SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

Using the principle of similar triangles, the height of the tree can be found by linear interpolation between the height of Suhas's eyes and the height of the building.

Let the ground level be the reference line. The height of Suhas's eyes is 1.8 m at the starting position (0 m). The tree is located at 7.5 m, and the building of height 11.25 m is located at 52.5 m (7.5 m + 45 m).

The formula for the height of the tree, h, is:

h = h_eyes + (h_building - h_eyes) × (d_tree / d_building)

Substituting the given values into the formula:

h = 1.8 + (11.25 - 1.8) × (7.5 / 52.5)

h = 1.8 + 9.45 × (1 / 7)

h = 1.8 + 1.35

h = 3.15 m

Explanation

Correct Answer: D

Given the equation: p² - 4p - 1 = 0

Dividing the entire equation by p: p - 4 - 1/p = 0 p - 1/p = 4

Squaring both sides of this equation: (p - 1/p)² = 4² p² - 2 + 1/p² = 16 p² + 1/p² = 18

The expression to be evaluated is: p² + 3p + 1/p² - 3/p

Rearranging the terms: (p² + 1/p²) + 3(p - 1/p)

Substituting the values of (p² + 1/p²) and (p - 1/p) into this expression: 18 + 3(4) = 18 + 12 = 30

Explanation

According to the exterior angle theorem of a triangle, the measure of an exterior angle is equal to the sum of the measures of its two interior opposite angles.

For triangle ABC, when side BC is produced to D, the exterior angle formed is ∠ACD. The two interior opposite angles are ∠ABC and ∠BAC. Therefore:

∠ACD = ∠ABC + ∠BAC

Substituting the given values:

127° = 35° + ∠BAC

∠BAC = 127° - 35°

∠BAC = 92°

Explanation

Ravi's total marks across the four unit tests are 24 + 22 + 20 + 24 = 90. His average score is calculated by dividing this total by the number of tests, which is 90 / 4 = 22.5. This total of 90 marks is the highest sum among all the students, which directly gives him the highest average score.

Explanation

Using the standard speed-distance-time relation, when an object covers the same distance at two different speeds s₁ and s₂ with a time difference Δt, the distance d is given by the formula:

d = (s₁ × s₂) / (s₂ - s₁) × Δt

In this problem, the two speeds are: s₁ = 15 km/h s₂ = 20 km/h

The time difference Δt is the difference between being 10 minutes late and 5 minutes early, which is: Δt = 10 + 5 = 15 minutes = 15/60 hours = 1/4 hour

Substituting these values into the formula:

d = (15 × 20) / (20 - 15) × 1/4 d = 300 / 5 × 1/4 d = 60 × 1/4 d = 15 km

Explanation

To find the value of the expression 4(sin⁶ A + cos⁶ A) - 6(sin⁴ A + cos⁴ A) + 8, we can use the method of value substitution, which is the quickest way to solve such trigonometric identity problems.

Let A = 0.

Substituting A = 0 into the expression: sin 0 = 0 cos 0 = 1

Expression = 4(0⁶ + 1⁶) - 6(0⁴ + 1⁴) + 8 Expression = 4(1) - 6(1) + 8 Expression = 4 - 6 + 8 Expression = 6

Alternatively, we can use the algebraic identities: sin⁶ A + cos⁶ A = 1 - 3 sin² A cos² A sin⁴ A + cos⁴ A = 1 - 2 sin² A cos² A

Substituting these identities into the expression: 4(1 - 3 sin² A cos² A) - 6(1 - 2 sin² A cos² A) + 8 = 4 - 12 sin² A cos² A - 6 + 12 sin² A cos² A + 8 = 4 - 6 + 8 = 6

The variable terms cancel out completely, yielding a constant value of 6.

Explanation

The total surface area (TSA) of a cylinder is given by the formula: TSA = 2πr(r + h)

Given that r + h = 42 m and TSA = 6336 m², we substitute these values into the formula to find the radius (r): 2 × (22/7) × r × 42 = 6336 264 × r = 6336 r = 24 m

Since the sum of the radius and height is 42 m, the height (h) is: h = 42 - 24 = 18 m

The curved surface area (CSA) of the cylinder is given by the formula: CSA = 2πrh

Substituting the values of r and h: CSA = 2 × (22/7) × 24 × 18 CSA = 19008 / 7 CSA ≈ 2715.43 m²

Explanation

Using the trigonometric identity for a right-angled triangle, where cos A = base / hypotenuse: Base = 63 Hypotenuse = 65

Using Pythagoras' theorem: Perpendicular² = Hypotenuse² - Base² Perpendicular² = 65² - 63² Perpendicular² = (65 - 63)(65 + 63) Perpendicular² = 2 × 128 = 256 Perpendicular = √256 = 16

We need to find the value of tan A + cot A: tan A + cot A = sin A / cos A + cos A / sin A = 1 / (sin A × cos A)

Using the values of sin A = 16/65 and cos A = 63/65: tan A + cot A = 1 / ((16/65) × (63/65)) tan A + cot A = 65² / (16 × 63) tan A + cot A = 4225 / 1008 tan A + cot A ≈ 4.19

Explanation

Given that cos θ = 4/5, we can find sin θ using the identity sin² θ + cos² θ = 1:

sin θ = √(1 - cos² θ) = √(1 - (4/5)²) = √(1 - 16/25) = √(9/25) = 3/5

The expression to evaluate is cot θ + tan θ:

cot θ + tan θ = (cos θ / sin θ) + (sin θ / cos θ)

By taking the common denominator, this simplifies to:

(cos² θ + sin² θ) / (sin θ cos θ) = 1 / (sin θ cos θ)

Substituting the values of sin θ and cos θ into this expression:

1 / ((3/5) × (4/5)) = 1 / (12/25) = 25/12

Explanation

The given expression is of the form (a³ - b³) / (a² + ab + b²), which simplifies to (a - b) using the algebraic identity. Substituting a = 143 and b = 139, we get 143 - 139 = 4.

Explanation

The selling price of an item after a discount is calculated using the formula:

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

Given: Marked Price = 28,500 Discount Percentage = 12%

Substituting these values into the formula:

Selling Price = 28,500 × (100 - 12) / 100 Selling Price = 28,500 × 88 / 100 Selling Price = 285 × 88 Selling Price = 25,080

Explanation

The relationship between the Selling Price (SP), the Discount Percentage (d), and the Discount Amount is given by the formula:

Discount Amount = SP × d / (100 - d)

Given that the selling price SP is 19,040 and the discount percentage d is 15%, we substitute these values into the formula:

Discount Amount = 19,040 × 15 / (100 - 15) Discount Amount = 19,040 × 15 / 85

Simplifying the fraction by dividing both the numerator and the denominator by 5:

Discount Amount = 19,040 × 3 / 17

Dividing 19,040 by 17 gives 1,120. Now, multiplying this result by 3:

Discount Amount = 1,120 × 3 = 3,360

Explanation

Let the cost price and the marked price of the article be 4x and 7x respectively.

A discount of 20% is given on the marked price. Selling Price = Marked Price × (100% - Discount%) Selling Price = 7x × 0.80 = 5.6x

Profit is the difference between the selling price and the cost price. Profit = Selling Price - Cost Price Profit = 5.6x - 4x = 1.6x

Given that the profit is ₹80: 1.6x = 80 x = 50

The cost price of the article is: Cost Price = 4x = 4 × 50 = ₹200

Explanation

The relationship between Cost Price (CP) and Marked Price (MP) when discount percentage (D%) and gain percentage (G%) are given is:

CP / MP = (100 - D%) / (100 + G%)

Substituting the given values: CP = 450 D% = 10% G% = 20%

450 / MP = (100 - 10) / (100 + 20) 450 / MP = 90 / 120 450 / MP = 3/4 MP = (450 × 4) / 3 MP = 150 × 4 = 600

The marked price of the cosmetic product is £600.

Explanation

Under the simple interest system with partial payments, the total accumulated debt at the end of the term is equated to the sum of the accumulated values of all installments at the end of that term.

The total debt of ₹4,000 accumulates interest for 3 years at 10% per annum: Total Debt = 4000 × (1 + 0.10 × 3) = 4000 × 1.3 = 5200

The first installment of 1500 is paid at the end of the first year, so it accumulates interest for the remaining 2 years: Value of 1st Installment = 1500 × (1 + 0.10 × 2) = 1500 × 1.2 = 1800

The second installment of 2500 is paid at the end of the second year, so it accumulates interest for the remaining 1 year: Value of 2nd Installment = 2500 × (1 + 0.10 × 1) = 2500 × 1.1 = 2750

Let X be the third installment paid at the end of the third year to clear the debt.

Equating the total debt to the sum of the installments at the end of the third year: 5200 = 1800 + 2750 + X 5200 = 4550 + X X = 5200 - 4550 = 650

Explanation

The fourth proportional x to three numbers a, b, and c satisfies the ratio a/b = c/x. This gives the formula: x = (b × c)/a

Using the given values a = 6, b = 8, and c = 12: x = (8 × 12)/6 x = 96/6 x = 16

Explanation

Correct Answer: D

The area of an equilateral triangle is given by the formula:

Area = (√3 / 4) × side²

Given that the side of the equilateral triangle is 12 cm, substituting this value into the formula gives:

Area = (√3 / 4) × 12² Area = (√3 / 4) × 144 Area = 36√3 cm²

Explanation

The work done by 3 men is equal to the work done by 5 women, which gives the efficiency relation: 3 Men = 5 Women 1 Man = 5/3 Women

To find the time taken by 9 men and 4 women to complete the same work, we convert their combined rate into equivalent women: 9 Men + 4 Women = 9 × (5/3 Women) + 4 Women = 15 Women + 4 Women = 19 Women

Using the work-equivalence formula M₁ × D₁ = M₂ × D₂: 5 Women × 57 days = 19 Women × D₂ days D₂ = (5 × 57) / 19 D₂ = 5 × 3 = 15 days

Explanation

The total production of Type R motorcycles from 2018 to 2021 is the sum of the production in each of these years: Total production = 65 + 37 + 50 + 28 = 180 thousand

The production of Type R motorcycles in the year 2020 is 50 thousand.

The central angle of the sector representing the production in 2020 in a pie chart is calculated using the formula: Central Angle = (Production in 2020 / Total Production) × 360°

Substituting the values into the formula: Central Angle = (50 / 180) × 360° = 50 × 2° = 100°

Explanation

We are given that sin θ cos θ = 1/√3.

Using the fundamental trigonometric identity: sin² θ + cos² θ = 1

We can express sin⁴ θ + cos⁴ θ using the algebraic identity a² + b² = (a + b)² - 2ab, where a = sin² θ and b = cos² θ: sin⁴ θ + cos⁴ θ = (sin² θ + cos² θ)² - 2 sin² θ cos² θ

Substituting the known values into this equation: sin⁴ θ + cos⁴ θ = 1² - 2 (sin θ cos θ)² sin⁴ θ + cos⁴ θ = 1 - 2 (1/√3)² sin⁴ θ + cos⁴ θ = 1 - 2 (1/3) sin⁴ θ + cos⁴ θ = 1 - 2/3 sin⁴ θ + cos⁴ θ = 1/3

Explanation

Since ΔABC ≅ ΔFDE, the corresponding parts of congruent triangles (CPCT) must be equal.

This gives the relation between the corresponding sides as AB = FD, BC = DE, and AC = FE. Since it is given that AB = 5 cm, we have DF = 5 cm.

Similarly, the corresponding angles are equal, which gives ∠A = ∠F, ∠B = ∠D, and ∠C = ∠E.

Using the angle sum property of ΔABC, the sum of all interior angles is 180°: ∠A + ∠B + ∠C = 180° 80° + 40° + ∠C = 180° 120° + ∠C = 180° ∠C = 60°

Since ∠E = ∠C, we get ∠E = 60°.

Explanation

We multiply the numerator and the denominator inside the square root by (1 + cos P):

√((1 + cos P) / (1 - cos P)) = √(((1 + cos P)(1 + cos P)) / ((1 - cos P)(1 + cos P)))

Using the algebraic identity (a - b)(a + b) = a² - b², the denominator simplifies as: (1 - cos P)(1 + cos P) = 1 - cos² P

Using the fundamental trigonometric identity sin² P + cos² P = 1, we substitute 1 - cos² P = sin² P:

= √((1 + cos P)² / sin² P)

Taking the square root of both the numerator and the denominator yields: (1 + cos P) / sin P

We can split this expression into two separate fractions: 1/sin P + cos P/sin P

Applying the reciprocal and quotient identities, 1/sin P = cosec P and cos P/sin P = cot P: cosec P + cot P

Explanation

According to the angle sum property of a triangle, the sum of all interior angles of a triangle is always equal to 180°.

Setting up the equation with the given angle expressions: (2y + 40°) + (5y - 60°) + (3y - 80°) = 180°

Combining the like terms: 10y - 100° = 180°

Solving for y: 10y = 180° + 100° 10y = 280° y = 28°

Explanation

Let the center of the larger circle with radius R = 4 cm be O₁ and the center of the smaller circle with radius r = 3 cm be O₂. Since the two circles touch internally, the distance between their centers is: O₁O₂ = R - r = 4 - 3 = 1 cm

To find the longest chord of the larger circle that is outside the inner circle, we find the chord of the larger circle that is tangent to the inner circle. To maximize the length of this chord, we must minimize its distance d from the center of the larger circle O₁.

For any tangent line to the inner circle, the distance from O₂ to the tangent line is r = 3 cm. The perpendicular distance d from O₁ to this tangent line is given by: d = r - O₁O₂ cos θ = 3 - 1 cos θ

where θ is the angle between the line of centers O₁O₂ and the perpendicular to the tangent line. To minimize d, we set cos θ = 1: d = 3 - 1 = 2 cm

The length of the chord of the larger circle (radius R = 4 cm) at a distance of d = 2 cm from its center is: Length = 2 √(R² - d²) Length = 2 √(4² - 2²) = 2 √12 = 4√3 cm

Explanation

To find the time when Anil and Bilal meet for the first time on the circular track, convert their speeds from km/h to m/s.

Anil's speed = 36 × 5/18 = 10 m/s Bilal's speed = 54 × 5/18 = 15 m/s

When two runners run in the same direction, their relative speed is the difference between their individual speeds.

Relative speed = 15 m/s - 10 m/s = 5 m/s

The time taken for them to meet for the first time on the track is the time required to cover a relative distance equal to one full lap of the track.

Time = Track length / Relative speed Time = 1500 / 5 = 300 seconds

Explanation

When a solid sphere is melted and recast into a solid cone, the volume remains conserved.

The volume of a sphere is given by the formula (4/3)πR³, where R is the radius of the sphere. The volume of a cone is given by the formula (1/3)πr²h, where r is the base radius and h is the height of the cone.

Equating the two volumes: (4/3)πR³ = (1/3)πr²h

Canceling out π/3 from both sides gives: 4R³ = r²h

Substituting the given values, R = 4 cm and h = 4 cm: 4 × 4³ = r² × 4

Dividing both sides by 4: 4³ = r² r² = 64 r = 8 cm

Explanation

Let CP₁ and CP₂ be the cost prices of the two goods, where CP₁ is sold at a 20% loss and CP₂ is sold at a 15% profit.

Since the selling prices are equal: 0.80 × CP₁ = 1.15 × CP₂

This gives the ratio of their cost prices: CP₁ / CP₂ = 1.15 / 0.80 = 115 / 80 = 23 / 16

The total cost price is CP₁ + CP₂ = ₹19,500. Using the ratio, the sum of parts is 23 + 16 = 39 parts.

The cost price of the goods sold at profit (CP₂) corresponds to 16 parts: CP₂ = (16 / 39) × 19,500 CP₂ = 16 × 500 = 8,000

Explanation

To find the combined ratio of X, Y, and Z, we make the term for Y common in both ratios.

Given: X:Y = 4:5 Y:Z = 7:6

Multiplying the first ratio by 7 and the second ratio by 5: X:Y = 28:35 Y:Z = 35:30

This gives the combined ratio: X:Y:Z = 28:35:30

The sum of the ratio parts is: 28 + 35 + 30 = 93

Using the total sum of ₹1488, the share of Z is: (30 / 93) × 1488 = 30 × 16 = ₹480

Explanation

Let Samita's income be S and Bhairav's income be B.

From the first condition: 65% of S = 91% of B 65 × S = 91 × B

Dividing both sides by 13: 5S = 7B S / B = 7 / 5

Let S = 7x and B = 5x.

From the second condition: (S + 1500) / (B + 500) = 3 / 2 (7x + 1500) / (5x + 500) = 3 / 2

Cross-multiplying to solve for x: 2 × (7x + 1500) = 3 × (5x + 500) 14x + 3000 = 15x + 1500 15x - 14x = 3000 - 1500 x = 1500

The actual combined income of Samita and Bhairav is: S + B = 7x + 5x = 12x 12 × 1500 = 18,000

Explanation

To find the year in which the production of variety C was the maximum, we observe the row corresponding to variety C across all the given years. The production values for variety C are 67 quintals in 2018, 89 quintals in 2019, 67 quintals in 2020, and 98 quintals in 2021. Comparing these values, the maximum production is 98 quintals, which occurred in the year 2021.

Explanation

To simplify the expression, we factor the numerator by grouping the terms: 10x² + 5x + 2xy + y = 5x(2x + 1) + y(2x + 1) = (2x + 1)(5x + y)

Substituting this back into the original expression gives: ((2x + 1)(5x + y)) / (5x + y) = 2x + 1

Substituting x = -1 into this simplified expression: 2(-1) + 1 = -2 + 1 = -1

Explanation

Using the algebraic identity for the difference of squares, a² - b² can be written as (a - b)(a + b).

Substituting this into the second fraction of the given expression: (a + b)² / (a² - b²) = (a + b)² / ((a - b)(a + b))

By canceling the common term (a + b) from both the numerator and the denominator, the second fraction simplifies to: (a + b) / (a - b)

Now, substituting this back into the original division expression: ((a + b) / (a - b)) ÷ ((a + b) / (a - b))

Any non-zero expression divided by itself equals 1.

Explanation

Using the division formula: Dividend = Divisor × Quotient + Remainder

The number is calculated as: Number = 78 × 280 + 0

By factoring the terms to check divisibility by 65: 78 = 13 × 6 280 = 5 × 56

Substituting these factors into the expression: Number = (13 × 6) × (5 × 56) Number = (13 × 5) × (6 × 56) Number = 65 × 336

Since the number is a multiple of 65, it is exactly divisible by 65, which leaves a remainder of 0.

Explanation

The prime numbers lying between 51 and 77 are 53, 59, 61, 67, 71, and 73.

The formula for the average of a set of numbers is: Average = Sum of all values / Total number of values

The total count of these prime numbers is 6.

First, calculate the sum of these prime numbers: Sum = 53 + 59 + 61 + 67 + 71 + 73 = 384

Now, compute the average by dividing the sum by the count: Average = 384 / 6 = 64

Explanation

Let CP be the cost price of the product.

The marked price (MP) is 5% above the cost price: MP = CP × (1 + 5/100) = 1.05 CP

The selling price (SP) is obtained after giving a 10% discount on the marked price: SP = MP × (1 - 10/100) = 0.90 MP

Substitute the value of MP into the SP equation: SP = 0.90 × 1.05 CP = 0.945 CP

Given that the selling price is ₹1890, we can set up the equation: 0.945 CP = 1890

Solving for CP: CP = 1890 / 0.945 = 2000

Explanation

By the Alternate Segment Theorem, the angle between the tangent AT and the chord AB through the point of contact A is equal to the angle subtended by the chord AB in the alternate segment, which is angle ACB. Since angle ACB is given as 65°, the angle BAT must also be 65°.

Alternatively, because AOC is a diameter, the angle in the semicircle angle ABC = 90°. In triangle ABC, the sum of angles is 180°, so angle CAB = 180° - 90° - 65° = 25°. Since the tangent AT is perpendicular to the diameter AC at the point of tangency, angle CAT = 90°. Therefore, angle BAT = angle CAT - angle CAB = 90° - 25° = 65°.

Explanation

To find the required percentage, first calculate the total number of laptops sold by Wholesaler 1 and Wholesaler 2 together on Thursday.

Laptops sold by Wholesaler 1 on Thursday = 24 Laptops sold by Wholesaler 2 on Thursday = 14 Total laptops sold by both on Thursday = 24 + 14 = 38

Next, find the number of laptops sold by Wholesaler 3 on Friday. Laptops sold by Wholesaler 3 on Friday = 19

Now, calculate the percentage: Percentage = (Total laptops sold by Wholesaler 1 and 2 on Thursday / Laptops sold by Wholesaler 3 on Friday) × 100% Percentage = (38 / 19) × 100% = 2 × 100% = 200%

Explanation

By the Pythagorean theorem, the relationship between the hypotenuse, base, and height of a right-angled triangle is given by:

hypotenuse² = base² + height²

Substituting the given values:

13² = 12² + height² 169 = 144 + height² height² = 169 - 144 height² = 25 height = 5 cm

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

Area = ½ × base × height

Substituting the base and height:

Area = ½ × 12 × 5 = 30 cm²

Explanation

Using the trigonometric identity cot(A) = tan(90° - A), the second term of the expression can be rewritten by substituting A = 40° - θ:

cot(40° - θ) = tan(90° - (40° - θ)) cot(40° - θ) = tan(90° - 40° + θ) cot(40° - θ) = tan(50° + θ)

Substituting this equivalence back into the original expression:

tan(50° + θ) - cot(40° - θ) = tan(50° + θ) - tan(50° + θ) = 0

Explanation

Using the successive percentage formula for net change:

Net percentage change = a + b + ab/100

Here, the initial salary increment is a = 40, and the subsequent salary reduction is b = -15.

Substituting these values into the formula:

Net change = 40 - 15 + (40 × (-15)) / 100 Net change = 25 - 600 / 100 Net change = 25 - 6 Net change = 19%

A positive result of 19% indicates a net increase in salary.

Explanation

The volume of a cylinder is given by the formula V = π r² h, where r is the radius and h is the height or length.

When the rod is recast into a wire, the volume of the material remains constant. Let r₁ and h₁ be the radius and length of the rod, and r₂ and h₂ be the radius and length of the wire.

π r₁² h₁ = π r₂² h₂

By canceling π from both sides, the relation simplifies to: r₁² h₁ = r₂² h₂

Substituting the given values r₁ = 5 cm, h₁ = 52 cm, and h₂ = 100 cm: 5² × 52 = r₂² × 100 25 × 52 = 100 r₂² 1300 = 100 r₂² r₂² = 13 r₂ = √13 cm

Explanation

Using the algebraic identity (y - 1/y)² = y² + 1/y² - 2, we substitute the given value y² + 1/y² = 74 into the equation:

(y - 1/y)² = 74 - 2 (y - 1/y)² = 72

Taking the square root of both sides gives:

y - 1/y = ±√72 = ±6√2

Since it is given that y > 1, the reciprocal 1/y must be less than 1. This means y is greater than 1/y, so the difference y - 1/y must be positive. Therefore, the positive value is chosen:

y - 1/y = 6√2

Explanation

The formula for the population after two successive percentage increases is: Present Population = Initial Population × (1 + r₁/100) × (1 + r₂/100)

Substituting the given values: Initial Population = 12,000 r₁ = 2.5% r₂ = 3%

Present Population = 12,000 × (1 + 2.5/100) × (1 + 3/100) Present Population = 12,000 × 1.025 × 1.03

We can calculate this step-by-step: 12,000 × 1.025 = 12,000 × 41/40 = 300 × 41 = 12,300 12,300 × 1.03 = 12,300 × 103/100 = 123 × 103 = 12,669

Explanation

Under simple interest, if a sum of money becomes N times itself in T years, the interest earned is N - 1 times the principal. Since simple interest is directly proportional to time, the relationship is given by the formula:

(N₁ - 1) / T₁ = (N₂ - 1) / T₂

Here, the sum becomes N₁ = 45 times in T₁ = 110 years. We need to find the time T₂ for the sum to become N₂ = 23 times.

Substituting these values into the formula:

(45 - 1) / 110 = (23 - 1) / T₂

44 / 110 = 22 / T₂

Solving for T₂:

T₂ = 22 × 110 / 44

T₂ = 55 years

Explanation

When two objects move toward each other, their relative speed is the sum of their individual speeds.

Relative speed = 70 + 56 = 126 kmph

To convert this speed into meters per second, multiply by 5/18: Relative speed = 126 × 5/18 = 35 m/s

The total distance to be covered for the trains to completely cross each other is the sum of their lengths: Total distance = 320 + 205 = 525 m

Using the formula Time = Distance / Speed: Time taken = 525 / 35 = 15 seconds

Explanation

Let the rates of work done per day by A, B, and C be a, b, and c respectively.

According to the given conditions: a = b + c a + b = 1/10 c = 1/50

Substituting c into the equation for the combined rate of A, B, and C: a + b + c = 1/10 + 1/50 a + b + c = 6/50 = 3/25

Since a = b + c, we can substitute a in the combined rate equation: (b + c) + b + c = 3/25 2(b + c) = 3/25 b + c = 3/50

Using the value of c = 1/50: b + 1/50 = 3/50 b = 3/50 - 1/50 b = 2/50 b = 1/25

Therefore, B alone can complete the task in 25 days.

Explanation

The relationship between the sum and difference of two terms is given by the algebraic identity (a + b)² = (a - b)² + 4ab.

By substituting a = 3x and b = 2y into the identity, we get: (3x + 2y)² = (3x - 2y)² + 4(3x)(2y) (3x + 2y)² = (3x - 2y)² + 24xy

Now, substitute the given values 3x - 2y = 3 and xy = 9 into this equation: (3x + 2y)² = 3² + 24 × 9 (3x + 2y)² = 9 + 216 (3x + 2y)² = 225

Taking the positive square root on both sides: 3x + 2y = √225 = 15

Explanation

To find the ratio of the total number of girls to the total number of boys in all subjects, calculate the individual numbers for each subject first.

The total number of students across all subjects is 110 + 70 + 50 + 40 + 80 + 90 = 440.

Calculate the number of girls in each subject: Hindi: 110 × 60% = 66 English: 70 × 50% = 35 Physics: 50 × 30% = 15 Chemistry: 40 × 80% = 32 Mathematics: 80 × 20% = 16 Computers: 90 × 40% = 36

Total number of girls = 66 + 35 + 15 + 32 + 16 + 36 = 200.

Calculate the total number of boys by subtracting the total number of girls from the total number of students: Total number of boys = 440 - 200 = 240.

The ratio of the total number of girls to the total number of boys is: 200 : 240 = 5 : 6.

Explanation

According to the BODMAS rule, the operations are performed in the order of Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, simplify the expression inside the innermost parentheses (): 5 × 7 = 35

Next, simplify the expression inside the curly braces {}: 25 × 49 ÷ 35 This can be written as: 25 × 49 / 35 = 25 × 7 / 5 = 5 × 7 = 35

Now, substitute this value back into the square brackets []: 24 ÷ 70 × 35 Evaluating from left to right: (24 / 70) × 35 = 24 × 35 / 70 = 24 × ½ = 12

Finally, multiply the terms outside the bracket: 18 × 12 = 216

Explanation

Initially, the 40-litre mixture contains 28 litres of milk and 12 litres of water. To make the water content 60% of the new mixture, the milk content (which remains constant at 28 litres) must represent the remaining 40% of the new mixture, making the total new mixture volume 70 litres. Therefore, the amount of water to be added is 70 minus 40, which equals 30 litres.