SSC CHSL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The rectangle with the largest area that can be inscribed in a circle of radius r is always a square.

The diagonal of this square is equal to the diameter of the circle, which is 2r.

If s is the side length of the square, then by Pythagoras' theorem: s² + s² = (2r)² 2s² = 4r² s² = 2r²

Given the radius r = 5: Area of the square = 2r² = 2 × 5² = 50 Area of the circle = πr² = π × 5² = 25π

The area of the remaining portion of the circle is the difference between the area of the circle and the area of the square: Remaining Area = 25π - 50 = 25(π - 2)

Explanation

Using the BODMAS rule, the order of operations is Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, simplify the of operations: 4 of 3 = 4 × 3 = 12 16 of 4 = 16 × 4 = 64

Substitute these values back into the expression: 12 ÷ 12 × 6 + 8 ÷ 64 × 8 - 4 ÷ 6 × 9

Next, perform the division and multiplication operations from left to right for each individual term: First term: 12 ÷ 12 × 6 = 1 × 6 = 6 Second term: 8 ÷ 64 × 8 = 1/8 × 8 = 1 Third term: 4 ÷ 6 × 9 = 2/3 × 9 = 6

Finally, combine the terms using addition and subtraction: 6 + 1 - 6 = 1

Explanation

Average Speed = Total Distance ÷ Total Time

The total distance covered by the cyclist is: D = 10 km + 15 km = 25 km

The time taken for the first leg of the journey is: t₁ = 10 km ÷ 15 km/hr = 2/3 hr

The time taken for the second leg of the journey is: t₂ = 15 km ÷ 10 km/hr = 3/2 hr

The total time taken for the entire ride is: T = 2/3 hr + 3/2 hr = 13/6 hr

Substituting these values into the average speed formula: Average Speed = 25 ÷ (13/6) = (25 × 6) ÷ 13 = 150 ÷ 13 ≈ 11.54 km/hr

Explanation

Let the tens digit of the two-digit number be x and the units digit be y. The value of the number is 10x + y.

According to the first condition, the sum of the digits is 9: x + y = 9

According to the second condition, reversing the digits gives the number 10y + x, which is 27 more than the original number: (10y + x) - (10x + y) = 27 9y - 9x = 27 y - x = 3

Adding these two equations: (x + y) + (y - x) = 9 + 3 2y = 12 y = 6

Substituting y = 6 into the first equation: x + 6 = 9 x = 3

The original number 10x + y is equal to 10(3) + 6, which is 36.

Explanation

The length of the common internal tangent t between two circles with radii r₁ and r₂ and distance between their centres d is given by the relation: t = √(d² - (r₁ + r₂)²)

For a stick of length 6 units to be used as the common internal tangent, the distance between the points of contact, which is the tangent segment t, cannot exceed the length of the stick. Therefore: t ≤ 6 √(d² - (r₁ + r₂)²) ≤ 6

Squaring both sides of the inequality: d² - (r₁ + r₂)² ≤ 36

Substituting the given values r₁ = 2 and r₂ = 3 into the inequality: d² - (2 + 3)² ≤ 36 d² - 25 ≤ 36 d² ≤ 61 d ≤ √61

Explanation

The volume of a cuboid is V = l × b × h. Since 1 litre is equal to 1000 cm³, the volume of the water purifier in cubic centimeters is: V = 10 × 1000 = 10000 cm³

Let the height of the cuboid be h. The breadth is 1/5 of the height: b = h/5 = 0.2h

The length is 1/3 of the sum of breadth and height: l = (b + h)/3 = (0.2h + h)/3 = 1.2h / 3 = 0.4h

Substituting these values into the volume formula: 10000 = 0.4h × 0.2h × h 10000 = 0.08h³ h³ = 10000 / 0.08 h³ = 125000 h = ∛125000 = 50 cm

Explanation

We can express the product sin A cos A in terms of tan A using the trigonometric relation: sin A cos A = (sin A cos A) / (sin² A + cos² A) = tan A / (1 + tan² A)

Given that tan A = √2 - 1, we calculate the denominator 1 + tan² A: tan² A = (√2 - 1)² = 2 - 2√2 + 1 = 3 - 2√2 1 + tan² A = 4 - 2√2 = 2√2(√2 - 1)

Now, substitute these values into the expression for sin A cos A: sin A cos A = (√2 - 1) / (2√2(√2 - 1)) = 1 / (2√2) = √2 / 4

Finally, we find the value of sin A cos A + tan A: sin A cos A + tan A = (√2 / 4) + (√2 - 1) = (√2 + 4(√2 - 1)) / 4 = (5√2 - 4) / 4

Explanation

Total number of students in the college is the sum of students in all subjects: 840 + 220 + 450 + 200 = 1710

The number of girl students in each subject is calculated using their respective percentages: Mathematics girls = 40% of 840 = 336 Chemistry girls = 70% of 220 = 154 Physics girls = 30% of 450 = 135 Biology girls = 80% of 200 = 160

The total number of girl students is the sum of girls in all subjects: 336 + 154 + 135 + 160 = 785

The overall percentage of girl students in the college is: (Total girl students / Total students) × 100 = (785 / 1710) × 100 ≈ 45.9%

Explanation

The total percentage of votes in the election is represented as 100%.

The first candidate received 40% of the votes, and the second candidate received 35% of the votes. This leaves the third candidate with the remaining percentage:

100% - (40% + 35%) = 25%

Given that the total number of votes in the election is 50,000, the number of votes received by the third candidate is calculated as:

25% of 50,000 = (25/100) × 50,000 = 12,500

Explanation

In the isosceles triangle ABC with AB = AC, the angles opposite to the equal sides are also equal, which gives angle ABC = angle ACB.

The total angle BAC is the sum of the adjacent angles BAD and CAD: angle BAC = angle BAD + angle CAD = 20° + 80° = 100°

The sum of the interior angles of triangle ABC is 180°: angle BAC + angle ABC + angle ACB = 180°

Substituting the known values and using the equality of the base angles: 100° + 2 × angle ABC = 180° 2 × angle ABC = 80° angle ABC = 40°

Explanation

The length of a direct common tangent to two circles is given by the formula:

Length = √(d² - (r₁ - r₂)²)

Here, d represents the distance between the centres of the two circles, and r₁ and r₂ represent the radii of the two circles where r₁ is greater than r₂.

Given: Distance between centres (d) = 40 cm Radius of the first circle (r₁) = 14 cm Radius of the second circle (r₂) = 5 cm

Substituting these values into the formula:

Length = √(40² - (14 - 5)²) Length = √(1600 - 9²) Length = √(1600 - 81) Length = √1519

Explanation

When individuals work together for the same duration, their wages are distributed in the ratio of their efficiencies, which is the inverse ratio of the time taken by each to complete the work individually.

Time taken by P = 12 days Time taken by Q = 9 days

Ratio of efficiency of P to Q = 1/12 : 1/9 = 9 : 12 = 3 : 4

The total earnings of ₹63,000 correspond to the sum of the ratio parts, which is 3 + 4 = 7 parts.

Value of 1 part = 63,000 ÷ 7 = ₹9,000

Share of P = 3 parts = 3 × 9,000 = ₹27,000

Explanation

The average cost per kg of the mixture is calculated using the weighted average formula:

Average Cost = (Quantity₁ × Price₁ + Quantity₂ × Price₂) / (Quantity₁ + Quantity₂)

Substituting the given values:

Quantity₁ = 10 kg, Price₁ = ₹88 Quantity₂ = 20 kg, Price₂ = ₹187

Average Cost = (10 × 88 + 20 × 187) / (10 + 20) = (880 + 3740) / 30 = 4620 / 30 = 154

This calculation yields a per kg cost of ₹154 for the entire mixture.

Explanation

The profit percentage for successive markup and discount is calculated using the formula:

Profit % = m - d - md/100

where m is the markup percentage and d is the discount percentage.

For Shopkeeper A: m = 25% d = 15% Profit % = 25 - 15 - (25 × 15)/100 Profit % = 10 - 3.75 = 6.25%

For Shopkeeper B: m = 20% d = 12% Profit % = 20 - 12 - (20 × 12)/100 Profit % = 8 - 2.4 = 5.6%

Comparing the two profit percentages, Shopkeeper A gets a better deal with a profit percentage of 6.25%.

Explanation

Since the time of travel for both the policeman and the thief is the same until the chase ends, the distance covered by each is directly proportional to their respective speeds.

The ratio of the speed of the policeman to the speed of the thief is: Speed ratio = 20 ÷ 16 = 5 ÷ 4

Therefore, the ratio of the distance covered by the policeman to the distance covered by the thief is also 5 ÷ 4.

The difference in their distance ratio units is: 5 - 4 = 1 unit

This difference of 1 unit represents the initial head start of the thief, which is 200 metres: 1 unit = 200 metres

The distance run by the thief before being overtaken is 4 units: Distance = 4 × 200 metres = 800 metres

Explanation

Let P's age at the time of her marriage be x years. Since she married 15 years ago, her present age is x + 15 years.

We are given that her present age is 1⅗ (which is 8/5) times her age at marriage: x + 15 = 8/5 × x

Solving for x: 5(x + 15) = 8x 5x + 75 = 8x 3x = 75 x = 25

Thus, P's present age is 25 + 15 = 40 years.

The present age of her son is ⅕ of her present age: Son's present age = 40 / 5 = 8 years.

Four years ago, their respective ages were: Son's age = 8 - 4 = 4 years P's age = 40 - 4 = 36 years

The fraction of her son's age to P's age 4 years ago is: 4 / 36 = 1/9

Explanation

Given the ratio x:y = 5:6, we can substitute x = 5 and y = 6 directly into the expression (2x + 3y) : (3x + 5y) to find the required ratio.

Evaluating the terms of the ratio: 2x + 3y = 2(5) + 3(6) = 10 + 18 = 28 3x + 5y = 3(5) + 5(6) = 15 + 30 = 45

This yields the ratio 28:45.

Explanation

Simple interest is constant for every equal interval of time. The difference in the amounts over the 2-year period from year 2 to year 4 represents the interest earned during those 2 years.

Simple interest for 2 years = ₹32,000 - ₹24,000 = ₹8,000 Simple interest for 1 year = ₹8,000 ÷ 2 = ₹4,000

The principal sum is the amount at 2 years minus the interest earned over those 2 years: Principal = ₹24,000 - ₹8,000 = ₹16,000

The annual rate of interest is: Rate = (Simple interest for 1 year ÷ Principal) × 100% Rate = (4,000 ÷ 16,000) × 100% = 25%

Explanation

By the Tangent-Secant Theorem, the relation between the tangent AB and the secant AD intersecting the circle at C and D is:

AB² = AC × AD

Substitute the given values AB = 42 cm and AC = 21 cm into the formula:

42² = 21 × AD 1764 = 21 × AD AD = 84 cm

The length of the secant segment AD is the sum of AC and CD:

AD = AC + CD 84 = 21 + CD CD = 63 cm

To find the ratio between AB and CD:

AB/CD = 42/63 = 2/3

Explanation

Applying BODMAS rules, we first simplify the terms inside the parentheses.

First, calculate the "of" operation: 16/15 of 3/8 = 16/15 × 3/8 = 2/5

Next, convert the mixed fraction to an improper fraction: 2 1/12 = 25/12

Then, solve the subtraction inside the parentheses: 1 - 25/12 = -13/12

Now substitute these simplified values back into the main expression: A = 60[6/25 ÷ 2/5 + 5 - (-13/12)]

Next, perform the division: 6/25 ÷ 2/5 = 6/25 × 5/2 = 3/5

Substituting this back gives: A = 60[3/5 + 5 + 13/12]

To add the terms inside the brackets, find a common denominator, which is 60: 3/5 + 5 + 13/12 = 36/60 + 300/60 + 65/60 = 401/60

Now multiply by 60: A = 60 × 401/60 = 401

The digits of A are 4, 0, and 1. The sum of the digits is: 4 + 0 + 1 = 5

Explanation

To find the values of A and D:

A = (9/16) ÷ (25/36) To divide by a fraction, multiply by its reciprocal: A = (9/16) × (36/25) Simplifying by dividing 36 and 16 by their common factor of 4: A = (9 × 9) / (4 × 25) = 81/100

D = (9 ÷ 16) ÷ (25 ÷ 36) Expressing the division operations inside the parentheses as fractions: D = (9/16) ÷ (25/36) Multiplying by the reciprocal: D = (9/16) × (36/25) D = 81/100

Since both A and D simplify to 81/100, they are equal to each other.

Explanation

The total surface area of a cube with edge length a is given by the formula: Total Surface Area = 6a²

For the large cube with an edge length of L = 4 cm, the total surface area is: Total Surface Area of large cube = 6 × 4² = 96 cm²

For one of the small cubes with an edge length of s = 2 cm, the total surface area is: Total Surface Area of small cube = 6 × 2² = 24 cm²

The ratio of the total surface area of one small cube to that of the large cube is: Ratio = (6 × s²) / (6 × L²) = s² / L² = 2² / 4² = 4 / 16 = 1 / 4

Explanation

Cost Price of 20 dozen bananas = ₹1200 Cost Price of 5 dozen bananas = (1200 / 20) × 5 = ₹300

Selling Price of 5 dozen bananas = ₹350

Gain = Selling Price - Cost Price = 350 - 300 = ₹50

Gain percentage = (Gain / Cost Price) × 100 = (50 / 300) × 100 = 50/3 % = 16 ⅔%

Explanation

The volume of a hemisphere is given by the formula:

V = (2/3) × π × r³

Given that the radius r = 5.5 cm and π = 3.14, we substitute these values into the formula:

V = (2/3) × 3.14 × (5.5)³

V = (2/3) × 3.14 × 166.375

V = 1044.835 ÷ 3

V ≈ 348.28 cm³

Rounding to the nearest integer, the volume of the hemisphere is 348 cm³.

Explanation

The curved surface area of a right circular cone is given by the formula πrl, where r is the radius and l is the slant height.

Substituting the given values into the formula: π × 4 × l = 20π l = 5 cm

The relation between the slant height (l), radius (r), and height (h) of the cone is: l² = r² + h² 5² = 4² + h² 25 = 16 + h² h² = 9 h = 3 cm

The volume of a right circular cone is given by the formula ⅓πr²h.

Substituting the radius and height: Volume = ⅓ × π × 4² × 3 Volume = 16π cm³

Explanation

The given function is f(p) = sinᵖ(2x) + cosᵖ(2x).

To find the value of the expression 6f(4) - 4f(6) + 10f(2), we first express f(2), f(4), and f(6) in terms of sin(2x) and cos(2x):

f(2) = sin²(2x) + cos²(2x) = 1

f(4) = sin⁴(2x) + cos⁴(2x) = (sin²(2x) + cos²(2x))² - 2sin²(2x)cos²(2x) = 1 - 2sin²(2x)cos²(2x)

f(6) = sin⁶(2x) + cos⁶(2x) = (sin²(2x) + cos²(2x))³ - 3sin²(2x)cos²(2x)(sin²(2x) + cos²(2x)) = 1 - 3sin²(2x)cos²(2x)

Substituting these values into the expression:

6f(4) - 4f(6) + 10f(2) = 6(1 - 2sin²(2x)cos²(2x)) - 4(1 - 3sin²(2x)cos²(2x)) + 10(1)

= 6 - 12sin²(2x)cos²(2x) - 4 + 12sin²(2x)cos²(2x) + 10

= 6 - 4 + 10

= 12

Explanation

Let R and r be the radii of the outer and inner circles, respectively.

The ratio of the circumferences of the two circles is equal to the ratio of their radii: 2πR / 2πr = R / r = 7 / 6

This gives R = 7x and r = 6x for some constant x.

The width of the circular path is the difference between the outer and inner radii: R - r = 45 7x - 6x = 45 x = 45

The radius of the inner circle is: r = 6x = 6 × 45 = 270 m

Explanation

Let the number be x.

According to the given condition, subtracting 12 from the number reduces it to 40% of its value: x - 12 = 40% of x x - 12 = 0.4x x - 0.4x = 12 0.6x = 12 x = 12 / 0.6 x = 20

Now, we find two-fifth of this number: 2/5 of 20 = 2/5 × 20 = 8

Explanation

Let the total number of votes recorded be V.

Since there are only two candidates, the winning candidate secures the remaining percentage of the votes: 100% - 40% = 60%

The majority by which the winning candidate defeats the losing candidate is the difference between their vote percentages: 60% - 40% = 20% of the total votes

We are given that this majority is equal to 298 votes: 20% of V = 298 (20/100) × V = 298 V = 298 × (100/20) V = 298 × 5 = 1490

Explanation

The simple interest earned over 1.5 years (from year 2 to year 3½) is the difference between the two amounts: Interest for 1.5 years = 2,130 - 1,860 = 270

The simple interest for 1 year is: Interest per year = 270 ÷ 1.5 = 180

The principal sum (P) is the amount at 2 years minus the interest accrued over those 2 years: P = 1,860 - 2 × 180 = 1,860 - 360 = 1,500

The rate of interest (R) per annum is: R = (Interest per year ÷ P) × 100 = (180 ÷ 1,500) × 100 = 12%

Explanation

Number of girl students in Mathematics is 40% of 840, which equals 336. Number of girl students in Chemistry is 70% of 220, which equals 154. The total number of girl students in Mathematics and Chemistry is 336 + 154 = 490.

Number of boy students in Physics is 70% of 450, which equals 315. Number of boy students in Biology is 20% of 200, which equals 40. The total number of boy students in Physics and Biology is 315 + 40 = 355.

The required ratio is 490 : 355. Dividing both terms by 5 gives 98 : 71.

Explanation

Correct Answer: D

Let the total work be the Least Common Multiple (LCM) of the days taken by the pairs, which are 36, 54, and 81.

LCM(36, 54, 81) = 324 units

Therefore, the total work is 324 units.

The daily work efficiency of each pair is: Efficiency of Reena + Riya = 324 / 36 = 9 units/day Efficiency of Riya + Geeta = 324 / 54 = 6 units/day Efficiency of Reena + Geeta = 324 / 81 = 4 units/day

Adding the efficiencies of all three pairs: 2 × (Efficiency of Reena + Riya + Geeta) = 9 + 6 + 4 = 19 units/day Efficiency of Reena + Riya + Geeta = 19 / 2 = 9.5 units/day

To find Reena's efficiency alone, subtract the efficiency of Riya and Geeta from the total combined efficiency: Efficiency of Reena = 9.5 - 6 = 3.5 units/day

The number of days Reena alone takes to complete the work is: Time = Total work / Efficiency of Reena Time = 324 / 3.5 = 648 / 7 ≈ 92.57 days

Rounding to the nearest whole number gives 92 days.

Explanation

Correct Answer: B

By the Intersecting Secants Theorem for a circle, when two secant segments are drawn to a circle from an exterior point, the product of the external segment and the entire secant segment is equal for both secants. This gives the relation:

PB × PA = PD × PC

Since the points are collinear, we have PA = PB + AB and PC = PD + CD. Substituting these into the formula:

PB × (PB + AB) = PD × (PD + CD)

Substitute the given values AB = 6 cm, CD = 3 cm, and PD = 5 cm into the equation:

PB × (PB + 6) = 5 × (5 + 3)

PB² + 6PB = 5 × 8

PB² + 6PB - 40 = 0

Solving this quadratic equation by factoring:

(PB + 10)(PB - 4) = 0

Since length must be a positive value, PB = 4 cm.

Explanation

We evaluate the expression using the BODMAS rule, which specifies the order of operations as Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

First, simplify the innermost brackets: (40 × 5) + (7 × 6) = 200 + 42 = 242

Next, substitute this value and calculate the first major term in square brackets: [3³ × 242 ÷ 11] = [27 × 22] = 594

Then, simplify the second term in parentheses: (2³ × 15 ÷ 24) = (8 × 15 ÷ 24) = 120 ÷ 24 = 5

Adding both simplified terms together: 594 + 5 = 599

Explanation

The angle formed by the intersection of the exterior angle bisectors of ∠B and ∠C in triangle ABC is given by the standard geometric formula:

∠P = 90° - ∠A/2

Substituting the given value of ∠A = 72° into the formula:

∠P = 90° - 72°/2 ∠P = 90° - 36° ∠P = 54°

Explanation

The area of a quadrilateral ABCD can be calculated by summing the areas of the two triangles ABD and BCD, which share the diagonal BD as their common base.

The formula for the area of a quadrilateral given a diagonal d and the perpendicular heights h₁ and h₂ from the opposite vertices to this diagonal is:

Area = ½ × d × (h₁ + h₂)

Substituting the given values: d = BD = 20 cm h₁ = 6 cm h₂ = 8 cm

Area = ½ × 20 × (6 + 8) Area = 10 × 14 Area = 140 cm²

Explanation

By setting up a system of linear equations where x is the cost of a pencil and y is the cost of a pen, we get 21x + 9y = 276 and 6x + 3y = 84. Solving these equations gives x = 8 and y = 12, so the cost of 1 pencil and 1 pen is 8 + 12 = ₹20.

Explanation

To find the greatest ratio, we convert each ratio into its decimal form. The ratio 7:4 is equal to 1.75, the ratio 8:3 is equal to 2.67, the ratio 2:3 is equal to 0.67, and the ratio 1:5 is equal to 0.2. Comparing these values, 2.67 is the highest decimal value, which makes 8:3 the greatest ratio.

Explanation

To divide a cuboid into 8 identical pieces using 3 cuts, each cut must be made parallel to one of the three pairs of opposite faces, passing through the midpoint of each dimension.

Every cut made parallel to a face of the cuboid creates two new identical surfaces of that same cross-sectional area. Thus, making one cut parallel to each of the three dimensions results in adding new surface area equivalent to another full set of the original cuboid's surface area.

The formula for the total surface area of all 8 pieces is: Total Surface Area = 4(lw + wh + hl)

Substituting the given dimensions, l = 20 cm, w = 12 cm, and h = 10 cm: Total Surface Area = 4 × (20 × 12 + 12 × 10 + 10 × 20) Total Surface Area = 4 × (240 + 120 + 200) Total Surface Area = 4 × 560 Total Surface Area = 2240 cm²

Explanation

The cost price of 15 kg of wheat is the selling price of 15 kg minus the total profit earned on 15 kg.

First, calculate the selling price of 15 kg of wheat: Selling price of 15 kg = (Total selling price ÷ Total weight) × 15 Selling price of 15 kg = (10260 ÷ 195) × 15 Selling price of 15 kg = 10260 ÷ 13 ≈ 789.23

Next, calculate the profit on 15 kg of wheat: Profit on 15 kg = Profit per kg × 15 Profit on 15 kg = 4.50 × 15 = 67.50

Now, find the cost price of 15 kg of wheat: Cost price of 15 kg = Selling price of 15 kg - Profit on 15 kg Cost price of 15 kg = 789.23 - 67.50 = 721.73

Rounding to the nearest integer gives 722.

Explanation

Let x = sin θ. The given quadratic equation 4sin²θ - 4√3 sin θ + 3 = 0 can be written in terms of x as: 4x² - 4√3 x + 3 = 0

This quadratic equation is a perfect square: (2x - √3)² = 0

Taking the square root on both sides: 2x - √3 = 0 2x = √3 x = √3/2

Substituting x = sin θ back into the equation: sin θ = √3/2

Since θ lies in the interval 0 < θ < 90°, we find the angle θ that satisfies this relation: θ = 60°

Explanation

Let the third number be 100.

Since the first number is 30% more than the third number, the first number is 100 + 30 = 130.

Since the second number is 40% more than the third number, the second number is 100 + 40 = 140.

The ratio of the second number to the first number is 140 : 130, which simplifies to 14 : 13.

Explanation

The formula for the length of the direct common tangent (L_DCT) between two circles with radii r₁ and r₂ and distance between their centres d is:

L_DCT = √(d² - (r₁ - r₂)²)

Substituting the given values, d = 60 cm, r₁ = 20 cm, and r₂ = 32 cm:

L_DCT = √(60² - (32 - 20)²) L_DCT = √(3600 - 12²) L_DCT = √(3600 - 144) L_DCT = √3456 cm

The formula for the length of the transverse common tangent (L_TCT) is:

L_TCT = √(d² - (r₁ + r₂)²)

Substituting the given values:

L_TCT = √(60² - (32 + 20)²) L_TCT = √(3600 - 52²) L_TCT = √(3600 - 2704) L_TCT = √896 cm

The ratio of the length of the direct common tangent to the length of the transverse common tangent is:

L_DCT : L_TCT = √3456 : √896 L_DCT : L_TCT = √(3456 / 896)

Dividing both the numerator and denominator by 128:

3456 / 128 = 27 896 / 128 = 7

Therefore:

L_DCT : L_TCT = √(27 / 7) = √27 : √7 = 3√3 : √7

Explanation

The number of articles that can be packed in the box is calculated by dividing the volume of the box by the volume of a single article.

Number of articles = Volume of box ÷ Volume of article

Substituting the given dimensions:

Number of articles = (63 × 42 × 21) ÷ (9 × 6 × 3)

By simplifying each dimension:

63 ÷ 9 = 7 42 ÷ 6 = 7 21 ÷ 3 = 7

This gives:

Number of articles = 7 × 7 × 7 = 343

Explanation

Since the distance covered is constant, the speed of the car is inversely proportional to the time taken.

Ratio of time taken = 16 : 12 = 4 : 3

Therefore, the ratio of speed = 3 : 4

Let the initial speed be 3x and the increased speed be 4x. The difference in speed is: 4x - 3x = 20 km/h x = 20 km/h

The initial speed is: 3 × 20 = 60 km/h

The total distance covered is: Distance = Speed × Time = 60 km/h × 16 hours = 960 km

Explanation

Production is directly proportional to the product of manpower and working hours. Let the original production be P, original manpower be M, and original working hours be H, so that P = M × H.

When manpower decreases by 25%, the new manpower is 0.75M. To restore the original production, the working hours must be increased by r% such that:

0.75M × H × (1 + r/100) = M × H

Dividing both sides by M × H gives:

0.75 × (1 + r/100) = 1

1 + r/100 = 1 / 0.75 = 4/3

r/100 = 4/3 - 1 = 1/3

r = 100/3 = 33 ⅓ %

Explanation

Applying the BODMAS rule, we have:

42 ÷ 3 + k × 3 - 22 ÷ 11 + 4 = 28

First, perform the division operations: 42 ÷ 3 = 14 22 ÷ 11 = 2

Substitute these values back into the equation: 14 + 3k - 2 + 4 = 28

Simplify the constants by addition and subtraction: 14 - 2 + 4 + 3k = 28 16 + 3k = 28

Solve for k: 3k = 28 - 16 3k = 12 k = 4

Explanation

In a 30-day month beginning with a Sunday, the Sundays fall on the 1st, 8th, 15th, 22nd, and 29th days of the month. Thus, there are exactly 5 Sundays and 25 remaining days (30 - 5 = 25).

Total attendance for the month = (5 × 410) + (25 × 230) = 2050 + 5750 = 7800

Average attendance per day = Total attendance / Total days = 7800 / 30 = 260

Explanation

The discount rate can be calculated using the standard percentage formula:

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

Substituting the given values:

Discount Rate = ((145 - 127.6) / 145) × 100

Discount Rate = (17.4 / 145) × 100

Discount Rate = 0.12 × 100 = 12%

Explanation

In △QRS and △QPR, ∠RQS is common and ∠QRS = ∠QPR is given. By the Angle-Angle similarity criterion, △QRS ~ △QPR.

For similar triangles, the ratio of corresponding sides is equal: QS/QR = QR/QP = RS/PR

Substituting the given values QS = 27 cm, QR = 36 cm, and RS = 18 cm: QS/QR = 27/36 = 3/4

Using this ratio to find the unknown sides: 36/QP = 3/4, which gives QP = 48 cm 18/PR = 3/4, which gives PR = 24 cm

Since S lies on PQ, the length of PS is: PS = QP - SQ = 48 - 27 = 21 cm

The perimeter of △PRS is: PR + RS + PS = 24 + 18 + 21 = 63 cm

The perimeter of △QSR is: QS + RS + QR = 27 + 18 + 36 = 81 cm

The ratio of the perimeter of △PRS to that of △QSR is: 63/81 = 7/9