SSC CHSL Tier 1 2013 Quantitative Aptitude Questions with Solutions

Explanation

To find the total number of prime factors in the given expression, we first express each base as a product of its prime factors.

The prime factorizations of the bases are: 25 = 5 × 5 = 5² 24 = 2 × 2 × 2 × 3 = 2³ × 3 26 = 2 × 13

Substituting these into the expression: (25)¹⁰ × (24)²⁰ × (26)²⁵ = (5²)¹⁰ × (2³ × 3)²⁰ × (2 × 13)²⁵

Using the laws of exponents, this simplifies to: 5²⁰ × 2⁶⁰ × 3²⁰ × 2²⁵ × 13²⁵ = 2⁸⁵ × 3²⁰ × 5²⁰ × 13²⁵

The total number of prime factors is the sum of all the exponents of the prime bases: 85 + 20 + 20 + 25 = 150

Alternatively, this can be calculated directly by multiplying the number of prime factors in each base by its respective exponent: For (25)¹⁰: 2 prime factors × 10 = 20 For (24)²⁰: 4 prime factors × 20 = 80 For (26)²⁵: 2 prime factors × 25 = 50

Summing these values gives: 20 + 80 + 50 = 150

Explanation

To find the value of the given expression, we first count the total number of decimal places in both the numerator and the denominator.

Total decimal places in the numerator = 3 (from 0.625) + 4 (from 0.0729) + 1 (from 28.9) = 8 Total decimal places in the denominator = 4 (from 0.0081) + 3 (from 0.025) + 1 (from 1.7) = 8

Since the total number of decimal places in both the numerator and denominator is equal, we can eliminate the decimals entirely to rewrite the expression as:

(625 × 729 × 289) / (81 × 25 × 17)

Now, we divide each corresponding term:

625 / 25 = 25 729 / 81 = 9 289 / 17 = 17

Multiplying these simplified factors yields:

25 × 9 × 17 = 225 × 17 = 3825

Explanation

The fundamental relation in division is Dividend = (Divisor × Quotient) + Remainder.

We are given that the remainder is 40.

Since the divisor is 5 times the remainder, we calculate the divisor as: Divisor = 5 × 40 = 200

Since the divisor is also 10 times the quotient, we calculate the quotient as: Quotient = Divisor / 10 = 200 / 10 = 20

Substituting these values back into the division formula yields: Dividend = (200 × 20) + 40 = 4000 + 40 = 4040

Explanation

The given expression can be simplified by applying the BODMAS rule, which dictates the order of operations: Brackets, Of, Division, Multiplication, Addition, Subtraction.

First, calculate the product inside the first parentheses: 7/8 × 4/5 = 7/10

Applying the of operator to this result with the first term: 1/7 of 7/10 = 1/7 × 7/10 = 1/10

Next, simplify the terms inside the second parentheses: 2/3 - 2/5 = (10 - 6)/15 = 4/15

Then, evaluate the final term of the expression: 3/4 of 2/5 = 3/4 × 2/5 = 3/10

Now, substitute these simplified values back into the expression: 1/10 + 4/15 + 3/10

Combine the fractions that have the same denominator: 1/10 + 3/10 = 4/10 = 2/5

Finally, add this result to the remaining fraction by converting it to have a common denominator of 15: 2/5 + 4/15 = 6/15 + 4/15 = 10/15

Dividing both the numerator and the denominator by their greatest common divisor, 5, gives the final simplified value: 2/3

Explanation

To simplify the expression, we rationalize each of the irrational terms step-by-step. The corrected form of the expression is:

x = 5 + 1/√5 + 1/(√5 + 5) + 3/(√5 - 5)

First, we rationalize the second term by multiplying the numerator and denominator by √5: 1/√5 = √5 / 5 = 4√5 / 20

Second, we rationalize the third term by multiplying the numerator and denominator by (5 - √5): 1/(√5 + 5) = 1/(5 + √5) = (5 - √5) / ((5 + √5)(5 - √5)) = (5 - √5) / (25 - 5) = (5 - √5) / 20 = 1/4 - √5 / 20

Third, we rationalize the fourth term by multiplying the numerator and denominator by (5 + √5): 3/(√5 - 5) = -3/(5 - √5) = -3(5 + √5) / ((5 - √5)(5 + √5)) = -3(5 + √5) / (25 - 5) = -3(5 + √5) / 20 = -15/20 - 3√5 / 20 = -3/4 - 3√5 / 20

Now, we sum the simplified second, third, and fourth terms: (4√5 / 20) + (1/4 - √5 / 20) + (-3/4 - 3√5 / 20) = (1/4 - 3/4) + (4√5 / 20 - √5 / 20 - 3√5 / 20) = -2/4 + (4 - 1 - 3)√5 / 20 = -1/2 + 0 = -1/2

Substituting this sum back into the main expression: x = 5 - 1/2 x = 9/2

Explanation

Using the rule of alligation on the proportion of Darjeeling tea in each blend:

In the first blend, the proportion of Darjeeling tea is 4 / (4 + 7) = 4/11.

In the second blend, the proportion of Darjeeling tea is 2 / (2 + 5) = 2/7.

In the final mixture, the desired proportion of Darjeeling tea is 6 / (6 + 13) = 6/19.

The absolute difference between the proportion of Darjeeling tea in the first blend and the final mixture is: |4/11 - 6/19| = (4 × 19 - 6 × 11) / (11 × 19) = (76 - 66) / 209 = 10/209

The absolute difference between the proportion of Darjeeling tea in the second blend and the final mixture is: |2/7 - 6/19| = |(2 × 19 - 6 × 7)| / (7 × 19) = |38 - 42| / 133 = 4/133

The ratio in which the first and second blends must be mixed is: (4/133) : (10/209)

Since 133 = 7 × 19 and 209 = 11 × 19, we multiply both terms by 19: (4/7) : (10/11)

Multiplying both terms by 77 to clear the denominators: (4 × 11) : (10 × 7) = 44 : 70

Dividing by the common factor of 2: 22 : 35

Explanation

Let the third number be 100.

The first number is 25% less than the third number, which is 100 - 25 = 75.

The second number is 40% less than the third number, which is 100 - 40 = 60.

To find what percent of the first number is the second number, we divide the second number by the first number and multiply by 100:

(60 / 75) × 100 = (4 / 5) × 100 = 80%

Explanation

Since the price of sugar decreases by 20%, the new price is 4/5 of the original price.

Because the total expenditure remains constant at ₹ 320, the quantity of sugar bought is inversely proportional to the price. This means the new quantity becomes 5/4 of the original quantity.

The increase in quantity is 1/4 of the original quantity.

Given that this increase is 5 kg: 1/4 × Original quantity = 5 kg Original quantity = 20 kg

The original rate of sugar is calculated by dividing the total expenditure by the original quantity: Original rate = ₹ 320 ÷ 20 kg = ₹ 16 per kg

Explanation

Let AB be the monument of height H and CD be the building of height h, separated by horizontal distance BD = a.

Let E be the projection of C on the monument AB, which gives AE = H - h and EC = a.

Let the angle of depression of the top of the building C from the top of the monument A be θ. This makes the angle of elevation of A from C equal to θ. In right triangle AEC: tan θ = AE/EC = (H - h)/a

Since the angles of depression are complementary, the angle of depression of the bottom of the building D from A is 90° - θ. This makes the angle of elevation of A from D equal to 90° - θ. In right triangle ABD: tan(90° - θ) = AB/BD = H/a cot θ = H/a

Using the trigonometric identity tan θ × cot θ = 1, we multiply the two expressions: ((H - h)/a) × (H/a) = 1 H(H - h)/a² = 1 H(H - h) = a²

Rearranging the terms yields the relation: H/a = a/(H - h)

Explanation

Applying the componendo and dividendo rule, if a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d).

Using this relation on the given equation: ((sec θ + tan θ) + (sec θ - tan θ)) / ((sec θ + tan θ) - (sec θ - tan θ)) = ((2 + √3) + (2 - √3)) / ((2 + √3) - (2 - √3))

2 sec θ / 2 tan θ = 4 / 2√3

sec θ / tan θ = 2 / √3

Since sec θ = 1/cos θ and tan θ = sin θ/cos θ, the left side simplifies to: (1/cos θ) / (sin θ/cos θ) = 2 / √3

1 / sin θ = 2 / √3

sin θ = √3 / 2

The angle θ in the first quadrant for which sin θ = √3 / 2 is 60°, which in circular measure is π/3.

Explanation

We are given: cos 43° = x / √(x² + y²)

Using the complementary angle relation: sin 47° = cos(90° - 47°) = cos 43° = x / √(x² + y²)

In a right-angled triangle with an angle of 47°, let the perpendicular be x and the hypotenuse be √(x² + y²). Using the Pythagorean theorem, the base of the triangle is: base = √((hypotenuse)² - (perpendicular)²) = √((x² + y²) - x²) = √(y²) = y

The tangent of the angle is the ratio of the perpendicular to the base: tan 47° = perpendicular / base = x / y

Explanation

The sum of the first m numbers is the product of their count and their average, which is m × n² = mn².

Similarly, the sum of the next n numbers is n × m² = nm².

The total sum of these (m + n) numbers is the sum of both groups: mn² + nm² = mn(n + m) = mn(m + n)

The average of these (m + n) numbers is the total sum divided by the total count: Average = mn(m + n) / (m + n) = mn

Explanation

Using the fundamental trigonometric identities: 1 + cot²α = cosec²α 1 + tan²α = sec²α

Substituting these identities into the given expression yields: 1/(1 + cot²α) + 1/(1 + tan²α) = 1/cosec²α + 1/sec²α

Using the reciprocal relations where 1/cosec α = sin α and 1/sec α = cos α, the expression becomes: sin²α + cos²α

According to the Pythagorean identity, sin²α + cos²α = 1. Therefore, the expression simplifies to 1.

Explanation

Substituting cos θ = 1/sec θ into the given equation yields: sec θ - 1/sec θ = 3/2

Let sec θ = x. The equation simplifies to: x - 1/x = 3/2

Multiplying both sides by 2x gives the quadratic equation: 2x² - 2 = 3x 2x² - 3x - 2 = 0

Factoring this quadratic equation: 2x² - 4x + x - 2 = 0 2x(x - 2) + 1(x - 2) = 0 (2x + 1)(x - 2) = 0

This gives: x = 2 or x = -½

Since θ is a positive acute angle (0° < θ < 90°), sec θ must be positive and greater than or equal to 1. Therefore, the negative root -½ is discarded, leaving sec θ equal to 2.

Explanation

The volume V of a pyramid is given by the formula: V = ⅓ × Base Area × Height

For a pyramid with an equilateral triangle base of side a, the base area is: Base Area = (√3/4) × a²

Substituting the given volume of 48√3 cm³ and height of 4 cm into the volume formula: 48√3 = ⅓ × ((√3/4) × a²) × 4

Simplifying the right side of the equation: 48√3 = (√3/3) × a²

Dividing both sides by √3: 48 = a² / 3

Multiplying both sides by 3: a² = 144

Taking the square root: a = 12 cm

Explanation

Correct Answer: B

When a right-angled triangle ABC with ∠ABC = 90° is rotated about a side, that side becomes the height of the resulting cone, and the other perpendicular side becomes the radius of the base.

In the first case, when the triangle is rotated about AB: Height of the first cone, h₁ = AB = 3 cm Radius of the first cone, r₁ = BC = 4 cm Volume of the first cone, V₁ = ⅓ × π × r₁² × h₁ = ⅓ × π × 4² × 3 = 16π cm³

In the second case, when the triangle is rotated about BC: Height of the second cone, h₂ = BC = 4 cm Radius of the second cone, r₂ = AB = 3 cm Volume of the second cone, V₂ = ⅓ × π × r₂² × h₂ = ⅓ × π × 3² × 4 = 12π cm³

The percentage x by which the first volume is more than the second volume is: x = ((V₁ - V₂) / V₂) × 100 x = ((16π - 12π) / 12π) × 100 x = (4/12) × 100 = 33 ⅓

Explanation

By Pythagoras' theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

height² + base² = hypotenuse²

Substituting the given values of base = 5 units and hypotenuse = 13 units:

height² + 5² = 13² height² + 25 = 169 height² = 169 - 25 height² = 144 height = 12 units

The formula for the area of a right-angled triangle is:

Area = ½ × base × height

Substituting the values of base and height:

Area = ½ × 5 × 12 = 30 sq. units

Explanation

The number of bricks required to construct a wall is calculated using the formula:

Number of bricks = Volume of the wall / Volume of a single brick

First, all dimensions are converted to centimeters to maintain consistency: Wall length = 9 m = 900 cm Wall height = 6 m = 600 cm Wall thickness = 20 cm

Volume of the wall = 900 cm × 600 cm × 20 cm = 10,800,000 cm³

Volume of one brick = 30 cm × 15 cm × 10 cm = 4,500 cm³

Substituting these values into the formula: Number of bricks = 10,800,000 / 4,500 = 2,400

Explanation

The volume of a sphere of radius R is given by: V = (4/3) × π × R³

Since the diameter of the sphere is 6 cm, its radius R is 3 cm. The volume of the sphere is: V = (4/3) × π × 3³ = 36π cm³

When the sphere is melted and drawn into a cylindrical wire of diameter 2 cm, its radius r is 1 cm. Let the length of the wire be h. The volume of the cylindrical wire is: V = π × r² × h = π × 1² × h = π × h

Since the volume remains constant during melting: Volume of wire = Volume of sphere π × h = 36π h = 36 cm

Explanation

The number of cubes that can fit along any dimension of a rectangular box is given by the ratio of that dimension to the edge length of the cube.

Given: Edge length of each solid cube = 0.2 m Dimensions of the box = 1 m by 1 m by 2 m

Number of cubes along the first 1 m side = 1 / 0.2 = 5 Number of cubes along the second 1 m side = 1 / 0.2 = 5 Number of cubes along the 2 m side = 2 / 0.2 = 10

Total number of cubes that can be packed in the box = 5 × 5 × 10 = 250.

Explanation

To find the number of circles that can be made from the wire, first convert the total length of the wire into centimeters to match the unit of the circumference. Since 1 meter is equal to 100 centimeters, the total length of the wire is 66 × 100 = 6600 cm.

The formula to find the number of circles is: Number of circles = Total length of wire ÷ Circumference of one circle

Substituting the values into the formula gives: Number of circles = 6600 ÷ 1.32 Number of circles = 6600 × 100 / 132 Number of circles = 660000 / 132 Number of circles = 5000

Explanation

In quadrilateral AQOR, the angles AQO and ARO are both 90° because OQ and OR are radii of the incircle perpendicular to the tangent lines AC and AB respectively. Since the sum of angles in a quadrilateral is 360°, we have:

∠QOR = 180° - ∠A = 180° - 60° = 120°

The angle ∠QPR is the angle subtended by the arc QR at the remaining part of the circle's circumference, while ∠QOR is the angle subtended by the same arc at the center of the circle. By the inscribed angle theorem:

∠QPR = ½ ∠QOR = ½ × 120° = 60°

Explanation

The formula for Simple Interest (SI) is SI = (P × R × T) / 100, where P is the principal sum, R is the rate of interest per annum, and T is the time period in years.

According to the given values: Simple Interest (SI) = x Rate of interest (R) = x% Time (T) = x years

Substituting these values into the formula: x = (P × x × x) / 100

Simplifying the equation: x = (P × x²) / 100

Rearranging the terms to solve for the principal sum P: P = (x × 100) / x² P = 100/x

Explanation

Since the circle is drawn with AB as the diameter and it intersects BC at D, the angle ∠ADB is subtended by the diameter on the circumference of the circle. This makes ∠ADB = 90°, which means AD is perpendicular to BC.

In the isosceles triangle ABC with AB = AC, the perpendicular AD from the vertex A to the base BC bisects the base BC. This is because the right-angled triangles ABD and ACD are congruent by the RHS criterion, as AB = AC and AD is a common side.

Therefore, BD = CD. Given that BD = 3 cm, the length of CD must also be 3 cm.

Explanation

At any given time, the angle of elevation of the sun is constant for objects in the same vicinity, which means the ratio of the height of an object to the length of its shadow is constant due to similar triangles.

Height of stick / Shadow of stick = Height of tower / Shadow of tower

Substituting the given values into the relation:

12 / 8 = Height of tower / 40

Height of tower = (12 × 40) / 8

Height of tower = 12 × 5 = 60 cm

Explanation

In a right-angled triangle, the angle subtended by the hypotenuse at the opposite vertex is 90°. According to the properties of circles, an angle of 90° is always subtended by the diameter of a circle at any point on its circumference. Since triangle ABC is right-angled at B, the hypotenuse CA must be the diameter of the circumcircle of the triangle. The circumcentre of a triangle is the center of its circumcircle. Since the center of a circle lies at the mid-point of its diameter, the circumcentre O is located at the mid-point of the hypotenuse CA.

Explanation

By the angle-side relationship in a triangle, the angle opposite to the longer side is always greater in measure.

In triangle PQR, the given side lengths are QR = 5 cm, PR = 4 cm, and PQ = 3 cm. Arranging the sides in descending order of length gives QR > PR > PQ.

The angle opposite to side QR is ∠P, the angle opposite to side PR is ∠Q, and the angle opposite to side PQ is ∠R. Matching the angles to their opposite sides preserves the inequality, yielding ∠P > ∠Q > ∠R.

Explanation

The given equations of the lines are x + y = 4 and 3x + 3y = 6.

By dividing the entire second equation, 3x + 3y = 6, by 3 on both sides, we get: x + y = 2

Comparing the coefficients of the two linear equations of the standard form a₁x + b₁y = c₁ and a₂x + b₂y = c₂, we have: a₁/a₂ = 1/1 = 1 b₁/b₂ = 1/1 = 1 c₁/c₂ = 4/2 = 2

Since the ratio of the coefficients satisfies the condition a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the two equations represent a system of parallel lines. Parallel lines have the same slope but different y-intercepts, which means they never cross each other.

Explanation

We are given: yz : zx : xy = 1 : 2 : 3

From the first two terms of the ratio: yz / zx = 1 / 2

Since z ≠ 0, we cancel the common factor z: y / x = 1 / 2 x = 2y

The expression x/yz + y/zx represents the division of the two terms, where the division operator (÷) is written as a plus sign (+). Evaluating the division: (x/yz) ÷ (y/zx) = (x/yz) × (zx/y)

Multiplying the fractions: (x × z × x) / (y × z × y) = x²z / y²z

Since z ≠ 0, we cancel z: x² / y² = (x/y)²

Substituting x = 2y: (2y / y)² = 2² = 4

Explanation

Using the algebraic identity (a - b)² = a² + b² - 2ab, we square both sides of the given equation:

x - 1/x = 2

(x - 1/x)² = 2²

x² + 1/x² - 2 = 4

x² + 1/x² = 6

Now, squaring both sides of this new equation using the identity (a + b)² = a² + b² + 2ab:

(x² + 1/x²)² = 6²

x⁴ + 1/x⁴ + 2 = 36

x⁴ + 1/x⁴ = 34

Explanation

To find the value of x² + 2x + 3, first rationalize the given expression for x:

x = 1 / (√2 + 1)

Multiplying the numerator and the denominator by (√2 - 1):

x = (√2 - 1) / ((√2 + 1)(√2 - 1)) x = (√2 - 1) / (2 - 1) x = √2 - 1

Rearranging the equation to isolate the radical:

x + 1 = √2

Squaring both sides of the equation:

(x + 1)² = (√2)² x² + 2x + 1 = 2 x² + 2x = 1

Adding 3 to both sides of the equation yields the required expression:

x² + 2x + 3 = 1 + 3 x² + 2x + 3 = 4

Explanation

Given the relation a + b + c = 4, we can write a + b = 4 - c.

Using the algebraic identity for the sum of cubes: a³ + b³ = (a + b)³ - 3ab(a + b)

Substituting a + b = 4 - c into this identity gives: a³ + b³ = (4 - c)³ - 3ab(4 - c) a³ + b³ = 64 - 48c + 12c² - c³ - 12ab + 3abc

Substituting this result back into the target expression: a³ + b³ + c³ - 12c² + 48c - 64 = (64 - 48c + 12c² - c³ - 12ab + 3abc) + c³ - 12c² + 48c - 64

By simplifying and canceling the common terms: 64 - 64 = 0 -48c + 48c = 0 12c² - 12c² = 0 -c³ + c³ = 0

The expression reduces to: 3abc - 12ab

Explanation

Using the work-equivalence relation: M₁ × D₁ = M₂ × D₂

Here, M₁ represents 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: M₁ = 8 D₁ = 40 M₂ = 8 + 2 = 10

8 × 40 = 10 × D₂ 320 = 10 × D₂ D₂ = 320 / 10 = 32

Explanation

To find the ratio x/y, we set up the given proportion:

(2x + 3y)/(3x + 5y) = 18/29

Cross-multiplying the terms gives:

29(2x + 3y) = 18(3x + 5y)

Expanding both sides:

58x + 87y = 54x + 90y

Grouping the terms of x on one side and y on the other:

58x - 54x = 90y - 87y

4x = 3y

Dividing both sides by 4y yields:

x/y = 3/4

Explanation

The speed of the car is 36 km/h. Converting this speed into m/s: Speed of car = 36 × 5/18 = 10 m/s

Initially, the bus is 40 m ahead of the car, and after 20 seconds, it is 60 m behind. The total relative distance covered by the car with respect to the bus during this time is: Relative distance = 40 + 60 = 100 m

Using the formula for relative speed: Relative speed = Relative distance / Time Relative speed = 100 / 20 = 5 m/s

Since the car overtakes the bus, the car is moving faster than the bus. Therefore, the relative speed is the difference between the speed of the car and the speed of the bus: Relative speed = Speed of car - Speed of bus 5 = 10 - Speed of bus Speed of bus = 5 m/s

Converting the speed of the bus back to km/h: Speed of bus = 5 × 18/5 = 18 km/h

Explanation

Let the quantities of the two varieties of tea blended be 5 kg and 3 kg.

The total cost price (CP) of 8 kg of the blended tea is: CP = (5 × 240) + (3 × 300) = 1200 + 900 = ₹ 2100

The total selling price (SP) of 8 kg of the blended tea at ₹ 315 per kg is: SP = 8 × 315 = ₹ 2520

The total gain made is: Gain = SP - CP = 2520 - 2100 = ₹ 420

Using the gain percentage formula: Gain % = (Gain / CP) × 100 Gain % = (420 / 2100) × 100 = 20%

Explanation

The profit in a partnership is distributed in the ratio of the equivalent capitals, which is the product of the capital invested and the time period of investment.

Let the initial capitals of A and B be 4x and 5x respectively.

For the first 3 months, A's capital is 4x and B's capital is 5x.

After 3 months: A withdraws 1/4 of his capital, so his remaining capital is 4x - (1/4 × 4x) = 3x, which is invested for the remaining 7 months. B withdraws 1/5 of his capital, so his remaining capital is 5x - (1/5 × 5x) = 4x, which is invested for the remaining 7 months.

The ratio of equivalent capitals of A and B is: A's equivalent capital = (4x × 3) + (3x × 7) = 12x + 21x = 33x B's equivalent capital = (5x × 3) + (4x × 7) = 15x + 28x = 43x

Thus, the profit-sharing ratio A : B is 33 : 43.

The total profit of ₹ 760 is divided into 33 + 43 = 76 equal parts.

A's share of profit is (33 / 76) × 760 = ₹ 330.

Explanation

The relation between cost price (CP) and printed price (MP) is given by the formula:

CP / MP = (100 - Discount%) / (100 + Profit%)

Substituting the given values into the formula:

CP / MP = (100 - 10) / (100 + 15)

CP / MP = 90 / 115

Dividing the numerator and the denominator by their greatest common divisor, which is 5:

CP / MP = 18 / 23

This gives the ratio of the cost price and the printed price as 18 : 23.

Explanation

Let the cost price of the article be ₹ x. Since the profit percentage is numerically equal to the cost price, the profit percentage is x%.

The formula for the selling price (SP) is: SP = CP × (1 + Profit%/100)

Substituting the given values: 39 = x × (1 + x/100) 39 = x + x²/100 x² + 100x - 3900 = 0

Factoring the quadratic equation yields: x² + 130x - 30x - 3900 = 0 x(x + 130) - 30(x + 130) = 0 (x - 30)(x + 130) = 0

Since the cost price must be a positive value, x = 30.

Explanation

When two items are sold at the same selling price, one at a gain of x% and the other at a loss of x%, there is always an overall loss given by the formula:

Loss % = x² / 100

Here, x = 20%, so:

Loss % = 20² / 100 = 400 / 100 = 4%

The total selling price of both tables is 720 × 2 = ₹ 1440.

Since the transaction results in a 4% loss, the total selling price represents 96% of the total cost price:

Total Cost Price = 1440 / (1 - 0.04) = 1440 / 0.96 = ₹ 1500

Total Loss = Total Cost Price - Total Selling Price = 1500 - 1440 = ₹ 60

Explanation

The simple interest (SI) on a principal of P for T years at a rate of R% per annum is given by the formula:

SI = (P × R × T) / 100

Substituting P = 1000, R = 10, and T = 4:

SI = (1000 × 10 × 4) / 100 = 400

The compound interest (CI) on the same principal is calculated as:

CI = P × ((1 + R/100)ⁿ - 1)

Substituting the values with n = 4 years:

CI = 1000 × ((1 + 10/100)⁴ - 1) CI = 1000 × (1.1⁴ - 1) CI = 1000 × (1.4641 - 1) CI = 1000 × 0.4641 = 464.10

The difference between the compound interest and the simple interest is:

Difference = CI - SI Difference = 464.10 - 400 = 64.10

Explanation

Using the relation between average and sum: Sum = Average × Number of values

The sum of all 15 numbers is: 15 × 7 = 105

The sum of the first 8 numbers is: 8 × 6.5 = 52

The sum of the last 8 numbers is: 8 × 8.5 = 68

Since there are 15 numbers in total, the middle number (the 8th number) is included in both the first 8 numbers and the last 8 numbers. Adding these two sums together counts the middle number twice. Subtracting the total sum of all 15 numbers from this combined sum gives the value of the middle number: (52 + 68) - 105 = 120 - 105 = 15

Explanation

Using the property that shifting the decimal point two places to the left inside a square root shifts the decimal point of the square root's value one place to the left:

√156.25 = 12.5 √1.5625 = 1.25 √0.015625 = 0.125

Evaluating the expression:

12.5 + 1.25 - 0.125 = 13.75 - 0.125 = 13.625

Explanation

Let the three numbers be x, y, and z. We are given the ratio of the second to the third number as y : z = 9 : 6 = 3 : 2, and the ratio of the first to the third number as x : z = 1 : 4.

To make the term for z common in both ratios, we multiply the terms of y : z by 2, which gives y : z = 6 : 4. Now, we can write the combined ratio of the three numbers as x : y : z = 1 : 6 : 4.

The sum of the ratio terms is 1 + 6 + 4 = 11. Since the actual sum of the three numbers is 121, the value of 1 part is 121 / 11 = 11.

The second number corresponds to 6 parts, which is 6 × 11 = 66.

Explanation

To solve the expression, we first count the decimal places in the numerator and the denominator.

The numerator has 3 + 3 + 3 = 9 decimal places. The denominator has 4 + 4 + 1 = 9 decimal places.

Since both have the same number of decimal places, we can eliminate the decimal points entirely. The expression inside the square root becomes:

(81 × 324 × 4624) / (15625 × 289 × 729 × 64)

Now, we take the square root of each individual term: √81 = 9 √324 = 18 √4624 = 68 √15625 = 125 √289 = 17 √729 = 27 √64 = 8

Substituting these square roots back into the fraction gives: (9 × 18 × 68) / (125 × 17 × 27 × 8)

We simplify the terms: 68 ÷ 17 = 4 9 ÷ 27 = 1/3 18 ÷ 3 = 6 4 ÷ 8 = 1/2 6 ÷ 2 = 3

This leaves: 3 / 125

To convert this fraction to a decimal, multiply both the numerator and the denominator by 8: (3 × 8) / (125 × 8) = 24 / 1000 = 0.024

Explanation

Let the two numbers be 3x and 5x, where x represents their highest common factor (H.C.F.).

The least common multiple (L.C.M.) of two numbers in the ratio a : b with an H.C.F. of x is given by the formula L.C.M. = a × b × x.

Substituting the given values into the formula: 15 × x = 75

Solving for x: x = 75 ÷ 15 = 5

Since x is the highest common factor, the H.C.F. of the two numbers is 5.

Explanation

To find how much percent B's salary is less than A's salary, we use the standard percentage relationship.

Let B's salary be 100. Since A's salary is 30% higher than B's salary, A's salary is: 100 + 30 = 130

The percentage by which B's salary is less than A's salary is calculated relative to A's salary: Percentage less = ((A - B) / A) x 100 = ((130 - 100) / 130) x 100 = (30 / 130) x 100 = 300 / 13 % = 23 1/13 %

This matches Option B.

Explanation

To find the percentage, first express both time durations in the same unit, which is seconds.

The duration of 1 minute 48 seconds in seconds is: 1 × 60 + 48 = 108 seconds

The duration of 1 hour in seconds is: 1 × 60 × 60 = 3600 seconds

Using the percentage formula: Percentage = (Given Value / Total Value) × 100

Substituting the calculated values into the formula: Percentage = (108 / 3600) × 100 = 108 / 36 = 3%

Explanation

To simplify the expression, we first find a common denominator by multiplying the two denominators: (√5 + √3)(√5 - √3) = (√5)² - (√3)² = 5 - 3 = 2

Next, we combine the numerators over this common denominator: ((1 + √2)(√5 - √3) + (1 - √2)(√5 + √3)) / 2

Expanding both products in the numerator: (1 + √2)(√5 - √3) = √5 - √3 + √10 - √6 (1 - √2)(√5 + √3) = √5 + √3 - √10 - √6

Adding the two expanded terms together, the -√3 and +√3 cancel out, and the +√10 and -√10 cancel out: (√5 - √3 + √10 - √6) + (√5 + √3 - √10 - √6) = 2√5 - 2√6

Dividing this result by the common denominator 2: (2√5 - 2√6) / 2 = √5 - √6

Explanation

To find the highest common factor (H.C.F.) of a set of fractions, the standard formula is:

H.C.F. of fractions = H.C.F. of numerators / L.C.M. of denominators

For the given fractions 1/2, 3/4, 5/6, 7/8, and 9/10:

First, calculate the H.C.F. of the numerators (1, 3, 5, 7, 9): H.C.F. (1, 3, 5, 7, 9) = 1

Next, calculate the least common multiple (L.C.M.) of the denominators (2, 4, 6, 8, 10) using prime factorization: 2 = 2¹ 4 = 2² 6 = 2 × 3 8 = 2³ 10 = 2 × 5

Taking the highest power of each prime factor: L.C.M. (2, 4, 6, 8, 10) = 2³ × 3 × 5 = 8 × 3 × 5 = 120

Substituting these results back into the formula yields: H.C.F. of fractions = 1/120