SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

The volume of the wall to be constructed is 800 cm x 600 cm x 22.5 cm = 1,08,00,000 cubic cm, while the volume of each brick is 25 cm x 11.25 cm x 6 cm = 1687.5 cubic cm. Dividing the wall's volume by the volume of one brick gives 1,08,00,000 / 1687.5 = 6400 bricks required.

Explanation

Since AB=AC in triangle ABC, the base angles are equal, so angle B = angle C = 50°. As AB=PQ, AC=PR (because AB=AC and PQ=PR) and BC=QR, triangles ABC and PQR are congruent by the SSS criterion, so the corresponding angle R equals angle C, which is 50°.

Explanation

The profit percentage when a trader uses less weight is calculated using the standard formula:

Profit % = (True Weight - False Weight) / False Weight × 100%

Here, the true weight of a kilogram is 1000 gm and the false weight used by the dealer is 900 gm.

Substituting these values into the formula:

Profit % = (1000 - 900) / 900 × 100%

Profit % = 100 / 900 × 100%

Profit % = 1/9 × 100%

Profit % = 11 1/9%

Explanation

Using the average formula: Total Marks = Average × Number of Subjects

Initially, the average marks for 4 subjects is 75: Total marks in 4 subjects = 75 × 4 = 300

When English is added, there are 5 subjects in total, and the new average is 70: Total marks in 5 subjects = 70 × 5 = 350

The score in English is the difference between the new total and the initial total: Marks in English = 350 - 300 = 50

Explanation

To find the greatest number, we express all the terms with a common root index, which is the LCM of 2, 3, 6, and 12, namely 12. This gives the values as the 12th root of 64, 81, 36, and 12, respectively, showing that the cube root of 3 (the 12th root of 81) is the greatest.

Explanation

Given the ratios P:Q = 10:11 and Q:R = 11:12, the common term Q has the same value of 11 in both ratios. Therefore, the combined ratio of the three quantities can be written directly as:

P : Q : R = 10 : 11 : 12

Assuming P = 10, Q = 11, and R = 12, the required sum terms are calculated as:

P + Q = 10 + 11 = 21 Q + R = 11 + 12 = 23 R + P = 12 + 10 = 22

Combining these terms gives the ratio P + Q : Q + R : R + P as 21:23:22.

Explanation

The number of two-digit numbers divisible by 3 or 5 is calculated using the principle of inclusion-exclusion: N(A ∪ B) = N(A) + N(B) - N(A ∩ B)

The two-digit numbers range from 10 to 99.

To find the number of two-digit numbers divisible by 3, N(A), we calculate the difference between the count of multiples of 3 up to 99 and the count of multiples of 3 up to 9: N(A) = floor(99/3) - floor(9/3) = 33 - 3 = 30

To find the number of two-digit numbers divisible by 5, N(B): N(B) = floor(99/5) - floor(9/5) = 19 - 1 = 18

To find the number of two-digit numbers divisible by both 3 and 5, which are the multiples of their lowest common multiple LCM(3, 5) = 15, N(A ∩ B): N(A ∩ B) = floor(99/15) - floor(9/15) = 6 - 0 = 6

Substituting these values into the inclusion-exclusion formula: N(A ∪ B) = 30 + 18 - 6 = 42

Explanation

Let the initial price of the article be 100.

When the price is raised by 45%, the price becomes: 100 × (1 + 45/100) = 145

Now, two successive discounts of 15% each are applied to this price.

The price after the first 15% discount is: 145 × (1 - 15/100) = 145 × 0.85 = 123.25

The price after the second 15% discount is: 123.25 × (1 - 15/100) = 123.25 × 0.85 = 104.7625

The overall change in the price of the article is: 104.7625 - 100 = 4.7625%

Since this result is positive, the price of the article is ultimately increased by 4.7625%.

Explanation

The largest right circular cone that can be carved out of a cube of edge a has a base diameter equal to the edge of the cube and a height equal to the edge of the cube.

Given the edge of the cube a = 10 cm: Base radius of the cone, r = a / 2 = 5 cm Height of the cone, h = a = 10 cm

Using the formula for the volume of a cone: V = (1/3)πr²h

Substituting the values: V = (1/3) × π × 5² × 10 V = 250π/3 cm³

Explanation

The standard formula for the difference (d) between compound interest and simple interest for a period of 2 years is given by:

d = P × (r/100)²

where P is the principal and r is the rate of interest per annum.

Substituting the given values: d = 1725 r = 25

1725 = P × (25/100)² 1725 = P × (1/4)² 1725 = P × 1/16 P = 1725 × 16 P = 27600

Thus, the principal is ₹27,600.

Explanation

Using the algebraic identity (x + y)² = x² + 2xy + y², let x = 2a and y = 5b. This gives:

(2a + 5b)² = 4a² + 20ab + 25b²

Rearranging the formula to solve for the required expression:

4a² + 25b² = (2a + 5b)² - 20ab

Substituting the given values, 2a + 5b = 12 and ab = 3, into the equation:

4a² + 25b² = 12² - 20 × 3

4a² + 25b² = 144 - 60

4a² + 25b² = 84

Explanation

When a quantity is successively increased by x% and then decreased by y%, the net percentage change is given by the formula:

Net Change % = x + y + xy/100

Substituting x = 20 for the increase and y = -20 for the decrease:

Net Change % = 20 - 20 + (20 × -20)/100 Net Change % = -400/100 Net Change % = -4%

The negative sign indicates a net decrease. Thus, the salary decreases by 4%.

Explanation

Using the work-rate formula, if two individuals take T days to complete a task together, and one takes S days alone, the time M taken by the other alone is given by:

1/M = 1/T - 1/S

Given that T = 18 days and S = 24 days, substituting these values into the formula yields:

1/M = 1/18 - 1/24

Finding the least common multiple of 18 and 24, which is 72, the equation becomes:

1/M = 4/72 - 3/72

1/M = 1/72

M = 72

Thus, Mona alone can complete the typing work in 72 days.

Explanation

The final population of the town after two years is calculated using the formula for successive percentage changes:

Final Population = P × (1 - R₁/100) × (1 + R₂/100)

Here: P = 80000 (initial population) R₁ = 8 (percentage decrease in the first year) R₂ = 5 (percentage increase in the second year)

Substituting these values into the formula:

Final Population = 80000 × (1 - 8/100) × (1 + 5/100) Final Population = 80000 × 0.92 × 1.05 Final Population = 73600 × 1.05 Final Population = 77280

Explanation

Using the trigonometric sum formula, sin(A + C) = sin A cos C + cos A sin C.

In a triangle ABC right-angled at B, the sum of the remaining angles A and C is 90°: A + C = 90°

Substituting this relation into the sum formula: sin A cos C + cos A sin C = sin(A + C) = sin 90° = 1

Alternatively, using the given value tan A = 1/√3, angle A is 30°. Since the triangle is right-angled at B, angle C is 90° - 30° = 60°. Substituting these values: sin 30° cos 60° + cos 30° sin 60° = (½ × ½) + (√3/2 × √3/2) = ¼ + ¾ = 1

Explanation

Let the number be x. The relation between the number and its square root is represented by the equation: x - √x = 2

Let √x = y, where y is a non-negative real number (y ≥ 0). Substituting this into the equation yields a quadratic equation: y² - y - 2 = 0

Factoring this quadratic equation: (y - 2)(y + 1) = 0

This gives the roots y = 2 and y = -1. Since y represents a principal square root, it cannot be negative, so we discard y = -1. Thus, y = 2.

Finding the value of x: x = y² = 2² = 4

Explanation

Dividing the given equation a² - 4a + 1 = 0 by a on both sides, we get: a - 4 + 1/a = 0 a + 1/a = 4

Squaring both sides of this equation: (a + 1/a)² = 4² a² + 2 + 1/a² = 16 a² + 1/a² = 14

The expression to be evaluated can be rearranged as: a² + a + 1/a + 1/a² = (a² + 1/a²) + (a + 1/a)

Substituting the values of (a² + 1/a²) and (a + 1/a): 14 + 4 = 18

Explanation

Using the identity x cubed minus one over x cubed equals the cube of (x minus one over x) plus three times (x minus one over x), substitute x minus one over x equals seven thirds. This gives (7/3) cubed plus 3 times 7/3, which is 343/27 plus 189/27, equal to 532/27, or 19 and 19/27.

Explanation

To find the overall average, we calculate the total marks of all students and divide by the total number of students. The total marks are (30 * 55) + (50 * 70) + (70 * 70) = 1650 + 3500 + 4900 = 10050, and the total number of students is 30 + 50 + 70 = 150. Dividing 10050 by 150 gives an average of 67.

Explanation

According to the divisibility rule of 11, a number is completely divisible by 11 if the difference between the sum of its digits at odd positions and the sum of its digits at even positions is either 0 or a multiple of 11.

For the number 28171, the digits at the odd positions from left to right (first, third, and fifth) are 2, 1, and 1, and the digits at the even positions (second and fourth) are 8 and 7.

Sum of digits at odd positions = 2 + 1 + 1 = 4 Sum of digits at even positions = 8 + 7 = 15 Difference = 15 - 4 = 11

Since the difference is 11, which is a multiple of 11, the number 28171 is divisible by 11.

Explanation

When interest is compounded half-yearly, the annual interest rate is halved and the number of conversion periods is doubled.

Rate of interest per half-year R = 12% / 2 = 6% Number of half-years in 1½ years n = 1.5 × 2 = 3 Principal P = ₹ 8,400

The formula for the total amount A is: A = P × (1 + R/100)ⁿ

Substituting the given values: A = 8400 × (1 + 6/100)³ A = 8400 × (1.06)³ A = 8400 × 1.191016 A = 10004.53

The compound interest is the difference between the total amount and the principal: Compound Interest = A - P Compound Interest = 10004.53 - 8400 = 1604.53 ≈ 1,604.50

Explanation

For two similar triangles, ABC and PQR, the ratio of their corresponding sides is equal. Therefore, the relationship between the corresponding sides of triangle ABC and triangle PQR is given by the formula:

AB/PQ = BC/QR

Substituting the given values into the formula:

AB/6.25 = 5/12.5

Solving for AB:

AB = 5 × 6.25 / 12.5

Since 12.5 is twice of 6.25, the ratio simplifies to:

AB = 5 × ½ = 2.5 cm

Explanation

Given the ratio x : y = 1 : 2, we can assign x = 1 and y = 2.

Substituting these values into the expression (2x + 4y) : (x + 4y):

Numerator = 2x + 4y = 2(1) + 4(2) = 2 + 8 = 10

Denominator = x + 4y = 1 + 4(2) = 1 + 8 = 9

Thus, the required ratio is 10 : 9.

Explanation

Correct Answer: D

To find the distance covered by the train, the formula is:

Distance = Speed × Time

First, convert the speed of the train from km/h to m/s by multiplying by the conversion factor 5/18:

Speed = 108 × 5/18 = 30 m/s

Now, substitute the speed and the given time of 11 seconds into the distance formula:

Distance = 30 m/s × 11 s = 330 m

Explanation

The radius r of the circular base is calculated from the circumference C using the formula C = 2 π r.

44 = 2 × (22/7) × r r = 7 m

The volume V of the solid cone is determined by the formula V = (1/3) π r² h, where h is the height of the cone. Substituting the radius r = 7 m and height h = 12 m into the equation:

V = (1/3) × (22/7) × 7² × 12 V = 22 × 7 × 4 V = 616 m³

Explanation

Let the total work be the Least Common Multiple of 16 and 12, which is 48 units.

The daily efficiency of the man is 48 / 16 = 3 units per day. The daily efficiency of the woman is 48 / 12 = 4 units per day.

They work on alternate days, starting with the man. In one 2-day cycle, the total work completed is 3 + 4 = 7 units.

In 6 complete cycles (which equals 12 days), the work completed is 6 × 7 = 42 units. The remaining work is 48 - 42 = 6 units.

On the 13th day, the man works and completes 3 units of work. The remaining work is 6 - 3 = 3 units.

On the 14th day, the woman works. Since she can complete 4 units in a full day, she will take 3/4 of a day to complete the remaining 3 units.

The total time taken is 13 + 3/4 days, which is equal to 55/4 days.

Explanation

The given side lengths of the triangle are 5 cm, 12 cm, and 13 cm. Since 5² + 12² = 25 + 144 = 169 = 13², these sides form a Pythagorean triple. This confirms that the triangle is a right-angled triangle, where the sides of length 5 cm and 12 cm are perpendicular to each other, and the side of length 13 cm is the hypotenuse.

The area of a right-angled triangle is calculated using the formula: Area = ½ × base × height

Substituting the perpendicular sides as base and height: Area = ½ × 5 × 12 = 30 cm²

Explanation

Using the complementary angle identity tan(90° - θ) = cot(θ), we can rewrite the terms of the expression:

tan 42° = tan(90° - 48°) = cot 48° tan 67° = tan(90° - 23°) = cot 23°

Substituting these values into the given expression:

tan 48° × tan 23° × tan 42° × tan 67° = tan 48° × tan 23° × cot 48° × cot 23°

Grouping the complementary terms and using the identity tan(θ) × cot(θ) = 1:

(tan 48° × cot 48°) × (tan 23° × cot 23°) = 1 × 1 = 1

Explanation

The total number of students enrolled in Coaching centre R is 300.

The percentage of students from school M enrolled in Coaching centre R is 34%. Total students from school M in Coaching centre R = 300 × 34% = 300 × 0.34 = 102

The percentage of female students in Coaching centre R is 35%. Total female students in Coaching centre R = 300 × 35% = 300 × 0.35 = 105

The percentage of these female students who are from school M is 20%. Female students from school M in Coaching centre R = 105 × 20% = 105 × 0.20 = 21

The number of male students from school M in Coaching centre R is obtained by subtracting the female students from school M from the total students from school M in Coaching centre R: Male students from school M in Coaching centre R = 102 - 21 = 81

Explanation

The average of three tests is calculated by dividing the sum of the marks by the total number of tests.

Average = (Sum of marks in all three tests) / 3

Let x be the marks Rohan needs to score in the third test.

(70 + 75 + x) / 3 = 70

Multiplying both sides by 3:

145 + x = 210

Subtracting 145 from both sides:

x = 210 - 145 x = 65

Explanation

To find the overall profit or loss percentage, the successive percentage change formula can be directly applied:

Net percentage change = x + y + xy/100

Here, the first transaction is a loss of 15%, so x = -15. The second transaction is a profit of 22.5%, so y = 22.5.

Substituting these values into the formula:

Net percentage change = -15 + 22.5 + (-15 × 22.5)/100 Net percentage change = 7.5 - 337.5/100 Net percentage change = 7.5 - 3.375 Net percentage change = 4.125%

A positive result indicates a net profit. Therefore, the overall profit is 4.125%.

Explanation

The selling price of an item after successive discounts is calculated using the formula:

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

Given: Marked Price = ₹4,875 First discount (d₁) = 28% Second discount (d₂) = 30%

Substituting the values into the formula:

Selling Price = 4,875 × (1 - 28/100) × (1 - 30/100) Selling Price = 4,875 × 0.72 × 0.70 Selling Price = 4,875 × 0.504 Selling Price = 2,457

Thus, the selling price of the toy is ₹2,457.

Explanation

Let the original sum of money be P. When a sum is increased by r%, the final amount S is given by the formula:

S = P × (1 + r/100)

Given that the final amount S is ₹19,320 and the rate of increase r is 15%, we substitute these values into the formula:

19320 = P × (1 + 15/100) 19320 = P × 1.15 P = 19320 / 1.15 P = 1932000 / 115 P = 16800

Thus, the original sum of money is ₹16,800.

Explanation

The relationship between the original price, successive discounts, and the final selling price is given by the formula:

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

Substituting the given values where the first discount d₁ = 25%, the second discount d₂ = 10%, and the Selling Price = ₹480:

480 = Original Price × (1 - 25/100) × (1 - 10/100) 480 = Original Price × 0.75 × 0.90 480 = Original Price × 0.675

To find the original price:

Original Price = 480 / 0.675 Original Price = 19200 / 27 Original Price ≈ 711.11

Rounding to the nearest rupee, the original price of the shirt is ₹711.

Explanation

By the definition of the arithmetic mean, the mean of three numbers is their sum divided by 3. Since the two sets of three numbers have the same mean, their sums must also be equal:

a/3 + b + c = b/3 + c/3 + x

Multiplying the entire equation by 3 to eliminate the denominators:

a + 3b + 3c = b + c + 3x

To find the value of 3x, we isolate 3x on one side of the equation by subtracting b and c from both sides:

3x = a + 3b - b + 3c - c

3x = a + 2b + 2c

Explanation

The weighted average formula directly gives the overall pass percentage of the class:

Pass percentage = (b × P_b + g × P_g) / (b + g)

Here, b and g represent the ratio components of boys and girls, which are 11 and 9. P_b and P_g are the pass percentages of boys and girls, which are 30% and 20% respectively.

Substituting these values into the formula:

Pass percentage = (11 × 30 + 9 × 20) / (11 + 9) Pass percentage = (330 + 180) / 20 Pass percentage = 510 / 20 = 25.5%

Explanation

The volume V of a cone with base radius r and height h is given by the formula V = ⅓πr²h.

Let the volumes of the two cones be V₁ and V₂, their base radii be r₁ and r₂, and their heights be h₁ and h₂. The ratio of their volumes is:

V₁/V₂ = (r₁/r₂)² × (h₁/h₂)

We are given V₁/V₂ = 11/16 and r₁/r₂ = 3/4. Substituting these values into the ratio formula:

11/16 = (3/4)² × (h₁/h₂)

11/16 = 9/16 × (h₁/h₂)

Multiplying both sides by 16/9 to isolate the height ratio:

h₁/h₂ = (11/16) × (16/9) = 11/9

Explanation

A number is divisible by 3 if the sum of its individual digits is divisible by 3.

For the number 2143251, the sum of the digits is: 2 + 1 + 4 + 3 + 2 + 5 + 1 = 18

Since 18 is divisible by 3 (18 ÷ 3 = 6), the number 2143251 is divisible by 3.

Explanation

The total number of boys in the school is calculated by adding the number of boys who love cricket and those who love soccer: 542 + 438 = 980

Similarly, the total number of girls in the school is calculated by adding the number of girls who love cricket and those who love soccer: 358 + 372 = 730

The difference between the total number of boys and the total number of girls is: 980 - 730 = 250

Explanation

The formula for the total surface area of a solid hemisphere of radius r is: Total Surface Area = 3πr²

The radius r is half of the diameter: r = 14 / 2 = 7 cm

Substituting r = 7 cm and π = 22/7 into the formula: Total Surface Area = 3 × (22/7) × 7² Total Surface Area = 3 × (22/7) × 49 Total Surface Area = 3 × 22 × 7 Total Surface Area = 462 cm²

Explanation

To find the time taken when A and B work alternately, first determine the total work by taking the Least Common Multiple (LCM) of their individual days.

LCM of 21 and 24 = 168 units

This means the total work is 168 units.

Efficiency of A = 168 ÷ 21 = 8 units per day Efficiency of B = 168 ÷ 24 = 7 units per day

In a two-day cycle where A works on the first day and B works on the second day, the total work completed is: 8 + 7 = 15 units

Dividing the total work by the work completed in one cycle: 168 ÷ 15 = 11 cycles with a remainder of 3 units

Since each cycle consists of 2 days, 11 cycles represent: 11 × 2 = 22 days

Work completed in 22 days is: 11 × 15 = 165 units

The remaining work is: 168 - 165 = 3 units

On the 23rd day, it is A's turn to work. Since A can complete 8 units in a full day, the time A takes to finish the remaining 3 units is: 3/8 day

Combining the days, the total time required to complete the work is 22 3/8 days.

Explanation

The area of a square is calculated using the formula A = s², where s represents the side length.

For the initial square with side s = 5 cm, the area is: A₁ = 5² = 25 cm²

When the side length is doubled, the new side length is 10 cm, and the new area is: A₂ = 10² = 100 cm²

The increase in area is the difference between the new area and the original area: Increase in area = A₂ - A₁ = 100 - 25 = 75 cm²

The number of times the area increases is determined by dividing this increase by the original area: Times increase = (A₂ - A₁) / A₁ = 75 / 25 = 3

Explanation

Average speed is defined as the total distance divided by the total time taken for the journey.

Let the total distance of the journey be 100 km.

The bus covers 60% of this distance, which is 60 km, at a speed of 60 km/h. Time taken for the first part of the journey = 60 / 60 = 1 hour.

The bus covers the remaining 40% of the distance, which is 40 km, at a speed of 70 km/h. Time taken for the second part of the journey = 40 / 70 = 4/7 hours.

Total time taken for the entire journey = 1 + 4/7 = 11/7 hours.

Average speed = Total Distance / Total Time = 100 / (11/7) = 700 / 11 ≈ 63.64 km/h.

Rounding to the nearest integer, the average speed is 64 km/h.

Explanation

The total production of all types of cars during the year 2013 is 70.

To find which type of car constituted 30% of this total, calculate 30% of 70: 30% of 70 = (30 ÷ 100) × 70 = 21

Looking at the column for the year 2013 in the table, the production of car type D1 is exactly 21.

Explanation

Correct Answer: D

Let C be the cost price of the manufacturer. The successive profit percentages are 11%, 15%, and 19%.

The relationship between the final selling price and the initial cost price is given by the formula: Final Price = C × (1 + P₁/100) × (1 + P₂/100) × (1 + P₃/100)

Substituting the given values: 17045 = C × 1.11 × 1.15 × 1.19 17045 = C × 1.519035 C = 17045 / 1.519035 C ≈ 11220.93

Considering only the integral part, the cost price of the manufacturer is ₹11,220.

Explanation

The average marks of the combined group of boys and girls is calculated using the weighted average formula:

Weighted Average = (N₁ × A₁ + N₂ × A₂) / (N₁ + N₂)

Here, N₁ is the number of girls (60), A₁ is the average marks of the girls (15), N₂ is the number of boys (40), and A₂ is the average marks of the boys (30).

Substituting these values into the formula:

Weighted Average = (60 × 15 + 40 × 30) / (60 + 40) Weighted Average = (900 + 1200) / 100 Weighted Average = 2100 / 100 = 21

Explanation

The algebraic identity for the expression x⁴ + x² + 1 is: x⁴ + x² + 1 = (x² + x + 1)(x² - x + 1)

Substitute the given values into this formula: 585 = (1 + x + x²) × 65

Solving for 1 + x + x²: 1 + x + x² = 585 ÷ 65 = 9

Explanation

To find the required sale in the 6th week, we use the formula for the average:

Average = Total Sale ÷ Number of Weeks

First, calculate the total sale required for all 6 weeks to achieve an average of ₹8,500: Total sale for 6 weeks = 8,500 × 6 = 51,000

Next, calculate the sum of the sales for the first 5 weeks: Total sale for 5 weeks = 8,435 + 8,927 + 8,855 + 9,230 + 8,562 = 44,009

Now, subtract the total sale of the first 5 weeks from the total required sale for 6 weeks to find the sale in the 6th week: Sale in the 6th week = 51,000 - 44,009 = 6,991

Explanation

The conversion formula from kilometers per hour (km/h) to meters per second (m/s) is: Speed in m/s = Speed in km/h × 5/18

First, calculate the difference between the speeds of Shyam and Ram in km/h: Difference = 90 km/h - 72 km/h = 18 km/h

Now, convert this difference into m/s using the conversion factor: Difference in m/s = 18 × 5/18 = 5 m/s

Explanation

The net selling price after successive discounts is calculated using the formula:

Selling Price = Marked Price × (1 - d₁/100) × (1 - d₂/100) × (1 - d₃/100)

Given: Marked Price = Rs 4,88,000 d₁ = 10% d₂ = 5% d₃ = 2%

Substituting the values into the formula:

Selling Price = 4,88,000 × (1 - 0.10) × (1 - 0.05) × (1 - 0.02) Selling Price = 4,88,000 × 0.90 × 0.95 × 0.98 Selling Price = 4,39,200 × 0.95 × 0.98 Selling Price = 4,17,240 × 0.98 Selling Price = 4,08,895.2