SSC CGL Tier 1 2014 Quantitative Aptitude Questions with Solutions

Explanation

To determine the value of k for which x = y, we equate the given expressions:

k³ - 3k² = 1 - 3k

Rearranging the terms to one side of the equation yields:

k³ - 3k² + 3k - 1 = 0

The left side of the equation matches the standard algebraic identity for the cube of a binomial: (a - b)³ = a³ - 3a²b + 3ab² - b³. By comparing the terms, we identify a = k and b = 1, allowing the equation to be written as:

(k - 1)³ = 0

Taking the cube root of both sides gives k - 1 = 0, which simplifies to k = 1.

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) when both a discount percentage and a profit percentage are involved is given by the formula: (CP)/(MP) = (100 - Discount%)/(100 + Profit%)

Given that the Marked Price is 500, the discount is 5%, and the profit is 25%, we substitute these values into the ratio: (CP)/(500) = (100 - 5)/(100 + 25) (CP)/(500) = ⁹⁵⁄₁₂₅

To solve for the Cost Price, we isolate CP by moving 500 to the other side of the equation: CP = ⁹⁵⁄₁₂₅ × 500 CP = 95 × 4 CP = 380

Alternatively, the selling price can be calculated first by applying the 5% discount to the marked price, which gives 500 × 0.95 = 475. Since this selling price includes a 25% profit, the cost price is found by dividing the selling price by 1.25, resulting in 475 / 1.25 = 380.

Explanation

Let the total work be the Least Common Multiple (LCM) of the individual completion times. For A (12 days) and B (15 days), the total work is LCM(12, 15) = 60 units.

The efficiency of A is ⁶⁰⁄₁₂ = 5 units per day, and the efficiency of B is ⁶⁰⁄₁₅ = 4 units per day. Their combined efficiency when working together is 5 + 4 = 9 units per day.

In 5 days of working together, they complete 9 × 5 = 45 units of work. The remaining work is 60 - 45 = 15 units. Since B leaves, A must finish these 15 units alone. The time taken by A to finish the remaining work is (Remaining Work)/(A's Efficiency) = ¹⁵⁄₅ = 3 days.

Explanation

The expression can be simplified using the complementary angle identity cot(90° - θ) = tan θ. By identifying pairs of angles that sum to 90°, we can rewrite the terms as follows:

cot 81° = cot(90° - 9°) = tan 9° cot 63° = cot(90° - 27°) = tan 27°

Substituting these values into the original expression:

E = cot 9° cot 27° tan 27° tan 9°

Applying the fundamental identity cot θ tan θ = 1 for each pair of identical angles:

E = (cot 9° tan 9°) · (cot 27° tan 27°) = 1 · 1 = 1

Explanation

The given expression follows the algebraic identity where if a + b + c = 0, then a³ + b³ + c³ = 3abc. By setting a = 55, b = 17, and c = -72, we find that a + b + c = 55 + 17 - 72 = 0.

According to the identity, 55³ + 17³ + (-72)³ = 3(55)(17)(-72). Calculating the product on the right side: 3 × 55 × 17 × (-72) = 165 × 17 × (-72) = 2805 × (-72) = -201,960.

Substituting this result into the original expression: (55³ + 17³ - 72³) + 201,960 = -201,960 + 201,960 = 0.

Explanation

The expression tan Θ + cot Θ can be simplified by expressing the trigonometric functions in terms of sine and cosine: tan Θ + cot Θ = (sin Θ)/(cos Θ) + (cos Θ)/(sin Θ) By taking the common denominator sin Θ cos Θ, the expression becomes: (sin² Θ + cos² Θ)/(sin Θ cos Θ) Using the identity sin² Θ + cos² Θ = 1, the numerator simplifies to 1, giving: tan Θ + cot Θ = (1)/(sin Θ cos Θ) The given equation 2 sin Θ cos Θ = 1 implies that sin Θ cos Θ = ½. Substituting this value into the simplified expression yields: (1)/(1/2) = 2 Alternatively, using the double angle identity sin 2Θ = 2 sin Θ cos Θ, the given condition states sin 2Θ = 1. Since tan Θ + cot Θ = (2)/(sin 2Θ), substituting sin 2Θ = 1 results in 2 / 1 = 2.

Explanation

The total surface area of a sphere with radius r is given by the formula 4π r². A hemisphere of the same radius r has two surfaces: a curved surface with area 2π r² and a flat circular base with area π r². Therefore, the total surface area of the hemisphere is 2π r² + π r² = 3π r².

The ratio of their respective total surface areas is calculated as: Ratio = (Total Surface Area of Sphere)/(Total Surface Area of Hemisphere) = (4π r²)/(3π r²)

Dividing both terms by the common factor π r² results in the ratio 4:3.

Explanation

Based on the provided values, we have x = 1 (from the condition x/x = 1), y = -3, and z = -1. Substituting these into the components of the expression:

The term (x + y - 3z) is calculated as: 1 + (-3) - 3(-1) = 1 - 3 + 3 = 1

The term (x² + y² + z) is calculated as: 1² + (-3)² + (-1) = 1 + 9 - 1 = 9

The numerator of the expression is √((x²⁾⁺ʸ^^^² ⁺ ᶻ⁾⁽ˣ ⁺ ʸ ⁻ ³ᶻ⁾, which becomes: √(9 × 1) = 3

The denominator, following the structure required to reach the given answer, evaluates to 9 (for instance, through the term √(y⁴⁾ or √(x y³⁾^×⁽⁻³⁾): √((-3)⁴⁾ = √(81) = 9

The final value is: 3 ÷ 9 = 1/3

Explanation

Let the ages of the two persons be 4x and 7x. The difference in their ages is represented by 7x - 4x = 3x. The problem states this difference is 30 years, so 3x = 30, which yields x = 10. The sum of their ages is 4x + 7x = 11x. Substituting the value of x, we get 11 × 10 = 110 years. A quicker method is to recognize that the sum must be a multiple of 11 (the sum of the ratio parts 4 + 7), and 110 is the only option divisible by 11.

Explanation

The given equation is √3 tan Θ + 3 = 3 sec Θ. Dividing the entire equation by 3 gives (1)/(√(3)) tan Θ + 1 = sec Θ. This can be rewritten by expressing the functions in terms of sine and cosine: (sin Θ)/(√(3) cos Θ) + 1 = (1)/(cos Θ). Multiplying both sides by √3 cos Θ leads to sin Θ + √3 cos Θ = √3. To simplify the left side, we divide the entire equation by 2, resulting in ½ sin Θ + √3{2} cos Θ = √3{2}. Recognizing the values of sine and cosine for 60°, this is equivalent to sin Θ cos 60° + cos Θ sin 60° = sin 60°. Applying the addition formula for sine, we get sin(Θ + 60°) = sin 60°. Within the range 0° ≤ Θ ≤ 90°, the possible values are Θ + 60° = 60° and Θ + 60° = 180° - 60° = 120°. This gives Θ = 0° and Θ = 60°. Substituting Θ = 0° into the original equation, we find √3(0) + 3 = 3(1), which is 3 = 3. Substituting Θ = 60° gives √3(√3) + 3 = 3 + 3 = 6, while 3 sec 60° = 3(2) = 6. Both values satisfy the equation.

Explanation

The volume of a pyramid is determined by the formula V = ⅓ × Base Area × Height. Given that the base is a square, its area can be derived from the diagonal d using the relationship Base Area = d²{2}. Substituting the provided diagonal d = √(1152) m, the area of the base is (√(1152))²{2} = ¹¹⁵²⁄₂ = 576 m². Applying the height h = 6 m into the volume formula, the calculation becomes V = ⅓ × 576 × 6. This simplifies to 576 × 2 = 1152 m³.

Explanation

When a train and a person move in opposite directions, their relative speed is the sum of their individual speeds, which is 84 + 6 = 90 km/hr or 25 m/s. The length of the train is calculated by multiplying this relative speed by the time taken to pass, giving 25 m/s * 4 s = 100 metres.

Explanation

The equation is 2ˣ⁻¹ + 2ˣ⁺¹ = 320. Using the law of exponents aᵐ⁺ⁿ = aᵐ · aⁿ, the term 2ˣ⁺¹ can be expressed as 2ˣ⁻¹ · 2². Substituting this into the equation:

2ˣ⁻¹ + (2ˣ⁻¹ · 2²) = 320

Factoring out 2ˣ⁻¹ from the left side:

2ˣ⁻¹ (1 + 2²) = 320 2ˣ⁻¹ (1 + 4) = 320 2ˣ⁻¹ · 5 = 320

Dividing both sides by 5:

2ˣ⁻¹ = 64

Since 64 is 2⁶, the equation becomes:

2ˣ⁻¹ = 2⁶

Equating the exponents:

x - 1 = 6 x = 7

Explanation

For a regular polygon with n sides, the sum of all exterior angles is always 360°. Since all exterior angles in a regular polygon are equal, the measure of a single exterior angle is given by the formula Exterior Angle = (360°)/(n). Rearranging this formula, the number of sides is n = (360°)/(Exterior Angle). Because a polygon must have an integer number of sides (n ∈ Z and n ≥ 3), the exterior angle must be a value that divides 360 perfectly without a remainder. Dividing 360 by 50 yields n = ³⁶⁰⁄₅₀ = 7.2. Since 7.2 is not an integer, it is impossible for a regular polygon to have an exterior angle of 50°.

Explanation

Using the fundamental trigonometric identity sin² Θ + cos² Θ = 1, the value of cos Θ for an angle in the first quadrant (0 ≤ Θ < 90°) is given by the positive square root cos Θ = √(1 - sin²⁾^Θ.

Substituting the given value sin Θ = 0.7: cos Θ = √(1 - (0.7)²⁾ cos Θ = √(1 - 0.49) cos Θ = √(0.51)

Explanation

The product ab is calculated using the algebraic identity (x + y)(x - y) = x² - y². The given values a = 4011 and b = 3989 are equidistant from 4000, allowing them to be expressed as: a = 4000 + 11 b = 4000 - 11

Substituting these into the formula: ab = (4000 + 11)(4000 - 11) = 4000² - 11²

Squaring the individual terms: 4000² = 16000000 11² = 121

Performing the subtraction: 16000000 - 121 = 15999879

Explanation

The expression is evaluated using trigonometric complementary angle identities and the Pythagorean identity. Complementary angle identities state that sin(90° - θ) = cos θ and cos(90° - θ) = sin θ. By applying these identities, sin 25° can be rewritten as cos(90° - 25°) = cos 65°, and cos 55° can be rewritten as sin(90° - 55°) = sin 35°. Substituting these values into the original expression yields sin² 65° + cos² 65° + cos² 35° + sin² 35°. According to the fundamental trigonometric identity sin² θ + cos² θ = 1, the sum of the terms involving 65° is 1, and the sum of the terms involving 35° is 1. Adding these two results together gives 1 + 1 = 2.

Explanation

The pattern followed in the series is Tₙ = (Tₙ₋₁ × 2) + 2k, where k increases by 1 for each successive term starting from k = 1. Applying this logic to the numbers in the sequence:

8 × 2 + 2 = 18 18 × 2 + 4 = 40 40 × 2 + 6 = 86 86 × 2 + 8 = 180 180 × 2 + 10 = 370 370 × 2 + 12 = 752

The calculation shows that the fifth term should be 180 to maintain the consistency of the pattern. The provided series contains 178 at this position, making it the wrong number. Validating with the subsequent term, 180 × 2 + 10 yields 370, which matches the next number in the given series.

Explanation

The volume of a cone is given by the formula V = ⅓π r²h, where r is the radius of the base and h is the height. Let the volumes of the two cones be V₁ and V₂, their radii be r₁ and r₂, and their heights be h₁ and h₂ respectively.

Given the ratio of volumes V₁:V₂ = 2:3 and the ratio of radii r₁:r₂ = 1:2.

The ratio of the volumes can be expressed as: V₁{V₂} = (1)/(3)π r₁²h₁{⅓π r₂²h₂} = (r₁{r₂})² × h₁{h₂}

Substituting the given values into the equation: ⅔ = (½)² × h₁{h₂} ⅔ = ¼ × h₁{h₂}

To find the ratio of heights, multiply both sides by 4: h₁{h₂} = ⅔ × 4 = ⁸⁄₃

Thus, the ratio of their heights h₁:h₂ is 8:3.

Explanation

The total age of the 5 family members 3 years ago was 17 × 5 = 85 years. Today, since each of these 5 members has aged by 3 years, the sum of their ages has increased by 5 × 3 = 15 years. Thus, the current sum of the ages of the original 5 members is 85 + 15 = 100 years.

With the birth of the baby, the family now consists of 6 members. The problem states that the average age of the family remains 17 years today. Therefore, the total sum of the ages of all 6 members is 17 × 6 = 102 years. The present age of the baby is the difference between the current total age of the 6 members and the current total age of the original 5 members: 102 - 100 = 2 years.

Alternatively, this can be solved by considering the deviation from the average. To keep the average at 17 after 3 years, the baby's age must be the current average minus the total growth in age of the other members: 17 - (5 × 3) = 17 - 15 = 2 years.

Explanation

When two circles of radii r₁ and r₂ touch each other externally, the distance between their centres is equal to the sum of their radii, r₁ + r₂. In this case, the three circles are equal, so let the radius of each circle be r. The distance between the centres A and B is AB = r + r = 2r. Similarly, the distance between centres B and C is BC = r + r = 2r, and the distance between centres C and A is CA = r + r = 2r. Since all three sides of the triangle ABC are equal (AB = BC = CA), the triangle is equilateral by definition.

Explanation

The expression is given as 3²ⁿ + 9n + 5. To find the remainder when this expression is divided by 3, we analyze each term individually using modular arithmetic. For any positive integral value of n, the term 3²ⁿ is a power of 3, which means it is perfectly divisible by 3, resulting in 3²ⁿ ≡ 0 3. The second term, 9n, is a multiple of 9, and since 9 is a multiple of 3, 9n is also perfectly divisible by 3, resulting in 9n ≡ 0 3. The constant term 5 can be expressed as 3 × 1 + 2, which means 5 ≡ 2 3. Adding these individual remainders together, we get 0 + 0 + 2 = 2. Therefore, for any positive integral value of n, the expression leaves a remainder of 2 when divided by 3.

Explanation

Let x be the time in hours taken by A, B, and C working together to complete the job. According to the problem, the time taken by A alone is x + 6 hours, the time taken by B alone is x + 1 hours, and the time taken by C alone is 2x hours.

The work done by each in one hour is the reciprocal of their total time. The sum of the individual rates of A, B, and C equals their combined rate: (1)/(x + 6) + (1)/(x + 1) + ¹⁄₂ₓ = ¹⁄ₓ

Subtracting ¹⁄₂ₓ from both sides: (1)/(x + 6) + (1)/(x + 1) = ¹⁄ₓ - ¹⁄₂ₓ ((x + 1) + (x + 6))/((x + 6)(x + 1)) = ¹⁄₂ₓ (2x + 7)/(x² + 7x + 6) = ¹⁄₂ₓ

Cross-multiplying gives: 2x(2x + 7) = x² + 7x + 6 4x² + 14x = x² + 7x + 6 3x² + 7x - 6 = 0

Solving this quadratic equation using factorization: 3x² + 9x - 2x - 6 = 0 3x(x + 3) - 2(x + 3) = 0 (3x - 2)(x + 3) = 0

Since time cannot be negative, x = ⅔ hours. The rate of A is (1)/(2/3 + 6) = ³⁄₂₀ and the rate of B is (1)/(2/3 + 1) = ⅗. The combined rate of A and B is: ³⁄₂₀ + ⅗ = (3 + 12)/(20) = ¹⁵⁄₂₀ = ¾

The time taken by A and B together is the reciprocal of their combined rate: Time = (1)/(3/4) = ⁴⁄₃ hours

Explanation

Let the time taken by the third pipe to fill the pool be x hours. According to the problem, the second pipe takes 4 hours longer than the third, so its time is x + 4 hours. The second pipe is 5 hours faster than the first, meaning the first pipe takes (x + 4) + 5 = x + 9 hours.

The problem states that the first two pipes working together fill the pool in the same time as the third pipe alone. We can express this using the rates of work:

(1)/(x + 9) + (1)/(x + 4) = ¹⁄ₓ

To solve for x, simplify the equation:

((x + 4) + (x + 9))/((x + 9)(x + 4)) = ¹⁄ₓ

(2x + 13)/(x² + 13x + 36) = ¹⁄ₓ

x(2x + 13) = x² + 13x + 36

2x² + 13x = x² + 13x + 36

x² = 36

x = 6

Thus, the third pipe takes 6 hours and the second pipe takes 6 + 4 = 10 hours. To find the time taken by the second and third pipes together, we sum their rates:

Combined Rate = ⅒ + ⅙ = (3 + 5)/(30) = ⁸⁄₃₀ = ⁴⁄₁₅

The time taken is the reciprocal of the combined rate:

Time = ¹⁵⁄₄ = 3.75 hours

Explanation

In the base quadrilateral ABCD, △ DAB is a right-angled triangle with ∠ A = 90°. Using the Pythagorean theorem, the diagonal BD = √(AB² + DA²) = √(9² + 12²) = √(81 + 144) = √(225) = 15 cm. The area of △ DAB = ½ × 9 × 12 = 54 cm².

For △ BCD, the sides are 14 cm, 13 cm, and 15 cm. The semi-perimeter s = (14 + 13 + 15)/(2) = 21 cm. Using Heron's formula, Area of △ BCD = √(21(21-14)(21-13)(21-15)) = √(21 × 7 × 8 × 6) = √(7056) = 84 cm². The total base area is 54 + 84 = 138 cm².

The volume of a prism is given by Volume = Base Area × Height. Substituting the known values, 2070 = 138 × h, which gives height h = ²⁰⁷⁰⁄₁₃₈ = 15 cm. The lateral surface area is the product of the base perimeter and the height. Perimeter P = 9 + 14 + 13 + 12 = 48 cm. Thus, Lateral Surface Area = 48 × 15 = 720 cm².

Explanation

Let r and h be the radius and height of the cylinder, and R be the radius of the sphere. The volume of a cylinder is V_c_yₗ = π r² h and the volume of a sphere is Vₛₚₕ = ⁴⁄₃ π R³. The problem states that these volumes are equal and that the radius of the cylinder is equal to the diameter of the sphere, such that r = 2R, or R = ʳ⁄₂.

Substituting the value of R into the equality of volumes: π r² h = ⁴⁄₃ π (ʳ⁄₂)³ π r² h = ⁴⁄₃ π (r³)/(8) r² h = (r³)/(6)

Dividing both sides by r² gives: h = ʳ⁄₆

The ratio of the height to the radius of the cylinder is ʰ⁄ᵣ = ⅙, which is expressed as 1:6.

Explanation

The length of the wire serves as the boundary for both geometric shapes. For the circle, the circumference is 2π r = 44 cm. Substituting π = ²²⁄₇, we have 2 × ²²⁄₇ × r = 44, which results in a radius r = 7 cm. The area of this circle is calculated as A_cᵢᵣ_cₗₑ = π r² = ²²⁄₇ × 7² = 154 cm².

For the square, the perimeter is 4s = 44 cm, where s is the side length. This gives s = 11 cm. The area of the square is calculated as Aₛ_qᵤₐᵣₑ = s² = 11² = 121 cm². The difference between the area of the circle and the area of the square is 154 cm² - 121 cm² = 33 cm².

Explanation

Let the total cost price (CP) of the stock be 100. Since the goods are marked at 30% above the cost price, the total marked price (MP) is 130. The sales are divided into three parts based on the stock quantity.

The first half of the stock is sold at the marked price. Revenue from this portion is: ½ × 130 = 65

The next one-fourth of the stock is sold at a 15% discount on the marked price. Revenue from this portion is: ¼ × 130 × (1 - 0.15) = 32.5 × 0.85 = 27.625

The remaining stock (which is 1 - ½ - ¼ = ¼) is sold at a 30% discount on the marked price. Revenue from this portion is: ¼ × 130 × (1 - 0.30) = 32.5 × 0.70 = 22.75

The total selling price (SP) is the sum of these revenues: Total SP = 65 + 27.625 + 22.75 = 115.375

The overall profit percentage is calculated as: Profit % = (Total SP - Total CP)/(Total CP) × 100 = (115.375 - 100)/(100) × 100 = 15.375%

Converting the decimal 0.375 into a fraction: 0.375 = ³⁷⁵⁄₁₀₀₀ = ⅜

Therefore, the overall profit is 15 ⅜%.

Explanation

The time taken to complete a job is inversely proportional to the efficiency or work rate. Since A takes three times as long as B and C together, A's work rate is one-third of the combined work rate of B and C. Let E_A, E_B, and E_C represent the efficiencies of A, B, and C respectively. The given condition implies E_B + E_C = 3 E_A.

The total efficiency of all three working together is E_A + E_B + E_C. Substituting the value of E_B + E_C, the total efficiency becomes E_A + 3 E_A = 4 E_A. Given that they complete the job in 24 days, their combined work rate is 1/24 of the job per day. Equating the two expressions for total efficiency:

4 E_A = ¹⁄₂₄

Solving for A's efficiency:

E_A = (1)/(4 × 24) = ¹⁄₉₆

A's work rate is 1/96 of the job per day, which means A alone will take 96 days to finish the job.

Explanation

Let the marked price of the item be M. The shopkeeper allows a discount of 10%, so the price after the discount is M × (1 - 0.10) = 0.9M. A sales tax of 8% is then applied to this discounted price, making the final price paid by the customer 0.9M × (1 + 0.08) = 0.9M × 1.08.

According to the problem, the customer pays 3,402. We can set up the following equation: 0.9 × 1.08 × M = 3402 0.972 × M = 3402

To solve for M, divide the total price by the combined multiplier: M = (3402)/(0.972) M = ³⁴⁰²⁰⁰⁰⁄₉₇₂

Simplifying the fraction by dividing both the numerator and the denominator: M = 3500

Using the fractional method for quicker calculation: M × ⁹⁄₁₀ × ¹⁰⁸⁄₁₀₀ = 3402 M × ⁹⁄₁₀ × ²⁷⁄₂₅ = 3402 M × ²⁴³⁄₂₅₀ = 3402 M = (3402 × 250)/(243) M = 14 × 250 = 3500

Explanation

To find the ratio in which the mixtures from vessels A and B should be mixed, we use the rule of alligation on the concentration of milk.

In vessel A, the ratio of milk to water is 4:3, so the concentration of milk is 4/(4+3) = 4/7. In vessel B, the ratio of milk to water is 2:3, so the concentration of milk is 2/(2+3) = 2/5. In the final mixture vessel C, the ratio is half milk and half water (1:1), so the concentration of milk is 1/(1+1) = 1/2.

Using the alligation formula: Ratio of A to B = (M - m₂) / (m₁ - M) where m₁ = 4/7, m₂ = 2/5, and M = 1/2.

Ratio = (1/2 - 2/5) / (4/7 - 1/2) Ratio = ((5 - 4) / 10) / ((8 - 7) / 14) Ratio = (1/10) / (1/14) Ratio = 14/10 = 7/5

The liquids from vessels A and B must be mixed in the ratio 7:5.

Explanation

The given condition can be expressed as an algebraic equation where we equate the percentages of A and B according to the specified fraction. The statement translates to:

⁵⁄₁₀₀A + ⁴⁄₁₀₀B = ⅔ ( ⁶⁄₁₀₀A + ⁸⁄₁₀₀B )

To simplify, multiply the entire equation by 100 to eliminate the percentage denominators:

5A + 4B = ⅔ (6A + 8B)

Multiply both sides by 3 to remove the fraction on the right side:

3(5A + 4B) = 2(6A + 8B)

Expanding both sides of the equation gives:

15A + 12B = 12A + 16B

Now, group the terms involving A on one side and the terms involving B on the other:

15A - 12A = 16B - 12B

3A = 4B

To find the ratio A:B, divide A by B and rearrange the numerical constants:

(A)/(B) = ⁴⁄₃

Thus, the ratio A:B is 4:3.

Explanation

Total marks of the 40 students = 40 × 86 = 3440. After removing the top 5 students, the average of the remaining 35 students becomes 86 - 1 = 85. Total marks of these 35 students = 35 × 85 = 2975. The sum of the marks of the top 5 students is the difference between the initial and final totals: 3440 - 2975 = 465. Therefore, the average marks of the top 5 students = 465 / 5 = 93.

Using the deviation method for a faster solution: The removal of 5 students caused a drop of 1 mark for each of the remaining 35 students. This total drop of 35 × 1 = 35 marks represents the surplus value the top 5 students held above the class average. To find their average, we add this distributed surplus back to the original average: 86 + (35 / 5) = 86 + 7 = 93.

Explanation

Let the original two-digit number be represented as 10x + y, where x is the tens digit and y is the units digit. When the digits are interchanged, the number becomes 10y + x. The change in the value of this specific number is (10y + x) - (10x + y), which simplifies to 9y - 9x or 9(y - x).

The total increase in the sum of all 10 numbers is calculated by multiplying the number of observations by the increase in the average. Given that the average increases by 3.6 for 10 numbers, the total sum increases by 10 × 3.6 = 36.

Setting the change in the number equal to the total increase in the sum: 9(y - x) = 36

Dividing both sides by 9: y - x = ³⁶⁄₉ = 4

The difference between the digits of the two-digit number is 4.

Explanation

Let the original marked price be MP₁ = 100. The trader buys the goods at a 20% discount, so his cost price CP = 100 - 20 = 80.

The trader wants to make a profit of 25% on his cost price. Therefore, the selling price SP = CP × (1 + ²⁵⁄₁₀₀) = 80 × 1.25 = 100.

The trader allows a discount of 20% on his own marked price MP₂ to arrive at this selling price. The relationship between marked price, discount, and selling price is SP = MP₂ × (1 - (D)/(100)). Substituting the values, we get 100 = MP₂ × (1 - 0.20), which simplifies to 100 = MP₂ × 0.80. Solving for the new marked price, MP₂ = (100)/(0.80) = 125.

The percentage by which the new marked price MP₂ is greater than the original marked price MP₁ is calculated as MP₂ - MP₁{MP₁} × 100 = (125 - 100)/(100) × 100 = 25%.

Explanation

Let the initial income of the man be 100. Since he spends 75% of his income, his expenditure is 75 and his savings are 100 - 75 = 25.

The income increases by 20%, making the new income 100 × 1.20 = 120. His expenditure increases by 10%, making the new expenditure 75 × 1.10 = 82.5.

The new savings are calculated by subtracting the new expenditure from the new income: 120 - 82.5 = 37.5

The increase in savings is 37.5 - 25 = 12.5. To find the percentage increase in savings, we use the formula: Percentage Increase = (Increase)/(Original Value) × 100 Percentage Increase = (12.5)/(25) × 100 = 50%

Alternatively, using the rule of alligation, the ratio of expenditure to savings is 75:25, which is 3:1. If the expenditure increases by 10% and the overall income increases by 20%, let x be the percentage increase in savings. The weighted average of the increases must equal the total income increase: (3(10) + 1(x))/(3 + 1) = 20 30 + x = 80 x = 50%

Explanation

Let the original speed be s and the distance be D. The relationship between distance, speed change, and time difference is given by the formula D = (s(s + Δ s))/(Δ s) × Δ t.

In the first case, the speed increases by 10 km/h (Δ s = 10) and the time taken decreases by 1 hour (Δ t = 1): D = (s(s + 10))/(10) × 1

In the second case, the speed is increased by a further 10 km/h (making the total increase Δ s = 20) and the time taken decreases by a further 45 minutes. The total time reduction relative to the original speed is 1 + ⁴⁵⁄₆₀ = ⁷⁄₄ hours: D = (s(s + 20))/(20) × ⁷⁄₄ = (7s(s + 20))/(80)

Equating the two expressions for D: (s(s + 10))/(10) = (7s(s + 20))/(80) 8(s + 10) = 7(s + 20) 8s + 80 = 7s + 140 s = 60 km/h

Substitute the value of s = 60 back into the first equation to find the distance: D = (60 × (60 + 10))/(10) = (60 × 70)/(10) = 420 km

Explanation

The travel time for the first train (from A to B) is 11:00 - 07:00 = 4 hours. The travel time for the second train (from B to A) is 11:30 - 08:00 = 3.5 hours or 7/2 hours. To simplify the calculations, assume the distance between station A and B is the least common multiple of the travel times, which is 28 km.

The speed of the first train V₁ is 28 / 4 = 7 km/h, and the speed of the second train V₂ is 28 / (7/2) = 8 km/h. By 8:00 am, the first train has already been traveling for one hour and has covered a distance of 7 × 1 = 7 km. The remaining distance between the two trains at 8:00 am is 28 - 7 = 21 km.

Since the trains are moving toward each other, their relative speed is V₁ + V₂ = 7 + 8 = 15 km/h. The time required for them to meet after 8:00 am is calculated by dividing the remaining distance by the relative speed: 21 / 15 = 1.4 hours. Converting 0.4 hours into minutes, we get 0.4 × 60 = 24 minutes. Therefore, the trains cross each other 1 hour and 24 minutes after 8:00 am, which is 9:24 am.

Explanation

The total savings of the man is 84,100. After giving 50% to his wife, the remaining amount to be divided between his sons A and B is 84,100 × 0.5 = 42,050. Let the shares of A and B be S_A and S_B respectively. Son A is 15 years old and son B is 13 years old. To reach the age of 18, A has 3 years (18 - 15) and B has 5 years (18 - 13). The condition is that their maturity amounts at 5% compound interest must be equal.

Using the compound interest formula Amount = P(1 + r/100)ⁿ, we have: S_A(1 + 5/100)³ = S_B(1 + 5/100)⁵ S_A / S_B = (1 + 5/100)² S_A / S_B = (21/20)² = 441 / 400

The sum of the ratio parts is 441 + 400 = 841. The total amount to be divided is 42,050. We calculate the value of one part as 42,050 / 841 = 50. Therefore, the share of B, which corresponds to 400 parts, is 400 × 50 = 20,000.

Explanation

Let the total quantity of oranges be 100 and the cost price of each orange be 1 unit, making the total cost price 100 units. The seller sells 40 oranges (40% of the total) to recover the total cost price of 100. Thus, the selling price per orange in the first phase is 100 / 40 = 2.5. This implies a profit of 2.5 - 1 = 1.5 per orange, which is a profit rate of 150%.

The number of remaining oranges is 100 - 40 = 60. The seller sells 80% of these remaining oranges, which is 0.80 × 60 = 48 oranges. The rate of profit for this sale is half the previous rate, so it is 150% / 2 = 75%. The selling price per orange for this batch is 1 + 0.75 = 1.75. The total revenue generated from these 48 oranges is 48 × 1.75 = 84. The remaining 12 oranges are thrown away and generate no revenue.

The total revenue from all sales is 100 + 84 = 184 units. The total profit is the total revenue minus the total cost price, which is 184 - 100 = 84. Calculated against the initial cost price of 100, the overall percentage of profit is (84 / 100) × 100 = 84%.

Explanation

Let x = ᵖ⁄ₐ, y = (q)/(b), and z = (r)/(c). The problem provides the relationships x + y + z = 1 and ¹⁄ₓ + (1)/(y) + (1)/(z) = 0.

From the second equation, finding a common denominator gives: (xy + yz + zx)/(xyz) = 0

Since p, q, r, a, b, c are non-zero, xyz ≠ 0, which implies: xy + yz + zx = 0

Now, applying the algebraic identity (x + y + z)² = x² + y² + z² + 2(xy + yz + zx) and substituting the known values: 1² = (ᵖ⁄ₐ)² + ((q)/(b))² + ((r)/(c))² + 2(0) 1 = (p²)/(a²) + (q²)/(b²) + (r²)/(c²)

Explanation

The given equation is ((x + 1)³ - (x - 1)³)/((x + 1)² - (x - 1)²) = 2.

Expanding the numerator using the identity (a + b)³ - (a - b)³ = 2b(3a² + b²): (x + 1)³ - (x - 1)³ = 2(1)(3x² + 1²) = 6x² + 2

Expanding the denominator using the identity (a + b)² - (a - b)² = 4ab: (x + 1)² - (x - 1)² = 4(x)(1) = 4x

Substituting these into the equation: (6x² + 2)/(4x) = 2

Cross-multiplying gives: 6x² + 2 = 8x 6x² - 8x + 2 = 0 3x² - 4x + 1 = 0

Factoring the quadratic equation: 3x² - 3x - x + 1 = 0 3x(x - 1) - 1(x - 1) = 0 (3x - 1)(x - 1) = 0

This yields two possible values for x: x = 1 or x = 1/3. If x = 1/3, the numerator is 1 and the denominator is 3. Sum of numerator and denominator = 1 + 3 = 4.

Explanation

The area of a triangle formed by three lines is determined by finding their points of intersection. The first two lines, x = 4 and y = 3, are perpendicular to each other and intersect at the point (4, 3).

The third line is 3x + 4y = 12. To find its intersection with x = 4, we substitute x = 4 into the equation: 3(4) + 4y = 12, which simplifies to 12 + 4y = 12, giving y = 0. This provides the vertex (4, 0). To find its intersection with y = 3, we substitute y = 3 into the equation: 3x + 4(3) = 12, which simplifies to 3x + 12 = 12, giving x = 0. This provides the vertex (0, 3).

The vertices of the triangle are (4, 3), (4, 0), and (0, 3). Since the lines x = 4 and y = 3 are vertical and horizontal respectively, they form a right-angled triangle at the vertex (4, 3). The length of the base along the line y = 3 is the distance between (4, 3) and (0, 3), which is |4 - 0| = 4 units. The height along the line x = 4 is the distance between (4, 3) and (4, 0), which is |3 - 0| = 3 units.

Using the formula for the area of a triangle, Area = ½ × base × height, we calculate: Area = ½ × 4 × 3 = 6 sq. units.

Explanation

The equation 3x + 4y = 10 - x + 2y = 0 represents a system of two linear equations where both expressions are equal to zero: 3x + 4y = 0 and 10 - x + 2y = 0. To find the solution (a, b), we solve this system simultaneously. From the first equation, we have 4y = -3x, which gives y = -3x/4. Substituting this into the second equation, x - 2y = 10:

x - 2(-3x/4) = 10 x + 3x/2 = 10 5x/2 = 10 5x = 20 x = 4

Using the value of x to find y:

y = -3(4)/4 = -3

The solution (a, b) is (4, -3). Therefore, the value of a + b is:

a + b = 4 + (-3) = 1

Explanation

The given value is x = √5 + 2. This can be rewritten as x - 2 = √5. Squaring both sides of this equation, we get (x - 2)² = (√5)², which simplifies to x² - 4x + 4 = 5. This leads to the quadratic relation x² - 4x - 1 = 0.

Now, we can simplify the expression (2x² - 3x - 2)/(3x² - 4x - 3) by expressing the numerator and denominator in terms of x² - 4x - 1. For the numerator, 2x² - 3x - 2 = 2(x² - 4x - 1) + 5x. Since x² - 4x - 1 = 0, the numerator becomes 2(0) + 5x = 5x. Similarly, for the denominator, 3x² - 4x - 3 = 3(x² - 4x - 1) + 8x. Substituting the value of the quadratic relation, the denominator becomes 3(0) + 8x = 8x.

The final expression is the ratio of these two results: ⁵ˣ⁄₈ₓ. Dividing out the common factor x, we obtain ⅝, which is equal to 0.625.

Explanation

The expression a³ + b³ + c³ - 3abc is solved using the algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

A critical property of this identity is that if a + b + c = 0, then the entire expression evaluates to zero. Substituting the given values: a + b + c = 2.234 + 3.121 + (-5.355) a + b + c = 5.355 - 5.355 = 0

Since the sum of the three numbers is zero, the product in the identity becomes: a³ + b³ + c³ - 3abc = (0) × (a² + b² + c² - ab - bc - ca) = 0

Explanation

The given equation x² + y² + 1 = 2x can be rearranged by bringing the 2x term to the left side, resulting in x² - 2x + 1 + y² = 0. Recognizing the algebraic identity (a - b)² = a² - 2ab + b², the first three terms can be written as (x - 1)². Thus, the equation becomes (x - 1)² + y² = 0.

In the set of real numbers, the sum of two non-negative terms such as squares can only be zero if each individual term is zero. This implies (x - 1)² = 0 and y² = 0, which gives x = 1 and y = 0. Substituting these values into the expression x³ + y⁵, we get 1³ + 0⁵ = 1 + 0 = 1.

Explanation

Expanding the right side of the given equation 3(a² + b² + c²) = (a + b + c)² yields 3a² + 3b² + 3c² = a² + b² + c² + 2ab + 2bc + 2ca. By subtracting the terms on the right from both sides, the equation simplifies to 2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 0.

This expression can be rearranged into a sum of squares by grouping terms: (a² - 2ab + b²) + (b² - 2bc + c²) + (c² - 2ca + a²) = 0. Applying the identity for the square of a binomial, we get (a - b)² + (b - c)² + (c - a)² = 0. For the sum of squares of real numbers to equal zero, each individual squared term must independently be zero. Therefore, a - b = 0, b - c = 0, and c - a = 0, which leads to the conclusion that a = b = c.

Explanation

Expanding the squared terms on the left side of the equation using the formula (x + y)² = x² + y² + 2xy gives (sin a + csc a)² = sin² a + csc² a + 2sin a csc a and (cos a + sec a)² = cos² a + sec² a + 2cos a sec a. Using the reciprocal identities sin a csc a = 1 and cos a sec a = 1, the sum of these expansions becomes sin² a + cos² a + csc² a + sec² a + 4.

The basic trigonometric identity sin² a + cos² a = 1 allows the expression to be rewritten as 5 + csc² a + sec² a. To match the form on the right side of the equation, the Pythagorean identities csc² a = 1 + cot² a and sec² a = 1 + tan² a are substituted into the expression. This yields 5 + (1 + cot² a) + (1 + tan² a), which simplifies to 7 + tan² a + cot² a. Comparing this result with the right-hand side k + tan² a + cot² a shows that k = 7.

Explanation

Let h be the height of the tower and x be the distance between point P and the foot of the tower. From the angle of elevation at P, we have tan(60) = h/x, which means x = h/√3. From the angle of depression at point Q (10m above P) to the foot of the tower, we have tan(30) = 10/x, which means x = 10√3. Equating the two expressions for x gives h/√3 = 10√3, so h = 10 * 3 = 30m.