SSC CHSL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The volume of wood in a covered box is the difference between the outer volume and the inner volume of the box.

Since the box is covered on all sides, the outer dimensions are calculated by adding twice the thickness of the wood to each of the inner dimensions.

Thickness of the wood = 5.5 cm Twice the thickness = 2 × 5.5 = 11 cm

Outer length = 128 + 11 = 139 cm Outer width = 90 + 11 = 101 cm Outer height = 25 + 11 = 36 cm

Inner Volume = 128 × 90 × 25 = 288000 cm³

Outer Volume = 139 × 101 × 36 = 505404 cm³

Volume of the wood = Outer Volume - Inner Volume Volume of the wood = 505404 - 288000 = 217404 cm³

Explanation

Correct Answer: D

The curved surface area of a cylinder is given by the formula CSA = 2πRH, and its volume is given by V = πR²H.

Let the larger cylinder have radius R_L and height H_L, and the smaller cylinder have radius R_S and height H_S.

The ratio of their radii is given as R_L : R_S = 3 : 1.

The curved surface area of the larger cylinder is 350% of the curved surface area of the smaller cylinder: 2π R_L H_L = 3.5 × 2π R_S H_S R_L H_L = 3.5 R_S H_S

Substituting R_L = 3R_S into the equation: 3R_S H_L = 3.5 R_S H_S 3H_L = 3.5 H_S H_S / H_L = 3 / 3.5 = 6/7

The volume of the smaller cylinder as a fraction of the larger cylinder is: V_S / V_L = (π R_S² H_S) / (π R_L² H_L) = (R_S / R_L)² × (H_S / H_L)

Substituting the known ratios: V_S / V_L = (1/3)² × (6/7) = (1/9) × (6/7) = 6/63 = 2/21

To find this as a percentage: (2/21) × 100% ≈ 9.52%

Explanation

Let the third number be represented by the value 100.

The first number is 30% more than the third number: 100 + (30% of 100) = 130

The second number is 60% more than the third number: 100 + (60% of 100) = 160

The ratio of the first number to the second number is: 130 : 160

Dividing both parts of the ratio by their greatest common divisor, which is 10, gives the simplified ratio: 13 : 16

Explanation

Let the cost price (CP) of the item be 100.

Since the marked price (MP) is 1.4 times the cost price: MP = 1.4 × 100 = 140

To earn a profit of 23.2%, the selling price (SP) must be: SP = 100 × (1 + 23.2/100) = 123.2

The absolute discount allowed on the marked price is: Discount = MP - SP = 140 - 123.2 = 16.8

The discount percentage is calculated relative to the marked price: Discount % = (Discount / MP) × 100 Discount % = (16.8 / 140) × 100 = 12%

Explanation

A number is divisible by 9 if the sum of its digits is divisible by 9.

For the number 6428A83, the sum of the digits is: 6 + 4 + 2 + 8 + A + 8 + 3 = 31 + A

For 31 + A to be divisible by 9, the sum must be equal to the nearest multiple of 9 greater than or equal to 31, which is 36.

31 + A = 36 A = 36 - 31 = 5

Explanation

The perimeter of a rectangle is given by the formula P = 2(L + B), where L is the length and B is the breadth.

Given that the perimeter P is 110 m: 2(L + B) = 110 L + B = 55

We are also given that the length exceeds the breadth by 15 m: L = B + 15 B = L - 15

Substituting the expression for B into the perimeter equation: L + (L - 15) = 55 2L - 15 = 55 2L = 70 L = 35

Thus, the length of the park is 35 m.

Explanation

Let the initial income be 100.

Since the person saves 45% of his income: Initial savings = 45 Initial expenditure = 100 - 45 = 55

Next year, the income increases by 200%: New income = 100 + 200% of 100 = 300

The savings are increased to five times the previous year's savings: New savings = 5 × 45 = 225

New expenditure = New income - New savings = 300 - 225 = 75

Percentage change in expenditure = ((75 - 55) ÷ 55) × 100% = (20 ÷ 55) × 100% = 36.36% ≈ 36.4%

The expenditure has increased by 36.4%.

Explanation

The simple interest SI is the difference between the final amount A and the principal P: SI = A - P SI = 8,928 - 7,200 = 1,728

Using the standard simple interest formula: SI = (P × R × T) / 100

Substituting the values into the formula: 1,728 = (7,200 × 8 × T) / 100 1,728 = 72 × 8 × T 1,728 = 576 × T T = 1,728 / 576 T = 3 years

Explanation

The lateral surface area of a cube is given by the formula LSA = 4a², where a is the side length of the cube.

Given: 4a² = 144

Solving for the side length a: a² = 36 a = 6 cm

The volume of a cube is given by the formula V = a³.

Substituting the side length a = 6 cm: V = 6³ = 216 cm³

Explanation

The length of the direct common tangent t between two circles of radii R and r that touch each other externally is given by the formula:

t = 2√(R × r)

Given that the length of the common tangent t is 12 cm and the radius of the bigger circle R is 18 cm, substituting these values into the formula gives:

12 = 2√(18 × r)

Dividing both sides by 2:

6 = √(18 × r)

Squaring both sides:

36 = 18 × r

r = 2 cm

Explanation

The surface area of a cuboid with length l, width w, and height h is given by the formula: Total Surface Area = 2(lw + wh + lh)

The areas of the three adjacent faces of the cuboid are given as: lw = 5 cm² wh = 15 cm² lh = 27 cm²

Substituting these values directly into the formula: Total Surface Area = 2 × (5 + 15 + 27) Total Surface Area = 2 × 47 Total Surface Area = 94 cm²

Explanation

According to the intersecting chords theorem, when two chords AB and CD of a circle intersect internally at a point L, the products of the segments of the chords are equal:

AL × LB = CL × LD

Substituting the given values into the formula:

9 × 7 = CL × 3

63 = CL × 3

CL = 63 ÷ 3

CL = 21 cm

Explanation

Let the total number of enrolled voters be x.

Since 10% of the voters did not cast their votes, the polled votes are 90% of x, which is 0.9x.

Out of these, 5% of the votes were invalid, making the valid votes 95% of the polled votes: Valid votes = 0.95 × 0.9x = 0.855x

The winning candidate received 52.5% of the valid votes. Therefore, the losing candidate received: 100% - 52.5% = 47.5% of the valid votes.

The winner's majority is the difference between the winner's share and the loser's share: 52.5% - 47.5% = 5% of the valid votes.

We are given that this majority is equal to 1710 votes: 0.05 × 0.855x = 1710 0.04275x = 1710 x = 1710 / 0.04275 x = 40000

Explanation

CP of the mixture is calculated from the selling price (SP) and the profit percentage: CP = (SP × 100) / (100 + Profit %) CP = (36 × 100) / (100 + 25) CP = (36 × 100) / 125 CP = ₹28.8 per kg

Applying the rule of alligation to find the mixing ratio: Ratio of cheaper tea to dearer tea = (Dearer Price - Mean Price) / (Mean Price - Cheaper Price) Ratio = (31 - 28.8) / (28.8 - 27) Ratio = 2.2 / 1.8 Ratio = 11 / 9

Explanation

Let the angles of the triangle be 2x, 4x, and 6x. Since the sum of the angles in a triangle is 180°: 2x + 4x + 6x = 180° 12x = 180° x = 15°

Therefore, the three angles of the triangle are: A = 2 × 15° = 30° B = 4 × 15° = 60° C = 6 × 15° = 90°

Using the Sine Rule, the relationship between the side lengths, their opposite angles, and the circumradius R is: a/sin A = b/sin B = c/sin C = 2R

With a circumradius R = 12 cm, the diameter of the circumcircle is 2R = 24 cm.

Calculating the lengths of the individual sides: a = 2R sin A = 24 × sin 30° = 24 × ½ = 12 cm b = 2R sin B = 24 × sin 60° = 24 × (√3)/2 = 12√3 cm c = 2R sin C = 24 × sin 90° = 24 × 1 = 24 cm

The side lengths of the triangle are 12 cm, 12√3 cm, and 24 cm.

Explanation

The area of a circle is given by the formula A = πr², where r is the radius of the circle.

Substituting the given area of 154 m² and using the standard approximation of π as 22/7: 154 = (22/7) × r² r² = 154 × 7/22 r² = 49 r = √49 = 7 m

The circumference of a circle is calculated using the formula C = 2πr.

Substituting the radius r = 7 m: C = 2 × (22/7) × 7 C = 44 m

Explanation

The circumference of the circle is given as 2πr = 88 cm. Using π = 22/7, we find the radius r: 2 × 22/7 × r = 88 r = 14 cm

Since P is a point on the circle on the segment OC, the length OP represents the radius of the circle, so OP = 14 cm. The length of OC is: OC = OP + PC = 14 + 6 = 20 cm

Since OD exceeds OC by 1 cm: OD = OC + 1 = 21 cm

In the right-angled triangle COD (with ∠COD = 90°), we apply the Pythagoras theorem to find the length of CD: CD² = OC² + OD² CD² = 20² + 21² CD² = 400 + 441 = 841 CD = √841 = 29 cm

The question asks for half of the length of CD: CD/2 = 29/2 = 14.5 cm

Explanation

The relationship between Marked Price (MP), Cost Price (CP), Profit percentage (P), and Discount percentage (D) is given by the formula:

MP/CP = (100 + P) / (100 - D)

Substituting the given values of P = 10 and D = 10:

MP/CP = (100 + 10) / (100 - 10) MP/CP = 110/90 MP/CP = 11/9 ≈ 1.22

Explanation

Correct Answer: C

Given that cot A = 7, we can represent this in a right-angled triangle where the adjacent side is 7 and the opposite side is 1.

Using the Pythagorean theorem, the hypotenuse is: √(7² + 1²) = √50 = 5√2

From this, we find the values of sin A and cos A: sin A = 1 / (5√2) cos A = 7 / (5√2)

Now, we substitute these values into the given expression: (5 cos A + 4 sin A) / (cos² A + 7 sin² A + 6 sin A)

First, calculate the numerator: 5(7 / (5√2)) + 4(1 / (5√2)) = (35 + 4) / (5√2) = 39 / (5√2)

Next, calculate the denominator: (7 / (5√2))² + 7(1 / (5√2))² + 6(1 / (5√2)) = 49/50 + 7/50 + 6 / (5√2) = 56/50 + 6 / (5√2) = 28/25 + 6 / (5√2)

Now, divide the numerator by the denominator: (39 / (5√2)) ÷ (28/25 + 6 / (5√2))

To simplify, multiply both the numerator and the denominator by 125√2: Numerator: (39 / (5√2)) × 125√2 = 39 × 25 = 975 Denominator: (28/25) × 125√2 + (6 / (5√2)) × 125√2 = 140√2 + 150

Evaluating this using the approximation √2 ≈ 1.414: Denominator ≈ 140 × 1.414 + 150 = 197.96 + 150 = 347.96 Value ≈ 975 / 347.96 ≈ 2.8

Rounding 2.8 to the nearest integer yields 3.

Explanation

To find the intersection point (p, q), solve the system of simultaneous linear equations:

3p - 2q + 23 = 0 -2p + q - 13 = 0

From the second equation, express q in terms of p: q = 2p + 13

Substituting this expression for q into the first equation: 3p - 2(2p + 13) + 23 = 0 3p - 4p - 26 + 23 = 0 -p - 3 = 0 p = -3

Substituting p = -3 back into the expression for q: q = 2(-3) + 13 q = 7

Now, calculate the value of 3p + q using p = -3 and q = 7: 3p + q = 3(-3) + 7 = -9 + 7 = -2

Explanation

Let the incomes of the three families be I₁, I₂, and I₃.

Since their expenditures are 80%, 85%, and 75% of their respective incomes, their savings percentages are: Savings of first family = 100% - 80% = 20% of I₁ Savings of second family = 100% - 85% = 15% of I₂ Savings of third family = 100% - 75% = 25% of I₃

The ratio of their savings is given as 8 : 9 : 20. Therefore, we can write: 20% of I₁ : 15% of I₂ : 25% of I₃ = 8 : 9 : 20

To find the ratio of their incomes, we divide each part of the savings ratio by its corresponding saving percentage: I₁ : I₂ : I₃ = 8 / 20% : 9 / 15% : 20 / 25% I₁ : I₂ : I₃ = 8 / 0.20 : 9 / 0.15 : 20 / 0.25 I₁ : I₂ : I₃ = 40 : 60 : 80 I₁ : I₂ : I₃ = 2 : 3 : 4

The sum of the ratio terms is 2 + 3 + 4 = 9.

The total income of the three families is ₹72,000. Thus, the income of the second family is: Income of second family = (3 / 9) × 72,000 = 1/3 × 72,000 = 24,000

Explanation

Using the work equivalence formula M₁ × D₁ × H₁ = M₂ × D₂ × H₂, where M represents the number of men, D represents the number of days, and H represents the number of working hours per day:

30 × 18 × 5 = 20 × D₂ × 7

2700 = 140 × D₂

D₂ = 2700 ÷ 140

D₂ = 135 ÷ 7

D₂ = 19 2/7

Explanation

The simple interest earned over a period is the difference between the total amount and the principal.

Simple Interest for 6 years = ₹5,420 - ₹3,260 = ₹2,160

Since simple interest remains constant every year, the interest earned per year is: Simple Interest for 1 year = ₹2,160 / 6 = ₹360

The interest earned in 4 years is: Simple Interest for 4 years = 4 × ₹360 = ₹1,440

The total amount after 4 years is the sum of the principal and the interest earned in 4 years: Amount after 4 years = ₹3,260 + ₹1,440 = ₹4,700

Explanation

Applying the BODMAS rules, we evaluate the expression term-by-term.

For the first term: 24 ÷ 3 of 1/6 The operator "of" takes precedence over division. 3 of 1/6 = 3 × 1/6 = ½ Now perform the division: 24 ÷ ½ = 24 × 2 = 48

For the second term: 4/5 ÷ 9/8 × 3/5 Perform the division first, then the multiplication: 4/5 ÷ 9/8 = 4/5 × 8/9 = 32/45 32/45 × 3/5 = 32/75

For the third term: 4/7 ÷ 3/14 of 3/14 Evaluate "of" first: 3/14 of 3/14 = 9/196 Then perform the division: 4/7 ÷ 9/196 = 4/7 × 196/9 = (4 × 28)/9 = 112/9

Combining these terms as specified in the original expression: First term × Second term ÷ Third term 48 × 32/75 ÷ 112/9 This simplifies to: 48 × 32/75 × 9/112

To calculate this efficiently, we can simplify the common factors: Both 48 and 112 are divisible by 16, leaving 3 and 7 respectively: 3 × 32/75 × 9/7 Now, 3 and 75 can be simplified by dividing both by 3, leaving 1 and 25: 32/25 × 9/7 Multiplying the remaining numerators and denominators: (32 × 9) / (25 × 7) = 288/175

Explanation

Average = Sum of values ÷ Number of values

From the data for the year 2013 (represented by the fourth data column): Number of employees in Organisation Y = 160 Number of employees in Organisation Z = 212 Number of employees in Organisation T = 126

Sum of employees in Y, Z, and T = 160 + 212 + 126 = 498

Average number of employees = 498 ÷ 3 = 166

Explanation

The area of the playground leveled by the roller in one complete revolution is equal to its curved surface area. The formula for the curved surface area of a cylindrical roller is:

Curved Surface Area = 2 × π × r × h

Given: Radius, r = 49 cm Length, h = 200 cm Number of revolutions, N = 700 Value of π = 22/7

The total area of the playground is the curved surface area of the roller multiplied by the total number of revolutions:

Total Area = 2 × π × r × h × N Total Area = 2 × (22/7) × 49 × 200 × 700 Total Area = 2 × 22 × 7 × 200 × 700 Total Area = 308 × 140000 Total Area = 43120000 cm²

Expressing this value in exponential form:

Total Area = 4312 × 10⁴ cm²

Explanation

The angle subtended by the arc SP at the centre O is ∠SOP = 104°.

By the circle theorem, the angle subtended by the same arc SP at any point on the remaining part of the circle is half the angle subtended at the centre. Thus, the angle at the circumference is: ∠SQP = ½ × ∠SOP = ½ × 104° = 52°

Since PT is a tangent to the circle at P and PQ is the diameter, the angle between the tangent and the diameter is a right angle: ∠TPQ = 90°

In the right-angled triangle TPQ, the sum of the angles is 180°: ∠PTQ = 180° - ∠TPQ - ∠PQT ∠PTQ = 180° - 90° - 52° = 38°

Explanation

The percentage by which Somu's salary is more than Ramu's salary is calculated using the percentage comparison formula: Percentage increase = (r / (100 - r)) × 100 %

Substituting r = 30 into the formula: Percentage increase = (30 / (100 - 30)) × 100 % Percentage increase = (30 / 70) × 100 % Percentage increase = 3/7 × 100 % Percentage increase = 300/7 % ≈ 42.86%

Explanation

The perpendicular distance from the center of a circle with radius r to a chord of length L is given by the formula d = √(r² - (L/2)²).

Here, the half-lengths of the chords PQ and RS are 6 cm and 10 cm respectively. Since the distance between the parallel chords is 4 cm and the sum of their distances from the center exceeds 4 cm, the chords must lie on the same side of the center.

The distances of the chords PQ and RS from the center are d₁ = √(r² - 36) and d₂ = √(r² - 100).

Using the relation d₁ - d₂ = 4: √(r² - 36) - √(r² - 100) = 4

Rearranging the terms: √(r² - 36) = 4 + √(r² - 100)

Squaring both sides: r² - 36 = 16 + r² - 100 + 8√(r² - 100) 48 = 8√(r² - 100) 6 = √(r² - 100)

Squaring both sides again: 36 = r² - 100 r² = 136 r = √136 = 2√34 cm

The diameter of the circle is 2r = 2 × 2√34 = 4√34 cm.

Explanation

To find the value of (4 sin θ - 3 cos θ) / (4 sin θ + 3 cos θ), we can divide both the numerator and the denominator by cos θ. This converts the expression in terms of tan θ, because tan θ = sin θ / cos θ.

The expression simplifies to: (4 tan θ - 3) / (4 tan θ + 3)

Substituting the given value of tan θ = 7/4 into the expression: (4 × (7/4) - 3) / (4 × (7/4) + 3) = (7 - 3) / (7 + 3) = 4/10 = 2/5

Explanation

For similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides:

Perimeter(△XYZ) / Perimeter(△PQR) = XY / PQ

Substituting the given values:

16 / 9 = XY / 3.6

XY = (16 × 3.6) / 9

XY = 16 × 0.4

XY = 6.4 cm

Converting 6.4 to a mixed fraction:

XY = 6 + 4/10 = 6 2/5 cm

Explanation

Using the BODMAS rule, simplify the expression by first resolving the innermost parentheses:

First, evaluate the multiplication inside the first parenthesis: (3/7) × 35 = 3 × 5 = 15

Next, evaluate the second parenthesis: 19 × 3 = 57

Then, evaluate the third parenthesis: 21/2 - 5/2 = 16/2 = 8

Substitute these values back into the main expression: 469 ÷ [270 ÷ 15 + 57 - 8]

Perform the division inside the square brackets: 270 ÷ 15 = 18

Now, perform the addition and subtraction inside the square brackets: 18 + 57 - 8 = 75 - 8 = 67

Finally, perform the division: 469 ÷ 67 = 7

Explanation

The total distance of a journey is the sum of the distances covered in each individual part. The distance formula is:

Distance = Speed × Time

For the first part of the journey: Speed = 68 km/h Time = 22.5 minutes = 22.5 / 60 hours = 0.375 hours Distance₁ = 68 × 0.375 = 25.5 km

For the second part of the journey: Speed = 72 km/h Time = 12 minutes = 12 / 60 hours = 0.2 hours Distance₂ = 72 × 0.2 = 14.4 km

The total distance of the journey is: Total Distance = Distance₁ + Distance₂ = 25.5 + 14.4 = 39.9 km

Rounding 39.9 km to the nearest integer yields 40 km.

Explanation

By the rule of allegation, the ratio of the number of employees in the first group to the second group is calculated from their respective salary deviations from the average:

First group (N₁) deviation: 3,950 - 2,150 = 1,800 Second group (N₂) deviation: 2,150 - 1,850 = 300

The ratio of the number of employees in the two groups is: N₁ : N₂ = 300 : 1,800 = 1 : 6

Given that N₁ = 12, we can find N₂: N₂ = 12 × 6 = 72

The total number of employees in the institution is: N₁ + N₂ = 12 + 72 = 84

Explanation

The volume of a cube with side length s is V = s³, and its surface area is A = 6s².

Let the side lengths of the two cubes be s₁ and s₂. The ratio of their volumes is: V₁/V₂ = s₁³/s₂³ = 1/125

Taking the cube root on both sides gives the ratio of their side lengths: s₁/s₂ = 1/5

The ratio of their surface areas is: A₁/A₂ = 6s₁²/(6s₂²) = (s₁/s₂)² = (1/5)² = 1/25

Explanation

Using the work formula for two individuals working together, the time taken T is given by:

T = (a × b) / (a + b)

Substituting the given values, where A completes the work in a = 9 days and B completes the work in b = 15 days:

T = (9 × 15) / (9 + 15) T = 135 / 24

Simplifying the fraction by dividing both the numerator and the denominator by 3:

T = 45 / 8

Converting this improper fraction into a mixed fraction:

T = 5 5/8 days

Explanation

Since CD is the diameter of the circle and the radius is 6 cm, the length of CD is 2 × 6 = 12 cm.

AD is a tangent to the circle at point D, which means the radius (or diameter CD) is perpendicular to the tangent AD at the point of contact D. Therefore, angle ADC = 90°, making triangle ADC a right-angled triangle at D.

Applying Pythagoras' theorem in triangle ADC: AC² = AD² + CD²

By the tangent-secant theorem, the relation between the tangent AD and the secant AC (which cuts the circle at B and C) is: AD² = AB × AC

Substituting AD² into the Pythagorean equation gives: AC² = (AB × AC) + CD²

Let AC = x. Given that AB = 7 cm and CD = 12 cm, substitute these values into the equation: x² = 7x + 12² x² - 7x - 144 = 0

Solving this quadratic equation by factoring: (x - 16)(x + 9) = 0

Since the length of AC must be a positive value, we have: x = 16 cm

Explanation

Using the relation between Marked Price (MP) and Cost Price (CP): MP / CP = (100 + Gain%) / (100 - Discount%)

Given that CP = 1092, Gain% = 36, and Discount% = 22, we substitute these values into the formula: MP / 1092 = (100 + 36) / (100 - 22) MP / 1092 = 136 / 78

Simplifying the ratio: MP / 1092 = 68 / 39

Solving for MP: MP = 1092 × 68 / 39 MP = 28 × 68 MP = 1904

Explanation

To find the number of small cubes that can fit inside the large cubical box, the dimensions must be in the same unit. The edge of the large cubical box is 3 m, which is equal to 300 cm since 1 m = 100 cm. The edge of each small cube is 15 cm.

The number of small cubes that can fit along any single edge of the large box is 300 ÷ 15 = 20.

Since the box is three-dimensional, the total number of cubes that can fit inside the large box is the product of the number of cubes along its length, width, and height:

Number of cubes = 20 × 20 × 20 = 20³ = 8000

Explanation

The successive percentage change formula is given by:

Net percentage change = x + y + xy/100

Here, the number is first decreased by 18%, so x = -18. The resulting number is then increased by 25%, so y = 25.

Substituting these values into the formula:

Net percentage change = -18 + 25 + (-18 × 25)/100 Net percentage change = 7 - 450/100 Net percentage change = 7 - 4.5 Net percentage change = 2.5%

Since the result is positive, it represents an overall increase of 2.5%.

Explanation

The gain percentage for a dishonest dealer who sells at cost price using a false weight is given by the formula:

Gain Percentage = ((True Weight - False Weight) / False Weight) × 100%

In this case, the true weight of 1 kg is equal to 1000 grams, and the false weight used by the dealer is 850 grams.

Substituting these values into the formula:

Gain Percentage = ((1000 - 850) / 850) × 100%

Gain Percentage = (150 / 850) × 100%

Gain Percentage = 3/17 × 100%

Gain Percentage = 300/17 %

Gain Percentage ≈ 17.647%

Rounding to two decimal places gives 17.65%.

Explanation

Let R be the number of rings in the trunk, A be the age of the tree in years, and V be the value of the tree in ₹.

The age of the tree is directly proportional to the number of rings: A ∝ R A₂ / A₁ = R₂ / R₁

Substituting the given values A₁ = 50, R₁ = 40, and R₂ = 30: A₂ / 50 = 30 / 40 A₂ = 50 × (3 / 4) = 37.5 years

The value of the tree is directly proportional to its age: V ∝ A V₂ / V₁ = A₂ / A₁

Substituting V₁ = 50000, A₁ = 50, and A₂ = 37.5: V₂ / 50000 = 37.5 / 50 V₂ = 50000 × 0.75 = 37500

Explanation

Let the cost price of one pen be CP and the selling price of one pen be SP.

According to the given condition, the selling price of 100 pens is equal to the cost price of 140 pens: 100 × SP = 140 × CP

Rearranging the terms gives the ratio of the selling price to the cost price: SP/CP = 140/100 = 7/5

The formula for profit percentage is: Profit % = ((SP - CP)/CP) × 100 = (SP/CP - 1) × 100

Substituting the ratio into the formula: Profit % = (7/5 - 1) × 100 = 2/5 × 100 = 40%

Explanation

Correct Answer: D

Let the rate of the stream be v km/h. The speed of the man in still water is given as 18 km/h.

The downstream speed is 18 + v km/h, and the upstream speed is 18 - v km/h.

For a constant distance, speed is inversely proportional to time. Since the upstream time is twice the downstream time, the downstream speed must be twice the upstream speed.

18 + v = 2 × (18 - v)

18 + v = 36 - 2v

3v = 18

v = 6

The rate of the stream is 6 km/h.

Explanation

By the given condition of similarity between triangle BMD and triangle ABC, the ratio of the corresponding sides is equal: BM/AB = BD/AC

Substitute the given lengths into this relation: AB = 6 units AC = 10 units BM = 5 units

This gives: 5/6 = BD/10

Solving for BD: BD = (5 × 10) / 6 BD = 50/6 BD = 25/3 units

Explanation

The formula for the length of the transverse common tangent (L) of two circles with radii r₁ and r₂ and the distance between their centers (d) is:

L = √(d² - (r₁ + r₂)²)

Substitute the given values r₁ = 6 cm, r₂ = 3 cm, and L = 8 cm into the formula:

8 = √(d² - (6 + 3)²)

Squaring both sides of the equation:

64 = d² - 9² 64 = d² - 81 d² = 64 + 81 d² = 145 d = √145 cm

Explanation

Simple interest is directly proportional to time when the principal and the rate of interest remain constant.

For an amount to double, the interest earned must equal the principal (P). This means SI₁ = P, which takes T₁ = 7 years.

For the amount to become four times, the interest earned must be three times the principal (3P). This means SI₂ = 3P, which takes T₂ years.

Using the proportional relationship SI₁ / SI₂ = T₁ / T₂:

P / 3P = 7 / T₂

1 / 3 = 7 / T₂

T₂ = 3 × 7 = 21 years

Explanation

Let the number of £2, £5, and £10 coins be x, y, and z respectively.

Given the ratio of £2 and £5 coins is 5 : 2, we can represent them as: x = 5k y = 2k for some integer k.

The total number of coins is 50: x + y + z = 50 5k + 2k + z = 50 7k + z = 50

The total value of the coins is £150: 2x + 5y + 10z = 150 2(5k) + 5(2k) + 10z = 150 20k + 10z = 150 2k + z = 15

Subtracting the equation 2k + z = 15 from 7k + z = 50: 5k = 35 k = 7

Substituting k = 7 into 2k + z = 15: 2(7) + z = 15 z = 1

Explanation

When the price of an article increases by R%, the percentage reduction in consumption required to keep the expenditure constant is given by the formula:

Percentage reduction = (R ÷ (100 + R)) × 100%

Substituting the given value of price increase, R = 35:

Percentage reduction = (35 ÷ (100 + 35)) × 100% Percentage reduction = (35 ÷ 135) × 100% Percentage reduction = (7 ÷ 27) × 100% Percentage reduction = 700 / 27 % Percentage reduction = 25 25/27 %

Explanation

To find the unique solution of the two lines, solve the system of linear equations: x + y = 10 -3x + y = 2

Subtracting the second equation from the first equation: (x + y) - (-3x + y) = 10 - 2 4x = 8 x = 2

Substituting x = 2 into the first equation: 2 + y = 10 y = 8

Thus, the unique solution is the point P(2, 8).

Next, the point of intersection of the line x + y = 10 with the x-axis is found by setting y = 0: x + 0 = 10 x = 10

So, the point of intersection with the x-axis is Q(10, 0).

Using the distance formula, the distance d between P(2, 8) and Q(10, 0) is: d = √((10 - 2)² + (0 - 8)²) d = √(8² + (-8)²) d = √(64 + 64) d = √128 d = 8√2