SSC CHSL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

Each letter of the word is represented first by its row number and next by its column number.

For the letter T, looking at Matrix II, the row is 7 and the column is 7, yielding the code 77.

For the letter O, looking at Matrix II, the row is 6 and the column is 9, yielding the code 69.

For the letter M (which corresponds to the letter N in the misspelled target word TONB), looking at Matrix I, the row is 4 and the column is 3, yielding the code 43.

For the letter B, looking at Matrix I, the row is 2 and the column is 2, yielding the code 22.

Explanation

The expression is (49)³/² + (49)⁻³/².

Using the laws of exponents, we can simplify each term by expressing 49 as 7²: (49)³/² = (7²)³/² = 7³ = 343 (49)⁻³/² = (7²)⁻³/² = 7⁻³ = 1/7³ = 1/343

Adding these two values together: 343 + 1/343 = (343 × 343 + 1) / 343

Since 343 × 343 = 117649, the numerator becomes: 117649 + 1 = 117650

This yields the simplified value: 117650/343

Explanation

To find the total number of prime factors in the given expression, we must first express all the composite base numbers as a product of prime numbers.

The given expression is: 4¹¹ × 5⁵ × 3² × 13²

Here, the base 4 is a composite number, which can be written as 2². The other bases (5, 3, and 13) are already prime numbers.

Substituting 4 = 2² into the expression: (2²)¹¹ × 5⁵ × 3² × 13² = 2²² × 5⁵ × 3² × 13²

Now that all bases (2, 5, 3, and 13) are prime numbers, the total number of prime factors is the sum of their exponents: 22 + 5 + 2 + 2 = 31

Explanation

To rationalize the denominator of the given expression, multiply both the numerator and the denominator by the conjugate of the denominator, which is √5 - 2:

15 / (√5 + 2) = (15 × (√5 - 2)) / ((√5 + 2) × (√5 - 2))

Apply the algebraic identity (a + b)(a - b) = a² - b² to the denominator:

(√5 + 2)(√5 - 2) = (√5)² - 2² = 5 - 4 = 1

Now, expand the numerator:

15 × (√5 - 2) = 15√5 - 30

Dividing the simplified numerator by the denominator:

(15√5 - 30) / 1 = 15√5 - 30

Explanation

To solve this expression, we can use the standard algebraic identity: (a + b)² - (a - b)² = 4ab

Let a = 943 and b = 864. The numerator of the given expression becomes: (943 + 864)² - (943 - 864)² = 4 × 943 × 864

Now we rewrite the terms in the denominator: 1886 = 2 × 943 1728 = 2 × 864

This means the denominator is: 1886 × 1728 = (2 × 943) × (2 × 864) = 4 × 943 × 864

Substituting the simplified numerator and denominator back into the equation: x = (4 × 943 × 864) / (4 × 943 × 864)

Both the numerator and the denominator are identical, so they cancel out completely: x = 1

Explanation

Two triangles are mathematically defined as similar if their corresponding angles are equal and their corresponding sides are proportional to each other. When two triangles are similar, the ratio of their corresponding sides remains constant, which means their side lengths can either be equal (resulting in congruent triangles with a scale factor of 1) or unequal. Thus, the corresponding sides may or may not be equal to each other, making all the given conditions valid for similar triangles.

Explanation

The area of an equilateral triangle can be directly calculated from its height h using the formula:

Area = h² / √3

Substituting the given height h = 2√3 cm into the formula:

Area = (2√3)² / √3 Area = 12 / √3 Area = 4√3 cm²

Alternatively, the height h of an equilateral triangle with side length a is given by h = (√3/2)a. Equating (√3/2)a = 2√3 gives the side length a = 4 cm. The area of the equilateral triangle is given by (√3/4)a², which evaluates to (√3/4) × 4² = 4√3 cm².

Explanation

The 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 = 10 and b = 20:

Net percentage increase = 10 + 20 + (10 × 20)/100 % Net percentage increase = 30 + 200/100 % Net percentage increase = 30 + 2 % Net percentage increase = 32%

Explanation

Let the three numbers be represented as 3x, 6x, and 8x.

The product of these three numbers is given as 9216: 3x × 6x × 8x = 9216 144x³ = 9216

Solving for x³: x³ = 9216 / 144 x³ = 64 x = 4

The sum of the three numbers is: 3x + 6x + 8x = 17x

Substituting the value of x: 17 × 4 = 68

Explanation

The ratio of profits in a partnership is given by the product of the investment and the time period of investment.

Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Let Rohit be person 1 and Simran be person 2. Rohit's investment (I₁) = Rs 75000 Rohit's time period (T₁) = 12 months Simran's investment (I₂) = Rs 60000 Simran's time period = T₂ months Profit ratio = 3 : 1

Substituting the values into the formula: (75000 × 12) ÷ (60000 × T₂) = 3 ÷ 1

Simplifying the equation: (5 × 12) ÷ (4 × T₂) = 3 60 ÷ (4 × T₂) = 3 15 ÷ T₂ = 3 T₂ = 5 months

Simran invested her money for 5 months. Therefore, the number of months after which Simran joined the business is: 12 - 5 = 7 months

Explanation

Total price of 50 chocolates is Rs 1000.

The average price of 30 chocolates is Rs 25, so their total price is: 30 × 25 = Rs 750

The remaining number of chocolates is: 50 - 30 = 20

The total price of these remaining 20 chocolates is: 1000 - 750 = Rs 250

Therefore, the average price of the remaining chocolates is: 250 / 20 = Rs 12.5

Explanation

The total simple interest earned on a sum of money is calculated as the sum of the interests earned during the respective periods.

Let the principal sum be P. The rate of interest is 10% per annum for the first 5 years and 15% per annum for the next 3 years.

The total interest percentage accrued over the 8 years is: (10% × 5) + (15% × 3) = 50% + 45% = 95%

According to the problem, this total interest is equal to Rs 47500: 95% of P = 47500 P × 95 / 100 = 47500 P = 47500 × 100 / 95 P = 500 × 100 P = 50000

The principal sum is Rs 50000.

Explanation

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

CP = SP × 100 / (100 - Loss%)

Substituting the given values, SP = Rs 1386 and Loss% = 23%:

CP = 1386 × 100 / (100 - 23)

CP = 1386 × 100 / 77

CP = 18 × 100 = 1800

Explanation

The marked price of the computer is Rs 48000.

After the first discount of 13%, the intermediate selling price becomes: 48000 × (1 - 0.13) = 48000 × 0.87 = Rs 41760

The final price to be offered to the customer is Rs 39672.

The additional discount amount required on the intermediate price of Rs 41760 is: 41760 - 39672 = Rs 2088

The additional discount percentage is: (2088 / 41760) × 100 = 5%

Explanation

In Statement I, representing the term as the cube root of 125, we have ∛125 = 5. LHS = √144 × √36 = 12 × 6 = 72 RHS = ∛125 × √121 = 5 × 11 = 55 Since 72 is greater than 55, Statement I is false.

In Statement II, representing the term as the cube root of 216, we have ∛216 = 6. LHS = √324 + √49 = 18 + 7 = 25 RHS = ∛216 × √9 = 6 × 3 = 18 Since 25 is greater than 18, Statement II is true.

Explanation

Let the total capacity of the tank be the Least Common Multiple (LCM) of 20 and 35, which is 140 units.

The efficiency of Pipe X is 140 ÷ 20 = 7 units per hour. The efficiency of Pipe Y is 140 ÷ 35 = 4 units per hour.

Since they are opened on alternate hours starting with Pipe Y, the work done in a 2-hour cycle is: Work done in first hour (by Y) = 4 units Work done in second hour (by X) = 7 units Total work done in one 2-hour cycle = 4 + 7 = 11 units

To find the number of complete cycles, we divide the total capacity by the work done in one cycle: 140 ÷ 11 = 12 cycles with a remainder of 8 units.

The time taken for 12 complete cycles is 12 × 2 = 24 hours. The work completed is 12 × 11 = 132 units, leaving 140 - 132 = 8 units to be filled.

In the 25th hour, Pipe Y is opened and fills 4 units. The remaining capacity is 8 - 4 = 4 units, and the total time elapsed is 25 hours.

In the next hour, Pipe X is opened to fill the remaining 4 units. Since the efficiency of Pipe X is 7 units per hour, the time taken by Pipe X is 4/7 hours.

The total time taken to fill the tank is 25 + 4/7 = 179/7 hours.

Explanation

To convert the speed from km/hr to m/s, multiply the speed by 5/18: Speed = 54 × 5/18 = 15 m/s

Using the standard relationship: Time = Distance / Speed

Substitute the given values into the formula: Time = 450 / 15 = 30 seconds

Explanation

The diagonal of a rectangle forms a right-angled triangle with its two adjacent sides. According to the Pythagoras theorem: d² = l² + w²

Here, the diagonal (d) is 25 cm and one side (l) is 7 cm. Substituting these values into the formula: 25² = 7² + w² 625 = 49 + w² w² = 625 - 49 w² = 576 w = 24 cm

The other side of the rectangle is 24 cm.

The perimeter of a rectangle is given by the formula: Perimeter = 2 × (l + w) Perimeter = 2 × (7 + 24) Perimeter = 2 × 31 Perimeter = 62 cm

Explanation

For a regular polygon with n sides, the measure of each exterior angle is given by the formula 360°/n.

Substituting n = 5: Exterior angle = 360°/5 = 72°

Since the interior angle and exterior angle at any vertex of a polygon sum to 180°: Interior angle = 180° - 72° = 108°

The difference between the interior angle and the exterior angle is: 108° - 72° = 36°

Explanation

The volume of a cube is calculated using the formula V = s³, where s represents the length of the side of the cube.

Substituting the given side length of 3.5 cm into the formula:

V = (3.5)³ V = 3.5 × 3.5 × 3.5 V = 42.875 cm³

Explanation

In triangle XYZ, which is right-angled at Y, the sum of all interior angles is 180°. Given that ∠Y = 90° and ∠X = 60°, we can find the measure of angle Z using the angle sum property of triangles: ∠Z = 180° - ∠Y - ∠X = 180° - 90° - 60° = 30°

We need to evaluate the expression sec Z + 2 / √3. Substituting ∠Z = 30° into the expression gives: sec 30° + 2 / √3

Using the standard trigonometric ratio, sec 30° = 2 / √3. Thus, the expression becomes: 2 / √3 + 2 / √3 = 4 / √3

Explanation

Applying the BODMAS rules step-by-step:

1. Simplify the innermost parentheses: (b - a - b) = -a.
2. Simplify the braces: {b - (-a)} = {b + a}.
3. Simplify the square bracket: [b - (a + b) - (b + a) + 2a] = [b - a - b - b - a + 2a] = [-b].
4. Finally, subtract this result from the outermost term: b - [-b] = 2b (Option A).

Explanation

Using the algebraic identity (x + y)² + (x - y)² = 2(x² + y²), we can simplify the given expression by choosing x = 200 and y = 13.

Here, 213 = 200 + 13 and 187 = 200 - 13.

Substituting these values into the identity:

213² + 187² = (200 + 13)² + (200 - 13)²

This simplifies to:

2 × (200² + 13²) = 2 × (40000 + 169)

Multiplying the terms:

2 × 40169 = 80338

Explanation

Using the algebraic identity for the square of a binomial:

(x - y)² = x² + y² - 2xy

Substitute the given values x - y = 5 and x² + y² = 45 into the formula:

5² = 45 - 2xy

25 = 45 - 2xy

Rearranging the terms to solve for 2xy:

2xy = 45 - 25

2xy = 20

Dividing by 2 gives:

xy = 10

Explanation

The sum of all interior angles of a triangle is always 180°. Let the largest, smallest, and third angles of the triangle be represented by L, S, and T respectively.

Given that the largest angle L = 80°.

The smallest angle S is 40° less than the largest angle: S = 80° - 40° = 40°

Using the triangle angle sum formula: L + S + T = 180° 80° + 40° + T = 180° 120° + T = 180° T = 60°

Explanation

According to the Basic Proportionality Theorem, also known as Thales's Theorem, if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

In triangle ABC, the line DE is drawn parallel to the base BC, intersecting the sides AB and AC at points D and E respectively. This parallel line divides the sides AB and AC proportionally, which mathematically yields the relation AD/DB = AE/EC.

Explanation

Let the original price of 1 kg of sugar be x.

The price increases by 30%, which is equal to ₹48. Using the percentage relation: 30% of x = 48 (30/100) × x = 48 x = (48 × 100) / 30 x = 160

The original price of the sugar is ₹160.

The new price is the original price plus the increase: New Price = 160 + 48 = 208

Explanation

Let the three numbers be x, y, and z.

The ratio of the first number to the second number is: x : y = 3 : 5

The ratio of the second number to the third number is: y : z = 4 : 7

To combine these ratios, we make the term representing y equal in both ratios by taking the least common multiple of 5 and 4, which is 20.

Multiplying the first ratio by 4: x : y = 12 : 20

Multiplying the second ratio by 5: y : z = 20 : 35

Thus, the combined ratio of the three numbers is: x : y : z = 12 : 20 : 35

Let the three numbers be 12k, 20k, and 35k. The sum of the three numbers is given as 67: 12k + 20k + 35k = 67 67k = 67 k = 1

The second number is: 20k = 20 × 1 = 20

Explanation

In a partnership, profit is distributed in the ratio of the product of the capital invested and the duration of the investment.

A invested Rs 1800 for the entire 12 months: A's equivalent investment = 1800 × 12 = 21600

B joined after 4 months, which means B invested Rs 3600 for 8 months: B's equivalent investment = 3600 × 8 = 28800

C joined after 6 months, which means C invested Rs 1200 for 6 months: C's equivalent investment = 1200 × 6 = 7200

The profit-sharing ratio among A, B, and C is: A : B : C = 21600 : 28800 : 7200 = 3 : 4 : 1

The sum of the ratio parts is 3 + 4 + 1 = 8.

Since the total profit is Rs 1760, the value of 1 ratio part is 1760 ÷ 8 = 220.

A's share in the profit represents 3 ratio parts, which equals 3 × 220 = 660.

Explanation

The average of any arithmetic progression, including consecutive integers, is the middle value, which is equal to the average of the first and the last term.

For 8 consecutive integers starting with n, the terms are n, n+1, ..., n+7.

Average of these 8 integers = (n + n + 7)/2 = (2n + 7)/2

Given that the average is 23/2: (2n + 7)/2 = 23/2 2n + 7 = 23 2n = 16 n = 8

The first three integers are 8, 9, and 10.

The average of the first three consecutive integers is simply their middle term, which is 9.

Explanation

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

Where SI is the simple interest, P is the principal sum, R is the annual rate of interest, and T is the time period in years.

Converting the given time of 8 months into years: T = 8/12 = 2/3 years

Substituting the given values (SI = Rs 340, R = 6%) into the formula: 340 = (P × 6 × (2/3)) / 100 340 = (P × 4) / 100 340 = P / 25 P = 340 × 25 P = 8500

Explanation

Let the cost price (CP) of the wire be ₹100.

Since the trader gains 60% on the cost price, the profit is ₹60.

The selling price (SP) is: SP = CP + Profit = 100 + 60 = ₹160

The profit percentage on the selling price is calculated as: Profit Percentage on SP = (Profit / SP) × 100

Substituting the values: Profit Percentage on SP = (60 / 160) × 100 = (3 / 8) × 100 = 37.5%

Explanation

Discount = Marked Price - Selling Price Discount = 520 - 384.8 = 135.2

Discount percentage = (Discount / Marked Price) × 100 Discount percentage = (135.2 / 520) × 100 = 26%

Explanation

By factoring out 5 from the expression, we get: 5 × [1/(√2 + 1) + 1/(√3 + √2) + 1/(√4 + √3) + ... + 1/(√121 + √120)]

We can rationalize each term in the series by multiplying the numerator and denominator by its conjugate: 1/(√(n+1) + √n) = (√(n+1) - √n) / ((√(n+1) + √n)(√(n+1) - √n)) = √(n+1) - √n

Applying this rationalization to each individual term inside the bracket gives: 1/(√2 + 1) = √2 - 1 1/(√3 + √2) = √3 - √2 1/(√4 + √3) = √4 - √3 ... 1/(√121 + √120) = √121 - √120

Substituting these rationalized values back into the expression yields a telescoping sum where intermediate terms cancel each other out: 5 × [(√2 - 1) + (√3 - √2) + (√4 - √3) + ... + (√121 - √120)] 5 × (√121 - 1)

Since √121 = 11, the calculation simplifies to: 5 × (11 - 1) = 5 × 10 = 50

Explanation

When two people work together for the same duration to complete a task, their wages are distributed in the ratio of their efficiencies, which is the inverse ratio of the time they take to complete the work individually.

Ratio of time taken by Ram and Shyam = 5 : 9

Therefore, the ratio of their efficiencies and wage distribution is 9 : 5.

Total parts in the ratio = 9 + 5 = 14 parts

Total wage amount = ₹ 5600

Value of 1 part = 5600 / 14 = ₹ 400

The difference between their shares in ratio parts is 9 - 5 = 4 parts.

Difference in the amount received = 4 × 400 = ₹ 1600

Explanation

The time taken for a journey is given by the relation: Time = Distance ÷ Speed

Let the one-side distance be d km. The time taken to cover this distance by train is d/40 hours, and the time taken to walk back the same distance is d/8 hours.

Since the total time taken for the entire journey is 12 hours, we can write the equation: d/40 + d/8 = 12

To solve for d, take the common denominator of 40: (d + 5d) / 40 = 12 6d / 40 = 12 6d = 480 d = 80

Explanation

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

Area = (√3/4) × a²

Substituting the given side length of 8 cm into this formula:

Area = (√3/4) × 8² Area = (√3/4) × 64 Area = 16√3

Explanation

The area of a circle is given by the formula A = πr², where r is the radius of the circle.

Substituting the given area: πr² = 346.5 (22/7) × r² = 346.5 r² = (346.5 × 7) / 22 r² = 110.25 r = √110.25 = 10.5 cm

The circumference of a circle is given by the formula C = 2πr.

Substituting the value of the radius: C = 2 × (22/7) × 10.5 C = 2 × 22 × 1.5 C = 66 cm

Explanation

The total surface area of a cube with side length a is given by the formula:

Total Surface Area = 6 × a²

Given that the side length a = 7.5 cm:

Total Surface Area = 6 × (7.5)²

Total Surface Area = 6 × 56.25

Total Surface Area = 337.5 cm²

Explanation

We know that the standard trigonometric value of tan 45° is 1.

Substituting this value into the given expression: tan 45° + 1 / √2 = 1 + 1 / √2

By taking the common denominator of √2, we can combine the terms: 1 + 1 / √2 = (√2 + 1) / √2

This simplifies to (√2 + 1) / √2.

Explanation

In a right-angled triangle PQR where angle Q = 90°, the sides satisfy the Pythagorean theorem.

Given cosec P = PR / QR = 17 / 15, where PR is the hypotenuse and QR is the opposite side to angle P.

Using the Pythagorean relation PQ² + QR² = PR²: PQ = √(PR² - QR²) PQ = √(17² - 15²) = √(289 - 225) = √64 = 8

For angle R, the opposite side is PQ and the hypotenuse is PR. The sine of angle R is given by: sin R = PQ / PR = 8 / 17

Explanation

The prime factorization of 69120 is: 69120 = 2⁹ × 3³ × 5¹

For a number to be a perfect cube, the exponent of each prime factor in its prime factorization must be a multiple of 3.

The exponent of 2 is 9, which is a multiple of 3. The exponent of 3 is 3, which is a multiple of 3. The exponent of 5 is 1, which is not a multiple of 3. The nearest multiple of 3 greater than 1 is 3.

To make the exponent of 5 a multiple of 3, the number must be multiplied by 5²: 5² = 25

Multiplying 69120 by 25 yields 1728000, which is the cube of 120 (120³).

Explanation

Let d be the original number of days of the tour. Since the total budget for the tour is Rs. 360, the original daily expenditure is 360/d.

If the tour is extended by 4 days, the new duration becomes d + 4 days, making the new daily expenditure 360/(d + 4).

The difference in daily expenditure is Rs. 3, which gives the equation: 360/d - 360/(d + 4) = 3

Dividing both sides of the equation by 3: 120/d - 120/(d + 4) = 1

Taking the common denominator and simplifying: 120 × (d + 4 - d) / (d(d + 4)) = 1 480 / (d(d + 4)) = 1 d(d + 4) = 480 d² + 4d - 480 = 0

Factoring the quadratic equation: d² + 24d - 20d - 480 = 0 d(d + 24) - 20(d + 24) = 0 (d + 24)(d - 20) = 0

Since the number of days must be a positive value, we find d = 20.

Explanation

The algebraic identity for the sum of cubes is: r³ + s³ = (r + s)³ - 3rs(r + s)

Substituting the given values r³ + s³ = 0 and r + s = 6: 0 = 6³ - 3rs(6) 0 = 216 - 18rs 18rs = 216 rs = 12

The expression to be evaluated is: 1/r + 1/s = (r + s) / rs

Substituting the values of r + s = 6 and rs = 12: 1/r + 1/s = 6 / 12 = 0.5

Explanation

To find the y-intercept of the linear equation 59x + 14y - 112 = 0, substitute x = 0 into the equation:

59(0) + 14y - 112 = 0

14y = 112

y = 112 / 14

y = 8

This gives the y-intercept as 8.

Explanation

When two circles touch externally, the distance between their centres is equal to the sum of their radii:

d = r₁ + r₂

Here, the distance between the centres d is 12 cm and the radius of one circle r₁ is 7 cm. Substituting these values into the formula:

12 = 7 + r₂ r₂ = 5 cm

The radius of the other circle is 5 cm. Therefore, its diameter is:

Diameter = 2 × r₂ Diameter = 2 × 5 = 10 cm

Explanation

For a triangle with side lengths 9 cm, 12 cm, and 15 cm, we can verify if it is a right-angled triangle using the Pythagorean theorem:

9² + 12² = 81 + 144 = 225 15² = 225

Since 9² + 12² = 15², the given triangle is a right-angled triangle with a hypotenuse of 15 cm.

For any right-angled triangle, the center of the circumcircle (circumference) lies at the midpoint of the hypotenuse. Consequently, the circumradius (R) is equal to half the length of the hypotenuse:

R = Hypotenuse / 2 = 15 / 2 = 7.5 cm

Explanation

To find how much percent X is less than Y, use the percentage decrease formula:

Percentage less = ((Y - X) / Y) × 100

Substitute the given values X = 600 and Y = 800 into the formula:

Percentage less = ((800 - 600) / 800) × 100 = (200 / 800) × 100 = ¼ × 100 = 25%

Explanation

Let the three numbers be 2x, 3x, and 5x.

The sum of these three numbers is: 2x + 3x + 5x = 10x

Given that the sum of the three numbers is 275, we can write: 10x = 275 x = 27.5

The largest among the three numbers is represented by the largest ratio component, which is 5x. Substituting the value of x: 5 × 27.5 = 137.5

Explanation

In any partnership business, the profit shared among partners is directly proportional to the product of their capital invested and the time period of investment. The relation is expressed as:

Profit = Capital × Time

This can be rearranged to find the time period:

Time = Profit / Capital

Let the capitals of S, T, and U be in the ratio C_S : C_T : C_U = 3 : 4 : 6, and their profit shares be in the ratio P_S : P_T : P_U = 1 : 2 : 3.

The ratio of their investment time periods T_S : T_T : T_U is:

T_S : T_T : T_U = (P_S / C_S) : (P_T / C_T) : (P_U / C_U)

Substituting the given ratios:

T_S : T_T : T_U = (1/3) : (2/4) : (3/6)

Simplifying the fractions:

T_S : T_T : T_U = (1/3) : (1/2) : (1/2)

To convert this into an integer ratio, multiply each term by the least common multiple of the denominators 3 and 2, which is 6:

T_S : T_T : T_U = (1/3 × 6) : (1/2 × 6) : (1/2 × 6) = 2 : 3 : 3