), the angle of elevation θ must be 30°.

Explanation

For a period of 2 years, the relationship between Compound Interest (CI) and Simple Interest (SI) at a rate of R% is given by the formula:

CI = SI ( 1 + (R)/(200) )

Given that the Simple Interest for 2 years is Rs. 50 and the rate is 5% per annum, we can substitute these values directly:

CI = 50 ( 1 + ⁵⁄₂₀₀ ) CI = 50 ( 1 + ¹⁄₄₀ ) CI = 50 × ⁴¹⁄₄₀ CI = (5 × 41)/(4) CI = ²⁰⁵⁄₄ = 51.25

Alternatively, the difference between CI and SI for 2 years is the interest earned on the first year's simple interest. Since the SI for 2 years is Rs. 50, the interest for the first year is 50 / 2 = 25. The interest on this Rs. 25 at a 5% rate is 25 × 0.05 = 1.25. Adding this extra interest to the total simple interest gives 50 + 1.25 = 51.25.

Explanation

To find the value of r, square and add the two given equations: (r sin θ)² + (r cos θ)² = (√3)² + (1)² r² (sin² θ + cos² θ) = 3 + 1 Using the trigonometric identity sin² θ + cos² θ = 1, we get: r² (1) = 4 r = √4 = 2

To find the value of θ, divide the first equation by the second equation: (r sin θ)/(r cos θ) = √3{1} tan θ = √3 In the given range 0° ≤ θ ≤ 90°, the tangent function equals √3 when θ = 60°. Substituting r = 2 and θ = 60° back into the original equations confirms the results: 2 sin 60° = 2(√3/2) = √3 and 2 cos 60° = 2(1/2) = 1.

Explanation

To find the time taken by B alone, we first determine the total work by taking the Least Common Multiple (LCM) of the given time periods. The LCM of 12 days and 30 days is 60 units, which represents the total work.

The efficiency of A and B working together is calculated as 60 / 12 = 5 units per day. The efficiency of A working alone is 60 / 30 = 2 units per day. By subtracting A efficiency from the combined efficiency of A and B, we find B efficiency: 5 - 2 = 3 units per day.

The time taken by B to complete the work alone is the total work divided by B efficiency, which is 60 / 3 = 20 days. Alternatively, using the work rate formula 1/B = 1/Total - 1/A, we get 1/B = 1/12 - 1/30 = (5 - 2) / 60 = 3/60 = 1/20. Thus, B requires 20 days.

Explanation

The length of the diagonal d of a cube with side a is given by the formula d = a√3. According to the question, a√3 = √(192). Squaring both sides of the equation yields 3a² = 192, which simplifies to a² = 64. Taking the square root, we find the side of the cube a = 8 cm. The volume V of a cube is calculated using the formula V = a³. Substituting the value of the side, V = 8³ = 512 cm³.

Explanation

The sale price of an article is determined by subtracting the discount amount from the listed price. Using the standard formula for selling price SP = MP × (1 - ᵈ⁄₁₀₀), where MP is the marked or listed price and d is the discount percentage:

SP = 230 × (1 - ¹²⁄₁₀₀) SP = 230 × ⁸⁸⁄₁₀₀ SP = 230 × 0.88 = 202.4

Alternatively, the discount value can be calculated as 12% of 230, which equals 0.12 × 230 = 27.6. Subtracting this discount from the listed price yields 230 - 27.6 = 202.4.

Explanation

The map distance is determined by the ratio of the map scale to the actual distance. Using the scale where 1.5 cm represents 18 km, the map distance d for a real-world distance of 72 km is calculated using the formula for direct proportionality:

d = (Map Scale Distance)/(Actual Scale Distance) × Distance Driven

Substituting the known values into the equation:

d = (1.5)/(18) × 72

To solve this efficiently, first divide 72 by 18 to find how many scale units are covered:

72 ÷ 18 = 4

Then, multiply this factor by the map scale measurement:

d = 1.5 × 4 = 6 cm

Explanation

When the profit earned by selling an article at one price is equal to the loss incurred by selling it at another price, the cost price (CP) is the arithmetic mean of the two selling prices. This is derived from the basic definitions of profit and loss:

Profit = SP₁ - CP Loss = CP - SP₂

Given that Profit = Loss: SP₁ - CP = CP - SP₂ 2 × CP = SP₁ + SP₂ CP = (SP₁ + SP₂)/(2)

Substituting the given selling prices SP₁ = 524 and SP₂ = 452: CP = (524 + 452)/(2) CP = ⁹⁷⁶⁄₂ CP = 488

The cost price of the article is Rs. 488.

Explanation

Let s represent the side of the square and r represent the radius of the circle. The area of a square is defined by the formula s², while the area of a circle is defined by the formula π r². Since the problem states that the circle and the square have the same area, we can equate the two expressions:

s² = π r²

To determine the ratio of the side of the square to the radius of the circle, we divide both sides of the equation by r²:

(s²)/(r²) = π

By taking the square root of both sides, we find the relationship between the side and the radius:

ˢ⁄ᵣ = √(π)

This expression represents the ratio s:r as √(π) : 1.

Explanation

To calculate the time taken by a train to pass a stationary telegraph post, the distance to be covered is equal to the length of the train, which is 75 m. First, the speed must be converted from km/hr to m/s using the conversion factor ⁵⁄₁₈.

Speed = 60 × ⁵⁄₁₈ = (10 × 5)/(3) = ⁵⁰⁄₃ m/s

Using the formula Time = (Distance)/(Speed):

Time = (75)/(50){3} = (75 × 3)/(50) = ²²⁵⁄₅₀ = 4.5 seconds

Explanation

From the given equation 2x - ²⁄ₓ = 1, we can factor out 2 to obtain 2(x - ¹⁄ₓ) = 1. Dividing both sides by 2 gives x - ¹⁄ₓ = ½.

To find the value of x³ - (1)/(x³), we apply the algebraic identity a³ - b³ = (a - b)³ + 3ab(a - b). Substituting a = x and b = ¹⁄ₓ, the expression becomes:

x³ - (1)/(x³) = (x - ¹⁄ₓ)³ + 3(x)(¹⁄ₓ)(x - ¹⁄ₓ)

x³ - (1)/(x³) = (x - ¹⁄ₓ)³ + 3(x - ¹⁄ₓ)

Now, substitute the derived value x - ¹⁄ₓ = ½ into the identity:

x³ - (1)/(x³) = (½)³ + 3(½)

x³ - (1)/(x³) = ⅛ + ³⁄₂

To perform the addition, convert the fractions to a common denominator of 8:

⅛ + ¹²⁄₈ = ¹³⁄₈

Explanation

The algebraic identity for the square of a binomial is (x + y)² = x² + 2xy + y². Applying this formula to the expression (a + ¹⁄ₐ)², where x = a and y = ¹⁄ₐ, we get:

(a + ¹⁄ₐ)² = a² + 2(a)(¹⁄ₐ) + (¹⁄ₐ)²

(a + ¹⁄ₐ)² = a² + 2 + (1)/(a²)

We are given that (a + ¹⁄ₐ)² = 3. Substituting this value into the expanded identity results in:

3 = a² + 2 + (1)/(a²)

By isolating the terms a² + (1)/(a²), we subtract 2 from both sides of the equation:

a² + (1)/(a²) = 3 - 2 = 1

Explanation

The circumcentre of a triangle is defined as the point of intersection of the perpendicular bisectors of its sides. According to the perpendicular bisector theorem, any point located on the perpendicular bisector of a line segment is equidistant from the endpoints of that segment. Since the circumcentre lies on the perpendicular bisectors of all three sides, it is equidistant from all three vertices of the triangle. This equal distance is known as the circumradius R, and it allows a circle, called the circumcircle, to be drawn passing through all the vertices with the circumcentre as its center.

Explanation

Let the three angles of the triangle be x, y, and z. The problem states that two angles are equal, so let y = z. The third angle x is defined as half the sum of these two equal angles, which can be expressed as x = (y + z)/(2). Substituting z = y into this equation, we get x = (y + y)/(2) = ²ʸ⁄₂ = y. Since x = y and y = z, it follows that x = y = z. In any triangle, the sum of the interior angles is 180°, which gives x + y + z = 180°. Substituting the equal values, 3x = 180°, which results in x = 60°. Because all three angles are equal to 60°, the triangle is equilateral.

Explanation

The expression sin⁴θ - cos⁴θ can be factored using the algebraic identity for the difference of squares, a² - b² = (a - b)(a + b). Let a = sin²θ and b = cos²θ:

sin⁴θ - cos⁴θ = (sin²θ - cos²θ)(sin²θ + cos²θ)

According to the fundamental trigonometric identity, sin²θ + cos²θ = 1. Substituting this value into the equation simplifies the expression:

sin⁴θ - cos⁴θ = (sin²θ - cos²θ) × 1

The problem states that sin²θ - cos²θ = 1/4. By substituting this value into the simplified expression, we find:

sin⁴θ - cos⁴θ = 1/4

Explanation

The number sought must be a multiple of the Least Common Multiple (LCM) of the divisors plus the common remainder. The prime factorization of the divisors is 5 = 5, 6 = 2 × 3, 7 = 7, and 8 = 2³. Thus, LCM(5, 6, 7, 8) = 2³ × 3 × 5 × 7 = 840. The required number takes the form 840k + 3, where k is a positive integer. We need this value to be divisible by 9, which can be expressed as 840k + 3 ≡ 0 9. Simplifying the coefficient 840 modulo 9 by finding the sum of its digits (8+4+0=12, and 12 9 = 3), the expression becomes 3k + 3 ≡ 0 9. Testing small integer values for k, we find that if k = 1, 3(1) + 3 = 6, which is not divisible by 9. If k = 2, 3(2) + 3 = 9, which is perfectly divisible by 9. Substituting k = 2 back into the general form gives 840 × 2 + 3 = 1680 + 3 = 1683.

Explanation

Let the number of wickets taken before the last match be x. Since the bowling average was 12.4 runs/wicket, the total runs conceded before the last match is 12.4x. In the final match, the cricketer takes 5 wickets for 22 runs, making the new total of runs 12.4x + 22 and the new total of wickets x + 5.

The new bowling average decreased by 0.4, so the new average is 12.4 - 0.4 = 12.0. Using the formula for bowling average:

(12.4x + 22)/(x + 5) = 12

12.4x + 22 = 12(x + 5)

12.4x + 22 = 12x + 60

0.4x = 38

x = (38)/(0.4) = ³⁸⁰⁄₄ = 95

Alternatively, this can be solved using the rule of allegation. The average of the last match is 22 / 5 = 4.4. Comparing the previous average (12.4) and the last match average (4.4) with the combined average (12.0), the ratio of wickets taken is (12.0 - 4.4) : (12.4 - 12.0). This simplifies to 7.6 : 0.4, or 19 : 1. Since the 1 unit represents the 5 wickets taken in the last match, the 19 units represent 19 × 5 = 95 wickets.

Explanation

The given expression 4b²c² - (b² + c² - a²)² can be written as a difference of squares in the form X² - Y², where X = 2bc and Y = b² + c² - a². Using the algebraic identity X² - Y² = (X + Y)(X - Y), the expression factors as:

[2bc + (b² + c² - a²)] [2bc - (b² + c² - a²)]

Rearranging the terms within the brackets to form perfect squares:

[(b² + c² + 2bc) - a²] [a² - (b² + c² - 2bc)] = [(b + c)² - a²] [a² - (b - c)²]

Applying the difference of squares identity again to both terms:

[(b + c) + a][(b + c) - a] · [a + (b - c)][a - (b - c)] = (a + b + c)(b + c - a)(a + b - c)(a - b + c)

The four individual factors of the expression are f₁ = a + b + c, f₂ = b + c - a, f₃ = a + b - c, and f₄ = a - b + c. To find the sum of these factors, we add them together:

Sum = (a + b + c) + (b + c - a) + (a + b - c) + (a - b + c)

Grouping the like variables:

Sum = (a - a + a + a) + (b + b + b - b) + (c + c - c + c) Sum = 2a + 2b + 2c Sum = 2(a + b + c)

Explanation

The given equation is { ½(a - b) }² + ab = p(a + b)². Expanding the left-hand side, we apply the power to the terms inside the braces:

¼(a - b)² + ab = p(a + b)²

Using the algebraic identity (a - b)² = a² - 2ab + b², the equation becomes:

¼(a² - 2ab + b²) + ab = p(a + b)²

Distributing the ¼ across the terms in the parentheses:

¼a² - ²⁄₄ab + ¼b² + ab = p(a + b)²

¼a² - ½ab + ¼b² + ab = p(a + b)²

Combining the like terms -½ab and ab results in +½ab:

¼a² + ½ab + ¼b² = p(a + b)²

Factoring out ¼ from the left side:

¼(a² + 2ab + b²) = p(a + b)²

Since a² + 2ab + b² = (a + b)², the equation simplifies to:

¼(a + b)² = p(a + b)²

By comparing the coefficients of (a + b)² on both sides, we find that p = ¼.

Explanation

Let the three angles of the triangle be x, y, and z. Given that the sum of two angles is x + y = 116° and their difference is x - y = 24°. To find the individual values of these two angles, we use the sum and difference formulas: x = (Sum + Difference)/(2) = (116° + 24°)/(2) = (140°)/(2) = 70° y = (Sum - Difference)/(2) = (116° - 24°)/(2) = (92°)/(2) = 46°

The sum of all interior angles in a triangle is always 180°. Therefore, the third angle z is calculated as: z = 180° - (x + y) = 180° - 116° = 64°

The three angles of the triangle are 70°, 46°, and 64°. By comparing these measures, the smallest angle is 46°.

Explanation

In Δ ABC, since DE ∥ BC, the triangles Δ ADE and Δ ABC are similar by the AA (Angle-Angle) similarity criterion because ∠ A is common and ∠ ADE = ∠ ABC as corresponding angles. According to the property of similar triangles, the ratio of corresponding sides is constant, giving the relation (BC)/(DE) = (AB)/(AD).

The length of AD is calculated by subtracting BD from the total length of AB, such that AD = AB - BD = 7 - 3 = 4 cm. Substituting the values of AB and AD into the ratio, we get BC : DE = 7 : 4. To express this ratio in the form provided in the options, we can divide both parts of the ratio by 2, which yields 3.5 : 2.

Explanation

Using the trigonometric identity for complementary angles, sin(90° - θ) = cos θ and cos(90° - θ) = sin θ. In the numerator, since 63° + 27° = 90°, we can express sin 27° as sin(90° - 63°) = cos 63°. Thus, the numerator becomes sin² 63° + cos² 63°. Applying the fundamental identity sin² θ + cos² θ = 1, the value of the numerator is 1.

In the denominator, since 17° + 73° = 90°, we can express cos 73° as cos(90° - 17°) = sin 17°. The denominator becomes cos² 17° + sin² 17°, which also equals 1 based on the same trigonometric identity. The fraction simplifies to 1 / 1, resulting in a value of 1.

Explanation

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

Substituting the given values into the formula: P = 6250 R = 8

D = 6250 × (⁸⁄₁₀₀)² D = 6250 × ⁶⁴⁄₁₀₀₀₀

To simplify the calculation, we can reduce the fraction: D = 6250 × ¹⁶⁄₂₅₀₀ D = ⁶²⁵⁰⁄₂₅₀₀ × 16 D = 2.5 × 16 D = 40

Alternatively, the simple interest for the first year is 6250 × 0.08 = 500. The difference between CI and SI after two years is the interest calculated on this first-year interest at the same rate: 500 × 0.08 = 40.

Explanation

The length of a shadow s of a vertical pillar of height h when the angle of elevation of the sun is θ is given by the formula s = h cot θ.

For the initial angle of elevation ₁ = 30° and pillar height h = 50√3 m, the shadow length s₁ is calculated as: s₁ = 50√3 × cot 30° = 50√3 × √3 = 150 m.

When the angle of elevation is raised to ₂ = 60°, the new shadow length s₂ becomes: s₂ = 50√3 × cot 60° = 50√3 × (1)/(√(3)) = 50 m.

The decrease in the length of the shadow is the difference between the two lengths: Δ s = s₁ - s₂ = 150 - 50 = 100 m.

Explanation

The work-rate relationship is governed by the formula (M₁ D₁)/(W₁) = (M₂ D₂)/(W₂), where M is the number of workers (cats), D is the time duration (days), and W is the amount of work completed (mice killed). Substituting the provided values into the formula gives (100 × 100)/(100) = (4 × D₂)/(4). Simplifying both sides of the equation leads to 100 = D₂. This result signifies that the individual rate of work per cat is one mouse per 100 days; therefore, four cats working simultaneously will still take 100 days to finish their respective tasks and kill four mice in total.

Explanation

According to the Basic Proportionality Theorem (Thales's Theorem), if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. Thus, in Δ ABC with DE ∥ BC, we have the relation:

(AD)/(AB) = (AE)/(AC)

We are given the ratio (AD)/(BD) = ⅗. To find the ratio of AD to the whole side AB, we sum the components:

AB = AD + BD = 3 + 5 = 8 units

Substituting the known values into the proportionality equation:

⅜ = (AE)/(5.6)

Solving for AE:

AE = (3 × 5.6)/(8) AE = 3 × 0.7 AE = 2.1 cm

Explanation

When two circles touch each other internally, the distance between their centers is equal to the difference between their radii. If the radius of the larger circle is denoted by R and the radius of the smaller circle is denoted by r, the distance d between the centers is given by the formula d = R - r.

Given that the radius of the greater circle R = 6 cm and the distance between the centers d = 2 cm, the equation becomes: 2 = 6 - r

Rearranging the terms to solve for r: r = 6 - 2 r = 4 cm

Explanation

The total duration of office hours from 10 AM to 5 PM is 7 hours. To find the ratio, both quantities must be in the same units. Converting the total office hours into minutes, we get 7 × 60 = 420 minutes. The lunch interval is given as 30 minutes. The ratio of the lunch interval to the total period of office hours is calculated as 30 : 420. Dividing both terms by 30 simplifies the ratio to 1 : 14.

Explanation

The amount to be paid by Sunil is the Selling Price (SP) after deducting the discount from the Marked Price (MP). Given the Marked Price is Rs. 1500 and the discount rate is 15%, the calculation is as follows:

Selling Price = MP × (100 - Discount Percentage)/(100)

SP = 1500 × (100 - 15)/(100)

SP = 1500 × ⁸⁵⁄₁₀₀

SP = 15 × 85

SP = 1275

Alternatively, the discount amount is 15% of 1500, which is 1500 × 0.15 = 225. Subtracting this discount from the total price gives 1500 - 225 = 1275. Thus, Sunil has to pay Rs. 1275 to the bookseller.

Explanation

The relationship between Sale Price (SP), Cost Price (CP), and Profit (P) is given by the formula CP = SP - P.

Given the ratio of profit to sale price is 1:5, we can assume P = 1 unit and SP = 5 units. Substituting these values into the formula:

CP = 5 - 1 = 4 units

The ratio between the sale price and the cost price is therefore:

SP : CP = 5 : 4

Explanation

To find the percentage of 1 day that 36 minutes represents, first determine the total number of minutes in a day. Since 1 day consists of 24 hours and each hour contains 60 minutes, the total duration is 24 × 60 = 1440 minutes. The percentage is calculated using the standard formula:

Percentage = (Value)/(Total) × 100

Substituting the given values: Percentage = ³⁶⁄₁₄₄₀ × 100

Simplifying the calculation: Percentage = ³⁶⁄₁₄₄ × 10

Since 36 × 4 = 144, the fraction ³⁶⁄₁₄₄ simplifies to ¼: Percentage = ¼ × 10 = 2.5%

Explanation

The distance covered by an object is given by the formula D = S × T, where S is speed and T is time. When time is kept constant, the distance covered is directly proportional to the speed.

The problem states that the speed is reduced by ⅕th, which means the reduction in speed is Δ S = ⅕S. Consequently, the reduction in distance over the same time period T is Δ D = Δ S × T.

Substituting the given values into this relationship: 200 = ⅕S × 25 200 = 5S S = ²⁰⁰⁄₅ = 40 km/hr