SSC CHSL Tier 1 2015 Quantitative Aptitude Questions with Solutions

Explanation

The single discount equivalent to a series of successive discounts can be calculated by finding the final remaining percentage of the price. If the initial price is 100%, the remaining percentage after successive discounts of d₁%, d₂%, and d₃% is given by:

Remaining Percentage = 100 × (1 - d₁/100) × (1 - d₂/100) × (1 - d₃/100)

Substituting the given values of 20%, 10%, and 5%:

Remaining Percentage = 100 × (1 - 0.20) × (1 - 0.10) × (1 - 0.05) Remaining Percentage = 100 × 0.80 × 0.90 × 0.95 Remaining Percentage = 100 × 0.72 × 0.95 Remaining Percentage = 72 × 0.95 Remaining Percentage = 68.4%

The equivalent single discount is the total reduction from the original 100%:

Equivalent Single Discount = 100% - 68.4% = 31.6%

Explanation

For any two numbers with sum S and difference D, their product P can be directly calculated using the algebraic relation:

P = (S² - D²)/4

Substituting the given values, S = 75 and D = 25:

P = (75² - 25²)/4

Using the difference of squares identity, a² - b² = (a + b)(a - b):

P = (75 + 25)(75 - 25)/4 P = (100 × 50)/4 P = 5000/4 P = 1250

Explanation

Simplifying each term of the given expression:

First term: 2 - 1/3 = (6 - 1)/3 = 5/3 Second term: 2 - 3/5 = (10 - 3)/5 = 7/5 Third term: 2 - 5/7 = (14 - 5)/7 = 9/7 ... Last term: 2 - 997/999 = (1998 - 997)/999 = 1001/999

Substituting these simplified fractions back into the product:

(5/3) × (7/5) × (9/7) × ... × (1001/999)

This forms a telescoping product where the numerator of each fraction cancels with the denominator of the succeeding fraction: The numerator 5 of the first term cancels with the denominator 5 of the second term. The numerator 7 of the second term cancels with the denominator 7 of the third term. This pattern continues until the denominator 999 of the last term is canceled by the numerator of the preceding term.

After all cancellations, only the denominator of the first term (3) and the numerator of the last term (1001) remain.

Thus, the final simplified value of the expression is 1001/3.

Explanation

For annual compounding over 2 years at an interest rate of 18% per annum: Effective rate of interest = 18 + 18 + (18 × 18)/100 = 39.24%

For half-yearly compounding over 2 years, the interest is compounded every six months at a rate of 9% per half-year for 4 compounding periods. Effective rate after the first year (2 periods of 9%) = 9 + 9 + (9 × 9)/100 = 18.81% Effective rate after two years (2 periods of 18.81%) = 18.81 + 18.81 + (18.81 × 18.81)/100 = 37.62 + 3.538161 = 41.158161%

The difference between the two effective rates is: 41.158161% - 39.24% = 1.918161%

This difference in interest corresponds to 960: Principal × 1.918161% = 960 Principal = (960 × 100) / 1.918161 ≈ 50000

Explanation

Let the six numbers arranged in decreasing order be a, b, c, d, e, and f.

The sum of the first five numbers is calculated by multiplying their average by 5: a + b + c + d + e = 5 × 30 = 150

Similarly, the sum of the last five numbers is: b + c + d + e + f = 5 × 25 = 125

Subtracting the second equation from the first equation eliminates the common terms b, c, d, and e: (a + b + c + d + e) - (b + c + d + e + f) = 150 - 125 a - f = 25

The difference between the first number and the last number is 25.

Explanation

For ₹1, 4 bananas are bought. Cost Price (CP) of 1 banana = ₹¼

The desired gain is 33 ⅓% = 1/3.

Selling Price (SP) of 1 banana = CP × (1 + gain) SP = ¼ × (1 + 1/3) SP = ¼ × 4/3 SP = ₹⅓

Since the selling price of 1 banana is ₹⅓, the number of bananas that must be sold for ₹1 is: 1 / (1/3) = 3

Explanation

The initial ratio of milk to water in the 20-liter mixture is 3 : 1. The total parts of the ratio are 3 + 1 = 4. The initial quantity of water is (1 / 4) × 20 = 5 liters. The initial quantity of milk is (3 / 4) × 20 = 15 liters.

Since only milk is added to the mixture, the quantity of water remains constant at 5 liters. Let x be the amount of milk added to the mixture. The new ratio of milk to water is 4 : 1, which gives the equation: (15 + x) / 5 = 4 / 1 15 + x = 20 x = 5 liters

Explanation

In a cyclic trapezium ABCD where sides AD and BC are parallel, consecutive interior angles along the transversal AB are supplementary, which gives ∠DAB + ∠ABC = 180°.

Additionally, in any cyclic quadrilateral, the opposite angles are supplementary, which gives ∠DAB + ∠BCD = 180°.

By comparing these two equations, we obtain ∠BCD = ∠ABC.

Given that ∠ABC = 75°, it follows directly that ∠BCD = 75°.

Explanation

Simplify the expression by rationalizing the denominator: 1/√28 = 1/√(4 × 7) = 1/(2√7)

Multiply the numerator and denominator by √7 to simplify the division: 1/(2√7) × √7/√7 = √7/(2 × 7) = √7/14

Substitute the given value of √7 = 2.646: 2.646 / 14 = 0.189

Explanation

Multiplying the given equation by 2 on both sides: 2(a² + b² + c² - ab - bc - ca) = 0 (a² - 2ab + b²) + (b² - 2bc + c²) + (c² - 2ca + a²) = 0 (a - b)² + (b - c)² + (c - a)² = 0

Since the sum of squares of real numbers is zero if and only if each individual term is zero: a - b = 0 → a = b b - c = 0 → b = c c - a = 0 → c = a

This gives a = b = c, which yields the ratio a:b:c as 1:1:1.

Explanation

X can complete the entire work in 24 days, meaning X's rate of work is 1/24 of the work per day.

The complete work was finished in 16 days. Since X worked from the beginning to the end, X worked for the entire 16 days.

The fraction of work completed by X in 16 days is: 16 × 1/24 = 2/3

Y joined X after 4 days, which means Y worked for: 16 - 4 = 12 days

The remaining fraction of the work completed by Y is: 1 - 2/3 = 1/3

Since Y completes 1/3 of the work in 12 days, the number of days Y alone would take to finish the complete work is: 12 ÷ 1/3 = 36 days

Explanation

The total area of a hemispherical bowl to be painted on both the inside and the outside is equal to the sum of its inner and outer curved surface areas.

Inner curved surface area = 2πr² Outer curved surface area = 2πr² Total surface area to be painted = 4πr²

Given that the radius r = 3.5 cm: Total surface area = 4 × (22/7) × 3.5 × 3.5 = 154 sq. cm.

The cost of painting is ₹5 per 10 sq. cm, which means the rate is 5/10 = ₹0.5 per sq. cm.

Total cost of painting = 154 × 0.5 = ₹77.

Explanation

The ratio of the circumference to the diameter of a circle is given as 22/7.

The circumference C is given as 1 4/7 m, which is equal to 11/7 m.

Using the relation C = (22/7) × d, where d is the diameter of the circle:

11/7 = (22/7) × d

Solving for d:

d = (11/7) × (7/22) = 11/22 = ½ m

The radius r of the circle is half of its diameter:

r = d/2 = (½) / 2 = ¼ m

Explanation

Using the standard algebraic identity: a⁴ + a² + 1 = (a² + a + 1)(a² - a + 1)

Given that a² + a + 1 = 0, substituting this value into the identity yields: a⁴ + a² + 1 = 0 × (a² - a + 1) = 0

Explanation

The expression inside the outer square root simplifies to the square root of 6, which is 6 to the power of 1/2. Taking the square root of this value yields 6 to the power of 1/4.

Explanation

Let the total number of eggs bought by the man be x.

Since 10% of the eggs are rotten, the remaining good eggs are 90% of x, which is 0.9x.

The man gives 80% of this remainder to his neighbours, meaning he retains 20% of the remainder.

According to the given information, the number of eggs left with him is 36. We can set up the equation: 20% of 0.9x = 36 0.2 × 0.9x = 36 0.18x = 36 x = 36 / 0.18 = 200

Explanation

Applying componendo and dividendo to the first equation: ((x + 1) + (x - 1)) / ((x + 1) - (x - 1)) = (a + b) / (a - b) 2x / 2 = (a + b) / (a - b) x = (a + b) / (a - b)

Applying componendo and dividendo to the second equation: ((1 - y) + (1 + y)) / ((1 - y) - (1 + y)) = (b + a) / (b - a) 2 / (-2y) = (a + b) / (b - a) -1 / y = -(a + b) / (a - b) y = (a - b) / (a + b)

Multiplying x and y: xy = ((a + b) / (a - b)) × ((a - b) / (a + b)) = 1

Thus, the denominator is: 1 + xy = 1 + 1 = 2

The numerator is: x - y = (a + b) / (a - b) - (a - b) / (a + b) x - y = ((a + b)² - (a - b)²) / ((a - b)(a + b)) x - y = 4ab / (a² - b²)

Substituting these into the given expression: (x - y) / (1 + xy) = (4ab / (a² - b²)) / 2 = 2ab / (a² - b²)

Explanation

Let the actual bill amount be B.

The discount offered is 15% of the actual bill, and the value of this discount is given as ₹54.

The relationship can be represented by the formula: Discount = (Discount Percentage / 100) × Actual Bill

Substituting the given values into the formula: 54 = (15 / 100) × B

Rearranging the equation to solve for B: B = (54 × 100) / 15

Simplifying the fraction by dividing both 54 and 15 by their common divisor 3: B = (18 × 100) / 5

B = 18 × 20

B = 360

Explanation

Given the trigonometric relation: cosec θ = cot² θ

Using the standard identity cot² θ = cosec² θ - 1, we can substitute it into the given equation: cosec θ = cosec² θ - 1 cosec² θ - cosec θ - 1 = 0

Letting x = cosec θ, we get the quadratic relation: x² - x - 1 = 0

We want to find the value of the expression: E = cosec⁴ θ - 2 cosec³ θ + cot² θ

Since cot² θ = cosec θ = x, we can rewrite the expression in terms of x: E = x⁴ - 2x³ + x

We can factor this polynomial by grouping terms or performing algebraic division with x² - x - 1: E = x²(x² - x - 1) - x³ + x² + x E = x²(x² - x - 1) - x(x² - x - 1) E = (x² - x)(x² - x - 1)

Substituting x² - x - 1 = 0 into the factored expression: E = (x² - x) × 0 = 0

Explanation

Using the algebraic identity: x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - (xy + yz + zx))

First, find the value of x² + y² + z² using the identity: (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

Substitute the given values into this equation: 6² = x² + y² + z² + 2(10) 36 = x² + y² + z² + 20 x² + y² + z² = 16

Now, substitute these values back into the main identity: x³ + y³ + z³ - 3xyz = 6 × (16 - 10) x³ + y³ + z³ - 3xyz = 6 × 6 = 36

Explanation

The sum of the 12 numbers is calculated as 12 × 15 = 180.

The sum of the first two numbers is 2 × 14 = 28.

The sum of the remaining 10 numbers is the difference between the total sum and the sum of the first two numbers, which is 180 - 28 = 152.

The average of these remaining 10 numbers is 152 / 10 = 15.2.

Expressing 15.2 as a mixed fraction gives 15 ⅕.

Explanation

To carve out a right circular cone from a cube of edge a wasting the minimum amount of wood, the height and the base diameter of the cone must both be equal to the edge of the cube.

Given the edge of the cube is 4.2 dm: Height of the cone (h) = 4.2 dm Radius of the base of the cone (r) = 4.2 / 2 = 2.1 dm

The formula for the volume of a right circular cone is: V = ⅓ × π × r² × h

Substituting the values into the formula using π = 22/7: V = ⅓ × (22/7) × (2.1)² × 4.2 V = ⅓ × (22/7) × 4.41 × 4.2 V = 13.86 × 1.4 V = 19.404 cu.dm

Explanation

To find the total distance, we use the formula for average speed when a journey is split into two equal distance halves:

Average Speed = 2 × v₁ × v₂ / (v₁ + v₂)

Here, v₁ = 60 km/h and v₂ = 45 km/h.

Average Speed = 2 × 60 × 45 / (60 + 45) = 5400 / 105 = 360/7 km/h

The total time of travel is 5 hours and 15 minutes, which is equal to 5 + 15/60 = 21/4 hours.

Using the distance formula: Total Distance = Average Speed × Total Time

Total Distance = (360/7) × (21/4) = 90 × 3 = 270 km

Explanation

For any triangle, the sum of internal angles is A + B + C = 180°. Dividing this equation by 2, we obtain (A + B)/2 = 90° - C/2. Taking the tangent of this angle gives tan((A + B)/2) = tan(90° - C/2) = cot(C/2). Since cot(C/2) is not mathematically equivalent to sec(C/2) for general angles, the relation tan((A + B)/2) = sec(C/2) is incorrect.

Explanation

Real numbers are divided into two main categories: rational numbers and irrational numbers. An irrational number, such as √2 or π, is a real number that cannot be written in the form of a simple fraction p/q, where p and q are integers and q is not equal to 0. Consequently, not all real numbers are rational.

Explanation

The plate has a square base of length x cm and a thickness (width) of 1 mm, which is equal to 0.1 cm.

The volume of the plate is: Volume = length × breadth × thickness = x × x × 0.1 = 0.1x² cubic cm

The weight of the plate is calculated using the relation: Weight = Volume × Density

Substituting the given values: 4725 = 0.1x² × 8.4 4725 = 0.84x² x² = 4725 / 0.84 x² = 5625 x = √5625 x = 75

Explanation

The number x leaves a remainder of 3 when divided by 8, 12, and 16. Therefore, the general form of x is given by the formula: x = k × LCM(8, 12, 16) + 3

To find the LCM of 8, 12, and 16, we look at their prime factorization: 8 = 2³ 12 = 2² × 3 16 = 2⁴

Thus, LCM(8, 12, 16) = 2⁴ × 3 = 48.

Substituting this into the general formula: x = 48k + 3

We are given that x is divisible by 7. Using modular arithmetic: 48k + 3 ≡ 0 (mod 7)

Since 48 leaves a remainder of 6 (or -1) when divided by 7: -k + 3 ≡ 0 (mod 7) k ≡ 3 (mod 7)

The least positive integer value for k is 3. Substituting k = 3 back into the expression for x: x = 48 × 3 + 3 = 144 + 3 = 147

Explanation

Let the total maximum marks be T. The difference between the two marks percentages is 32% - 20% = 12%. The difference in the actual marks obtained in these two scenarios is 42 - (-30) = 72 marks. Since 12% of the total marks is equivalent to 72 marks, 1% of the total marks is equal to 72 / 12 = 6 marks. The first candidate obtained 20% marks and failed by 30 marks. Converting this failing margin of 30 marks into a percentage gives 30 / 6 = 5%. Therefore, the minimum pass percentage is 20% + 5% = 25%.

Explanation

To convert a pure recurring decimal to a vulgar fraction, write the repeating digits in the numerator and place as many nines in the denominator as there are repeating digits.

The recurring decimal is 0.3939... where the block of two digits 39 repeats infinitely.

This can be expressed as: 39/99

Dividing both the numerator and denominator by their greatest common divisor, 3, simplifies the fraction: 39 ÷ 3 = 13 99 ÷ 3 = 33

This gives the simplified vulgar fraction as 13/33.

Explanation

Using standard trigonometric identities, we can simplify each term of the given expression:

First, we know that 1/csc²θ = sin²θ, so the first term becomes 9sin²θ.

Second, the identity 1 + tan²θ = sec²θ simplifies the denominator of the third term. Since 1/sec²θ = cos²θ, the third term 5/(1 + tan²θ) simplifies to 5cos²θ.

Substituting these back into the expression:

9sin²θ + 4cos²θ + 5cos²θ

Combining the cosine terms:

9sin²θ + 9cos²θ

Factoring out 9:

9(sin²θ + cos²θ)

Since sin²θ + cos²θ = 1, the expression simplifies to:

9 × 1 = 9

Explanation

The measure of each interior angle of a regular polygon with n sides is given by the formula ((n - 2) × 180°) / n. For a regular hexagon, the number of sides n is 6. Substituting this value into the formula:

((6 - 2) × 180°) / 6 = (4 × 180°) / 6 = 720° / 6 = 120°

Alternatively, the sum of all exterior angles of any polygon is 360°. For a regular hexagon, each exterior angle is 360° / 6 = 60°. Since the interior and exterior angles at each vertex form a linear pair, each interior angle is 180° - 60° = 120°.

Explanation

The volume of a regular tetrahedron with edge length a is given by the formula:

V = (a³ × √2) / 12

Substituting the given value of edge length a = 1 cm into the formula:

V = (1³ × √2) / 12 = (1/12) × √2 cu.cm

Explanation

From the given equations: x/a = b - c y/b = c - a z/c = a - b

Let p = x/a, q = y/b, and r = z/c.

Adding these three variables: p + q + r = (b - c) + (c - a) + (a - b) = 0

According to the algebraic identity, if the sum of three terms is zero (p + q + r = 0), then the sum of their cubes is equal to three times their product: p³ + q³ + r³ = 3pqr

Substituting the values of p, q, and r back into the identity: (x/a)³ + (y/b)³ + (z/c)³ = 3(x/a)(y/b)(z/c) = 3xyz/abc

Explanation

In a right-angled triangle, the cosine function is defined as: cos θ = Base / Hypotenuse = p / √(p² + q²)

This identifies: Base = p Hypotenuse = √(p² + q²)

Using Pythagoras' theorem: Hypotenuse² = Base² + Perpendicular² (√(p² + q²))² = p² + Perpendicular² p² + q² = p² + Perpendicular² Perpendicular² = q² Perpendicular = q

The tangent function is defined as: tan θ = Perpendicular / Base = q / p

Explanation

Given the equation: √y = 4x

Squaring both sides of the equation yields: y = 16x²

To find the value of x²/y, substitute y = 16x² into the expression: x²/y = x² / 16x²

Dividing the numerator and the denominator by x² gives: x²/y = 1/16

Explanation

A rhombus has all four sides equal in length, so AB = BC. This makes triangle ABC an isosceles triangle, which means the angles opposite to those equal sides are also equal, so ∠BAC = ∠BCA.

The sum of angles in triangle ABC is 180°. Given ∠ABC = 60°, the angle sum calculation is: ∠ABC + ∠BAC + ∠BCA = 180° 60° + 2 × ∠BAC = 180° 2 × ∠BAC = 120° ∠BAC = 60°

Since all angles in triangle ABC are equal to 60°, triangle ABC is an equilateral triangle. In an equilateral triangle, all three sides are equal, which means AB = BC = AC. Because the diagonal AC is given as 6 cm, the length of each side of the rhombus is also 6 cm.

Explanation

Let the two trains meet after a time duration of t hours.

The distance covered by the faster train starting from point A is: d₁ = 75t

The distance covered by the slower train starting from point B is: d₂ = 50t

The difference in the distance travelled by the two trains is given as 175 km: d₁ - d₂ = 175 75t - 50t = 175 25t = 175 t = 7 hours

The total distance between A and B is the sum of the distances covered by both trains during this time: Total Distance = d₁ + d₂ Total Distance = 75t + 50t Total Distance = 125t Total Distance = 125 × 7 = 875 km

Explanation

For two successive discounts of d₁% and d₂%, the single equivalent discount is calculated using the formula:

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

Substituting d₁ = 20 and d₂ = 10 into the formula:

Equivalent Discount = (20 + 10 - (20 × 10)/100)% Equivalent Discount = 30 - 2 Equivalent Discount = 28%

Explanation

At the beginning of Year 1, Rahim deposits x. At the end of Year 1, with 5% interest, the accumulated amount is: x × (1 + 5/100) = 1.05x = 21/20 x

At the beginning of Year 2, another x is deposited. The total principal becomes: 21/20 x + x = 41/20 x

At the end of Year 2, this total principal grows with 5% interest to: 41/20 x × 21/20 = 861/400 x

At the beginning of Year 3, another x is deposited. The total principal becomes: 861/400 x + x = 1261/400 x

At the end of Year 3, this total principal grows with 5% interest to: 1261/400 x × 21/20 = 26481/8000 x

Explanation

The percentage 66⅔% can be written as the fraction 2/3.

First, calculate 66⅔% of 312: (2/3) × 312 = 2 × 104 = 208

Next, find the amount by which this value exceeds 200: 208 - 200 = 8

Explanation

To find the time taken when working together, we calculate their combined daily work rate. With B completing the work in 6 days, the individual daily rates are:

Daily work of A = 1/24 Daily work of B = 1/6 Daily work of C = 1/12

Combined daily work = 1/24 + 1/6 + 1/12 = 1/24 + 4/24 + 2/24 = 7/24

The time taken to complete the work together is the reciprocal of this combined rate:

24/7 = 3 3/7 days

Explanation

Let the daily work efficiency of B be x. Since A does three times the work of B in one day, the daily work efficiency of A is 3x.

Together, their combined daily efficiency is 3x + x = 4x.

They complete 2/5 of the work in 9 days. This gives the relation: 9 × 4x = 2/5 36x = 2/5 x = 2 / (5 × 36) x = 1/90

Since B's one-day work is 1/90, the total number of days B needs to complete the work alone is 1 / (1/90) = 90 days.

Explanation

Let the length and breadth of the rectangular base be 3x and 2x respectively. The height of the prism is h = 12 cm.

The total surface area of a right prism is given by the formula: Total Surface Area = 2 × Base Area + Perimeter of Base × height

The base is a rectangle, so: Base Area = length × breadth = 3x × 2x = 6x² Perimeter of Base = 2(length + breadth) = 2(3x + 2x) = 10x

Substitute these values into the total surface area formula: 288 = 2(6x²) + 10x × 12 288 = 12x² + 120x

Dividing the entire equation by 12: x² + 10x - 24 = 0 (x + 12)(x - 2) = 0

Since the dimensions must be positive, we get x = 2.

Using x = 2, we find the dimensions of the base: Length = 3 × 2 = 6 cm Breadth = 2 × 2 = 4 cm Base Area = 6 × 4 = 24 cm²

The volume of a prism is given by: Volume = Base Area × height Volume = 24 × 12 = 288 cm³

Explanation

Correct Answer: B

Squaring both sides of the given equation x/y = (a + 2) / (a - 2) yields:

x²/y² = (a + 2)² / (a - 2)²

Expanding the terms on the right side:

x²/y² = (a² + 4a + 4) / (a² - 4a + 4)

Applying the componendo and dividendo rule to find (x² - y²) / (x² + y²):

(x² - y²) / (x² + y²) = ((a² + 4a + 4) - (a² - 4a + 4)) / ((a² + 4a + 4) + (a² - 4a + 4))

Simplifying the numerator and the denominator:

(x² - y²) / (x² + y²) = 8a / (2a² + 8)

Factoring out 2 from both the numerator and the denominator:

(x² - y²) / (x² + y²) = 4a / (a² + 4)

Explanation

By the rule of alligation, the ratio of the quantities sold at 8% profit and 18% profit is determined by comparing their individual profit percentages to the overall profit percentage.

Ratio of quantity sold at 8% profit to quantity sold at 18% profit is: (18 - 14) : (14 - 8) = 4 : 6 = 2 : 3

The sum of the ratio parts is 2 + 3 = 5.

The quantity sold at 8% profit is: (2 / 5) × 1000 = 400 kg

Explanation

To find the quotient when 10¹⁰⁰ is divided by 5⁷⁵, express the division as:

Quotient = 10¹⁰⁰ ÷ 5⁷⁵

Since 10 can be factored as 2 × 5, we can rewrite the numerator using the laws of exponents:

10¹⁰⁰ = (2 × 5)¹⁰⁰ = 2¹⁰⁰ × 5¹⁰⁰

Substituting this back into the division gives:

Quotient = (2¹⁰⁰ × 5¹⁰⁰) ÷ 5⁷⁵

Using the division rule of exponents, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, simplify the powers of 5:

Quotient = 2¹⁰⁰ × 5¹⁰⁰⁻⁷⁵ Quotient = 2¹⁰⁰ × 5²⁵

To match the structure of the options, split 2¹⁰⁰ into 2⁷⁵ × 2²⁵:

Quotient = 2⁷⁵ × 2²⁵ × 5²⁵

Using the multiplication rule of exponents for different bases with the same power, aⁿ × bⁿ = (a × b)ⁿ, combine 2²⁵ and 5²⁵:

Quotient = 2⁷⁵ × (2 × 5)²⁵ Quotient = 2⁷⁵ × 10²⁵

Explanation

The ratio of syrup to water is 3:1, which means there are 3 parts of syrup out of a total of 4 parts in the mixture. Calculating the percentage, we get (3 / 4) * 100 = 75%. Therefore, the percentage of syrup in the mixture is 75%.

Explanation

Given that x = y = z, we can substitute y and z with x in both the numerator and the denominator of the expression.

Substituting into the numerator: (x + y + z)² = (x + x + x)² = (3x)² = 9x²

Substituting into the denominator: x² + y² + z² = x² + x² + x² = 3x²

Dividing the numerator by the denominator: 9x² / 3x² = 3

Explanation

The cost price of the article is ₹1500. Since it is sold at 25% above the cost price, the gross profit is calculated as: Gross Profit = 25% of ₹1500 = 25/100 × 1500 = ₹375

The man must pay ₹75 as tax on the article, which directly reduces his earnings. This gives the net profit: Net Profit = Gross Profit - Tax = ₹375 - ₹75 = ₹300

The net profit percentage is determined relative to the original cost price: Net Profit Percentage = (Net Profit / Cost Price) × 100 = (300 / 1500) × 100 = 20%

Explanation

Given the equation a/b + b/a = 2, we can clear the denominators by multiplying both sides by ab:

a² + b² = 2ab

Rearranging the terms by bringing 2ab to the left side:

a² - 2ab + b² = 0

The expression on the left is a perfect square, which simplifies to:

(a - b)² = 0

Taking the square root of both sides gives:

a - b = 0