SSC CHSL Tier 1 2010 Quantitative Aptitude Questions with Solutions

Explanation

Downstream speed is calculated by dividing the distance by the downstream time. The downstream time of 48 minutes is converted to hours as 48/60 hours, which simplifies to 4/5 hours. Downstream speed (D) = 12 / (4/5) = 15 km/hr.

Upstream speed is calculated by dividing the distance by the upstream return time. The return time of 1 hour 20 minutes equals 80 minutes, which is converted to hours as 80/60 hours, simplifying to 4/3 hours. Upstream speed (U) = 12 / (4/3) = 9 km/hr.

The speed of the sailor in still water is the average of the downstream and upstream speeds: Speed in still water = (D + U) / 2 = (15 + 9) / 2 = 12 km/hr.

Explanation

The purity of gold is measured in carats, where 24-carat gold represents 100% pure gold.

The percentage of pure gold in a given carat rating is calculated using the formula: Percentage of pure gold = (Given Carat / 24) × 100

Substituting 22-carat into the formula: Percentage of pure gold = (22 / 24) × 100 Percentage of pure gold = (11 / 12) × 100 Percentage of pure gold = 1100 / 12 Percentage of pure gold = 275 / 3 Percentage of pure gold = 91 ⅔ %

Explanation

Given the relation: x + 1/x = 3

Multiplying both sides by x yields: x² + 1 = 3x

By correcting the typographical representation of 1/3 in the numerator of the target expression to 1/x, the expression to evaluate is: (x³ + 1/x) / (x² - x + 1)

To simplify the numerator, express it as a single fraction: x³ + 1/x = (x⁴ + 1)/x

Squaring both sides of the given relation: (x + 1/x)² = 3² x² + 2 + 1/x² = 9 x² + 1/x² = 7

Multiplying this equation by x² gives: x⁴ + 1 = 7x²

Substituting x⁴ + 1 = 7x² back into the numerator: x³ + 1/x = 7x²/x = 7x

For the denominator, substitute x² + 1 = 3x into the expression: x² - x + 1 = (x² + 1) - x = 3x - x = 2x

Dividing the simplified numerator by the simplified denominator: 7x / 2x = 7/2

Since x is non-zero, the variables cancel out to leave the constant value of 7/2.

Explanation

By the power of a power rule of exponents, (xᵃ)ᵇ = x⁽a × b).

Applying this rule to the given expression, we multiply the exponents: 2 × (-2) × (-1) = 4

Substituting this back into the base: (½)⁴ = 1/16

Explanation

The percentage reduction in consumption required to keep the total expenditure unchanged when the price increases is given by the formula:

Reduction percentage = (r / (100 + r)) × 100%

Here, r represents the percentage increase in the price of sugar, which is 25%. Substituting this value into the formula:

Reduction percentage = (25 / (100 + 25)) × 100% = (25 / 125) × 100% = (1 / 5) × 100% = 20%

Explanation

Under simple interest, the interest earned each year remains constant.

The simple interest for 1 year is the difference between the amounts at the end of 4 years and 3 years: Interest for 1 year = ₹925 - ₹850 = ₹75

The simple interest accumulated over 3 years is: Interest for 3 years = 3 × ₹75 = ₹225

The principal sum is obtained by subtracting the 3-year interest from the 3-year amount: Principal = ₹850 - ₹225 = ₹625

Explanation

Let the total capacity of the cistern be the least common multiple of 6 and 8, which is 24 units.

The filling rate of pipe A is 24 / 6 = 4 units per hour. The filling rate of pipe B is 24 / 8 = 3 units per hour.

When both pipes are opened together, their combined rate is 4 + 3 = 7 units per hour. In 2 hours, the amount of the cistern filled by both pipes is 7 × 2 = 14 units.

The remaining part of the cistern to be filled is 24 - 14 = 10 units.

Since pipe A is closed, pipe B fills this remaining part alone. The time taken by B to fill the remaining 10 units is 10 / 3 = 3 ⅓ hours.

Explanation

To find the ratio of the sides in integer terms, we multiply each part of the given ratio 1/3 : 1/4 : 1/5 by the least common multiple of their denominators. The LCM of 3, 4, and 5 is 60.

Multiplying each term by 60 gives: (1/3 × 60) : (1/4 × 60) : (1/5 × 60) = 20 : 15 : 12

Let the lengths of the sides be 20x, 15x, and 12x. The sum of these sides equals the perimeter of the triangle: 20x + 15x + 12x = 94 47x = 94 x = 2

The smallest side corresponds to the smallest ratio term, which is 12. Length of the smallest side = 12x = 12 × 2 = 24 cm.

Explanation

For any odd number of consecutive integers, the average is always equal to the middle term of the series. For seven consecutive integers, the middle term is the fourth term, which is given as 26.

To find the smallest (first) term, we subtract 1 step-by-step for three terms preceding the middle term: First term = 26 - 3 = 23

Alternatively, let the seven consecutive positive integers be represented as x, x + 1, x + 2, x + 3, x + 4, x + 5, and x + 6. The average of these seven numbers is: (7x + 21) / 7 = x + 3

Equating this to the given average of 26: x + 3 = 26 x = 23

Explanation

The 50% increase in the price on Rs. 24 represents an additional amount of: 24 × 50% = Rs. 12

This additional cost of Rs. 12 is equivalent to the price of the 4 eggs that can no longer be bought. Therefore, the present price of 4 eggs is Rs. 12.

The present price of 1 egg is: 12 ÷ 4 = Rs. 3

The present rate of eggs per dozen (12 eggs) is: 3 × 12 = Rs. 36

Explanation

The terms of the given sequence follow the pattern of adding 1 to the cube of the term's position. The formula for the nth term of this sequence is:

aₙ = n³ + 1

Applying this formula to find the 6th term of the sequence where n = 6:

a₆ = 6³ + 1 = 216 + 1 = 217

Explanation

To find how much percent B's income is less than A's income when A's income is r% more than B's, we use the standard percentage reduction formula:

Percentage less = (r ÷ (100 + r)) × 100%

Substituting r = 20 into the formula:

Percentage less = (20 ÷ (100 + 20)) × 100% Percentage less = (20 ÷ 120) × 100% Percentage less = (1 ÷ 6) × 100% Percentage less = 50/3% = 16 ⅔%

Explanation

Let the total work be 40 units. Since X can complete the work in 40 days, X's efficiency is 1 unit per day.

In 8 days, X completes: 8 days × 1 unit/day = 8 units of work.

The remaining work is: 40 - 8 = 32 units.

Since Y completes this remaining work in 16 days, Y's efficiency is: 32 / 16 = 2 units per day.

The combined efficiency of X and Y is: 1 + 2 = 3 units per day.

Therefore, the time taken by X and Y together to complete the total work is: 40 / 3 = 13⅓ days.

Explanation

For successive discount percentages d₁, d₂, and d₃, the final price multiplier of the marked price is calculated using the relation: Multiplier = (1 - d₁/100) × (1 - d₂/100) × (1 - d₃/100)

The net equivalent discount is given by: Net Discount = (1 - Multiplier) × 100%

For the successive discount series of 30%, 20%, and 10%: Multiplier = (1 - 0.30) × (1 - 0.20) × (1 - 0.10) Multiplier = 0.70 × 0.80 × 0.90 Multiplier = 0.504

Net Discount = (1 - 0.504) × 100% = 49.6%

This yields the lowest multiplier of 0.504, which translates to the maximum possible net discount for the customer.

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) when profit and discount percentages are given is:

MP / CP = (100 + Profit %) / (100 - Discount %)

Given: Cost Price (CP) = Rs. 200 Profit % = 35% Discount % = 25%

Substituting these values into the formula:

MP / 200 = (100 + 35) / (100 - 25)

MP / 200 = 135 / 75

Reducing the fraction on the right-hand side by dividing the numerator and denominator by 15:

MP / 200 = 9 / 5

MP = (9 / 5) × 200

MP = 9 × 40 = 360

Explanation

The formula to calculate the number of workers is:

Number of workers = Total weekly emoluments ÷ Average weekly emolument

Substituting the given values into the formula:

Number of workers = 1534 ÷ 118 = 13

The total number of workers in the factory is 13.

Explanation

The net percentage change in the area of a rectangle when its length increases by a% and its breadth increases by b% is given by the successive percentage formula:

Net percentage increase = a + b + (a × b)/100

Substituting the given values, a = 20 and b = 25:

Net percentage increase = 20 + 25 + (20 × 25)/100 = 45 + 500/100 = 45 + 5 = 50%

Explanation

Let the length of the train be L meters and its speed be v meters per second.

When the train crosses a bridge, the total distance covered is equal to the length of the train plus the length of the bridge. Therefore, the speed of the train can be determined by the difference in the distances covered divided by the difference in the time taken:

v = (300 - 240) / (21 - 18) v = 60 / 3 v = 20 m/s

To convert this speed from m/s to km/hr, we multiply by 18/5:

Speed = 20 × 18/5 = 72 km/hr

Explanation

We are given the relations A = (1/4)B and B = (1/2)C.

From the first relation, the ratio of A to B is A:B = 1:4. From the second relation, the ratio of B to C is B:C = 1:2.

To find the combined ratio A:B:C, we make the value representing B equal in both ratios. We can multiply both terms of the ratio B:C by 4: B:C = 4:8

Now, we can combine the ratios directly because the term for B is 4 in both: A:B:C = 1:4:8

Explanation

The circumference of the circular park is given by the formula 2πr = 176 m, where r is the inner radius of the park.

Using π = 22/7: 2 × 22/7 × r = 176 r = 176 × 7 / 44 r = 28 m

Since the road is 7 m wide and runs outside the park, the outer radius R is: R = r + 7 = 28 + 7 = 35 m

The area of the road is the difference between the area of the outer circle and the inner circle: Area of road = πR² - πr² = π(R² - r²) = π(R - r)(R + r)

Substituting the values: Area of road = 22/7 × (35 - 28) × (35 + 28) Area of road = 22/7 × 7 × 63 Area of road = 22 × 63 = 1386 m²

Explanation

Profit percentage is calculated using the formula: Profit % = (Profit / Cost Price) × 100

Given that the ratio of the cost price (CP) to the selling price (SP) is 10:11, we can assume: CP = 10 units SP = 11 units

The profit earned is: Profit = SP - CP = 11 - 10 = 1 unit

Substituting these values into the formula: Profit % = (1 / 10) × 100 = 10%

Explanation

The relationship between Marked Price, Selling Price, and Discount is given by the formula:

Marked Price = Selling Price / (1 - Discount Rate)

Given that the selling price is Rs. 629 and the discount is 15%:

Marked Price = 629 / (1 - 0.15) Marked Price = 629 / 0.85 Marked Price = 62900 / 85 Marked Price = 740

Explanation

The relation between the present population and the past population with a constant annual growth rate is given by the formula:

A = P × (1 + R/100)ⁿ

where A is the present population, P is the population n years ago, R is the annual growth rate, and n is the number of years.

Given: A = 67600 R = 4% n = 2

Substitute the values into the formula:

67600 = P × (1 + 4/100)² 67600 = P × (1 + 1/25)² 67600 = P × (26/25)² 67600 = P × (676/625)

Rearranging the equation to solve for P:

P = (67600 × 625) / 676 P = 100 × 625 P = 62500

Explanation

With the average age of A and B being 30 years, their sum of ages is A + B = 60. With the average age of B and C being 26 years, their sum of ages is B + C = 52. Subtracting the second equation from the first gives the difference between the ages of A and C as A - C = 8 years.

Explanation

Let the working capacity of a man be M and that of a woman be W.

Since the total work completed in both cases is the same, the total capacity required to complete the work is equal:

20 × W × 16 = 16 × M × 15

Dividing both sides of the equation by 16:

20 × W = 15 × M

Dividing both sides by 5:

4 × W = 3 × M

To find the ratio of the working capacity of a man to that of a woman (M : W):

M / W = 4 / 3

This gives the ratio of 4:3.

Explanation

Given the equation: a + 1/a + 1 = 0

Multiplying the entire equation by a: a² + 1 + a = 0 a² + a + 1 = 0

Using the standard algebraic identity: a³ - 1 = (a - 1)(a² + a + 1)

Substituting the value of a² + a + 1 = 0 into the identity: a³ - 1 = (a - 1) × 0 a³ - 1 = 0

We need to find the value of the expression: a⁴ - a

Factoring out a from the expression: a(a³ - 1)

Substituting the value of a³ - 1 = 0 into this expression gives a × 0 = 0.

Explanation

The volume of a cone is given by the formula V = ⅓ π r² h, where r is the radius of the base and h is the height of the cone.

When the height h is increased by 100%, the new height h' becomes h + 100% of h, which is 2h.

Substituting this new height into the volume formula while keeping the radius constant, the new volume V' is:

V' = ⅓ π r² (2h) = 2 (⅓ π r² h) = 2V

The percentage increase in the volume is:

Percentage Increase = ((V' - V) / V) × 100% = ((2V - V) / V) × 100% = 100%

Explanation

We are given the inequality: 7 < 2x - 3 < 17

Adding 3 to all parts of the inequality gives: 10 < 2x < 20

Dividing all parts by 2 gives: 5 < x < 10

Since x is a perfect square integer, the only perfect square that lies strictly between 5 and 10 is 9.

Explanation

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

Let the rates of interest of the two banks be R₁ and R₂. The difference in simple interest is given by:

(P × R₁ × T) / 100 - (P × R₂ × T) / 100 = 2.50

Factor out the common terms:

(P × T / 100) × (R₁ - R₂) = 2.50

Substitute the given values P = 500 and T = 2 into the equation:

(500 × 2 / 100) × (R₁ - R₂) = 2.50

10 × (R₁ - R₂) = 2.50

R₁ - R₂ = 2.50 / 10 = 0.25%

Explanation

The given expression can be written as a mixed fraction: 999 998/999 × 999

Using the property of mixed fractions, A B/C = A + B/C, the expression becomes: (999 + 998/999) × 999

Applying the distributive law of multiplication: = 999 × 999 + (998/999) × 999 = 999 × 999 + 998

To simplify the multiplication, write 999 as (1000 - 1): = (1000 - 1) × 999 + 998 = 999000 - 999 + 998 = 999000 - 1 = 998999

Explanation

When the cost price of N articles is equal to the selling price of M articles, the profit percentage is calculated using the formula:

Profit percentage = (N - M) / M × 100%

Substituting the given values, where N = 24 and M = 18:

Profit percentage = (24 - 18) / 18 × 100%

Profit percentage = 6/18 × 100%

Profit percentage = ⅓ × 100%

Profit percentage = 33 ⅓%

Explanation

Let the cost price (CP) of the article be x. Since the profit percentage is numerically equal to the cost price, the profit percentage is also x%.

Using the formula for selling price (SP) when there is a profit: SP = CP × (1 + Profit%/100)

Substituting the given values into the formula: 75 = x × (1 + x/100)

Multiplying both sides by 100 to clear the fraction: 7500 = 100x + x²

Rearranging this into a standard quadratic equation: x² + 100x - 7500 = 0

Factorizing the quadratic equation: x² + 150x - 50x - 7500 = 0 x(x + 150) - 50(x + 150) = 0 (x - 50)(x + 150) = 0

This gives two values for x, which are x = 50 or x = -150. Since the cost price cannot be negative, the cost price of the article is Rs. 50.

Explanation

Comparing the given values, we have 0.9, (0.9)² = 0.81, the square root of 0.9 which is approximately 0.9487, and 0.9 recurring (0.9) which is mathematically equal to 1. Since 1 is the largest among these values, 0.9 is the largest number.

Explanation

The volume of water collected on the roof equals the volume of the cylindrical vessel when it is just full.

The volume of a cylinder is given by the formula: V = πr²h

Given the diameter of the vessel is 2 m, the radius r is 1 m. The height h is 3.5 m. Substituting these values: V = (22/7) × 1² × 3.5 = 11 m³

Let the depth of the rainfall be d meters. The area of the roof is: A = 22 × 20 = 440 m²

The volume of rainwater on the roof is given by: V = A × d 11 = 440 × d d = 11 / 440 = 1/40 m

Converting the rainfall depth into centimeters: d = (1/40) × 100 = 2.5 cm

Explanation

The expression n³ - n can be factored as: n³ - n = n(n² - 1) = (n - 1)n(n + 1)

This expression represents the product of three consecutive integers: (n - 1), n, and (n + 1).

Among any three consecutive integers, at least one integer must be even, making it divisible by 2. Additionally, exactly one of the three consecutive integers must be a multiple of 3, making it divisible by 3.

Since 2 and 3 are coprime, their product, 2 × 3 = 6, must always divide the product of these three consecutive integers. Thus, n³ - n is always divisible by 6.

Explanation

The increase in simple interest is directly proportional to the increase in the interest rate over the given time period.

Increase in Simple Interest = P × ΔR × T / 100

Here, the increase in simple interest is ₹540, the increase in rate ΔR is 2.5% per annum, and the time T is 3 years. Substituting these values into the formula:

540 = P × 2.5 × 3 / 100 540 = P × 7.5 / 100 P = 54000 / 7.5 P = 7200

Explanation

Let the number to be added to each value be x.

For the resulting numbers to be in proportion, the ratio of the first two must equal the ratio of the last two: (6 + x)/(14 + x) = (18 + x)/(38 + x)

Cross-multiplying to solve for x: (6 + x)(38 + x) = (14 + x)(18 + x) 228 + 44x + x² = 252 + 32x + x²

Subtracting x² from both sides: 228 + 44x = 252 + 32x

Grouping the terms with x on one side and constant terms on the other: 12x = 24 x = 2

Substituting x = 2 back into the expression yields the terms 8, 16, 20, and 40, which satisfy the proportion since 8/16 = 20/40 = ½.

Explanation

Using the algebraic identity: a³ + b³ = (a + b)(a² - ab + b²)

This can be rewritten as: (a³ + b³) ÷ (a² - ab + b²) = a + b

By setting a = 0.08 and b = 0.02: a³ = 0.08 × 0.08 × 0.08 b³ = 0.02 × 0.02 × 0.02 a² = 0.08 × 0.08 ab = 0.08 × 0.02 = 0.0016 b² = 0.02 × 0.02

Substituting these values into the identity, the expression simplifies to: a + b = 0.08 + 0.02 = 0.1

Explanation

For x = √3 + √2, the value of 1/x is obtained by rationalizing the denominator: 1/x = 1/(√3 + √2) = √3 - √2

Adding x and 1/x: x + 1/x = (√3 + √2) + (√3 - √2) = 2√3

Squaring both sides of the equation: (x + 1/x)² = (2√3)² x² + 2 + 1/x² = 12 x² + 1/x² = 12 - 2 = 10

Explanation

Let the number of boys be 300 and the number of girls be 200, which satisfies the given ratio of 3:2. The total number of students in the school is 300 + 200 = 500.

Since 20% of the boys are scholarship holders, the percentage of boys who do not get the scholarship is 100% - 20% = 80%. Number of boys without the scholarship = 80% of 300 = 240.

Similarly, since 25% of the girls are scholarship holders, the percentage of girls who do not get the scholarship is 100% - 25% = 75%. Number of girls without the scholarship = 75% of 200 = 150.

The total number of students who do not get the scholarship is 240 + 150 = 390.

The percentage of students who do not get the scholarship is (390 / 500) × 100 = 78%.

Explanation

The relationship between the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers is given by the formula: First Number × Second Number = HCF × LCM

Substituting the given values (where the LCM is 225): 75 × Second Number = 15 × 225

Solving for the second number: Second Number = (15 × 225) / 75 = 225 / 5 = 45

Explanation

The rate of flow through a pipe is proportional to its cross-sectional area, which depends on the square of its diameter. When the diameter is doubled from d to 2d, the cross-sectional area increases by a factor of 4, meaning the pipe can drain the tank four times faster. Therefore, the time taken is reduced to one-fourth of the original time, which is 10 minutes.

Explanation

In the initial mixture of 60 litres, the ratio of milk to water is 2:1.

The quantity of milk is calculated as 60 × (2/3) = 40 litres.

The quantity of water is 60 × (1/3) = 20 litres.

Let x be the amount of water added to make the ratio of milk to water 1:2. Since only water is added, the quantity of milk remains 40 litres, and the new quantity of water becomes 20 + x litres.

Using the new ratio: 40 / (20 + x) = 1/2

Cross-multiplying gives: 80 = 20 + x x = 60 litres

Explanation

The area of an equilateral triangle with side length a is given by the formula:

Area = (√3/4) × a²

Substituting the given area of 4√3 cm² into the formula:

4√3 = (√3/4) × a²

Dividing both sides by √3:

4 = a² / 4

Multiplying both sides by 4:

a² = 16

Taking the square root of both sides gives the length of each side:

a = 4 cm

Explanation

Using the work-rate formula, the combined filling rate of the three taps is the sum of their individual rates:

1/A + 1/B + 1/C = 1/T

Here, the combined time T is 10 minutes, tap A alone takes 30 minutes, and tap B alone takes 40 minutes. Substituting these values into the formula:

1/30 + 1/40 + 1/C = 1/10

To find the rate of tap C, isolate 1/C:

1/C = 1/10 - 1/30 - 1/40

Taking the least common multiple of 10, 30, and 40, which is 120, rewrite the fractions with a common denominator:

1/C = 12/120 - 4/120 - 3/120

1/C = (12 - 4 - 3)/120

1/C = 5/120

1/C = 1/24

Taking the reciprocal, tap C alone takes 24 minutes to fill the cistern.

Explanation

To find the number of days taken by 1 man, 1 woman, and 1 boy working together, we use the work-rate relation:

Days = D / ( (m₂/m₁) + (w₂/w₁) + (b₂/b₁) )

Where D is the number of days given for the individual groups (88 days), m₁, w₁, b₁ are the initial counts of men, women, and boys (1, 2, and 3), and m₂, w₂, b₂ are the final counts of men, women, and boys working together (1, 1, and 1).

Substituting the values:

Days = 88 / ( 1/1 + ½ + ⅓ )

Days = 88 / ( 11/6 )

Days = (88 × 6) / 11

Days = 8 × 6 = 48

Explanation

The diagonals of a rhombus bisect each other at right angles. Let the side of the rhombus be s = 10 cm, and the two diagonals be d₁ = 16 cm and d₂.

Using the relation between the side and the diagonals of a rhombus: s² = (d₁/2)² + (d₂/2)²

Substituting the given values: 10² = (16/2)² + (d₂/2)² 100 = 8² + (d₂/2)² 100 = 64 + (d₂/2)² (d₂/2)² = 36 d₂/2 = 6 d₂ = 12 cm

The area of the rhombus is calculated using the formula: Area = ½ × d₁ × d₂ Area = ½ × 16 × 12 = 96 cm²

Explanation

The cow is tied to the corner of a rectangular field, so the grazing area forms a sector of a circle with a central angle of 90° and a radius equal to the length of the rope, 14 m.

The formula for the area of a sector is: Area = (θ/360°) × πr²

Substituting θ = 90°, r = 14, and π = 22/7: Area = (90/360) × (22/7) × 14 × 14 Area = ¼ × 22 × 2 × 14 Area = 11 × 14 Area = 154

Explanation

The net percentage change in the area of a rectangle when its length and breadth are changed by x% and y% respectively is given by the formula:

Net Percentage Change = x + y + (x × y)/100

Since both the length and the breadth are increased by 50%, we substitute x = 50 and y = 50 into the formula:

Net Percentage Change = 50 + 50 + (50 × 50)/100 = 100 + 2500/100 = 100 + 25 = 125%

Alternatively, if the original length is L and the original breadth is B, the original area is L × B. When both dimensions are increased by 50%, the new length becomes 1.5L and the new breadth becomes 1.5B. The new area is 1.5L × 1.5B = 2.25 × (L × B), which represents an increase of 1.25 times the original area, or 125%.

Explanation

Let a be the side of the cube and r be the radius of the sphere.

The surface area of the cube is 6a² and that of the sphere is 4πr². Since they are equal: 6a² = 4πr² a = r √(2π/3)

The volume of the cube is V_c = a³ and the volume of the sphere is Vₛ = (4/3) π r³. Substituting the value of a into the volume ratio: V_c / Vₛ = a³ / ((4/3) π r³) V_c / Vₛ = (r √(2π/3))³ / ((4/3) π r³) V_c / Vₛ = (2π/3) √(2π/3) / ((4/3) π) V_c / Vₛ = ½ √(2π/3) V_c / Vₛ = √(2π / 12) V_c / Vₛ = √(π / 6)

This gives the ratio √π : √6.