SSC CHSL Tier 1 2011 Quantitative Aptitude Questions with Solutions

Explanation

Let the larger original number be x and the smaller original number be y. Since the two numbers are unequal, we have x > y.

Half of the smaller number is y/2. Subtracting this from both x and y gives the two resulting numbers: x - y/2 and y - y/2 = y/2

Since x > y, the resulting number x - y/2 is larger than y/2. According to the given condition, this larger resulting number is five times the smaller resulting number: x - y/2 = 5 × (y/2)

Solving for x: x = 5y/2 + y/2 x = 6y/2 x = 3y

The ratio of the larger original number to the smaller original number is: x/y = 3/1

Explanation

The square root function f(x) = √x is a strictly increasing function for all non-negative real numbers. This means that if x is less than y, then √x is less than √y. Since 2 is less than 3, which is less than 4, it directly follows that √2 is less than √3, which is less than √4. Evaluating the terms, √2 is approximately 1.414, √3 is approximately 1.732, and √4 is exactly 2, which makes √4 the largest value.

Explanation

Let the age of A at the time of marriage be x years. Since A got married 8 years ago, A's present age is x + 8 years.

According to the problem, A's present age is 1¼ times his age at the time of marriage. This can be written as: x + 8 = 1.25x 8 = 0.25x x = 32

Thus, A's present age is 32 + 8 = 40 years.

The age of A's son is 1/10 times A's present age: Son's age = (1/10) × 40 = 4 years.

Explanation

An integer K satisfies two conditions. First, K leaves a remainder of 1 when divided by 3, which means K can be written in the form 3n + 1 for some integer n. Second, K + 1 leaves a remainder of 0 when divided by 5, which means K + 1 is a multiple of 5.

When K is 64, dividing 64 by 3 gives 21 with a remainder of 1, satisfying the first condition. Further, K + 1 becomes 65, which is exactly divisible by 5 (5 × 13 = 65) with a remainder of 0, satisfying the second condition.

Explanation

To find the maximum number of animals in each flock, we calculate the Highest Common Factor (HCF) of the number of cows and sheep.

The prime factorizations of the numbers are: 945 = 3³ × 5 × 7 2475 = 3² × 5² × 11

Taking the lowest power of the common prime factors: HCF(945, 2475) = 3² × 5 = 45

This gives the maximum number of animals in each flock as 45.

The total number of flocks required is the sum of the flocks of cows and sheep: Number of cow flocks = 945 ÷ 45 = 21 Number of sheep flocks = 2475 ÷ 45 = 55 Total flocks = 21 + 55 = 76

Explanation

The interior angle of a regular polygon with n sides is given by the formula I = 180° - 360°/n.

Let the number of sides of the two polygons be 5x and 4x, where x is a positive integer.

Since the polygon with more sides has a larger interior angle, the difference between their interior angles is: (180° - 360° / 5x) - (180° - 360° / 4x) = 6° 360° / 4x - 360° / 5x = 6°

Dividing the entire equation by 6: 60 / 4x - 60 / 5x = 1 15 / x - 12 / x = 1 3 / x = 1 x = 3

Substituting the value of x, the number of sides for the two polygons are: 5 × 3 = 15 4 × 3 = 12

Explanation

The volume V of a regular tetrahedron with side length a is given by the formula:

V = a³ / (6√2)

Substituting the given side length a = 12 cm into the formula:

V = 12³ / (6√2) V = 1728 / (6√2) V = 288 / √2 V = 144√2 cu. cm.

Explanation

The volume V of a truncated conical bucket (frustum of a cone) with height h and end radii R and r is given by the formula:

V = (1/3) × π × h × (R² + r² + R × r)

Substituting the given values h = 45 cm, R = 28 cm, r = 7 cm, and π = 22/7 into the formula:

V = (1/3) × (22/7) × 45 × (28² + 7² + 28 × 7)

V = 15 × (22/7) × (784 + 49 + 196)

V = 15 × (22/7) × 1029

V = 15 × 22 × 147

V = 330 × 147

V = 48510 cubic centimetres

Explanation

Let the common radius of the base of the cone, hemisphere, and cylinder be r. Since the hemisphere stands on the same base, its height is equal to its radius, r. Because all three solids have the same height, the height h of the cone and the cylinder must also be equal to r.

The formulas for the volumes of the three solids are: Volume of the cone = ⅓πr²h = ⅓πr³ Volume of the hemisphere = ⅔πr³ Volume of the cylinder = πr²h = πr³

Taking the ratio of their volumes: Ratio = ⅓πr³ : ⅔πr³ : πr³

Simplifying by dividing each term by πr³ gives: Ratio = ⅓ : ⅔ : 1

Multiplying each term by 3 to obtain integer values gives: Ratio = 1 : 2 : 3

Explanation

Area of the square = s² = 484 cm² Side of the square s = √484 = 22 cm

Length of the wire is equal to the perimeter of the square: Perimeter = 4 × s = 4 × 22 = 88 cm

When the same wire is bent into a circle, its circumference is equal to the length of the wire: Circumference = 2πr = 88 cm 2 × (22/7) × r = 88 r = 14 cm

The area of the circle is: Area = πr² = (22/7) × 14 × 14 = 616 cm²

Explanation

The area of a parallelogram is given by the formula:

Area = Base × Altitude

Let the greater side be 5x and the smaller side be 4x. The altitude on the greater side is 20 units. Let the altitude on the smaller side be h.

Equating the area calculated from both sides:

Greater side × Altitude on greater side = Smaller side × Altitude on smaller side 5x × 20 = 4x × h

Dividing both sides by x:

5 × 20 = 4 × h 100 = 4 × h h = 25 units

Explanation

The total area of the canvas required to make the tent is the sum of the curved surface area of the cylindrical part and the curved surface area of the conical part.

Radius of the base, r = diameter / 2 = 105/2 m = 52.5 m Height of the cylindrical part, h = 3 m Slant height of the conical part, l = 63 m

Curved surface area of the cylindrical part = 2πrh Curved surface area of the conical part = πrl

Total area of the canvas = 2πrh + πrl = πr(2h + l)

Substituting the given values into the formula: Total area = (22/7) × (105/2) × (2 × 3 + 63) Total area = 11 × 15 × (6 + 63) Total area = 165 × 69 Total area = 11385 m²

Explanation

To find the time taken by A and B together, let the total work be the least common multiple of 12, 8, and 6, which is 24 units.

The daily work rate of B and C is 24 / 12 = 2 units. The daily work rate of C and A is 24 / 8 = 3 units. The daily work rate of A, B, and C together is 24 / 6 = 4 units.

To find the individual work rate of A, subtract the rate of B and C from the total rate: 4 - 2 = 2 units.

To find the individual work rate of B, subtract the rate of C and A from the total rate: 4 - 3 = 1 unit.

The combined daily work rate of A and B is: 2 + 1 = 3 units.

The number of days A and B together will take to complete the total work is: 24 / 3 = 8 days.

Explanation

The time taken to complete a work is inversely proportional to the efficiency of the person.

Let the efficiency of A be 100. Since B is 50% more efficient than A, the efficiency of B is 150.

The ratio of the efficiencies of A and B is: Efficiency of A : Efficiency of B = 100 : 150 = 2 : 3

The ratio of the time taken by A and B to complete the work is the inverse of their efficiency ratio: Time taken by A : Time taken by B = 3 : 2

Given that A takes 9 days, the time taken by B (T) is calculated using the proportion: 3/2 = 9/T T = 9 × 2 / 3 = 6 days

Explanation

The single discount equivalent to two successive discounts of d₁% and d₂% is calculated using the formula d₁ + d₂ - (d₁ × d₂)/100. Substituting the given values d₁ = 10 and d₂ = 20 into the formula yields 10 + 20 - (10 × 20)/100, which simplifies to 30 - 2 = 28%.

Explanation

The net profit or loss percentage when there is a successive markup of x% and a discount of y% is given by the successive percentage formula:

Net Percentage Change = x + y + (xy)/100

In this case, the markup percentage x is +40 and the discount percentage y is -20.

Substituting these values into the formula:

Net Percentage Change = 40 - 20 + (40 × (-20))/100 Net Percentage Change = 20 - 800/100 Net Percentage Change = 20 - 8 Net Percentage Change = 12

Since the final value is positive, the dealer registers a gain of 12%.

Explanation

Given that 120% of a is equal to 80% of b: 120/100 × a = 80/100 × b 1.2 a = 0.8 b 3 a = 2 b b/a = 3/2

To find the value of (b + a)/(b - a), we can divide both the numerator and the denominator by a: ((b/a) + 1) / ((b/a) - 1)

Substituting the value of b/a = 3/2 into the expression: ((3/2) + 1) / ((3/2) - 1) = (5/2) / (1/2) = 5

Explanation

In the first mixture of 20 litres, the ratio of spirit to water is 3:7. Total parts = 3 + 7 = 10 Amount of spirit = (3/10) × 20 = 6 litres Amount of water = (7/10) × 20 = 14 litres

In the second mixture of 36 litres, the ratio of spirit to water is 7:5. Total parts = 7 + 5 = 12 Amount of spirit = (7/12) × 36 = 21 litres Amount of water = (5/12) × 36 = 15 litres

When both mixtures are combined: Total amount of spirit = 6 + 21 = 27 litres Total amount of water = 14 + 15 = 29 litres

The ratio of spirit to water in the new mixture is 27:29.

Explanation

The average x̄ of n numbers x₁, x₂, ..., xₙ is given by the formula: x̄ = (∑ xᵢ) / n, where the summation runs from i = 1 to n.

Multiplying both sides by n gives the sum of the observations: ∑ xᵢ = n x̄

We want to find the value of the summation of the deviations from the mean: ∑ (xᵢ - x̄) = ∑ xᵢ - ∑ x̄

Since x̄ is a constant value independent of the index i, summing it n times yields: ∑ x̄ = n x̄

Substituting these relations back into the expression: ∑ (xᵢ - x̄) = n x̄ - n x̄ = 0

Explanation

The new average can be directly calculated by finding the net change in the sum of the numbers and dividing it by the total count of numbers.

The formula for the new average is: New average = Old average + (Net change in sum / Total numbers)

The net change in the sum of the six numbers is: (3 × 2) + (3 × -4) = 6 - 12 = -6

The change in the average is: -6 / 6 = -1

Thus, the new average is: 32 - 1 = 31

Explanation

The cost price of the article is represented by a and the selling price is represented by b.

Since b is 200% of a, the relationship is: b = (200/100) × a = 2a

The profit made on the article is: Profit = Selling Price - Cost Price = b - a = 2a - a = a

The percentage of profit on the cost price is calculated using the formula: Profit Percentage = (Profit / Cost Price) × 100 Profit Percentage = (a / a) × 100 = 100%

Explanation

Let the cost price of 1 mango be ₹1.

The cost price of 400 mangoes is ₹400.

Since the selling price of 400 mangoes is equal to the cost price of 320 mangoes, the selling price of 400 mangoes is ₹320.

Loss = Cost Price - Selling Price = 400 - 320 = 80

Loss percentage = (Loss / Cost Price) × 100 = (80 / 400) × 100 = 20%

Explanation

Let the number of brand B shirts ordered in the original order be x. Let the price of each shirt of brand B be p. Since the price of each shirt of brand A is twice that of brand B, the price of each shirt of brand A is 2p.

The original cost of the bill is: Original Bill = 4 × 2p + x × p = (8 + x)p

When the numbers of the two brands are interchanged, the customer gets x shirts of brand A and 4 shirts of brand B. The new cost of the bill is: New Bill = x × 2p + 4 × p = (2x + 4)p

The problem states that the interchanged order increased the bill by 40%: New Bill = 1.4 × Original Bill (2x + 4)p = 1.4 × (8 + x)p

Dividing both sides by p: 2x + 4 = 11.2 + 1.4x 0.6x = 7.2 x = 12

The original order had 4 shirts of brand A and 12 shirts of brand B. The ratio of brand A shirts to brand B shirts in the original order is: 4 : 12 = 1 : 3

Explanation

Initial volume of the solution is 6 litres, which contains 30% alcohol and therefore 70% water.

Volume of water = 70% of 6 litres = 0.70 × 6 = 4.2 litres

When 1 litre of pure alcohol is added, the new total volume of the solution becomes: 6 + 1 = 7 litres

Since only alcohol is added, the quantity of water remains unchanged at 4.2 litres.

The percentage of water in the new solution is: (Volume of water / New total volume) × 100 = (4.2 / 7) × 100 = 60%

Explanation

Let the speed of the current be c km/hour. Since the speed of the boat in still water is four times the speed of the current, the speed of the boat in still water is 4c km/hour.

The downstream speed of the boat is: v_d = 4c + c = 5c km/hour

The upstream speed of the boat is: v_u = 4c - c = 3c km/hour

The total time taken for the round trip of 30 km downstream and 30 km upstream is 8 hours. The formula relating distance, speed, and time is: Distance / Downstream speed + Distance / Upstream speed = Total time

Substituting the values: 30 / 5c + 30 / 3c = 8 6 / c + 10 / c = 8 16 / c = 8 c = 16 / 8 = 2

Explanation

The difference between the compound interest and simple interest on a principal sum P for a period of 2 years at an annual interest rate r% is given by the formula:

D = P × (r/100)²

Substituting the given values, where D = 1 and r = 4:

1 = P × (4/100)²

Simplifying the fraction:

1 = P × (1/25)²

1 = P × (1/625)

P = 625

Explanation

Given the equation x² + 2 = 2x, we can rewrite it as: x² - 2x + 2 = 0

We want to find the value of the expression x⁴ - x³ + x² + 2. We can perform polynomial division of the target expression by the given quadratic equation:

x⁴ - x³ + x² + 2 = x²(x² - 2x + 2) + x(x² - 2x + 2) + 1(x² - 2x + 2) x⁴ - x³ + x² + 2 = (x² - 2x + 2)(x² + x + 1)

Since x² - 2x + 2 = 0, we substitute this value into the factored expression:

0 × (x² + x + 1) = 0

Explanation

Let 2ˣ = 3ʸ = 6⁻ᶻ = k, where k is a constant.

From this relationship, we can express each base in terms of k: 2 = k⁽1/x) 3 = k⁽1/y) 6⁻¹ = k⁽1/z), which simplifies to 6 = k⁽-1/z)

We know the mathematical relationship between the bases: 2 × 3 = 6

Substituting the exponential terms of k: k⁽1/x) × k⁽1/y) = k⁽-1/z)

Using the laws of exponents to add the powers on the left side: k⁽1/x + 1/y) = k⁽-1/z)

By equating the exponents from both sides: 1/x + 1/y = -1/z

Adding 1/z to both sides: 1/x + 1/y + 1/z = 0

Explanation

Given the equation: 1/(x + y) = 1/x + 1/y

By taking the least common multiple on the right-hand side, the equation becomes: 1/(x + y) = (x + y)/xy

Cross-multiplying the terms: xy = (x + y)² xy = x² + 2xy + y²

Rearranging the terms by subtracting xy from both sides yields: x² + xy + y² = 0

The algebraic identity for the difference of cubes is: x³ - y³ = (x - y)(x² + xy + y²)

Substituting the value of x² + xy + y² = 0 into the identity: x³ - y³ = (x - y) × 0 = 0

Explanation

The given equation is a² + b² + c² = ab + bc + ca. Multiplying both sides by 2 and rearranging the terms gives:

2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 0

This can be grouped as:

(a² - 2ab + b²) + (b² - 2bc + c²) + (c² - 2ca + a²) = 0

(a - b)² + (b - c)² + (c - a)² = 0

Since a, b, and c are real numbers, the square of any real number is non-negative. For the sum of non-negative terms to be zero, each term must be equal to zero:

a - b = 0, which means a = b b - c = 0, which means b = c c - a = 0, which means c = a

This yields the condition a = b = c.

Substituting a = b and c = b into the expression (a + c)/b:

(b + b)/b = 2b/b = 2

Explanation

Given the equations: x/a = b - c y/b = c - a z/c = a - b

Let P = x/a, Q = y/b, and R = z/c.

Summing these variables gives: P + Q + R = (b - c) + (c - a) + (a - b) = 0

Using the algebraic identity where if P + Q + R = 0, then P³ + Q³ + R³ = 3PQR.

Substituting the variables back into the identity yields: (x/a)³ + (y/b)³ + (z/c)³ = 3(x/a)(y/b)(z/c) (x/a)³ + (y/b)³ + (z/c)³ = 3xyz/abc

Explanation

Let the diagonals AC and BD of the quadrilateral ABCD intersect at point O at right angles. This forms four right-angled triangles: AOB, BOC, COD, and DOA.

Applying Pythagoras' theorem to triangles AOB and COD: AB² = OA² + OB² CD² = OC² + OD²

Adding these two equations gives: AB² + CD² = OA² + OB² + OC² + OD²

Similarly, applying Pythagoras' theorem to triangles BOC and DOA: BC² = OB² + OC² DA² = OA² + OD²

Adding these two equations gives: BC² + DA² = OA² + OB² + OC² + OD²

Equating the two sums yields: AB² + CD² = BC² + DA²

Explanation

Let the tangents drawn at the extremities A and B of the diameter AB intersect the tangent at C at points Q and R respectively. Thus, Q lies on the tangent at A, and R lies on the tangent at B.

QA and QC are tangents drawn from the external point Q to the circle with centre P. Since the line segment joining the centre of a circle to an external point bisects the angle subtended by the points of contact at the centre, we have: ∠APQ = ∠CPQ = ½ ∠APC

Similarly, RB and RC are tangents drawn from the external point R to the circle. Thus, the line segment PR bisects ∠BPC: ∠BPR = ∠CPR = ½ ∠BPC

Since AB is a diameter of the circle, points A, P, and B lie on a straight line. Therefore, the sum of the angles on the straight line is 180°: ∠APC + ∠BPC = 180°

Substituting the bisected angles into this equation gives: 2 ∠CPQ + 2 ∠CPR = 180° 2 (∠CPQ + ∠CPR) = 180° ∠CPQ + ∠CPR = 90°

Since ∠QPR is the sum of ∠CPQ and ∠CPR, we have: ∠QPR = 90°

Explanation

Since O is the in-centre of △ABC, the line OB is the angle bisector of ∠ABC, which means ∠OBD = ½ ∠ABC.

It is given that OD ⊥ BC, which means ∠ODB = 90°.

In the right-angled △OBD, the sum of the internal angles is 180°: ∠BOD + ∠OBD + ∠ODB = 180°

Substituting the known values: 15° + ∠OBD + 90° = 180° ∠OBD = 75°

Using the angle bisector relation: ∠ABC = 2 × ∠OBD = 2 × 75° = 150°

Explanation

According to the Alternate Segment Theorem, the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.

Here, PAT is the tangent at A and AB is the chord. The angle ∠BAT between the tangent AT and the chord AB is equal to the angle ∠ACB subtended by the chord AB in the alternate segment. ∠ACB = ∠BAT = 75°

In triangle ABC, the sum of all interior angles is 180°: ∠ABC + ∠BAC + ∠ACB = 180°

Substituting the given value ∠BAC = 45° and the derived value ∠ACB = 75°: ∠ABC + 45° + 75° = 180° ∠ABC + 120° = 180° ∠ABC = 60°

Explanation

By applying the Midpoint Theorem in triangle ABD, the line segment PX connects the midpoints of sides AB and AD. Therefore, PX is parallel to BD and its length is half of the length of BD: PX = ½ BD

Similarly, by applying the Midpoint Theorem in triangle CBD, the line segment QY connects the midpoints of sides BC and CD. Therefore, QY is parallel to BD and its length is half of the length of BD: QY = ½ BD

Since both PX and QY are equal to half of BD, they are equal to each other, which means PX = QY. This directly gives the ratio PX : QY = 1 : 1.

Explanation

Let x = 2 cos θ - sin θ = 1/√2 and y = 2 sin θ + cos θ.

By squaring and adding these two expressions: x² + y² = (2 cos θ - sin θ)² + (2 sin θ + cos θ)² x² + y² = (4 cos² θ - 4 cos θ sin θ + sin² θ) + (4 sin² θ + 4 sin θ cos θ + cos² θ) x² + y² = 5 (cos² θ + sin² θ)

Using the trigonometric identity cos² θ + sin² θ = 1: x² + y² = 5

Substituting x = 1/√2 into this relation: (1/√2)² + y² = 5 ½ + y² = 5 y² = 5 - ½ y² = 9/2 y = ±3/√2

Since θ is in the first quadrant (0° < θ < 90°), both sin θ and cos θ are positive. This makes the sum y = 2 sin θ + cos θ strictly positive. y = 3/√2

Explanation

Applying componendo and dividendo to the given equation: (sin θ + cos θ) / (sin θ - cos θ) = 3 / 1

((sin θ + cos θ) + (sin θ - cos θ)) / ((sin θ + cos θ) - (sin θ - cos θ)) = (3 + 1) / (3 - 1)

2 sin θ / 2 cos θ = 4 / 2

tan θ = 2

The expression to be evaluated can be factored as: sin⁴ θ - cos⁴ θ = (sin² θ - cos² θ)(sin² θ + cos² θ)

Since sin² θ + cos² θ = 1, this simplifies to: sin⁴ θ - cos⁴ θ = sin² θ - cos² θ

We can express this in terms of tan θ by factoring out cos² θ: sin² θ - cos² θ = cos² θ (tan² θ - 1)

Using the identity cos² θ = 1 / sec² θ = 1 / (1 + tan² θ): sin² θ - cos² θ = (tan² θ - 1) / (1 + tan² θ)

Substituting tan θ = 2 into the expression: (2² - 1) / (1 + 2²) = (4 - 1) / (1 + 4) = 3/5

Explanation

The given expression represents the product of the tangent of angles from 1° to 89°: P = tan 1° · tan 2° · tan 3° · ... · tan 89°

Using the trigonometric identity for complementary angles: tan(90° - θ) = cot θ

The terms from the second half of the product can be rewritten as: tan 89° = tan(90° - 1°) = cot 1° tan 88° = tan(90° - 2°) = cot 2° ... tan 46° = tan(90° - 44°) = cot 44°

Now, pairing the terms from the beginning and the end of the product gives: P = (tan 1° · tan 89°) · (tan 2° · tan 88°) · ... · (tan 44° · tan 46°) · tan 45°

Substituting the cotangent equivalents: P = (tan 1° · cot 1°) · (tan 2° · cot 2°) · ... · (tan 44° · cot 44°) · tan 45°

Since tan θ · cot θ = 1 and tan 45° = 1, each paired term evaluates to 1: P = 1 · 1 · ... · 1 · 1 = 1

Explanation

Let the height of each pole be h. Let the point on the road be at a distance x from the pole with the 60° angle of elevation. Consequently, the distance of this point from the second pole with the 30° angle of elevation is 100 - x.

Using the trigonometric ratio for the first pole: tan(60°) = h / x √3 = h / x x = h / √3

Using the trigonometric ratio for the second pole: tan(30°) = h / (100 - x) 1 / √3 = h / (100 - x) 100 - x = h√3

Substituting the value of x into this equation: 100 - h / √3 = h√3 100 = h√3 + h / √3 100 = h(3 + 1) / √3 100 = 4h / √3 h = 25√3

Explanation

Using the fundamental trigonometric identity sec²θ - tan²θ = 1, we can express sec²θ as 1 + tan²θ.

Substituting this expression into the given equation: (1 + tan²θ) + tan²θ = 7 1 + 2 tan²θ = 7 2 tan²θ = 6 tan²θ = 3

Taking the square root on both sides for the interval 0° ≤ θ ≤ 90° where the tangent function is non-negative: tan θ = √3

Since tan 60° = √3, we find θ = 60°.

Explanation

The percentage of passed students in the medical stream is calculated using the formula:

Percentage of passed students = (Number of passed students / Number of appeared students) × 100

From the table, in the year 2003, the number of medical students who appeared is 384 and the number of students who passed is 326. Substituting these values into the formula:

Percentage = (326 / 384) × 100 ≈ 84.89%

Rounding to the nearest whole percentage gives 85%.

Explanation

To find the percentage of total students appearing in 2004 who appeared in the commerce stream, we first calculate the total number of students who appeared in all streams in 2004.

Total appeared students in 2004 = Engineering + Medical + Management + Commerce Total appeared students in 2004 = 310 + 395 + 106 + 1180 = 1991

The number of students who appeared in the commerce stream in 2004 is 1180.

The required percentage is calculated as: Percentage = (Students in commerce / Total students) × 100 Percentage = (1180 / 1991) × 100 ≈ 59.27%

Rounding this value to the nearest tenth gives approximately 59.3%.

Explanation

To find the year with the minimum number of appeared students across all streams, we calculate the total appeared students for each year:

In 2001, the total is 324 + 469 + 96 + 1467 = 2356.

In 2002, the total is 386 + 430 + 74 + 1246 = 2136.

In 2003, the total is 284 + 384 + 124 + 1387 = 2179.

In 2004, the total is 310 + 395 + 106 + 1180 = 1991.

In 2005, the total is 426 + 424 + 92 + 1562 = 2504.

In 2006, the total is 380 + 465 + 78 + 1374 = 2297.

The year 2004 recorded the lowest total of 1991 students appearing across all streams.

Explanation

Evaluating the given expressions by interpreting the hyphens as decimal points:

(0.1)² = 0.01 √(0.0121) = 0.11 0.12 is already in decimal form √(0.0004) = 0.02

Comparing these values: 0.12 > 0.11 > 0.02 > 0.01

This shows that 0.12 is the largest value among them.

Explanation

For any integer n ≥ 4, the factorial n! is divisible by 8 because it contains both 2 and 4 as factors. For instance, 4! = 24, which is a multiple of 8. Therefore, the terms from 4! to 100! all leave a remainder of 0 when divided by 8.

The remainder of the entire sum when divided by 8 depends only on the sum of the first three terms, which are 1!, 2!, and 3!:

1! + 2! + 3! = 1 + 2 + 6 = 9

Dividing 9 by 8 gives a remainder of 1.

Explanation

The last digit of powers of 3 repeats in a cycle of 4. 3¹ ends in 3 3² ends in 9 3³ ends in 7 3⁴ ends in 1

For any exponent n, the units digit of 3ⁿ is determined by finding the remainder when n is divided by 4. Since 40 is exactly divisible by 4 (40 = 4 × 10 with a remainder of 0), the units digit of 3⁴⁰ corresponds to the fourth position in the cycle, which is 1.

This can also be calculated directly using exponential properties: 3⁴⁰ = (3⁴)¹⁰ = 81¹⁰

Any positive integer ending in 1 raised to any power will always result in a number that has 1 as its last digit.

Explanation

The total work is constant and is given by the product of the number of men and the number of days. Using the formula:

M₁ × D₁ = M₂ × D₂

where M₁ is the initial number of men, D₁ is the initial number of days, M₂ is the final number of men, and D₂ is the final number of days.

Substituting the given values into the formula: M₁ = 30 men D₁ = 18 days M₂ = 30 + 6 = 36 men

30 × 18 = 36 × D₂ 540 = 36 × D₂ D₂ = 540 ÷ 36 D₂ = 15 days

Explanation

Total work can be assumed as the Least Common Multiple (LCM) of the individual times taken by A, B, and C.

LCM(12, 24, 36) = 72 units

The daily efficiency of each person is: Efficiency of A = 72 ÷ 12 = 6 units/day Efficiency of B = 72 ÷ 24 = 3 units/day Efficiency of C = 72 ÷ 36 = 2 units/day

Combined efficiency of A, B, and C working together is: 6 + 3 + 2 = 11 units/day

The number of days taken by them together to complete the work is: 72 ÷ 11 = 6 6/11 days

Explanation

The formula relating the Marked Price (MP), Selling Price (SP), and Discount Rate (D) is:

MP = SP ÷ (1 - D)

Substituting SP = 740 and D = 7.5% (or 0.075) into the formula:

MP = 740 ÷ (1 - 0.075) MP = 740 ÷ 0.925 MP = 740000 ÷ 925 MP = 800