SSC CHSL Tier 1 2011 Quantitative Aptitude Questions with Solutions

Explanation

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

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

Substituting the given values of d₁ = 25 and d₂ = 5:

Equivalent discount = 25 + 5 - (25 × 5 / 100) Equivalent discount = 30 - 1.25 Equivalent discount = 28.75%

Explanation

The relationship between Cost Price (CP), List Price (LP), Discount percentage, and Gain percentage is given by the direct formula:

LP / CP = (100 + Gain%) / (100 - Discount%)

Substituting the given values of Gain% = 8% and Discount% = 10%:

LP / CP = (100 + 8) / (100 - 10) LP / CP = 108 / 90 LP / CP = 6 / 5

The percentage hike in the cost price to determine the list price is:

Percentage Hike = (LP - CP) / CP × 100 Percentage Hike = (6 - 5) / 5 × 100 Percentage Hike = 1/5 × 100 = 20%

Explanation

Let us apply the rule of alligation on the proportion of milk in both vessels.

The fraction of milk in vessel A is 4/7. The fraction of milk in vessel B is 2/5. The required fraction of milk in the final mixture is 1/2.

Applying the alligation formula: Ratio of Vessel A to Vessel B = |Milk in B - Required Milk| : |Required Milk - Milk in A| Ratio = |2/5 - 1/2| : |1/2 - 4/7| Ratio = |4/10 - 5/10| : |7/14 - 8/14| Ratio = 1/10 : 1/14

Multiplying both terms by 70 to simplify the fractions: Ratio = 7 : 5

Explanation

Let the present ages of Sumit and Prakash be 2x and 3x respectively.

The difference in their present ages is given as 6 years: 3x - 2x = 6 x = 6

Thus, Sumit's present age is 2 × 6 = 12 years, and Prakash's present age is 3 × 6 = 18 years.

After 6 years, Sumit's age will be 12 + 6 = 18 years, and Prakash's age will be 18 + 6 = 24 years.

The ratio of their ages after 6 years is 18:24, which simplifies to 3:4.

Explanation

The selling price of the mixture is ₹9 per litre, on which a profit of 20% is made.

Cost price of the mixture = Selling Price / (1 + Profit Rate) = 9 / 1.2 = ₹7.5 per litre.

The cost price of pure milk is ₹10 per litre, and the cost price of water is taken as ₹0 per litre.

By the rule of alligation: Ratio of milk to water = (Mean Cost Price - Cost Price of Water) : (Cost Price of Pure Milk - Mean Cost Price) Ratio of milk to water = (7.5 - 0) : (10 - 7.5) = 7.5 : 2.5 = 3 : 1.

Explanation

When two men are replaced by two new persons, the average age of the 8 men increases by 3 years. This means the total age of the group increases by: 8 × 3 = 24 years

The sum of the ages of the two replaced men is: 30 + 34 = 64 years

To cause a net increase of 24 years in the total age, the sum of the ages of the two new persons must be: 64 + 24 = 88 years

Therefore, the average age of the two new persons is: 88 / 2 = 44 years

Explanation

The sum of the present ages of the 10 family members is: 10 × 20 = 200 years

Since the youngest member is 10 years old, the birth took place 10 years ago. Just before the birth, there were 9 members in the family.

To find the sum of the ages of these 9 members 10 years ago, we subtract the youngest member's age and 10 years of aging for each of the other 9 members from the current total: 200 - 10 - (9 × 10) = 100 years

The average age of the 9 members just before the birth of the youngest member was: 100 / 9 = 11 1/9 years (approximately 11.11 years)

Explanation

The average of any arithmetic progression, including consecutive even numbers, is equal to the midpoint of the sequence. For four consecutive even numbers, the average of 15 is the exact midpoint between the two middle terms.

Since consecutive even numbers differ by 2, the two middle numbers must be 15 - 1 = 14 and 15 + 1 = 16.

Thus, the four consecutive even numbers are 12, 14, 16, and 18. The second highest number in this sequence is 16.

Explanation

Selling price (SP) of each chair is 120.

The cost price (CP₁) of the first chair sold at 25% gain is calculated as: CP₁ = SP / (1 + Gain/100) = 120 / 1.25 = 96

The cost price (CP₂) of the second chair sold at 25% loss is calculated as: CP₂ = SP / (1 - Loss/100) = 120 / 0.75 = 160

Total Cost Price = CP₁ + CP₂ = 96 + 160 = 256 Total Selling Price = 120 + 120 = 240

Total loss in Rupees = Total Cost Price - Total Selling Price = 256 - 240 = 16

Explanation

Let the cost price of the book be x.

Selling the book at a 10% loss means the initial selling price is 90% of the cost price, which is 0.9x.

Selling the book at a 10% profit means the new selling price is 110% of the cost price, which is 1.1x.

The difference between these two selling prices is ₹108. This gives the relation:

1.1x - 0.9x = 108

0.2x = 108

x = 108 ÷ 0.2

x = 540

The cost price of the book is ₹540.

Explanation

Let the cost price of the cycle for A be x.

Since A sells the cycle to B at a profit of 5%, the cost price for B is: x × (1 + 5/100) = 1.05x

B then sells the cycle to C at a profit of 10%. The selling price for B, which is the price paid by C, is: 1.05x × (1 + 10/100) = 1.05x × 1.10 = 1.155x

We are given that C pays ₹ 2310 for the cycle. Therefore, we can set up the equation: 1.155x = 2310

Solving for x: x = 2310 / 1.155 x = 2310000 / 1155 x = 2000

Explanation

Let the two parts of the number be x and y.

According to the first condition, 80% of the first part is 3 more than 60% of the second part: 0.8x = 0.6y + 3 8x - 6y = 30 4x - 3y = 15 ... (1)

According to the second condition, 80% of the second part is 6 more than 90% of the first part: 0.8y = 0.9x + 6 8y - 9x = 60 9x - 8y = -60 ... (2)

To solve this system of linear equations, multiply equation (1) by 8 and equation (2) by 3: 32x - 24y = 120 27x - 24y = -180

Subtracting the second modified equation from the first: (32x - 24y) - (27x - 24y) = 120 - (-180) 5x = 300 x = 60

Substituting x = 60 into equation (1): 4(60) - 3y = 15 240 - 3y = 15 3y = 225 y = 75

The original number is the sum of the two parts: x + y = 60 + 75 = 135

Explanation

Let the original quantity of wheat obtained for one rupee be Q grams.

Since the price of wheat is reduced by 10%, the new price is 90% (or 9/10) of the original price.

The quantity of wheat obtained for a constant expenditure is inversely proportional to its price. Therefore, the new quantity obtained for one rupee is 10/9 of the original quantity Q.

The difference between the new quantity and the original quantity is given as 50 grams: (10/9) × Q - Q = 50 Q × (10/9 - 1) = 50 Q × (1/9) = 50 Q = 450

Explanation

The total time duration of the journey from 6:00 a.m. to 4:30 p.m. on the same day is computed by dividing it into two intervals. From 6:00 a.m. to 12:00 noon is 6 hours, and from 12:00 noon to 4:30 p.m. is 4.5 hours. Summing these values gives a total travel time of 10.5 hours.

Using the standard formula:

Distance = Speed × Time

Substitute the speed of 40 km/h and the total time of 10.5 hours into the formula:

Distance = 40 km/h × 10.5 hours = 420 km

Explanation

To find the average speed, we calculate the total distance traveled and divide it by the total time. The total distance is (6 hours * 5 km/h) + (3 hours * 6 km/h) = 30 km + 18 km = 48 km, and the total time is 6 hours + 3 hours = 9 hours. This gives an average speed of 48/9 = 16/3 = 5 1/3 km/h, which corresponds to option B.

Explanation

Simple interest remains constant each year. The difference in the amounts over the given periods represents the interest accumulated during that time.

Time difference = 4 - 2.5 = 1.5 years Interest for 1.5 years = 1067.20 - 1012 = 55.20 Interest per year = 55.20 / 1.5 = 36.80

Interest for 2.5 years = 36.80 × 2.5 = 92

Principal (P) = Amount in 2.5 years - Interest for 2.5 years = 1012 - 92 = 920

Rate of interest (R) = (Interest per year / Principal) × 100 R = (36.80 / 920) × 100 = 4%

Explanation

Under annual compound interest, the amount at the end of the third year is the amount at the end of the second year plus the interest earned on it during the third year.

Let the amount after two years be A₂ = 1440 and the amount after three years be A₃ = 1728.

The relation between consecutive years is: A₃ = A₂ × (1 + r/100)

Substitute the given values into the formula: 1728 = 1440 × (1 + r/100)

Divide both sides by 1440: 1.2 = 1 + r/100

Subtract 1 from both sides: 0.2 = r/100

Multiply by 100: r = 20

Explanation

Let the dimensions of the rectangular box be x, 2x, and 3x.

The total surface area of a cuboid is given by the formula: Total Surface Area = 2(lb + bh + hl)

Substituting the dimensions into the formula: 2(x × 2x + 2x × 3x + 3x × x) = 88 2(2x² + 6x² + 3x²) = 88 2(11x²) = 88 22x² = 88 x² = 4 x = 2

Thus, the dimensions of the box are 2 cm, 4 cm, and 6 cm.

The volume of the rectangular box is calculated using the formula: Volume = l × b × h Volume = 2 × 4 × 6 = 48 cm³

Explanation

In triangle ABC, let the base be BC = 6√3 cm. The two base angles are angle B = 30° and angle C = 60°. The third angle A must be 180° - (30° + 60°) = 90°, which makes ABC a right-angled triangle at A.

Using trigonometric ratios, the other two sides are: AB = BC × sin(60°) = 6√3 × (√3 / 2) = 9 cm AC = BC × sin(30°) = 6√3 × ½ = 3√3 cm

The area of triangle ABC can be expressed using either the two legs or the base and the height (h): Area = ½ × AB × AC = ½ × BC × h

Substituting the known values: 9 × 3√3 = 6√3 × h h = (27√3) / (6√3) = 4.5 cm

Explanation

Volume of a sphere of radius a is: V = 4/3 π a³

Volume of a right circular cone with base radius a and height h is: V = 1/3 π a² h

Since both volumes are equal, we set them equal to each other: 1/3 π a² h = 4/3 π a³

Dividing both sides by 1/3 π a² yields: h = 4a

Explanation

The volume of a cylinder is given by V = πr²h.

Let the radius r = 5x and the height h = 7x.

Substituting the given volume: 550 = (22/7) × (5x)² × 7x 550 = (22/7) × 25x² × 7x 550 = 22 × 25x³ 550 = 550x³ x³ = 1 x = 1

This gives the radius r = 5 cm and the height h = 7 cm.

The curved surface area of a cylinder is given by: CSA = 2πrh CSA = 2 × (22/7) × 5 × 7 CSA = 220 sq. cm.

Explanation

The area of an equilateral triangle with side length s is given by: Area = (√3/4)s²

If the side length is increased by 2, the new side length becomes s + 2. The new area is: New Area = (√3/4)(s + 2)²

The increase in area is the difference between the new area and the original area: Increase = (√3/4)(s + 2)² - (√3/4)s² Increase = (√3/4)((s + 2)² - s²) Increase = (√3/4)(s² + 4s + 4 - s²) Increase = (√3/4)(4s + 4) Increase = √3(s + 1)

Given that the increase in area is 3 + √3, we can equate the two expressions: √3(s + 1) = 3 + √3

Dividing both sides of the equation by √3: s + 1 = (3 + √3) / √3 s + 1 = √3 + 1 s = √3

The original length of each side of the triangle is √3 unit.

Explanation

Let the diameter of the Earth be dₑ and the diameter of the Moon be dₘ. According to the given condition, dₘ = dₑ / 4, which means dₑ / dₘ = 4.

The volume of a sphere with diameter d is given by the formula V = (π/6)d³. Since both the Earth and the Moon are assumed to be spherical, their volumes are directly proportional to the cubes of their respective diameters.

The ratio of the volume of the Earth (Vₑ) to that of the Moon (Vₘ) is calculated as: Vₑ / Vₘ = (dₑ / dₘ)³ = 4³ = 64

This yields the ratio of 64 : 1.

Explanation

A linear equation of the form x = k, where k is a constant, represents a locus of points that all share the same x-coordinate. On the Cartesian coordinate plane, this set of points forms a straight line that extends infinitely in the vertical direction. Since every point on this line is at a constant perpendicular distance of k units from the vertical axis, the resulting line is parallel to the y-axis.

Explanation

The algebraic identity for the given expression is: a³ + b³ + c³ - 3abc = ½(a + b + c)[(a - b)² + (b - c)² + (c - a)²]

Given the values a = 34, b = 33, and c = 33: a + b + c = 34 + 33 + 33 = 100 a - b = 34 - 33 = 1 b - c = 33 - 33 = 0 c - a = 33 - 34 = -1

Substituting these values into the formula: a³ + b³ + c³ - 3abc = ½ × 100 × [1² + 0² + (-1)²] = ½ × 100 × [1 + 0 + 1] = ½ × 100 × 2 = 100

Explanation

We are given the equation: (a² + b²)³ = (a³ + b³)²

Expanding both sides using algebraic identities: a⁶ + 3a⁴b² + 3a²b⁴ + b⁶ = a⁶ + 2a³b³ + b⁶

Subtracting a⁶ and b⁶ from both sides simplifies the equation to: 3a⁴b² + 3a²b⁴ = 2a³b³

Dividing the entire equation by a²b² (since a and b are non-zero to satisfy the denominators of the required expression): 3a² + 3b² = 2ab

Dividing both sides by ab: 3(a/b + b/a) = 2

Dividing by 3 gives: a/b + b/a = 2/3

Explanation

To find the value of the given expression, we divide both the numerator and the denominator by x²:

(x⁴ + 3x³ + 5x² + 3x + 1) ÷ x² = (x² + 1/x²) + 3(x + 1/x) + 5

(x⁴ + 1) ÷ x² = x² + 1/x²

We are given that: x + 1/x = 5

Squaring both sides of this equation: (x + 1/x)² = 5² x² + 2 + 1/x² = 25 x² + 1/x² = 23

Substituting these values into the simplified fraction: Numerator = 23 + 3(5) + 5 = 43 Denominator = 23

This gives the final simplified value of 43/23.

Explanation

In any triangle, the centroid, incenter, circumcenter, and orthocenter coincide if and only if the triangle is equilateral. When these four centers coincide at a single point, the line segment drawn from any vertex through this point to the opposite side acts simultaneously as the median, altitude, angle bisector, and perpendicular bisector.

If the altitude from vertex A to side BC is also the median, it divides the triangle into two right-angled triangles, ABD and ACD, where D is the midpoint of BC. Since BD = CD, AD is a common side, and the angles ADB and ADC are both 90°, the two triangles ABD and ACD are congruent by Side-Angle-Side symmetry. This congruence directly implies that the sides AB and AC are equal.

Applying this same geometric principle to the altitudes and medians from the other vertices establishes that BC = AB and AC = BC. Since AB = BC = AC, all three sides of the triangle are of equal length, which defines an equilateral triangle.

Explanation

The point of intersection O of the internal bisectors of ∠ABC and ∠ACB is the incenter of △ABC. The angle subtended by the incenter ∠BOC is related to the vertex angle ∠BAC by the standard relation:

∠BOC = 90° + ½ ∠BAC

Substituting the given value ∠BOC = 110° into the formula:

110° = 90° + ½ ∠BAC

Subtracting 90° from both sides:

½ ∠BAC = 20°

Multiplying by 2:

∠BAC = 40°

Explanation

Since AC is the diameter of the circumcircle of triangle ABC, the angle subtended by the semicircle at the circumference is a right angle. ∠ABC = 90°

Given that ∠CBE = 50°, we can calculate ∠ABE: ∠ABE = ∠ABC - ∠CBE = 90° - 50° = 40°

Angles subtended by the same arc AE at the circumference of the circle are equal. Therefore: ∠ACE = ∠ABE = 40°

Since the chord ED is parallel to the diameter AC, and EC is a transversal line intersecting them, the alternate interior angles are equal: ∠DEC = ∠ACE = 40°

Explanation

In any triangle ABC, the angle formed by the angle bisectors of ∠B and ∠C at the incenter O is given by the formula: ∠BOC = 90° + ∠BAC/2

Substituting the given value of ∠BOC into the formula: 110° = 90° + ∠BAC/2

Subtracting 90° from both sides: 20° = ∠BAC/2

Multiplying by 2: ∠BAC = 40°

Explanation

Using standard trigonometric identities, the terms inside the parentheses can be simplified as follows:

1/cos θ = sec θ 1/cot θ = tan θ

Substituting these into the given expression yields:

(sec θ + tan θ)(sec θ - tan θ)

Applying the algebraic identity (a + b)(a - b) = a² - b²:

sec² θ - tan² θ

According to the fundamental Pythagorean trigonometric identity, sec² θ - tan² θ = 1. Therefore, the numerical value of the expression is 1.

Explanation

Dividing the numerator and the denominator of the left-hand side of the given equation by cos θ:

(tan θ + 1) / (tan θ - 1) = 5/4

Applying the componendo and dividendo rule:

((tan θ + 1) + (tan θ - 1)) / ((tan θ + 1) - (tan θ - 1)) = (5 + 4) / (5 - 4)

2 tan θ / 2 = 9 / 1

tan θ = 9

Substituting the value of tan θ in the required expression:

(tan² θ + 1) / (tan² θ - 1) = (9² + 1) / (9² - 1)

(81 + 1) / (81 - 1) = 82 / 80 = 41 / 40

Explanation

Let P be the observation point on the ground, and G be the point on the ground vertically below the two planes. Let A be the lower plane at a height of 3125 m, and B be the higher plane directly above A.

In the right-angled triangle PGA, the angle of elevation of the lower plane A is 30 degrees. tan 30° = AG / PG 1/√3 = 3125 / PG PG = 3125 × √3 m

In the right-angled triangle PGB, the angle of elevation of the higher plane B is 60 degrees. tan 60° = BG / PG √3 = BG / (3125 × √3) BG = 3125 × 3 = 9375 m

The distance between the two planes at that instant is the difference between their heights: BG - AG = 9375 - 3125 = 6250 m