SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

In a circle, the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circumference. Here, the arc PA subtends the central angle ∠ POA = 120°. Point B lies on the circumference of the circle. Therefore, the angle subtended by arc PA at point B, which is ∠ PBA (or ∠ PBO), is half of the central angle ∠ POA. Using the formula ∠ PBO = ½ ∠ POA, we calculate ∠ PBO = ½ × 120° = 60°. Alternatively, since AOB is a straight line, ∠ POB = 180° - ∠ POA = 180° - 120° = 60°. In △ POB, since OP = OB (radii of the circle), the triangle is isosceles and ∠ OPB = ∠ OBP. Summing the angles gives 60° + 2∠ PBO = 180°, which results in ∠ PBO = 60°.

Explanation

Let h = 200 m be the altitude of the aeroplane. The two points lying on either side of the river are the banks of the river. The width of the river is the horizontal distance between these two points. The horizontal distance d of a point from the vertical projection of the observer is given by the formula d = h cot θ, where θ is the angle of depression.

For the point on the far bank with an angle of depression of 45°: d_fₐᵣ = 200 cot 45° = 200 × 1 = 200 m

For the point on the near bank with an angle of depression of 60°: dₙₑₐᵣ = 200 cot 60° = 200 × (1)/(√(3)) = (200)/(√(3)) m

The width of the river is the difference between the horizontal distances of the two banks from the aeroplane's vertical projection: Width = d_fₐᵣ - dₙₑₐᵣ Width = 200 - (200)/(√(3)) m

Explanation

The volume of the original solid sphere is conserved when it is melted and recast into a smaller sphere and a cylinder. According to the principle of conservation of volume, the volume of the large sphere equals the sum of the volumes of the new sphere and the cylinder.

The volume of a sphere is given by Vₛ = ⁴⁄₃ π r³ and the volume of a cylinder is V_c = π r² h.

Let the radius of the large sphere be R = 9 cm, the radius of the smaller sphere be r = 6 cm, and the height of the cylinder be h with radius r = 6 cm. The equation is:

⁴⁄₃ π R³ = ⁴⁄₃ π r³ + π r² h

Dividing the entire equation by π:

⁴⁄₃ (9³) = ⁴⁄₃ (6³) + (6²) h

⁴⁄₃ × 729 = ⁴⁄₃ × 216 + 36h

4 × 243 = 4 × 72 + 36h

972 = 288 + 36h

Subtracting 288 from both sides gives:

36h = 684

h = ⁶⁸⁴⁄₃₆

h = 19 cm

Explanation

The length of an arc s is given by the formula s = rθ, where r is the radius and θ is the angle subtended at the center. Let r₁ and r₂ be the radii of the first and second circles respectively, and s₁ and s₂ be their corresponding arc lengths. The problem states that r₂ = 3r₁ and s₁ = 2s₂.

For the first circle, the arc length is s₁ = r₁ × 30°. Using the relationship between the arc lengths, we find s₂ = s₁{2} = 30° × r₁{2} = 15° r₁.

Applying the arc length formula for the second circle, s₂ = r₂ × ₂, where ₂ is the angle subtended at the center of the second circle. Substituting r₂ = 3r₁ and the calculated value of s₂ into this equation gives 15° r₁ = 3r₁ × ₂. Solving for ₂ by dividing both sides by 3r₁ results in ₂ = 5°.

Explanation

In work and wages problems, the share of payment for each individual is directly proportional to the fraction of the total work they perform. The total work is represented as 1 unit. Given that A and B together complete 7/11 of the work, the remaining portion of the work done by C is calculated as:

Work done by C = 1 - ⁷⁄₁₁ = ⁴⁄₁₁

Since C performs 4/11 of the total work, C is entitled to 4/11 of the total payment of Rs. 550.

Share of C = ⁴⁄₁₁ × 550

Dividing 550 by 11 gives 50. Multiplying this by 4 results in:

Share of C = 4 × 50 = 200

Thus, the share of C is Rs. 200.

Explanation

In △ ADC, the height CD is perpendicular to AB, making △ ADC a right-angled triangle at D. According to the Pythagoras theorem, the length of the hypotenuse AC is calculated as AC = √(AD² + CD²). Substituting the given values, AC = √(8² + 6²) = √(64 + 36) = √(100) = 10 cm.

In △ ABC, P and Q are the midpoints of sides AB and BC respectively. By the Midpoint Theorem, the segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half its length. Therefore, the length of PQ is half the length of side AC, which gives PQ = ½ AC = ½ × 10 = 5 cm.

Explanation

The selling price after two successive discounts is calculated using the formula S.P. = M.P. × (1 - (d₁)/(100)) × (1 - (d₂)/(100)), where M.P. is the marked price and d₁, d₂ are the discount rates. Given the initial price of Rs. 500 and two successive discounts of 10% each, the calculation is 500 × (1 - ¹⁰⁄₁₀₀) × (1 - ¹⁰⁄₁₀₀). This simplifies to 500 × 0.9 × 0.9, which is 500 × 0.81 = 405. Alternatively, the effective discount rate for two successive discounts of 10% is 10 + 10 - (10 × 10)/(100) = 19%. The final selling price is 81% of the original price, resulting in 500 × 0.81 = 405.

Explanation

Let the cost price of the article be x. The formula for profit percentage is P% = (SP - CP)/(CP) × 100. According to the problem, the profit percentage when the article is sold for Rs. 78 is twice the profit percentage when it is sold for Rs. 69. Setting up the equation:

(78 - x)/(x) × 100 = 2 × ( (69 - x)/(x) × 100 )

Since the denominator x and the multiplier 100 are common to both sides, they can be eliminated to simplify the equation:

78 - x = 2(69 - x) 78 - x = 138 - 2x 2x - x = 138 - 78 x = 60

Explanation

The total number of villagers can be calculated by dividing the number of people below the poverty level by their percentage relative to the total population. Let x be the total number of villagers. Given that 14% of x is 574, the relationship is expressed as:

14% of x = 574

¹⁴⁄₁₀₀ × x = 574

To solve for x, rearrange the terms:

x = (574 × 100)/(14)

Performing the division:

574 ÷ 14 = 41

x = 41 × 100 = 4100

By using the percentage formula, the total population is found to be 4100.

Explanation

The time taken to complete a piece of work is inversely proportional to the rate of working or efficiency, provided the total work remains constant. This relationship is defined by the formula Work = Rate × Time. Since the work is the same for both A and B, the ratio of the days taken is the reciprocal of the ratio of their working rates. Given that the rate of working of A and B is in the ratio 2:3, the ratio of the time taken by them is ½ : ⅓. Multiplying both terms by 6 to clear the fractions results in a ratio of 3:2.

Explanation

Let the two numbers be 3x and 4x based on the given ratio. According to the problem, the sum of their squares is (3x)² + (4x)² = 400. This simplifies to 9x² + 16x² = 400, which results in 25x² = 400. Dividing both sides by 25, we find x² = 16, and since the numbers are positive, x = 4. The sum of the two numbers is 3x + 4x = 7x. Substituting the value of x, the sum is 7 × 4 = 28. Alternatively, one can observe that the sum of the numbers must be a multiple of 3 + 4 = 7. Among the provided choices, only 28 satisfies this condition.

Explanation

Let the selling price be 100. A 10 percent loss on the selling price means the loss is 10, making the cost price 110 (SP + Loss). The loss percentage on the cost price is calculated as (Loss / CP) * 100, which is (10 / 110) * 100 = 100/11% or 9 1/11%.

Explanation

The difference between the two percentages is calculated as follows: 82.5% - 62.5% = 20%

According to the definition provided, 1 percent is equal to 100 basis points. Therefore, to convert the percentage difference into basis points, the value is multiplied by the conversion factor: 20 × 100 = 2000

Thus, 82.5 percent is greater than 62.5 percent by 2000 basis points.

Explanation

The distance of an object can be calculated using the fundamental relation d = v × t, where v is the speed of sound and t is the time interval. Given that the speed of sound is 330 m/s and the time delay between the lightning flash and the thunder is 10 seconds, the total distance in metres is:

d = 330 m/s × 10 s = 3300 metres

Since the question asks for the distance in kilometres, the result is converted by dividing by 1000:

Distance in km = ³³⁰⁰⁄₁₀₀₀ = 3.3 km

Explanation

Using the algebraic identity for the square of a sum, (p + ¹⁄ₚ)² = p² + (1)/(p²) + 2(p)(¹⁄ₚ). This simplifies to (p + ¹⁄ₚ)² = p² + (1)/(p²) + 2. Given that p² + (1)/(p²) = 47, we substitute this value into the equation to get (p + ¹⁄ₚ)² = 47 + 2 = 49. Taking the square root of both sides yields p + ¹⁄ₚ = √(49) = 7.

Explanation

The third proportional c of two quantities a and b is defined by the continued proportion a : b = b : c, which leads to the formula c = (b²)/(a). In this problem, the given values are a = (x-y)² and b = (x²-y²)².

By applying the algebraic identity x² - y² = (x-y)(x+y), the expression for b can be expanded as: b = [(x-y)(x+y)]² = (x-y)²(x+y)²

Substituting these values into the formula for the third proportional: c = ([(x-y)²(x+y)²]²)/((x-y)²) c = ((x-y)⁴(x+y)⁴)/((x-y)²)

According to the laws of exponents, when dividing powers with the same base, the exponents are subtracted: c = (x-y)⁴⁻²(x+y)⁴ c = (x-y)²(x+y)⁴

Explanation

When two circles touch each other externally, the distance between their centres is equal to the sum of their radii. Let the radius of the first circle be r₁ and the radius of the second circle be r₂. The relationship is expressed by the formula d = r₁ + r₂, where d is the distance between the centres.

Given that the distance between the centres d is 7 cm and the radius of the first circle r₁ is 4 cm, the equation becomes:

7 = 4 + r₂

Solving for the unknown radius:

r₂ = 7 - 4 r₂ = 3 cm

Explanation

When a train crosses a telegraph post, the distance covered is equal to the length of the train. Using the formula for speed, v = d/t, where distance d = 240 meters and time t = 16 seconds:

Speed in m/s = 240 / 16 = 15 m/s

To convert the speed from meters per second to kilometers per hour, multiply the value by 18/5:

Speed in km/hr = 15 × (18/5) = 3 × 18 = 54 km/hr

Explanation

For any non-negative real numbers a and b, the Arithmetic Mean - Geometric Mean (AM-GM) Inequality states that (a + b)/(2) ≥ √(ab). In the given expression tan² x + cot² x, both terms are non-negative squares. Let a = tan² x and b = cot² x. Since cot x is the reciprocal of tan x, the product ab is tan² x · cot² x = 1.

Applying the AM-GM inequality: (tan² x + cot² x)/(2) ≥ √(tan² x · cot² x) (tan² x + cot² x)/(2) ≥ √1 tan² x + cot² x ≥ 2

The equality holds when a = b, which implies tan² x = cot² x. This condition is satisfied when tan² x = 1, such as at x = (π)/(4).

Alternatively, the expression can be evaluated by completing the square: tan² x + cot² x = (tan x - cot x)² + 2 tan x cot x tan² x + cot² x = (tan x - cot x)² + 2

Since the term (tan x - cot x)² is always greater than or equal to 0 for all real values of x where the functions are defined, the minimum value of the entire expression is 0 + 2 = 2.

Explanation

The decrease in profit corresponds directly to the increase in the discount offered on the marked price. Since the cost price of the watch is constant, the difference between a 7% discount and a 5% discount, which is 2% of the marked price, must equal the ₹ 15 reduction in earnings. Using the marked price (MP) as the base: 2% of MP = 15 ²⁄₁₀₀ × MP = 15 MP = (15 × 100)/(2) MP = 750

Explanation

The divisibility rule for 8 states that a number is divisible by 8 if the number formed by its last three digits is divisible by 8. In the number 63576x2, the last three digits are 6x2. This can be expressed as 600 + 10x + 2, which simplifies to 602 + 10x.

To find the least value of x, we check the divisibility of 602 + 10x by 8. Dividing 602 by 8 gives a remainder of 2, since 602 = (75 × 8) + 2. Therefore, the expression 2 + 10x must be divisible by 8.

Testing values for x: If x = 1, then 2 + 10(1) = 12, which is not divisible by 8. If x = 2, then 2 + 10(2) = 22, which is not divisible by 8. If x = 3, then 2 + 10(3) = 32, which is 8 × 4.

Since 32 is divisible by 8, the number 632 is divisible by 8, making the entire number 6357632 divisible by 8. Among the given options, 3 is the least value that satisfies the condition.

Explanation

The correct average can be calculated by adjusting the incorrect average for the difference in values. The error introduced was the difference between the wrongly recorded number and the actual number, which is 64 - 46 = 18. Since the number recorded was higher than the actual value, the sum was overstated by 18.

The impact on the average for the 10 numbers is the total error divided by the count, which is 18 / 10 = 1.8.

Correct average = Incorrect average - Adjustment Correct average = 50 - 1.8 = 48.2

Alternatively, the incorrect total sum is 50 × 10 = 500. Subtracting the error of 18 gives the correct sum as 500 - 18 = 482. The correct average is 482 / 10 = 48.2.

Explanation

Given the equation x + √5 = 5 + √y where x and y are positive integers, we can rearrange the terms to separate the rational and irrational parts: x - 5 = √y - √5. For the left side to be an integer, the right side must also be an integer. Since √5 is an irrational number, the expression √y - √5 can only result in an integer if it equals zero, because the difference of two square roots is rational only if the numbers are perfect squares or if the terms cancel out.

Setting the right side to zero gives √y - √5 = 0, which implies y = 5. Substituting y = 5 back into the original equation, we get x + √5 = 5 + √5, which simplifies to x = 5. Since x = 5 and y = 5 are both positive integers, they satisfy the given conditions.

Now, substitute x = 5 and y = 5 into the expression √x + y{x + √y}: √5 + 5{5 + √5} = 1

Because the numerator and the denominator are identical, the value of the expression is 1.

Explanation

The cube of 997 can be efficiently calculated using the algebraic identity (a - b)³ = a³ - 3a²b + 3ab² - b³. By expressing 997 as (1000 - 3), we substitute a = 1000 and b = 3 into the formula.

Calculation: 997³ = (1000 - 3)³ = 1000³ - 3 × 1000² × 3 + 3 × 1000 × 3² - 3³ = 1,000,000,000 - 9,000,000 + 27,000 - 27

Performing the arithmetic: 1,000,000,000 - 9,000,000 = 991,000,000 991,000,000 + 27,000 = 991,027,000 991,027,000 - 27 = 991,026,973

Explanation

The given equation a² + b² + c² = ab + bc + ca can be rearranged by multiplying both sides by 2, resulting in 2a² + 2b² + 2c² = 2ab + 2bc + 2ca. Moving all terms to one side gives 2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 0. This expression can be grouped into perfect squares as (a² - 2ab + b²) + (b² - 2bc + c²) + (c² - 2ca + a²) = 0, which simplifies to (a - b)² + (b - c)² + (c - a)² = 0. Since a, b, and c are side lengths (real numbers), the sum of their squares can only be zero if each individual term is zero. Therefore, a - b = 0, b - c = 0, and c - a = 0, which implies a = b = c. A triangle with three equal sides is an equilateral triangle.

Explanation

The area of a circle is determined by the formula A = π r². For the larger circle with radius Y, the area is given as W = π Y². For the smaller circle with radius X, the area is π X².

The problem defines W' as the difference between the area of the larger circle and the smaller circle: W' = π Y² - π X²

Substituting W for π Y² in the equation above: W' = W - π X²

Rearranging the equation to solve for the area of the smaller circle: π X² = W - W'

To find the ratio x/y, we divide the expression for X² by the expression for Y²: (X²)/(Y²) = (π X²)/(π Y²) = (W - W')/(W) (X²)/(Y²) = 1 - (W')/(W)

Taking the square root of both sides to find the ratio of the radii: (X)/(Y) = √(1 - (W')/(W))

Explanation

Given cos 21° = (x)/(y), we consider a right-angled triangle where the adjacent side (base) is x and the hypotenuse is y. Using the Pythagorean theorem, the opposite side (perpendicular) is √(y² - x²).

The required expression is csc 21° - cos 69°. By applying the complementary angle identity cos(90° - θ) = sin θ, we find that cos 69° = cos(90° - 21°) = sin 21°. The expression simplifies to csc 21° - sin 21°.

Using the ratios from the triangle: csc 21° = (hypotenuse)/(perpendicular) = (y)/(√(y² - x²)) sin 21° = (perpendicular)/(hypotenuse) = √(y² - x²){y}

Substituting these into the expression: (y)/(√(y² - x²)) - √(y² - x²){y} = y² - (√(y² - x²))²{y√(y² - x²)} = (y² - (y² - x²))/(y√(y² - x²)) = (x²)/(y√(y² - x²))

Explanation

The given equation a² + 1 = a can be rewritten as a² - a + 1 = 0. To find the value of a³, we apply the algebraic identity for the sum of cubes, x³ + y³ = (x + y)(x² - xy + y²). Setting x = a and y = 1 gives a³ + 1³ = (a + 1)(a² - a + 1). Substituting the value a² - a + 1 = 0 from the initial equation into the identity results in a³ + 1 = (a + 1) × 0. This simplifies to a³ + 1 = 0, which leads to a³ = -1.

Explanation

The radius r of the circular base is half of the diameter, so r = ²¹⁄₂ = 10.5 m. Given the height h = 14 m, the slant height l is found using the relation l = √(r² + h²). Substituting the values, l = √(10.5² + 14²) = √(110.25 + 196) = √(306.25) = 17.5 m.

The curved surface area CSA of the cone is calculated using the formula CSA = π rl. Substituting the given values, CSA = ²²⁄₇ × ²¹⁄₂ × ³⁵⁄₂ = 11 × 3 × ³⁵⁄₂ = 577.5 sq. m.

The total cost of colour washing is the product of the curved surface area and the rate per square metre. Total cost = 577.5 × 6 = 3465.

Explanation

Let the height of the vertical part of the post still standing be h and the length of the part that bent over be l. The total height of the post is H = h + l. When the top touches the ground, it forms a right-angled triangle where the base is 8√3 m and the angle with the horizontal is 30°.

Using the tangent ratio for the vertical height: tan 30° = (h)/(8√(3)) (1)/(√(3)) = (h)/(8√(3)) h = 8 m

Using the cosine ratio for the bent part (hypotenuse): cos 30° = 8√3{l} √3{2} = 8√3{l} l = 16 m

The total height of the telegraph post is the sum of these two parts: H = h + l = 8 + 16 = 24 m

Explanation

Given the linear equation x + 3y = -3x + y, we rearrange the terms to find the relationship between x and y. Adding 3x to both sides and subtracting 3y from both sides gives:

x + 3x = y - 3y 4x = -2y

Dividing both sides by 2, we get 2x = -y, which simplifies to the ratio (x)/(y) = -½.

The expression to be evaluated is (x²)/(2y²). This can be rewritten as:

½ · ((x)/(y))²

Substituting the value of the ratio (x)/(y) = -½ into the expression:

½ · (-½)² = ½ · ¼ = ⅛

Explanation

In a circle, the tangents drawn from an external point are perpendicular to the radii at the points of contact. Let P be the external point, O be the center of the circle, and A and B be the points of contact. The radii OA and OB are perpendicular to the tangents PA and PB respectively, making ∠ OAP = 90° and ∠ OBP = 90°.

In the quadrilateral OAPB, the sum of the interior angles is 360°. The relation is expressed as ∠ AOB + ∠ OAP + ∠ APB + ∠ OBP = 360°. Substituting the known values, we get 72° + 90° + ∠ APB + 90° = 360°. This simplifies to ∠ AOB + ∠ APB = 180°, which means the angle subtended by the chord at the center and the angle between the tangents are supplementary. Substituting the given angle at the center, ∠ APB = 180° - 72° = 108°.

Explanation

The Triangle Inequality Theorem states that for a triangle to be constructed, the sum of the lengths of any two sides must be strictly greater than the length of the third side. If the sides are a, b, and c, where c is the longest side, the condition a + b > c must be satisfied. For the measurements 8 cm, 15 cm, and 17 cm, the sum of the two shorter sides is 8 + 15 = 23 cm. Since 23 cm is greater than the longest side of 17 cm (23 > 17), a triangle can be formed. Additionally, these specific lengths satisfy the Pythagorean theorem because 8² + 15² = 64 + 225 = 289, which is equal to 17², indicating that they form a right-angled triangle.

Explanation

Given the equation sinθ + cscθ = 2, we can use the trigonometric identity cscθ = (1)/(sinθ) to rewrite the expression as sinθ + (1)/(sinθ) = 2.

Let sinθ = x. The equation becomes x + ¹⁄ₓ = 2. Multiplying the entire equation by x gives x² + 1 = 2x, which can be rearranged into the quadratic form x² - 2x + 1 = 0. This is a perfect square binomial (sinθ - 1)² = 0. Solving for sinθ yields sinθ = 1.

Since sinθ = 1, the value of cscθ is also ¹⁄₁ = 1. Substituting these values into the required expression sin⁷θ + csc⁷θ, we get: (1)⁷ + (1)⁷ = 1 + 1 = 2

Explanation

To find the number of cows, let H be the number of hens and C be the number of cows. Since each animal has one head, the first equation is H + C = 50. Hens have 2 feet and cows have 4 feet, so the second equation is 2H + 4C = 142.

A quick shortcut to find the number of four-legged animals (cows) in such problems is to assume all animals are two-legged. If all 50 animals were hens, the total number of feet would be 50 × 2 = 100. The problem states there are actually 142 feet, which is an excess of 142 - 100 = 42 feet. Since each cow contributes 2 more feet than a hen (4 - 2 = 2), the number of cows is found by dividing the excess feet by the difference in feet per animal.

Using the formula: Number of cows = (Total feet - (2 × Total heads))/(2) C = (142 - (2 × 50))/(2) C = (142 - 100)/(2) C = ⁴²⁄₂ = 21

Explanation

Let n be the total number of students in the class. Using the weighted average formula A = (n₁ A₁ + n₂ A₂)/(n₁ + n₂), where A is the combined average, n₁ and n₂ are the counts of the subgroups, and A₁ and A₂ are their respective averages:

a = (10b + (n - 10)c)/(n)

Multiply both sides by n to find the total height: na = 10b + nc - 10c

Rearrange the equation to group all terms involving n on one side: na - nc = 10b - 10c

Factor out n and 10 from their respective sides: n(a - c) = 10(b - c)

Solving for n gives: n = (10(b - c))/(a - c)

This expression defines the total number of students in the class.

Explanation

To solve this expression, rationalize each denominator by multiplying the numerator and the denominator by the conjugate of the denominator, applying the identity (a + b)(a - b) = a² - b².

The first term is rationalized as follows: 3√7{√5 + √2} × √5 - √2{√5 - √2} = 3√7(√5 - √2){5 - 2} = 3(√(35) - √(14)){3} = √(35) - √(14)

The second term is rationalized by treating the denominator as √7 + √2: 5√5{√7 + √2} × √7 - √2{√7 - √2} = 5√5(√7 - √2){7 - 2} = 5(√(35) - √(10)){5} = √(35) - √(10)

The third term is rationalized as follows: 2√2{√7 + √5} × √7 - √5{√7 - √5} = 2√2(√7 - √5){7 - 5} = 2(√(14) - √(10)){2} = √(14) - √(10)

Now, substitute these simplified terms back into the original expression: (√(35) - √(14)) - (√(35) - √(10)) + (√(14) - √(10)) √(35) - √(14) - √(35) + √(10) + √(14) - √(10)

Canceling the equal and opposite terms: (√(35) - √(35)) + (-√(14) + √(14)) + (√(10) - √(10)) = 0

Explanation

The given equation is (a+b-6)² + a²+b²+1+2b = 2ab+2a. Rearranging the terms by moving everything to the left side, we obtain (a+b-6)² + a²+b²+1-2ab-2a+2b = 0.

The expression a²+b²+1-2ab-2a+2b can be identified as a perfect square using the identity (x+y+z)² = x²+y²+z²+2xy+2yz+2zx. By setting x=a, y=-b, and z=-1, we see that (a-b-1)² = a²+(-b)²+(-1)²+2(a)(-b)+2(-b)(-1)+2(a)(-1) = a²+b²+1-2ab+2b-2a. This matches the second part of our equation exactly.

Substituting this back into the original equation gives (a+b-6)² + (a-b-1)² = 0. Since the sum of two squares of real numbers is zero if and only if each square is individually zero, we can set up the following system of linear equations: a+b-6 = 0 ⟹ a+b = 6 a-b-1 = 0 ⟹ a-b = 1

To find the value of a, we add the two equations together: (a+b) + (a-b) = 6 + 1 2a = 7 a = 3.5

Explanation

The payment for a contract is divided among the workers in proportion to the amount of work performed by each. Since A, B, and C work together for the same duration of 3 days, the payment is distributed according to the ratio of their individual efficiencies.

To determine these efficiencies, assume the total work is 15 units, which is the least common multiple of the individual and combined completion times (5, 15, and 3 days). The efficiency of A is ¹⁵⁄₅ = 3 units per day. The efficiency of B is ¹⁵⁄₁₅ = 1 unit per day. The combined efficiency of A, B, and C together is ¹⁵⁄₃ = 5 units per day.

The individual efficiency of C is found by subtracting the efficiencies of A and B from the combined total: Efficiency of C = 5 - (3 + 1) = 1 unit per day.

The total payment of Rs. 250 corresponds to the total efficiency of 5 units. The ratio of work done by A, B, and C is 3 : 1 : 1. C's share is calculated as: Amount for C = (Efficiency of C)/(Total Efficiency) × Total Payment Amount for C = ⅕ × 250 = 50

Explanation

The relationship between the List Price (LP) and the Cost Price (CP) involving both a profit percentage and a discount percentage is given by the formula:

LP = CP × (100 + Profit%)/(100 - Discount%)

Given that the Cost Price is Rs. 900, the desired profit is 20%, and the discount offered is 10%, we substitute these values into the formula:

LP = 900 × (100 + 20)/(100 - 10)

LP = 900 × ¹²⁰⁄₉₀

LP = 10 × 120 = 1200

Alternatively, the Selling Price (SP) needed to achieve a 20% profit on the Cost Price of Rs. 900 is calculated as 900 × 1.20 = 1080. Since a 10% discount is allowed on the List Price to arrive at this Selling Price, the relationship is 0.90 × LP = 1080. Solving for the List Price yields LP = (1080)/(0.90) = 1200.

Explanation

Let the cost price (CP) of one article be x. The total cost price of 28 articles is 28x. According to the problem, the selling price (SP) of 21 articles is equal to the cost price of 28 articles, which gives the equation 21 × SP = 28x. Solving for the selling price of a single article, we get SP = ²⁸ˣ⁄₂₁ = ⁴ˣ⁄₃.

Profit is determined by the difference between the selling price and the cost price: Profit = SP - CP = ⁴ˣ⁄₃ - x = ˣ⁄₃. To find the profit percentage, we use the formula (Profit)/(CP) × 100. Substituting the calculated values into the formula results in (x/3)/(x) × 100 = ⅓ × 100 = 33 ⅓%.

Explanation

The percentage increase is calculated using the formula: Percentage Increase = (New Value - Original Value)/(Original Value) × 100

The increase in the number of cars sold is: 51,300 - 41,800 = 9,500

Substituting the values into the formula: Percentage Increase = (9,500)/(41,800) × 100 Percentage Increase = (9,500)/(418) = ⁹⁵⁰⁰⁄₄₁₈ = ⁴⁷⁵⁰⁄₂₀₉

Dividing both numerator and denominator by 19: (4750 ÷ 19)/(209 ÷ 19) = ²⁵⁰⁄₁₁

Converting the improper fraction to a mixed fraction: 250 ÷ 11 = 22 with a remainder of 8 ²⁵⁰⁄₁₁ = 22 ⁸⁄₁₁%

Explanation

The effective speed of a boat traveling with the current, known as downstream speed, is the sum of the boat speed in still water and the speed of the stream. In this case, the downstream speed is 20 + 5 = 25 km/hr.

The time required to cover a distance is calculated using the standard formula: Time = (Distance)/(Speed)

Substituting the values into the formula: Time = ¹⁰⁰⁄₂₅ = 4 hours

Explanation

When a sum of money doubles itself, the simple interest earned is equal to the principal amount itself (SI = P). Using the simple interest formula SI = (P × R × T)/(100), where P is the principal, R is the rate of interest, and T is the time period in years, we can substitute the given values to find the rate.

P = (P × R × 8)/(100)

By canceling P from both sides, the equation becomes:

1 = (8R)/(100)

Rearranging to solve for R:

R = ¹⁰⁰⁄₈

R = 12.5%

The decimal 12.5% is equivalent to the mixed fraction 12 ½%.

Explanation

The sum of squares of real numbers is zero if and only if each individual term is equal to zero. This is because a² ≥ 0 for any real number a, and the sum of non-negative values can only result in zero if every value is zero.

Applying this principle to the given equation (x - 5)² + (y - 2)² + (z - 9)² = 0, we set each squared term to zero:

x - 5 = 0 ⟹ x = 5 y - 2 = 0 ⟹ y = 2 z - 9 = 0 ⟹ z = 9

Substituting these values into the required expression (x + y - z):

x + y - z = 5 + 2 - 9 x + y - z = 7 - 9 x + y - z = -2

Explanation

The given equation is x + ¹⁄ₓ = 3. To find the value of x⁸ + (1)/(x⁸), we use the algebraic property that if x + ¹⁄ₓ = k, then x² + (1)/(x²) = k² - 2. By applying this transformation repeatedly, we double the exponent in each step.

First, squaring the initial expression gives the value for the second power: x² + (1)/(x²) = 3² - 2 = 7

Next, squaring the result for the second power gives the value for the fourth power: x⁴ + (1)/(x⁴) = 7² - 2 = 47

Finally, squaring the result for the fourth power gives the value for the eighth power: x⁸ + (1)/(x⁸) = 47² - 2

Calculating the square of 47: 47² = (50 - 3)² = 2500 - 300 + 9 = 2209

Subtracting 2 from this result yields: 2209 - 2 = 2207

This iterative squaring process follows the standard power-doubling rule for algebraic identities of this form.

Explanation

In △ PQR, the sum of all interior angles must be 180°. Given that ∠ PQR = 40° and ∠ QRP = 60°, the third angle ∠ QPR is calculated as: ∠ QPR = 180° - (∠ PQR + ∠ QRP) ∠ QPR = 180° - (40° + 60°) = 80°

According to the circle theorem, the angle subtended by an arc at the centre of a circle is double the angle subtended by it at any point on the remaining part of the circle. Here, the arc QR subtends ∠ QPR at the circumference point P. Therefore, the angle subtended by arc QR at the centre is: Angle at center = 2 × ∠ QPR Angle at center = 2 × 80° = 160°

Explanation

Let O be the common center of the two concentric circles and AB be the chord of the larger circle that touches the smaller circle at point P. Because AB is tangent to the smaller circle, the radius OP is perpendicular to AB. In the right-angled triangle OPA, the hypotenuse OA is the radius of the larger circle (17 cm) and the altitude OP is the radius of the smaller circle (8 cm).

According to the Pythagoras theorem: AP² = OA² - OP² AP² = 17² - 8² AP² = 289 - 64 = 225 AP = √(225) = 15 cm

A perpendicular line from the center of a circle to a chord bisects that chord. Since OP is perpendicular to AB, P is the midpoint of the chord AB. Thus, the total length of the chord is: AB = 2 × AP AB = 2 × 15 = 30 cm

Explanation

The given ratio of the angles of the triangle is 1 : ⅔ : 3. To convert this into a simplified integer ratio, multiply each term by the denominator 3, resulting in 3 : 2 : 9. The sum of the components of this ratio is 3 + 2 + 9 = 14. Based on the angle sum property of a triangle, the sum of all interior angles is 180°. The smallest angle corresponds to the smallest part of the ratio, which is 2. The value of the smallest angle is calculated by the formula (Smallest Ratio Part)/(Sum of Ratio Parts) × 180°. Substituting the values, we get ²⁄₁₄ × 180° = ⅐ × 180° = (180°)/(7). Dividing 180 by 7 gives a quotient of 25 and a remainder of 5, which expressed as a mixed fraction is 25 (5°)/(7).

Explanation

The sum of the angles is given as a + β = 90° and their ratio is a : β = 2 : 1. We can determine the measure of each angle by dividing the total sum into three parts (2 + 1 = 3). Thus, a = ⅔ × 90° = 60° and β = ⅓ × 90° = 30°. The required ratio is cos a : cos β, which corresponds to cos 60° : cos 30°. Using standard trigonometric values, cos 60° = ½ and cos 30° = √3{2}. The ratio becomes ½ : √3{2}, which simplifies directly to 1 : √3.