SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The profit percentage earned by the shopkeeper remains the same regardless of the total quantity sold.

The formula for profit percentage when using a faulty weight is: Profit % = (True Weight - False Weight) / False Weight × 100%

In this case, the true weight is 1 kg, which is equal to 1000 g, and the false weight used by the shopkeeper is 930 g.

Substituting these values into the formula: Profit % = (1000 - 930) / 930 × 100% Profit % = 70 / 930 × 100% Profit % = 700 / 93 % Profit % = 7 49/93 %

Explanation

The equivalent single discount corresponding to two successive discounts of d₁% and d₂% is given by the formula:

Equivalent Discount = d₁ + d₂ - (d₁ × d₂)/100

Substituting the given values, d₁ = 20 and d₂ = 30:

Equivalent Discount = 20 + 30 - (20 × 30)/100 Equivalent Discount = 50 - 6 = 44%

Explanation

The percentage of qualified students over the applied number of students from University B in the given years is calculated using the formula:

Percentage = (Qualified / Applied) × 100

For the year 2016: (720 / 7500) × 100 = 9.6%

For the year 2018: (600 / 6500) × 100 ≈ 9.23%

For the year 2019: (1000 / 7200) × 100 ≈ 13.89%

For the year 2020: (1200 / 9000) × 100 ≈ 13.33%

Comparing these values, the highest percentage of qualified students over applied students from University B is 13.89% in the year 2019.

Explanation

Let the number be x. According to the question, we can write the equation as:

(4/5) × (5/6) × (1/8) × x = 4365

Simplifying the product of the fractions:

(4 × 5 × 1) / (5 × 6 × 8) × x = 4365 (20 / 240) × x = 4365 (1/12) × x = 4365 x = 4365 × 12 x = 52380

Now, we find 65% of this number:

65% of 52380 = (65 / 100) × 52380 = 0.65 × 52380 = 34047

Explanation

Using the trigonometric identity sec C = 1/cos C and tan C = sin C/cos C, the given expression can be simplified as:

sec C + sin C sec C = sec C + sin C × (1/cos C) = sec C + sin C/cos C = sec C + tan C

Given that ∠C = 45°, we substitute this value into the simplified expression:

sec 45° + tan 45° = √2 + 1 = 1 + √2

Explanation

Given the equation: 3y + 3/y = 8

Dividing both sides by 3 gives: y + 1/y = 8/3

Using the algebraic identity (a + b)² = a² + 2ab + b², squaring both sides of the equation yields: (y + 1/y)² = (8/3)² y² + 2 + 1/y² = 64/9

Subtracting 2 from both sides to isolate the term y² + 1/y²: y² + 1/y² = 64/9 - 2 y² + 1/y² = 46/9

Converting the improper fraction to a mixed fraction: 46/9 = 5 1/9

Explanation

The length of an arc of a circle is given by the formula L = (θ / 360°) × 2πr, where θ is the angle subtended by the arc at the centre and r is the radius of the circle.

Substituting the given values, θ = 90°, r = 14 cm, and π = 22/7, we get:

L = (90° / 360°) × 2 × (22 / 7) × 14 L = ¼ × 2 × 22 × 2 L = ¼ × 88 L = 22 cm

Explanation

For any real coefficients a and b, the trigonometric identity relates the sine and cosine combinations as follows:

(a sin θ + b cos θ)² + (a cos θ - b sin θ)² = a²(sin² θ + cos² θ) + b²(sin² θ + cos² θ) = a² + b²

In this problem, a = 4 and b = 5. Let the required value of (4cos θ - 5sin θ) be x. Substituting the given values into the identity:

3² + x² = 4² + 5²

9 + x² = 16 + 25

9 + x² = 41

x² = 32

x = √32 = 4√2

Explanation

The total surface area of a solid hemisphere is given by the formula 3πr², where r is the radius of the hemisphere.

The diameter of the solid hemisphere is given as 14 cm, which means the radius r is 14 / 2 = 7 cm.

Substituting the values into the formula:

Total Surface Area = 3 × (22/7) × 7² Total Surface Area = 3 × (22/7) × 7 × 7 Total Surface Area = 3 × 22 × 7 Total Surface Area = 462 cm²

Explanation

Using the work equivalence relation: M₁ × D₁ × H₁ = M₂ × D₂ × H₂

Here: M₁ = 30 women, D₁ = 18 days, H₁ = 5 hours per day M₂ = 21 women, H₂ = 8 hours per day, and D₂ is the required number of days

Substituting these values into the formula: 30 × 18 × 5 = 21 × D₂ × 8

2700 = 168 × D₂

D₂ = 2700 / 168

Simplifying the fraction by dividing the numerator and the denominator by 12: D₂ = 225 / 14

Converting to a mixed fraction: D₂ = 16 1/14 days

Explanation

For a circle of diameter 10 cm, the radius R is 5 cm.

A perpendicular drawn from the center of the circle to a chord bisects the chord. Since the length of each chord is 6 cm, the half-length of each chord is 3 cm.

In the right-angled triangle formed by the radius R, the perpendicular distance from the center to the chord d, and the half-length of the chord:

R² = d² + 3² 5² = d² + 9 25 = d² + 9 d² = 16 d = 4 cm

Since the two parallel chords are of equal length and lie on opposite sides of the center, the total distance between them is the sum of their individual distances from the center:

Distance = d + d = 4 + 4 = 8 cm

Explanation

Let the two numbers be x and y.

We are given: x³ + y³ = 10234 x + y = 34

Using the algebraic identity for the sum of cubes: x³ + y³ = (x + y)((x + y)² - 3xy)

Substituting the given values: 10234 = 34 × (34² - 3xy) 10234 = 34 × (1156 - 3xy) 301 = 1156 - 3xy 3xy = 855 xy = 285

The square of the difference of the two numbers is: (x - y)² = (x + y)² - 4xy (x - y)² = 34² - 4 × 285 (x - y)² = 1156 - 1140 (x - y)² = 16 x - y = 4 (assuming x > y)

The positive difference between the cubes of the two numbers is: x³ - y³ = (x - y)(x² + xy + y²) x³ - y³ = (x - y)((x + y)² - xy)

Substituting the values: x³ - y³ = 4 × (34² - 285) x³ - y³ = 4 × (1156 - 285) x³ - y³ = 4 × 871 x³ - y³ = 3484

Explanation

Using the fundamental trigonometric identity cosec² θ - cot² θ = 1, we factor it as: (cosec θ - cot θ)(cosec θ + cot θ) = 1

Substituting the given value cosec θ - cot θ = 7/2 into this identity: (7/2)(cosec θ + cot θ) = 1 cosec θ + cot θ = 2/7

Adding the two equations cosec θ - cot θ = 7/2 and cosec θ + cot θ = 2/7: 2 cosec θ = 7/2 + 2/7 2 cosec θ = (49 + 4) / 14 2 cosec θ = 53/14 cosec θ = 53/28

Explanation

Correct Answer: D

To find the population of the city at the end of 2013, we apply the formula for successive percentage change:

P_final = Pᵢnitial × (1 + r₁/100) × (1 + r₂/100) × (1 - r₃/100)

Substituting the given values: Pᵢnitial = 12,50,000 r₁ = 2% (increase in 2011) r₂ = 2% (increase in 2012) r₃ = 1% (decrease in 2013)

P_final = 12,50,000 × (1 + 2/100) × (1 + 2/100) × (1 - 1/100) P_final = 12,50,000 × (102/100) × (102/100) × (99/100)

Simplifying the fractions: P_final = 12,50,000 × (51/50) × (51/50) × (99/100) P_final = 12,50,000 × (2601 / 2500) × (99 / 100) P_final = 12,50,000 × (2601 × 99) / 2,50,000

Dividing 12,50,000 by 2,50,000 gives 5: P_final = 5 × 2601 × 99 P_final = 5 × 257499 P_final = 12,87,495

Explanation

The weight of the new student can be found by adding the total increase in the weight of the entire class to the weight of the replaced student. Since the average weight of 30 students increased by 200 g (0.2 kg), the total weight increased by 30 multiplied by 0.2, which is 6 kg. Therefore, the weight of the new student is 48.5 kg plus 6 kg, which equals 54.5 kg.

Explanation

The perimeter of a sector of a circle is given by the formula: P = 2r + l

where r is the radius of the circle and l is the arc length of the sector.

Given that the radius r = 5 cm and the perimeter P = 15.5 cm, we can find the arc length l: 15.5 = 2 × 5 + l 15.5 = 10 + l l = 5.5 cm

The area of a sector A is related to its radius and arc length by the formula: A = ½ × r × l

Substituting the values of r and l into the formula: A = ½ × 5 × 5.5 A = 13.75 cm²

Explanation

The relation between the discount, discount percentage, and marked price is given by the formula:

Discount = (Discount Percentage / 100) × Marked Price

In this problem, the discount percentage is 30% and the actual discount amount received by Gunter is ₹1,110. Let the marked price of the gift be M.

Using the formula:

1,110 = (30 / 100) × M

To find M, rearrange the equation:

M = (1,110 × 100) / 30

M = 1,110 × (10 / 3)

Dividing 1,110 by 3 gives 370:

M = 370 × 10

M = 3,700

Explanation

For a number to be divisible by 12, it must be divisible by both 3 and 4.

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. The last two digits of the number 3x7440 are 40, which is divisible by 4 since 40 ÷ 4 = 10. Thus, the number is divisible by 4 for any digit x.

A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 3x7440 is: 3 + x + 7 + 4 + 4 + 0 = 18 + x

For this sum to be divisible by 3, the digit x must be a multiple of 3. The possible single-digit values for x are 0, 3, 6, and 9.

The least of these values is 0.

Explanation

Cost price of the item for S is ₹37,800.

S sells the item to P at an 8% profit, so the cost price for P is: 37,800 × (1 + 8/100) = 37,800 × 1.08 = ₹40,824

P then sells the item to Q at a 4% loss. The selling price for P is: 40,824 × (1 - 4/100) = 40,824 × 0.96 = ₹39,191.04

Rounding to the nearest rupee, the price at which P sells the item is ₹39,191.

Explanation

For an isosceles right-angled triangle with equal sides of length x, the hypotenuse is x√2.

The perimeter P of the triangle is given by the formula: P = 2x + x√2 = x(2 + √2)

Given that the perimeter is 30 m: x(2 + √2) = 30 x = 30 / (2 + √2)

Rationalizing the denominator: x = 30(2 - √2) / (4 - 2) = 15(2 - √2)

The area A of the isosceles right-angled triangle is given by: A = ½ × x² A = ½ × [15(2 - √2)]² A = ½ × 225 × (4 + 2 - 4√2) A = 225 × (3 - 2√2)

Using the approximation √2 ≈ 1.4142: A ≈ 225 × (3 - 2 × 1.4142) A ≈ 225 × (3 - 2.8284) A ≈ 225 × 0.1716 A ≈ 38.61 m²

Rounding 38.61 to the nearest integral value yields 39.

Explanation

First, calculate the cost price (CP) of the mixture using the selling price (SP) and the gain percentage: CP = (SP × 100) / (100 + Gain%)

Substituting the given values: CP = (55 × 100) / (100 + 10) = 5500 / 110 = ₹50 per kg

Now, apply the rule of alligation to find the ratio in which the two varieties of rice must be mixed: Ratio of quantity of dearer rice to cheaper rice = (Mean Price - Cost of Cheaper Rice) / (Cost of Dearer Rice - Mean Price)

Substituting the prices: Ratio = (50 - 45) / (52 - 50) = 5 / 2

Let x be the required quantity of rice costing ₹52 per kg. The quantity of rice costing ₹45 per kg is 35 kg. x / 35 = 5 / 2 x = (35 × 5) / 2 x = 175 / 2 = 87.5 kg

Explanation

The area of the square is given as s² = 30.25 cm². Taking the square root, the side of the square is s = √30.25 = 5.5 cm.

Since the wire is bent to form a square, the total length of the wire is equal to the perimeter of the square: Perimeter = 4s = 4 × 5.5 = 22 cm

When the same wire is bent to form a circle, the circumference of the circle must equal the length of the wire: 2πr = 22 2 × (22/7) × r = 22 r = 7/2 = 3.5 cm

Using the radius, the area of the circle is calculated as: Area = πr² = (22/7) × 3.5 × 3.5 = 38.50 cm²

Explanation

Let the marked price be MP and the cost price be CP.

The selling price (SP) is 3/7 of the marked price: SP = 3/7 × MP

Selling the article at this price results in a loss of 30%, which means: SP = CP × (1 - 0.30) = 7/10 × CP

Equating both expressions for SP: 3/7 × MP = 7/10 × CP

Rearranging the terms to find the ratio of MP to CP: MP / CP = (7/10) × (7/3) = 49 / 30

Explanation

Correct Answer: D

Using the successive percentage change formula, the net profit percentage is given by the formula: Net Percentage = a + b + (a × b)/100

Here, the markup percentage a is +60% and the discount percentage b is -20% because a discount reduces the price.

Substituting these values: Net Percentage = 60 - 20 + (60 × (-20))/100 Net Percentage = 40 - 12 Net Percentage = 28%

Alternatively, if the cost price is 100, the marked price is 160. A 20% discount on 160 is 32, which results in a selling price of 128, representing a 28% profit.

Explanation

Total work can be assumed as the Least Common Multiple (LCM) of 18 and 24, which is 72 units.

Efficiency of A = 72 / 18 = 4 units per day Efficiency of B = 72 / 24 = 3 units per day

Combined efficiency of A and B = 4 + 3 = 7 units per day

Work completed by both in 5 days = 5 × 7 = 35 units

Remaining work = 72 - 35 = 37 units

Fraction of work left = 37 / 72

Explanation

The cost price of 1000 gm of potato is ₹20. This means the cost price of 1 gm of potato is 20 / 1000 = ₹0.02.

Since Ranjeet uses a false weight of 800 gm instead of 1 kg, the actual cost of the potato he sells is the cost of 800 gm.

Actual cost price for Ranjeet = 800 × 0.02 = ₹16.

He sells this quantity for ₹25, which is his selling price.

Selling price = ₹25.

The actual profit earned is Selling Price - Actual Cost Price = 25 - 16 = ₹9.

The actual profit percentage is (Profit / Actual Cost Price) × 100 = (9 / 16) × 100 = 56.25%.

Explanation

Correct Answer: D

Using the trigonometric identity tan A = cot (90° - A), it is established that if tan A = cot B, then A + B = 90°.

Applying this relation to the given equation tan 2θ = cot (θ - 36°):

2θ + (θ - 36°) = 90°

3θ - 36° = 90°

3θ = 90° + 36°

3θ = 126°

θ = 42°

Since 2θ = 84°, which is less than 90°, the condition that 2θ is an acute angle is satisfied.

Explanation

The area of a triangle is given by the formula:

Area = ½ × base × height

In triangle ABC, the altitude BD is perpendicular to the base AC. The length of the base AC is the sum of the segments AD and CD:

AC = AD + CD = 6 + 4 = 10 cm

Using the height BD = 12 cm, the area is calculated as:

Area = ½ × 10 × 12 = 60 cm²

Explanation

Using the algebraic identity for the square of a trinomial: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substituting the given values: 17² = 115 + 2(ab + bc + ca) 289 = 115 + 2(ab + bc + ca) 2(ab + bc + ca) = 289 - 115 = 174

The expression to evaluate is: (a + b)² + (b + c)² + (c + a)²

Expanding each squared term gives: (a² + 2ab + b²) + (b² + 2bc + c²) + (c² + 2ca + a²) = 2(a² + b² + c²) + 2(ab + bc + ca)

Substituting the known values into this expression: 2(115) + 174 = 230 + 174 = 404

Explanation

In a right-angled triangle ABC with angle ABC = 90°, the side AC is the hypotenuse.

According to Pythagoras' theorem: AC² = AB² + BC²

Since the triangle is isosceles with AB = BC, the relation simplifies to: AC² = 2 BC²

Given that AC = 12 cm, substituting this value yields: 12² = 2 BC² 144 = 2 BC² BC² = 72 BC = √72 = 6√2 cm

Explanation

The difference between compound interest and simple interest for a period of 2 years is given by the formula:

Difference = P × (R / 100)²

where P is the principal amount and R is the annual rate of interest.

Substituting the given values: Principal, P = 1,56,250 Rate, R = 8%

Difference = 1,56,250 × (8 / 100)² Difference = 1,56,250 × (64 / 10,000) Difference = 15.625 × 64 Difference = 1,000

Explanation

Let a = 379 and b = 276.

The given expression can be written in algebraic form as: ((a + b)² + (a - b)²) / (a² + b²)

Using the standard algebraic identity: (a + b)² + (a - b)² = (a² + 2ab + b²) + (a² - 2ab + b²) = 2(a² + b²)

Substituting this identity back into the expression: 2(a² + b²) / (a² + b²) = 2

The terms (a² + b²) in the numerator and denominator cancel out, which simplifies the entire expression to 2.

Explanation

Apply the BODMAS rule to solve the given equation:

25(4 + 3p) ÷ 5 - 3 × 8 = 26

First, simplify the division term: 5(4 + 3p) - 3 × 8 = 26

Next, perform the multiplication: 5(4 + 3p) - 24 = 26

Add 24 to both sides of the equation: 5(4 + 3p) = 50

Divide both sides by 5: 4 + 3p = 10

Subtract 4 from both sides: 3p = 6

Divide by 3 to solve for p: p = 2

Explanation

Using the trigonometric identity tan(90° - θ) = cot θ, the given equation can be rewritten as:

cot θ = 2/√3

Since tan θ is the reciprocal of cot θ:

tan θ = √3/2

Substituting this value into the expression 2√3 tan θ + 1 gives:

2√3 × (√3/2) + 1 = 3 + 1 = 4

Explanation

The radius of the circle is half of its diameter, so OS = 16/2 = 8 cm.

A tangent to a circle is perpendicular to the radius at the point of contact. Therefore, the angle OSQ is 90°, making triangle OSQ a right-angled triangle with the hypotenuse OQ.

Applying the Pythagorean theorem to right-angled triangle OSQ: OQ² = OS² + QS²

Substituting the known values: 17² = 8² + QS² 289 = 64 + QS² QS² = 225 QS = 15 cm

Explanation

If three numbers a, b, and c are in continuous proportion, then the ratio of the first to the second is equal to the ratio of the second to the third, which is written as a/b = b/c. This gives the formula for the third proportional c as c = b² / a.

Substituting the given values, where a = 8 and b = 20, we get: c = 20² / 8 c = 400 / 8 c = 50

Explanation

The weighted average formula for two groups is:

Weighted Average = (n₁ × A₁ + n₂ × A₂) / (n₁ + n₂)

Here, the first group has n₁ = 25 students with an average score of A₁ = 60. The remaining group has n₂ = 40 - 25 = 15 students with an average score of A₂ = 80.

Substituting these values into the formula:

Weighted Average = (25 × 60 + 15 × 80) / 40

Weighted Average = (1500 + 1200) / 40

Weighted Average = 2700 / 40 = 67.5

Explanation

If four numbers a, b, c, and d are in proportion, then a/b = c/d. For the given numbers 8, P, 16, and 18:

8/P = 16/18

Simplifying the fraction on the right side:

8/P = 8/9

This gives P = 9.

The third proportion of two numbers a and b is a third number x such that a/b = b/x, which gives the formula x = b²/a.

To find the third proportion of P and 12, substitute P = 9 and b = 12 into the formula:

x = 12²/9

x = 144/9

x = 16

Explanation

To find the plant with the maximum average production, we calculate the total production for each plant over the five-year period from 2018 to 2022. Since the number of years is constant, the plant with the highest total production also has the highest average production.

The total production for Plant 4 is: 45 + 40 + 30 + 48 + 42 = 205 thousand

The average production for Plant 4 is: 205 / 5 = 41 thousand

Comparing the total production of all plants, Plant 1 has 200 thousand, Plant 2 has 192 thousand, and Plant 3 has 185 thousand. Plant 4 has the highest total of 205 thousand, which yields the maximum average production.

Explanation

The mean of a grouped frequency distribution is calculated using the formula:

Mean = ∑(x × f) / ∑f

where x represents the age in years and f represents the number of children (frequency).

First, calculate the total number of children (∑f): ∑f = 8 + 3 + 7 + 2 + 20 = 40

Next, calculate the sum of the product of age and number of children (∑(x × f)): ∑(x × f) = (5 × 8) + (6 × 3) + (7 × 7) + (8 × 2) + (9 × 20) ∑(x × f) = 40 + 18 + 49 + 16 + 180 = 303

Now, divide the sum of the products by the total number of children to find the mean age: Mean = 303 / 40 = 7.575

Rounding to two decimal places gives 7.57.

Explanation

Using the algebraic identity x² - y² = (x - y)(x + y), let x = 2a - 3b - c and y = a + 2b + c.

First, calculate x - y: x - y = (2a - 3b - c) - (a + 2b + c) x - y = a - 5b - 2c

Next, calculate x + y: x + y = (2a - 3b - c) + (a + 2b + c) x + y = 3a - b

Now, multiply the two resulting factors: (x - y)(x + y) = (a - 5b - 2c)(3a - b) = a(3a - b) - 5b(3a - b) - 2c(3a - b) = 3a² - ab - 15ab + 5b² - 6ac + 2bc = 3a² + 5b² - 16ab + 2bc - 6ac

Explanation

The speed of a boat in still water (x) is calculated using the formula:

x = (d + u) / 2

where d is the downstream speed and u is the upstream speed.

Given: Downstream speed, d = 12 km / 1 h = 12 km/h Upstream speed, u = 8 km / 1 h = 8 km/h

Substituting these values into the formula:

x = (12 + 8) / 2 = 20 / 2 = 10 km/h

Explanation

Using the fundamental trigonometric identity: sec² θ - tan² θ = 1

This can be factored as: (sec θ - tan θ)(sec θ + tan θ) = 1

Given that sec θ + tan θ = 5, substituting this value gives: sec θ - tan θ = 1/5

Adding these two equations: (sec θ + tan θ) + (sec θ - tan θ) = 5 + 1/5 2 sec θ = 26/5 sec θ = 13/5

Subtracting the two equations: (sec θ + tan θ) - (sec θ - tan θ) = 5 - 1/5 2 tan θ = 24/5 tan θ = 12/5

Using the trigonometric relation sin θ = tan θ / sec θ: sin θ = (12/5) / (13/5) = 12/13

Explanation

The net profit percentage resulting from a markup followed by a discount can be calculated directly using the successive percentage formula:

Net Change = a + b + (a × b)/100

Here, the price is first increased by 30% (a = 30) and then decreased by a discount of 15% (b = -15).

Substituting these values into the formula:

Net Change = 30 - 15 + (30 × -15)/100 Net Change = 15 - 4.5 Net Change = 10.5%

Explanation

The formula for the curved surface area of a cone is: CSA = π × r × l

Let the radius r = 5x and the slant height l = 7x, where x is a common multiplier.

Substitute the given values into the formula: 1347.5 = (22/7) × 5x × 7x 1347.5 = 22 × 5 × x² 1347.5 = 110x²

Solve for x²: x² = 1347.5 / 110 x² = 12.25

Take the square root of both sides to find x: x = √12.25 x = 3.5

Calculate the radius r: r = 5x = 5 × 3.5 = 17.5 cm

Explanation

Given the quadratic equation x² - 3.2x + 1 = 0, dividing both sides by x (since x > 1, x ≠ 0) yields: x + 1/x = 3.2

Using the algebraic identity (x - 1/x)² = (x + 1/x)² - 4, we substitute the value of x + 1/x: (x - 1/x)² = 3.2² - 4 (x - 1/x)² = 10.24 - 4 (x - 1/x)² = 6.24

Since x > 1, it follows that x > 1/x, which means x - 1/x must be positive: x - 1/x = √6.24 x - 1/x = √(16 × 0.39) x - 1/x = 4√0.39

Now, we calculate the required expression using the difference of squares: x² - 1/x² = (x + 1/x)(x - 1/x) x² - 1/x² = 3.2 × 4√0.39 x² - 1/x² = 12.8√0.39

Explanation

The simple interest earned is the difference between the total amount and the principal sum: SI = Amount - Principal = 15000 - 12000 = 3000

The formula for simple interest is: SI = (P × R × T) / 100

Substituting the values into the formula: 3000 = (12000 × R × 5) / 100 3000 = 600 × R R = 3000 / 600 R = 5

Explanation

Using the rule of alligation, the ratio in which the two metals must be mixed is determined by comparing their individual densities with the mean density of the mixture.

Let the relative density of metal A be 19, the relative density of metal B be 9, and the desired relative density of the mixture be 15.

The formula of alligation gives: Ratio of metal A to metal B = (Mean density - Density of metal B) / (Density of metal A - Mean density)

Substituting the given values: Ratio of A to B = (15 - 9) / (19 - 15) Ratio of A to B = 6 / 4 Ratio of A to B = 3 / 2

This simplifies to the ratio 3:2.

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)

Given that the profit percentage P is 20%, the discount percentage D is 10%, and the cost price CP is 540, we substitute these values into the formula:

MP / 540 = (100 + 20) / (100 - 10)

MP / 540 = 120 / 90

MP / 540 = 4 / 3

MP = 540 × 4 / 3

MP = 180 × 4

MP = 720

Explanation

Since ΔPQR ≅ ΔMNY, their corresponding sides and angles are equal.

By the corresponding parts of congruent triangles: NM = PQ = 8 cm ∠M = ∠P = 72° ∠N = ∠Q = 55° ∠Y = ∠R

Using the angle sum property of a triangle for ΔPQR: ∠P + ∠Q + ∠R = 180° 72° + 55° + ∠R = 180° 127° + ∠R = 180° ∠R = 180° - 127° = 53°

Since ∠Y = ∠R, we get ∠Y = 53°.