SSC CHSL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

The average of a set of numbers in an arithmetic progression (AP) is given by the formula:

Average = (First term + Last term) / 2

For the first 7 multiples of 7, the terms form an arithmetic progression where: First term = 7 × 1 = 7 Last term = 7 × 7 = 49

Substituting these values into the formula: Average = (7 + 49) / 2 = 56 / 2 = 28

Alternatively, for an odd number of terms in an AP, the average is always the middle term. With 7 terms, the middle term is the 4th term. The 4th multiple of 7 is 7 × 4 = 28.

Explanation

Let the first part be P₁ and the second part be P₂. The sum of the two parts is P₁ + P₂ = 8000.

Since the simple interest on the first part at 21% per annum is equal to the simple interest on the second part at 35% per annum for the same duration: P₁ × 21% = P₂ × 35% P₁ × 21 = P₂ × 35 P₁ / P₂ = 35 / 21 = 5 / 3

The ratio of the two parts is 5 : 3, and the sum of the ratio parts is 5 + 3 = 8.

Now, we calculate the individual parts: P₁ = (5 / 8) × 8000 = 5000 P₂ = (3 / 8) × 8000 = 3000

The simple interest on each part is: SI = (5000 × 21 × 1) / 100 = 1050

Explanation

Let the Cost Price (CP) of the glass be CP.

When the selling price is ₹1965, the loss is 25%. Using the relationship between Selling Price and Cost Price: Selling Price = CP × (100 - Loss%) / 100 1965 = CP × (100 - 25) / 100 1965 = CP × 75 / 100 1965 = CP × 3/4 CP = (1965 × 4) / 3 CP = 655 × 4 = 2620

When the new selling price is ₹3013: Profit = New Selling Price - CP Profit = 3013 - 2620 = 393

The profit percentage is given by: Profit % = (Profit / CP) × 100 Profit % = (393 / 2620) × 100 Profit % = 3930 / 262 Profit % = 15%

Explanation

The relation between Cost Price (CP) and Marked Price (MP) given the discount percentage (D) and profit percentage (P) is:

CP / MP = (100 - D) / (100 + P)

Given that MP = ₹6000, D = 20, and P = 60:

CP / 6000 = (100 - 20) / (100 + 60)

CP / 6000 = 80 / 160

CP / 6000 = ½

CP = 6000 × ½ = ₹3000

Explanation

We calculate each square root term individually:

Since 27² = 729, we have: √729 = 27

For the term with two decimal places: √7.29 = √(729/100) = 27/10 = 2.7

For the term with one decimal place, we use the approximation √10 ≈ 3.162277: √72.9 = √(729/10) = 27/√10 ≈ 27 / 3.162277 ≈ 8.538

Summing these three values: 27 + 8.538 + 2.7 = 38.238

Rounding to two decimal places gives approximately 38.23.

Explanation

Let the daily work rates of Rohan, Mohit, and Vikas be R, M, and V respectively.

Based on the given information: R + M = 1/8 M + V = 1/10 V + R = 1/12

Adding these three equations: 2 × (R + M + V) = 1/8 + 1/10 + 1/12

To find the sum of the fractions, we use the least common multiple (LCM) of 8, 10, and 12, which is 120: 2 × (R + M + V) = (15 + 12 + 10) / 120 2 × (R + M + V) = 37 / 120

Dividing by 2 gives the combined daily rate of all three working together: R + M + V = 37 / (120 × 2) = 37 / 240

Since the number of days required to complete the wall is the reciprocal of their combined daily work rate, the total time taken when they work together is 240/37 days.

Explanation

The distance between the school and the house can be directly calculated using the standard formula for speed and time differences:

Distance = (s₁ × s₂) / |s₁ - s₂| × Δt

Here, the initial speed s₁ is 5 km/hr. The increased speed s₂ is 5 + 4 = 9 km/hr. The total difference in arrival time Δt is 3 minutes late plus 3 minutes early, which equals 6 minutes. Expressed in hours, this is 6/60 = 1/10 hours.

Substituting these values into the formula:

Distance = (5 × 9) / (9 - 5) × (1/10) Distance = 45 / 4 × 1/10 Distance = 45 / 40 = 1.125 km

Explanation

Correct Answer: B

The area of a rhombus is given by the formula:

Area = ½ × d₁ × d₂

where d₁ and d₂ are the lengths of its two diagonals.

Substituting the given values into the formula:

84 = ½ × 7 × d₂

Multiplying both sides by 2:

168 = 7 × d₂

Dividing by 7:

d₂ = 168 / 7 = 24 cm

Explanation

The perimeter of a semicircle of radius r is given by the formula:

Perimeter = πr + 2r

Substituting the given radius r = 28 cm and using π ≈ 22/7:

Perimeter = (22/7 × 28) + (2 × 28) Perimeter = (22 × 4) + 56 Perimeter = 88 + 56 Perimeter = 144 cm

Explanation

The formula for the curved surface area of a hemisphere is 2πr², where r is the radius. Given the diameter of the hemisphere is 28 cm, the radius is half of the diameter, which is r = 14 cm.

Substituting the radius and π ≈ 22/7 into the formula: Curved Surface Area = 2 × (22/7) × 14 × 14 Curved Surface Area = 2 × 22 × 2 × 14 Curved Surface Area = 1232 cm²

Explanation

The value of sin 60° is equal to √3 / 2.

Substituting this into the given expression: sin 60° + 2 / √3 = √3 / 2 + 2 / √3

To add these fractions, we find a common denominator, which is 2√3. Multiply the numerator and denominator of the first term by √3 and the second term by 2: (√3 × √3) / (2 × √3) + (2 × 2) / (2 × √3) = 3 / 2√3 + 4 / 2√3

Adding the numerators together: (3 + 4) / 2√3 = 7 / 2√3

Explanation

In a right-angled triangle XYZ, right-angled at Y, the acute angles X and Z are complementary, meaning X + Z = 90°.

This gives Z = 90° - X.

Using trigonometric identities: cosec Z = cosec(90° - X) = sec X

Since sec X is the reciprocal of cos X: cosec Z = 1 / cos X

Given that cos X = 3/5, we substitute this value: cosec Z = 1 / (3/5) = 5/3

Explanation

For any positive odd integer n, the expression aⁿ + bⁿ is always divisible by (a + b).

Applying this to the second expression, 37³⁷ + 43³⁷, the exponent 37 is odd. Therefore, 37³⁷ + 43³⁷ is divisible by (37 + 43) = 80.

Now, let us analyze the first expression, 37³⁷ + 43⁵⁷, modulo 80.

Since 37 ≡ -43 (mod 80), we can rewrite the expression as: 37³⁷ + 43⁵⁷ ≡ (-43)³⁷ + 43⁵⁷ (mod 80)

Because 37 is an odd exponent, (-43)³⁷ = -43³⁷. This gives: -43³⁷ + 43⁵⁷ ≡ 43³⁷(43²⁰ - 1) (mod 80)

To check if this is divisible by 80, we check the divisibility of 43²⁰ - 1 by 5 and 16: Modulo 5: 43 ≡ 3 (mod 5). Thus, 43²⁰ ≡ 3²⁰ = (3⁴)⁵ ≡ 1⁵ ≡ 1 (mod 5). This means 43²⁰ - 1 is divisible by 5. Modulo 16: 43 ≡ 11 (mod 16). Thus, 43²⁰ ≡ 11²⁰ = (121)¹⁰ ≡ 9¹⁰ = (81)⁵ ≡ 1⁵ ≡ 1 (mod 16). This means 43²⁰ - 1 is divisible by 16.

Since 43²⁰ - 1 is divisible by both 5 and 16, it is divisible by their product, 80. Consequently, 37³⁷ + 43⁵⁷ is divisible by 80.

Both expressions share 80, or (43 + 37), as a common factor.

Explanation

The given expression can be simplified using the algebraic identity: a² - b² = (a - b)(a + b)

Let a = 61681 and b = 31681.

The expression becomes: (a² - b²) / 30000

Substituting the identity: (a - b)(a + b) / 30000

Now compute the terms: a - b = 61681 - 31681 = 30000 a + b = 61681 + 31681 = 93362

Substituting these values back into the expression: (30000 × 93362) / 30000 = 93362

Explanation

For a pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have an infinite number of solutions, the coefficients must satisfy the ratio condition:

a₁/a₂ = b₁/b₂ = c₁/c₂

From the given equations, we have: a₁ = 6, b₁ = 12, c₁ = 9 a₂ = 2, b₂ = g, c₂ = 3

Substituting these values into the condition:

6/2 = 12/g = 9/3

This simplifies to:

3 = 12/g

Solving for g:

g = 12/3 g = 4

Explanation

A perpendicular drawn from the center of a circle to a chord bisects the chord. Since ON is perpendicular to AB, N is the midpoint of AB.

AN = AB / 2 = 20 / 2 = 10 cm

In the right-angled triangle ONA, applying Pythagoras' theorem: OA² = AN² + ON² OA² = 10² + (2√11)² OA² = 100 + (4 × 11) OA² = 100 + 44 OA² = 144 OA = √144 = 12 cm

Explanation

The largest chord of any circle is its diameter. Therefore, the diameter of the given circle is 20.4 cm.

The radius of a circle is always exactly half of its diameter.

Radius = Diameter / 2 = 20.4 / 2 = 10.2 cm

Explanation

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

Percentage increase = x + y + xy/100

Substituting the given values x = 40 and y = 70 into the formula:

Percentage increase = 40 + 70 + (40 × 70)/100 Percentage increase = 110 + 2800/100 Percentage increase = 110 + 28 Percentage increase = 138%

Explanation

To find B's share, first simplify the ratio of A, B, and C by multiplying each term by 10, which is the least common multiple of the denominators 2 and 5.

Ratio of A : B : C = 1/2 : 1 : 3/5 Multiplying by 10 gives: A : B : C = 5 : 10 : 6

The sum of the ratio parts is 5 + 10 + 6 = 21.

Using the total amount of ₹6300, the share of B is: B's share = (10 / 21) × 6300 = 10 × 300 = 3000

Explanation

In a business partnership where the investment time periods are equal, the profit is distributed in the ratio of the investments made by the partners.

Ratio of investment of U and V = 184000 : 224000 = 184 : 224 = 23 : 28

Let the total profit earned by U and V together be P.

U's share of profit = (Investment of U / Total Investment) × P 20700 = (23 / (23 + 28)) × P 20700 = (23 / 51) × P P = (20700 × 51) / 23 P = 900 × 51 P = 45900

Explanation

The sum of the first n odd numbers is represented by the formula n². The average of these n numbers is the sum divided by the total number of terms, which is n² / n = n. For the first 19 odd numbers, the value of n is 19. Therefore, the average of the first 19 odd numbers is 19.

Explanation

The difference between the amounts at the end of 3 years and 2 years represents the simple interest earned in 1 year.

Simple Interest for 1 year = 1488 - 1392 = 96

Since simple interest remains constant each year, the simple interest for 2 years is: 96 × 2 = 192

The principal sum is the amount at the end of 2 years minus the simple interest earned over those 2 years: Principal = 1392 - 192 = 1200

The rate of interest per annum is calculated using the simple interest relation: Rate = (Simple Interest for 1 year × 100) / Principal Rate = (96 × 100) / 1200 = 8%

Explanation

The relation between Selling Price (SP), Cost Price (CP), and profit percentage is given by the formula:

CP = (SP × 100) / (100 + Profit %)

Substituting the given values into the formula:

CP = (2808 × 100) / (100 + 17) CP = 280800 / 117 CP = 2400

Explanation

The selling price of the chair is Rs 520 and the discount offered is Rs 40. The marked price is the sum of the selling price and the discount: Marked Price = 520 + 40 = Rs 560

The marked price is given as 40% more than the cost price. Letting CP represent the cost price: Marked Price = CP × 1.40 560 = CP × 1.40 CP = 560 / 1.40 = Rs 400

The profit made on the sale is the difference between the selling price and the cost price: Profit = 520 - 400 = Rs 120

The profit percentage is calculated with respect to the cost price: Profit Percentage = (Profit / Cost Price) × 100 Profit Percentage = (120 / 400) × 100 = 30%

Explanation

To determine the validity of Statement I, square both sides to compare their values: (2√3)² = 4 × 3 = 12 (3√2)² = 9 × 2 = 18 Since 12 is less than 18, 2√3 is less than 3√2. Thus, Statement I is false.

To determine the validity of Statement II, simplify the right-hand side of the inequality: 2√8 = 2 × √(4 × 2) = 2 × 2√2 = 4√2 Since 4√2 is equal to 2√8, they cannot be unequal. Thus, Statement II is false.

Since both statements are false, neither statement is true.

Explanation

Let the total work be the least common multiple of the individual days, which is LCM(30, 40, 50) = 600 units.

The efficiency of each person is: Efficiency of X = 600 / 30 = 20 units per day Efficiency of Y = 600 / 40 = 15 units per day Efficiency of Z = 600 / 50 = 12 units per day

If Y had not left 4 days before the completion of the work, Y would have completed an additional amount of work: 15 × 4 = 60 units

By adding this hypothetical work to the total work, we can assume that X, Y, and Z all worked together for the entire duration of D days: Total hypothetical work = 600 + 60 = 660 units

Using the combined efficiency of all three, the total number of days D is: D = 660 / (20 + 15 + 12) = 660 / 47 days

Explanation

To find the time taken, first convert the speed from km/hr to m/s:

Speed = 72 × 5/18 = 20 m/s

Using the formula:

Time = Distance / Speed

Substitute the given values into the formula:

Time = 1400 / 20 = 70 seconds

Explanation

By the Pythagorean theorem, the relationship between the diagonal d, length l, and breadth b of a rectangle is: d² = l² + b²

Substituting the given values d = 17 and b = 8: 17² = l² + 8² 289 = l² + 64 l² = 225 l = 15

The perimeter P of a rectangle is calculated using the formula: P = 2(l + b)

Substituting the values of length and breadth: P = 2(15 + 8) = 2 × 23 = 46 cm

Explanation

The area of a semicircle of radius r is given by the formula: Area = ½ × π × r²

Substituting the given area of 1925 cm² and using π = 22/7: ½ × (22/7) × r² = 1925 (11/7) × r² = 1925 r² = (1925 × 7) / 11 r² = 175 × 7 r² = 1225 r = 35 cm

The perimeter of a semicircle consists of its curved boundary plus its diameter: Perimeter = π × r + 2 × r Perimeter = (22/7) × 35 + 2 × 35 Perimeter = 110 + 70 Perimeter = 180 cm

Explanation

The surface area of a sphere is given by the formula: A = 4πr²

where r is the radius of the sphere.

Given that the surface area is 5544 cm², and using π ≈ 22/7, we can set up the equation: 5544 = 4 × (22/7) × r² 5544 = (88/7) × r² r² = (5544 × 7) / 88 r² = 63 × 7 r² = 441 r = √441 r = 21 cm

The diameter of the sphere is twice the radius: d = 2r = 2 × 21 = 42 cm

Explanation

In a triangle, the sum of all interior angles is always 180°. Since triangle ABC is right-angled at B, we have: Angle B = 90°

Using the angle sum property for the triangle: Angle A + Angle B + Angle C = 180° 45° + 90° + Angle C = 180° 135° + Angle C = 180° Angle C = 45°

Now, substitute the value of Angle C into the required expression: cosec C + 1 / √3 = cosec 45° + 1 / √3

Since cosec 45° = √2, the expression simplifies to: √2 + 1 / √3

By taking √3 as the common denominator: (√2 × √3 + 1) / √3 = (√6 + 1) / √3

Explanation

In the right-angled triangle ΔUVW with ∠V = 90°: cosec U = Hypotenuse / Opposite = UW / VW = 13/12

Using the trigonometric identity: cosec²U - cot²U = 1 cot²U = (13/12)² - 1 = 169/144 - 1 = 25/144 cot U = 5/12

In ΔUVW: cot U = Adjacent / Opposite = UV / VW 5/12 = UV / VW

Substitute UV = 2.5 cm into the relation: 5/12 = 2.5 / VW VW = (2.5 × 12) / 5 VW = 0.5 × 12 VW = 6 cm

Explanation

To simplify the expression √3 / (3 + √3), we divide both the numerator and the denominator by √3:

√3 / (3 + √3) = 1 / (√3 + 1)

Next, we rationalize the denominator by multiplying both the numerator and the denominator by (√3 - 1):

(1 / (√3 + 1)) × ((√3 - 1) / (√3 - 1)) = (√3 - 1) / ((√3)² - 1²) = (√3 - 1) / (3 - 1) = (√3 - 1) / 2

Substituting the given value of √3 = 1.7320 into the simplified expression:

(1.7320 - 1) / 2 = 0.7320 / 2 = 0.366

Explanation

The given equation is 9^√a + 40^√a = 41^√a. Let √a = x, which transforms the equation into 9ˣ + 40ˣ = 41ˣ.

Dividing both sides by 41ˣ yields (9/41)ˣ + (40/41)ˣ = 1. Since both base values 9/41 and 40/41 are less than 1, the function on the left side is strictly decreasing, meaning there is at most one real solution for x.

Using the Pythagorean triple 9, 40, and 41, we know that 9² + 40² = 1681 and 41² = 1681. Thus, 9² + 40² = 41² is a true mathematical identity, which directly gives the unique solution x = 2.

Equating x back to √a gives √a = 2. Squaring both sides of this equation results in a = 4.

Explanation

Let the price of a bottle be b and the price of a bag be g.

According to the first condition, the cost of 5 bottles is equal to the cost of 2 bags: 5b = 2g

Multiplying both sides of the equation by 3 gives: 15b = 6g

According to the second condition, the total cost of 15 bottles and 4 bags is Rs. 2,000: 15b + 4g = 2000

Substituting 15b = 6g into the second equation: 6g + 4g = 2000 10g = 2000 g = 200

The price of a single bag is Rs. 200.

Explanation

The Converse of Basic Proportionality Theorem, also known as Thales's Theorem, states that if a line intersects two sides of a triangle and divides them in the same ratio, then the line must be parallel to the third side of the triangle.

For a triangle ABC, if a line intersects AB at point D and AC at point E such that the ratio AD/DB is equal to AE/EC, then the line DE is parallel to the side BC.

Explanation

For an equilateral triangle, the relation between the circumradius R and the inradius r is given by the formula r = R / 2.

Given that the radius of the circumscribed circle is 14 cm, we substitute this value into the formula:

r = 14 / 2 = 7 cm

Explanation

From the given equation, we have: 35% of (x + y) = 40% of (x - y)

By cancelling the percentage sign and simplifying: 35(x + y) = 40(x - y)

Dividing both sides by 5: 7(x + y) = 8(x - y)

Expanding both sides: 7x + 7y = 8x - 8y

Grouping the terms of x on one side and y on the other: 8x - 7x = 7y + 8y x = 15y

To find what percentage x is of y, we use the formula: Percentage = (x / y) × 100

Substituting x = 15y into the formula: Percentage = (15y / y) × 100 = 15 × 100 = 1500

Explanation

Let the two numbers be 6x and 11x. The difference between the two numbers is represented by the difference in their ratio parts.

Difference = 11x - 6x = 5x

Given that the difference is 65: 5x = 65 x = 13

The smaller number corresponds to 6x.

Smaller number = 6 × 13 = 78

Explanation

Let the total quantity of the mixture be T ml.

Since milk constitutes 19% of the total mixture, we can write the relation as: 19% of T = 437 (19/100) × T = 437

Solving for the total quantity: T = (437 × 100) / 19 T = 23 × 100 T = 2300

Explanation

Assume the initial population of the town is 100.

After the first year, with an increase of 44%, the population becomes: 100 + 44 = 144

After the second year, with an increase of 75% on the first year's ending population, the population becomes: 144 + (0.75 × 144) = 144 + 108 = 252

The total net increase in population over the two-year period is: 252 - 100 = 152

This represents a total growth of 152% over the span of two years.

The average rate of increase in the population over this period is: Total percentage increase ÷ Number of years = 152% ÷ 2 = 76%

Explanation

Amount under compound interest after t years is given by the formula A = P × (1 + R/100)ᵗ.

Let the compounding factor for a 2-year period be k = (1 + R/100)².

For the first 2 years: 3645 = 2000 × k k = 3645 / 2000 = 729 / 400

The amount after 4 years is: A = P × (1 + R/100)⁴ = P × k²

Substituting the values: A = 2000 × (729 / 400)² A = 2000 × 531441 / 160000 A = 531441 / 80 A = 6643.0125

Explanation

Let the cost price of the table be CP.

Initially, Mohit sells the table at a profit of 25%, so the initial selling price is: SP₁ = 125% of CP = 1.25 CP

When the selling price is reduced by Rs 240, Mohit suffers a loss of 5%, making the new selling price: SP₂ = 95% of CP = 0.95 CP

The change in the selling price is given by the difference between the two selling prices: SP₁ - SP₂ = 240 1.25 CP - 0.95 CP = 240 0.30 CP = 240 CP = 240 ÷ 0.30 = 800

The initial selling price of the table is: SP₁ = 1.25 × 800 = 1000

Explanation

Let the marked price (MP) of the article be 100.

After the first discount of 25%, the price becomes: 100 × (1 - 0.25) = 75

After the second successive discount of 30%, the final selling price (SP) becomes: 75 × (1 - 0.30) = 75 × 0.7 = 52.5

The ratio of the marked price to the selling price is: MP : SP = 100 : 52.5

Multiplying both terms of the ratio by 2 to eliminate the decimal gives: MP : SP = 200 : 105

Dividing both terms by their greatest common divisor, which is 5, yields: MP : SP = 40 : 21

Explanation

Let the required number be represented by x. According to the given condition, subtracting 49 from the square of the number gives 576. This relationship can be expressed mathematically as:

x² - 49 = 576

To solve for x, add 49 to both sides:

x² = 576 + 49 x² = 625

Taking the positive square root of both sides gives the value of the number:

x = √625 = 25

Explanation

Let the total work of making a table be represented by the lowest common multiple of 40 and 60, which is 120 units.

The combined efficiency of P, Q, and R working together is 120 ÷ 40 = 3 units per minute.

The combined efficiency of P and Q working together is 120 ÷ 60 = 2 units per minute.

The efficiency of R alone is the difference between the two efficiencies: 3 - 2 = 1 unit per minute.

The time taken by R alone to complete the table is: 120 units ÷ 1 unit per minute = 120 minutes.

Explanation

When two trains of equal length L move in opposite directions with the same speed v, their relative speed is v + v = 2v. The total distance they must cover to completely cross each other is the sum of their lengths, which is L + L = 2L.

Using the speed, distance, and time relationship: Relative speed = Total distance / Time

Given that the length of each train L = 486 metres and the time taken is 27 seconds: 2v = (486 + 486) / 27 2v = 972 / 27 2v = 36 m/s v = 18 m/s

To convert the speed from m/s to km/hr, multiply by 18/5: v = 18 × 18/5 v = 324/5 v = 64.8 km/hr

Explanation

The perimeter P of a rectangle is given by the formula P = 2(l + b), where l is the length and b is the breadth. Given P = 44 cm and b = 10 cm, we can solve for the length:

44 = 2(l + 10) 22 = l + 10 l = 12 cm

The area of a rectangle is given by the formula Area = l × b. Using the length of 12 cm and breadth of 10 cm:

Area = 12 × 10 = 120 cm²

Explanation

The sum of the exterior angles of any convex polygon is always 360°. For a regular polygon, all exterior angles are equal in measure.

The formula for the measure of each exterior angle of a regular polygon with n sides is: Exterior Angle = 360° / n

Substituting the given number of sides, n = 9: Exterior Angle = 360° / 9 = 40°

This directly gives the measure of each exterior angle as 40°.

Explanation

The volume V of a right circular cylinder is calculated using the formula V = πr²h, where r is the radius of the cylinder and h is its height.

Given that the diameter of the cylinder is 21 cm, the radius r is 21/2 cm. The height h is given as 10 cm. Using π ≈ 22/7, the volume is:

V = (22/7) × (21/2) × (21/2) × 10

V = 11 × 3 × 21 × 5

V = 33 × 105

V = 3465 cm³