SSC CHSL Tier 1 2015 Quantitative Aptitude Questions with Solutions

Explanation

The relation between Marked Price (MP), Selling Price (SP), and Discount percentage is given by:

SP = MP × (1 - Discount%)

Substituting the given values into the formula:

1200 = MP × (1 - 0.20) 1200 = MP × 0.80 MP = 1200 ÷ 0.80 MP = 1500

Explanation

Total annual income is the sum of total annual expenditure and annual savings.

Total annual expenditure = (5 × 1200) + (7 × 1300) = 6000 + 9100 = ₹15100

Total annual income = Total annual expenditure + Annual savings = 15100 + 2900 = ₹18000

Monthly average income = Total annual income / 12 = 18000 / 12 = ₹1500

Explanation

Using the compound interest formula: A = P(1 + R/100)ⁿ

Substitute the given values: 1348.32 = 1200(1 + R/100)²

Divide both sides by 1200: (1 + R/100)² = 1348.32 / 1200 (1 + R/100)² = 1.1236

Taking the square root of both sides: 1 + R/100 = √1.1236 1 + R/100 = 1.06

Subtract 1 from both sides: R/100 = 0.06 R = 6%

Explanation

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

Given the ratio of the principal P to the simple interest SI for 5 years is 10 : 3, we can assume P = 10x and SI = 3x, with T = 5 years.

Substituting these values into the formula: 3x = (10x × R × 5) / 100

Simplifying the equation: 3x = (50x × R) / 100 3x = x × R / 2 3 = R / 2 R = 6%

Explanation

In the right-angled triangle ABC with B = 90 degrees, the hypotenuse AC is calculated using the Pythagorean theorem as √(8² + 15²) = 17 cm. The sine of angle C is the ratio of the opposite side to the hypotenuse, which is AB/AC = 8/17.

Explanation

Let CP be the cost price of the hand-cart.

When selling at a loss of 25%, the selling price is given by: SP₁ = CP × (1 - Loss/100) 720 = CP × (1 - 25/100) 720 = CP × ¾ CP = 720 × 4/3 = 960

To achieve a gain of 25%, the new selling price is given by: SP₂ = CP × (1 + Gain/100) SP₂ = 960 × (1 + 25/100) SP₂ = 960 × 5/4 = 1200

Explanation

Using the trigonometric identities sin(90° - θ) = cos θ and cos(90° - θ) = sin θ, the given equation simplifies to:

cos θ + cos θ = √2 sin θ

2 cos θ = √2 sin θ

Dividing both sides by sin θ gives:

2 cot θ = √2

cot θ = √2 / 2 = 1 / √2

Using the fundamental identity cosec² θ = 1 + cot² θ:

cosec² θ = 1 + (1 / √2)²

cosec² θ = 1 + ½ = 3/2

Taking the square root of both sides:

cosec θ = √(3/2)

Explanation

Let the cost price of the article be 100%.

The transition from a 20% loss to a 5% profit represents a total percentage change of: 5% - (-20%) = 25% of the cost price

This 25% difference in selling price is equal to the monetary increase of ₹200.

Using the unitary method: 25% of the cost price = ₹200 1% of the cost price = ₹200 ÷ 25 = ₹8 100% of the cost price = ₹8 × 100 = ₹800

Explanation

In a cyclic quadrilateral, the sum of opposite angles is always 180°. For the cyclic quadrilateral ABCD, the angle opposite to ∠A is ∠C, and the angle opposite to ∠B is ∠D.

Applying this theorem: ∠A + ∠C = 180° 50° + ∠C = 180° ∠C = 180° - 50° = 130°

Similarly: ∠B + ∠D = 180° 80° + ∠D = 180° ∠D = 180° - 80° = 100°

Explanation

The volume of a solid sphere is given by the formula 4/3 × π × r³, and the volume of a hollow sphere is given by 4/3 × π × (R³ - r³), where R is the external radius and r is the internal radius.

The total volume of the two melted solid metallic spheres of radii 1 cm and 6 cm is: V = 4/3 × π × 1³ + 4/3 × π × 6³ V = 4/3 × π × (1 + 216) V = 4/3 × π × 217

The hollow sphere has a thickness of 1 cm. If its external radius is R, its internal radius is r = R - 1. The volume of this hollow sphere is: V = 4/3 × π × (R³ - (R - 1)³)

Equating the volumes because the total metallic mass remains conserved during melting: 4/3 × π × 217 = 4/3 × π × (R³ - (R - 1)³) 217 = R³ - (R³ - 3R² + 3R - 1) 217 = 3R² - 3R + 1 3R² - 3R - 216 = 0 R² - R - 72 = 0 (R - 9)(R + 8) = 0

Since the radius must be a positive quantity, solving the equation yields R = 9.

Explanation

To find a number divisible by each of 12, 18, 21, and 32, we calculate their Lowest Common Multiple (LCM).

The prime factorizations are: 12 = 2² × 3 18 = 2 × 3² 21 = 3 × 7 32 = 2⁵

The LCM is the product of the highest power of each prime factor involved: LCM = 2⁵ × 3² × 7 = 32 × 9 × 7 = 2016

The required number must be a multiple of this LCM. Multiplying 2016 by 2 gives 4032, which lies within the specified range of 4000 and 5000.

Explanation

The measure of each interior angle of a regular polygon with n sides is calculated using the formula (n - 2) × 180° / n.

For an 8-sided regular polygon, substitute n = 8 into the formula:

(8 - 2) × 180° / 8 = 6 × 180° / 8 = 1080° / 8 = 135°

Alternatively, the sum of the exterior angles of any polygon is 360°. For a regular octagon, each exterior angle is 360° / 8 = 45°. Because the interior and exterior angles are supplementary, the interior angle is 180° - 45° = 135°.

Explanation

Let the total profit be P. Since 5% of the total profit goes to charity, the remaining 95% of the profit is distributed between A and B in the investment ratio of 3 : 2.

A's share is 3/(3 + 2) of the distributed profit.

A's share = 3/5 × 95% of P

Substituting the given value of A's share:

8,550 = 0.6 × 0.95 × P

8,550 = 0.57 × P

P = 8,550 / 0.57

P = 15,000

Explanation

We are given the equation: sin A - cos A = (√3 - 1)/2

Squaring both sides of this equation: (sin A - cos A)² = ((√3 - 1)/2)²

Using the algebraic identity (a - b)² = a² + b² - 2ab: sin²A + cos²A - 2 sin A cos A = (3 + 1 - 2√3)/4

Since sin²A + cos²A = 1, we can substitute this into the equation: 1 - 2 sin A cos A = (4 - 2√3)/4

Simplifying the fraction on the right side: 1 - 2 sin A cos A = 1 - (√3)/2

Subtracting 1 from both sides of the equation: -2 sin A cos A = -(√3)/2

Dividing both sides by -2 yields: sin A cos A = (√3)/4

Explanation

Let the total money the man had in the beginning be x.

He gives 50% of his money to his son and 30% to his daughter, which accounts for 80% of his total money.

The remaining money (the rest) is 100% - 80% = 20% of x. This is equal to 0.20x.

He donates 80% of this rest amount to a trust, which leaves him with 20% of the rest amount.

The money left with him is 20% of 0.20x: 0.20 × 0.20x = 0.04x

We are given that he is left with ₹16,000: 0.04x = 16,000 x = 16,000 / 0.04 x = 4,00,000

Explanation

Let the two numbers be 5x and 8x.

The difference between the two numbers is: 8x - 5x = 3x

According to the question, this difference is equal to 48: 3x = 48 x = 16

The smallest number is represented by 5x: 5 × 16 = 80

Explanation

The given expression is a product of cosine values from 1° to 180°: cos 1° × cos 2° × cos 3° × ... × cos 180°

Among the terms in this product, there exists the term cos 90°. Since cos 90° = 0, multiplying any sequence of real numbers by zero results in a product of zero.

cos 1° × cos 2° × ... × cos 90° × ... × cos 180° = cos 1° × cos 2° × ... × 0 × ... × cos 180° = 0

Explanation

The time taken by Janardan to complete 2/3 of the work is 10 days.

Using the unitary method, the time taken to complete the entire work (1 whole work) is: 10 × 3/2 = 15 days

The time taken to complete 3/5 of the same work is: 15 × 3/5 = 9 days

Explanation

The simple interest is calculated using the formula SI = P × R × T / 100, where P is the principal, R is the rate of interest per annum, and T is the time period in years.

First, determine the number of days from 5th January to 31st May, 2013: January: 26 days (31 - 5) February: 28 days (2013 is a non-leap year) March: 31 days April: 30 days May: 31 days

Total number of days = 26 + 28 + 31 + 30 + 31 = 146 days

Convert the time period into years: T = 146 / 365 = 2/5 years

Now, substitute the values into the formula: SI = 36000 × 9.5 × (2 / 5) / 100 SI = 360 × 9.5 × 2 / 5 SI = 72 × 19 SI = 1368

Explanation

By the Pythagorean theorem, for a right-angled triangle with perpendicular sides a and b and hypotenuse c: a² + b² = c²

Given the hypotenuse c = 39 cm, we have: a² + b² = 39² = 1521

The difference between the other two sides is: |a - b| = 21

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

Substituting the values: 21² = 1521 - 2ab 441 = 1521 - 2ab 2ab = 1521 - 441 2ab = 1080 ab = 540

The area of the right-angled triangle is given by: Area = ½ab

Substituting the value of ab: Area = ½ × 540 = 270 sq.cm

Explanation

Total work is given by the product of the number of men and the number of days they work. Since the work done in both scenarios is equal, we can set up the equation:

(x + 4)(x + 5) = (x - 5)(x + 20)

Expanding both sides of the equation:

x² + 9x + 20 = x² + 15x - 100

Subtracting x² from both sides simplifies the equation to:

9x + 20 = 15x - 100

Rearranging the terms to solve for x:

15x - 9x = 100 + 20 6x = 120 x = 20

Explanation

Since (2, 0) is a solution of the linear equation 2x + 3y = K, substituting x = 2 and y = 0 into the equation gives:

2(2) + 3(0) = K

Simplifying the equation:

4 + 0 = K K = 4

Explanation

In triangle ABC, D and E are the midpoints of AB and AC respectively. By the midpoint theorem, DE is parallel to BC and DE = ½ BC.

Since DE is parallel to BC, triangle ODE is similar to triangle OBC. The ratio of their corresponding sides is DE : BC = 1 : 2. Consequently, the ratio of their heights from O is also 1 : 2.

Let h be the height of triangle ABC corresponding to base BC. The distance between the parallel lines DE and BC is ½ h. Since the heights of ODE and OBC are in the ratio 1 : 2, the height of triangle ODE is 1/3 of this distance: height of ODE = 1/3 × ½ h = 1/6 h

The area of triangle ODE is: Area(ODE) = ½ × DE × height of ODE = ½ × (½ BC) × (1/6 h) = 1/24 × BC × h

Since Area(ABC) = ½ × BC × h: Area(ODE) = 1/12 × Area(ABC)

The ratio of areas of triangle ODE : triangle ABC is 1 : 12.

Explanation

When two trains run in opposite directions, their relative speed is the sum of their individual speeds, and the total distance to be covered to cross each other is the sum of their lengths.

Total distance = 150 m + 180 m = 330 m

Relative speed = 50 km/hr + 58 km/hr = 108 km/hr

To convert the relative speed from km/hr to m/s: Relative speed = 108 × 5/18 = 30 m/s

Time taken to cross each other = Total distance / Relative speed = 330 / 30 = 11 seconds

Explanation

Let the angle be x.

The complementary angle of x is 90° - x, and the supplementary angle of x is 180° - x.

According to the problem statement: 90° - x = (180° - x) ÷ 4

Multiplying both sides of the equation by 4: 360° - 4x = 180° - x

Grouping the like terms on either side of the equation: 360° - 180° = 4x - x 180° = 3x x = 60°

Explanation

Using the trigonometric identity for specific angles, we know that tan(π/3) = √3. Given the equation:

tan(π/2 - α/2) = √3

We can equate the angles:

π/2 - α/2 = π/3

Solving for α/2:

α/2 = π/2 - π/3 α/2 = π/6

Multiplying both sides by 2 gives:

α = π/3

Now, we calculate cos α:

cos α = cos(π/3) = ½

Explanation

The required number is of the form N = k × LCM(30, 36, 80) + 11, where k is an integer.

To find the LCM of 30, 36, and 80, we use their prime factorization: 30 = 2 × 3 × 5 36 = 2² × 3² 80 = 2⁴ × 5

LCM(30, 36, 80) = 2⁴ × 3² × 5 = 16 × 9 × 5 = 720

Thus, the number is of the form N = 720k + 11.

To find the number that lies between 1000 and 2000, we substitute k = 2: N = 720 × 2 + 11 = 1440 + 11 = 1451

Since 1451 lies in the given range of 1000 to 2000, it satisfies all the conditions.

Explanation

Let the whole number be N. According to the given condition: 51% of N = 714 (51/100) × N = 714 N = (714 × 100) / 51 N = 1400

Now, we calculate 25% of this number: 25% of N = (25/100) × 1400 25% of N = ¼ × 1400 25% of N = 350

Explanation

To arrange the fractions 4/3, -2/9, -7/8, and 5/12 in ascending order, we can convert each fraction to its decimal equivalent to compare their values:

-7/8 = -0.875 -2/9 ≈ -0.222 5/12 ≈ 0.417 4/3 ≈ 1.333

Arranging these values from the smallest to the largest on the real number line:

-0.875 < -0.222 < 0.417 < 1.333

Replacing the decimals with their corresponding fractions gives:

-7/8 < -2/9 < 5/12 < 4/3

Explanation

By the rule of alligation, the ratio of the number of boys to the number of girls is: (75 - 72) : (72 - 70) = 3 : 2

The total number of students is 50. The number of boys in the class is (3 / (3 + 2)) × 50 = (3 / 5) × 50 = 30.

Explanation

The relationship between Selling Price (SP), Marked Price (MP), and Discount Percentage is given by the formula:

SP = MP × (100 - Discount %) / 100

Substituting the given values:

255 = MP × (100 - 15) / 100 255 = MP × 85 / 100 MP = (255 × 100) / 85 MP = 3 × 100 MP = 300

Explanation

The volume of water poured into the tank is 630 litres. Since 1000 litres is equal to 1 cubic metre (1 m³), the volume of this added water in cubic metres is: Volume = 630 / 1000 = 0.63 m³

The length of the cuboidal tank is 2.1 m and its breadth is 1.5 m. Let the rise in the water level be h metres. The volume of the water added is equal to the product of the base area of the tank and the rise in height: Volume = Length × Breadth × Rise in height 0.63 = 2.1 × 1.5 × h 0.63 = 3.15 × h h = 0.63 / 3.15 = 0.20 metres

Explanation

Let the age of the new boy be x years. Since one boy leaves and another joins, the total number of boys in the class remains 30.

The change in the average age of the class is 16.9 - 17 = -0.1 years.

The total change in the sum of the ages is the number of boys multiplied by the change in the average: 30 × (-0.1) = -3 years

This means the sum of the ages decreased by 3 years. Therefore, the age of the new boy must be 3 years less than the age of the boy who left: x = 18 - 3 = 15 years

Explanation

The smallest prime number is 2. The greatest prime number less than 100 is 97. The difference between these two prime numbers is 97 - 2 = 95.

Explanation

The cost price of 12 kg of potatoes is calculated using the formula: Cost Price = (Selling Price × 100) / (100 + Gain%)

Substituting the given values: Cost Price of 12 kg = (63 × 100) / (100 + 5) = 6300 / 105 = ₹60

The cost price per kg of potatoes is: ₹60 / 12 = ₹5

The cost price of 50 kg of potatoes is: 50 × ₹5 = ₹250

Selling 50 kg of potatoes for ₹247.50 results in a loss because the selling price is less than the cost price: Loss = Cost Price - Selling Price = 250 - 247.50 = ₹2.50

The loss percent is calculated as: Loss Percent = (Loss / Cost Price) × 100 Loss Percent = (2.50 / 250) × 100 = 1%

Explanation

The sum of all interior angles of an n-sided polygon is given by the formula (n - 2) × 180°, which is equivalent to (n - 2) × 2 right angles.

Equating this formula to the given sum of 14 right angles: (n - 2) × 2 = 14 n - 2 = 7 n = 9

Explanation

Given the ratio a/b = 25/6, we can divide both the numerator and the denominator of the expression (a² - b²)/(a² + b²) by b² to express it in terms of the ratio a/b.

This yields the simplified form: ((a/b)² - 1) / ((a/b)² + 1)

Substituting the value of a/b = 25/6 into the formula: ((25/6)² - 1) / ((25/6)² + 1) = (625/36 - 1) / (625/36 + 1) = (625 - 36) / (625 + 36) = 589 / 661

Explanation

Let the time taken by both trains to meet be t hours.

The distance covered by the first train is 14t km, and the distance covered by the second train is 21t km.

The difference in the distances travelled by the two trains is given as 70 km: 21t - 14t = 70 7t = 70 t = 10 hours

The total distance between the two stations is the sum of the distances travelled by both trains: Total Distance = 14t + 21t = 35t Substituting t = 10: Total Distance = 35 × 10 = 350 km

Explanation

The selling price after successive discounts can be calculated directly using the formula:

Selling Price = List Price × (1 - d₁/100) × (1 - d₂/100)

Substituting the given values where the List Price is ₹300, the first discount d₁ is 15%, and the second discount d₂ is 10%:

Selling Price = 300 × (1 - 15/100) × (1 - 10/100) Selling Price = 300 × 0.85 × 0.90 Selling Price = 300 × 0.765 Selling Price = ₹229.50

Explanation

The volume of a prism is calculated using the formula: Volume = Area of the base × Height of the prism

The base of the prism is an isosceles triangle with two equal sides of 5 cm and a third side of 6 cm. Drawing an altitude from the apex to the third side bisects it into two segments of 3 cm each, forming a right-angled triangle.

By the Pythagorean theorem, the height of this triangular base is: √(5² - 3²) = √16 = 4 cm

The area of the triangular base is: ½ × base × height = ½ × 6 × 4 = 12 cm²

Multiplying the base area by the height of the prism (8 cm) yields the volume: Volume = 12 × 8 = 96 cm³

Explanation

Let x = √(6 + √(6 + √(6 + ...)))

Since the nested radical continues infinitely, we can substitute the inner infinite radical as x: x = √(6 + x)

Squaring both sides of the equation: x² = 6 + x

Rearranging this into a standard quadratic equation: x² - x - 6 = 0

Factoring the quadratic equation: (x - 3)(x + 2) = 0

This yields two possible values for x: x = 3 or x = -2

Since the square root of a positive real number is always positive, the value of x must be greater than zero. Therefore, x = 3.

Alternatively, an infinite nested radical of the form √(n + √(n + √(n + ...))) where n = k(k + 1) always evaluates to the larger consecutive integer, k + 1. Since 6 can be factored into consecutive integers as 2 × 3, the value of the expression is 3.

Explanation

To find the ratio A:B:C, we need to make the value of the common variable, A, equal in both given ratios.

We are given: A:B = 2:1 A:C = 1:3

By multiplying both terms of the second ratio, A:C, by 2, we get: A:C = (1 × 2) : (3 × 2) = 2:6

Now that the value of A is the same in both representations, we can combine them directly: A:B:C = 2:1:6

Explanation

The annual income from simple interest decreases because of the reduction in the interest rate. Let the principal sum deposited be P.

The decrease in the rate of interest per annum is: 5% - 3½% = 1.5% = 3/2%

Using the formula for simple interest, the reduction in annual income for a time period of 1 year is given by: Decrease in Income = P × (Decrease in Rate / 100) × Time

Substituting the given values into the formula: 105 = P × (1.5 / 100) × 1 105 = P × (3 / 200) P = (105 × 200) / 3 P = 35 × 200 P = 7000

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab, we can square both sides of the given equation to find the value of x² + 1/x².

Given: x + 1/x = 3

Squaring both sides: (x + 1/x)² = 3²

Expanding the left side: x² + 2 × x × 1/x + 1/x² = 9

Simplifying the terms: x² + 2 + 1/x² = 9

Subtracting 2 from both sides: x² + 1/x² = 9 - 2 x² + 1/x² = 7

Explanation

For a point (x, y) to lie on the graph of the linear equation y = x, its x-coordinate and y-coordinate must be equal. In the coordinates (1, 1), the value of x is 1 and the value of y is 1, which satisfies the equation 1 = 1. Therefore, the graph of the equation passes through this point.

Explanation

We use the complementary angle relation cos(90° - θ) = sin θ.

Since n = cos 70°, we can express n as: n = cos(90° - 20°) = sin 20°

Given m = cos 20°, we substitute these values into the expression m² + n²: m² + n² = (cos 20°)² + (sin 20°)² = cos² 20° + sin² 20°

Using the fundamental Pythagorean trigonometric identity sin² θ + cos² θ = 1 where θ = 20°: cos² 20° + sin² 20° = 1