SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Let the 7/2th of the number be represented as y.

The problem states that 9/4th of this value is 126. This relationship can be written as: (9/4) × y = 126

To find the value of y, multiply both sides by the reciprocal of 9/4: y = 126 × (4/9)

Simplifying the division: 126 ÷ 9 = 14

Now, multiply the result: y = 14 × 4 y = 56

Explanation

The ratio of the present ages of Ratna and Shantanu is 5:4, and after 19 years, the ratio becomes 10:9.

By comparing the ratio units: Ratna's ratio increases from 5 to 10, which is an increase of 5 units. Shantanu's ratio increases from 4 to 9, which is also an increase of 5 units.

Since the passage of 19 years corresponds to an increase of 5 units for both, we can directly establish that: 5 units = 19 years

Since Ratna's present age is represented by 5 units in the initial ratio, Ratna's present age is 19 years.

Explanation

To find the possible values for x, we solve the system of inequalities:

First inequality: 5 - 5x < 4 - x 5 - 4 < 5x - x 1 < 4x x > 1/4

Second inequality: 2 - x < 6 - 4x 4x - x < 6 - 2 3x < 4 x < 4/3

Combining both inequalities gives the valid range: 1/4 < x < 4/3

Substituting x = 1 into this range yields: 1/4 < 1 < 4/3

This is a true mathematical statement, showing that x = 1 satisfies both inequalities.

Explanation

Let the ten's digit of the number be t and the unit's digit be u. The number is represented as 10t + u.

According to the first condition, the ten's digit is greater than the unit's digit by 4: t - u = 4

For the number 73, the ten's digit is 7 and the unit's digit is 3. Their difference is 7 - 3 = 4, which satisfies this condition.

According to the second condition, subtracting 36 from the number results in the number with interchanged digits: 10t + u - 36 = 10u + t

Substituting the value 73 gives: 73 - 36 = 37

Since 37 is indeed the number formed by interchanging the digits of 73, the number is 73.

Explanation

Since both the policeman and the thief run for the same duration of time until the policeman catches the thief, the distances they cover are directly proportional to their respective speeds.

Ratio of distance covered by the policeman to that covered by the thief = Speed of policeman / Speed of thief = 12 / 7

Let the distance covered by the policeman be 12x and that by the thief be 7x.

The difference in the distance covered by them is equal to the initial distance separating them: 12x - 7x = 450 5x = 450 x = 90

The distance run by the thief before being overtaken: 7x = 7 × 90 = 630 metres

Explanation

When purchasing 5 items, they are grouped as one set of 2 items and one set of 3 items to avail of the respective discounts.

The marked price of 5 items is 5 × ₹900 = ₹4500.

For the purchase of 2 items, the discount is 6% on their marked price of ₹1800: Discount 1 = 6% of ₹1800 = ₹108

For the purchase of 3 items, the discount is 12% on their marked price of ₹2700: Discount 2 = 12% of ₹2700 = ₹324

The total discount obtained on all 5 items is ₹108 + ₹324 = ₹432.

The effective discount percentage is calculated as: Effective discount = (Total discount / Total marked price) × 100 Effective discount = (432 / 4500) × 100 = 9.6%

Explanation

According to the property of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides:

Area(ΔABC) / Area(ΔPQR) = (AB / PQ)²

Substituting the given values into the formula:

1296 / Area(ΔPQR) = (36 / 16)²

Simplifying the ratio of the sides:

36 / 16 = 9 / 4

Substituting this back into the equation:

1296 / Area(ΔPQR) = (9 / 4)²

1296 / Area(ΔPQR) = 81 / 16

Area(ΔPQR) = (1296 × 16) / 81

Since 1296 / 81 = 16:

Area(ΔPQR) = 16 × 16 = 256 sq cm

Explanation

The slope m of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Substituting the given points (6, -3) and (2, -1) into the formula:

m = (-1 - (-3)) / (2 - 6) m = (-1 + 3) / (-4) m = 2 / (-4) m = -½

Since parallel lines have equal slopes, any line parallel to the given line will also have a slope of -½.

Explanation

Let the Marked Price of the microwave oven be MP.

A discount of 15% means the selling price is 85% of the Marked Price: 0.85 × MP = 34000 MP = 34000 / 0.85 = 40000

A discount of 37.5% means the selling price is 62.5% of the Marked Price: New Selling Price = 0.625 × MP New Selling Price = 0.625 × 40000 = 25000

Alternatively, this can be solved directly using ratios: New Selling Price = 34000 × (62.5 / 85) New Selling Price = 34000 × (25 / 34) New Selling Price = 1000 × 25 = 25000

Explanation

An n-gonal prism has bases with n vertices each. For any n-gonal prism, the number of vertices V is given by V = 2n, and the number of edges E is given by E = 3n.

Given that the prism has 18 edges: 3n = 18 n = 6

Using this value of n, the number of vertices is calculated as: V = 2n = 2 × 6 = 12

Explanation

Let the number be x. Since increasing the number by 21 makes it 114% of itself, the increase of 21 represents the 14% increase in the value of the number.

14% of x = 21 (14 / 100) × x = 21 x = (21 × 100) / 14 x = 3 × 50 x = 150

Explanation

Let CP1 and CP2 be the cost prices of the mixer and the microwave oven, respectively.

Since the selling prices of both items are equal, we can relate their cost prices as: CP1 × (100 + 32)% = CP2 × (100 - 22)% CP1 × 132% = CP2 × 78% CP1 / CP2 = 78 / 132 = 13 / 22

The ratio of their cost prices CP1 : CP2 is 13 : 22.

The total ratio parts equal 13 + 22 = 35. Since 13 is the smaller ratio, CP1 represents the cheaper item.

Cost price of the cheaper item = (13 / 35) × 36750 = 13 × 1050 = 13650

Explanation

Area of the circle = πr²

Given, πr² = 64π r² = 64 r = 8 cm

The diameter of the circle is d = 2r = 2 × 8 = 16 cm.

Since the side of the square s is equal to the diameter of the circle: s = 16 cm

The area of the square is s² = 16² = 256 sq cm.

Explanation

The volume V of a hemisphere of radius r is given by the formula:

V = (2/3) × π × r³

Substituting the given radius r = 14 cm and taking π as 22/7:

V = (2/3) × (22/7) × 14 × 14 × 14

V = (2/3) × 22 × 2 × 196

V = (2/3) × 8624

V = 17248/3

V ≈ 5749.33 cubic cm

Explanation

Using the trigonometric identity cos(-θ) = cos(θ), we can rewrite the expression as:

cos(-11π/6) = cos(11π/6)

We can express 11π/6 as 2π - π/6. Applying the periodic property of the cosine function, cos(2π - θ) = cos(θ):

cos(2π - π/6) = cos(π/6)

The standard value of cos(π/6) is √3/2. This gives the value of x as √3/2.

Explanation

The standard trigonometric identity for the tangent of the difference of two angles is given by the formula:

tan(A - B) = (tanA - tanB) / (1 + tanA tanB)

Comparing this formula directly with the given equation:

x = (tanA - tanB) / (1 + tanAtanB)

It follows immediately that x = tan(A - B).

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab and the relation tan A × cot A = 1, the given expression can be written as: x = tan²A + cot²A + 2 tan A cot A x = (tan A + cot A)²

By expressing tan A and cot A in terms of sin A and cos A, we get: tan A + cot A = sin A / cos A + cos A / sin A tan A + cot A = (sin²A + cos²A) / (sin A cos A)

Using the trigonometric identity sin²A + cos²A = 1, this simplifies to: tan A + cot A = 1 / (sin A cos A) tan A + cot A = sec A cosec A

Substituting this relation back into the expression for x gives: x = (sec A cosec A)² x = sec²A cosec²A

Explanation

To find the percentage of students who chose Engineering who are girls, we use the formula:

Percentage = (Number of girls who chose Engineering / Total students who chose Engineering) × 100

From the data table, the number of girls who chose Engineering is 30, and the number of boys who chose Engineering is 60.

Total students who chose Engineering = 60 + 30 = 90

Percentage of Engineering students who are girls = (30 / 90) × 100 = (1 / 3) × 100 = 33.33%

Explanation

To find the number of cars manufactured in the months of April and May, we subtract the cumulative production at the end of March from the cumulative production at the end of May.

Cumulative production up to May = 2860 Cumulative production up to March = 1720

Production in April and May = Cumulative production up to May - Cumulative production up to March Production in April and May = 2860 - 1720 = 1140

Explanation

To find the coefficient of x in the expression (x + 9)(8 - 5x), expand the product of the two binomials:

(x + 9)(8 - 5x) = x(8 - 5x) + 9(8 - 5x) = 8x - 5x² + 72 - 45x

Combine the like terms containing x:

= -5x² + (8 - 45)x + 72 = -5x² - 37x + 72

The term containing x is -37x, which makes the coefficient of x equal to -37.

Explanation

To find the total calories burned in the week, first calculate the total distance jogged by adding the distance for each day of the week.

Total distance = 4 + 2.5 + 3.5 + 2 + 4.5 + 2 + 4 = 22.5 km

Next, determine the number of calories burned per kilometer jogged. Since 400 calories are burned for every 5 km jogged, the rate is:

Calorie burn rate = 400 / 5 = 80 calories per km

Finally, multiply the total distance jogged by the calorie burn rate to find the total calories burned during the week:

Total calories burned = 22.5 × 80 = 1800 calories

Explanation

The total yearly expense is calculated by summing all the individual expense items: Total expenses = 5 + 5 + 2 + 5 + 4 = 21

The combined expense of Labour and Interest is: Labour and Interest = 5 + 5 = 10

To find the percentage of Labour and Interest with respect to the total expenses, use the formula: Percentage = (Labour and Interest / Total expenses) × 100 Percentage = (10 / 21) × 100 ≈ 47.62%

Explanation

The compound interest formula is: A = P(1 + R/100)ⁿ

Where: A is the final amount (₹12321) P is the principal sum (₹10000) n is the number of years (2) R is the annual interest rate

Substituting these values into the formula: 12321 = 10000(1 + R/100)²

Dividing both sides by 10000: 1.2321 = (1 + R/100)²

Taking the square root on both sides: √1.2321 = 1 + R/100 1.11 = 1 + R/100

Subtracting 1 from both sides: R/100 = 0.11 R = 11%

Explanation

Using the algebraic identity: (x - y)² = x² + y² - 2xy

Substitute the given values, x - y = -9 and xy = -20, into the formula: (-9)² = x² + y² - 2(-20)

81 = x² + y² + 40

Subtracting 40 from both sides yields: x² + y² = 81 - 40 x² + y² = 41

Explanation

The distance between two points when two different speeds and the difference in travel time are given is calculated using the formula:

Distance = (v₁ × v₂) / |v₁ - v₂| × Δt

Here, the speed of the first car v₁ = 36 km/hr, the speed of the second car v₂ = 54 km/hr, and the difference in time Δt = 3.5 hours.

Substituting these values into the formula:

Distance = (36 × 54) / (54 - 36) × 3.5 Distance = (36 × 54) / 18 × 3.5 Distance = 2 × 54 × 3.5 Distance = 108 × 3.5 Distance = 378 km

Explanation

Let the tens digit of the two-digit number be x and the units digit be y.

The product of the digits is x × y = 72. Since x and y must be single-digit integers, the only possible pairs of digits that satisfy this condition are 8 and 9. This means the number must be either 89 or 98.

Adding 9 to 89 gives 98, which is the number formed by interchanging the digits of 89. This satisfies both conditions of the problem.

Explanation

To find the values x can take, solve the compound inequality by breaking it into two separate inequalities.

First inequality: 4x - 5(2x - 1) > 2x + 3 4x - 10x + 5 > 2x + 3 -6x + 5 > 2x + 3 -8x > -2 x < 1/4

Second inequality: 2x + 3 > 2 - 3x 5x > -1 x > -1/5

Combining these two results gives the range for x: -1/5 < x < 1/4

Expressing the boundaries as decimals: -0.2 < x < 0.25

The value 0 is the only option that falls within this interval.

Explanation

To find the average bonus in rupees, first calculate the total bonus earned by all categories of employees:

For 1 Manager: Annual salary = 60 lakhs = ₹6,000,000 Bonus per manager = 60% of ₹6,000,000 = ₹3,600,000 Total bonus for managers = 1 × ₹3,600,000 = ₹3,600,000

For 5 Executives: Annual salary = 16 lakhs = ₹1,600,000 Bonus per executive = 30% of ₹1,600,000 = ₹480,000 Total bonus for executives = 5 × ₹480,000 = ₹2,400,000

For 1 Trainee: Annual salary = 2 lakhs = ₹200,000 Bonus per trainee = 20% of ₹200,000 = ₹40,000 Total bonus for trainees = 1 × ₹40,000 = ₹40,000

Total bonus of all employees = ₹3,600,000 + ₹2,400,000 + ₹40,000 = ₹6,040,000

Total number of employees = 1 + 5 + 1 = 7

Average bonus = Total bonus / Total number of employees Average bonus = ₹6,040,000 ÷ 7 ≈ ₹862,857.14

Explanation

To find the pair of years for which the total exports from the three companies together are equal, calculate the sum of exports of Company A, Company B, and Company C for each year:

Total exports in 2011 = 3000 + 2000 + 4000 = 9000 lakh rupees

Total exports in 2012 = 4000 + 2000 + 3000 = 9000 lakh rupees

The total exports for both 2011 and 2012 are equal.

Explanation

The given series 91 + 92 + 93 + ... + 110 is an arithmetic progression.

The formula for the sum of an arithmetic progression is: Sₙ = (n / 2) × (a + l)

In this formula, n represents the number of terms, a represents the first term, and l represents the last term.

For the given series: First term, a = 91 Last term, l = 110 Number of terms, n = 110 - 91 + 1 = 20

Substituting these values into the sum formula: S₂₀ = (20 / 2) × (91 + 110) S₂₀ = 10 × 201 S₂₀ = 2010

Explanation

To find the value of x for which the two expressions are equal, we set them equal to each other:

15x + 4 = 13x - 2

Subtracting 13x from both sides of the equation gives:

2x + 4 = -2

Subtracting 4 from both sides yields:

2x = -6

Dividing both sides by 2, we find the value of x:

x = -3

Explanation

When a journey covers the same distance at two different speeds, x and y, the average speed for the entire trip is calculated using the harmonic mean formula:

Average Speed = 2xy / (x + y)

Substituting the given speeds of x = 70 km/h and y = 30 km/h:

Average Speed = (2 × 70 × 30) / (70 + 30) Average Speed = 4200 / 100 Average Speed = 42 km/h

Explanation

To find the total profit or loss of the company over the 5 years from 2011 to 2015, we sum the profit and loss values for each year:

Total = (-25) + (-20) + (-5) + 20 + (-20)

Combining the negative and positive values: Total = -70 + 20 Total = -50 crore

Since the resulting sum is negative, it represents a net loss of ₹50 crore.

Explanation

To find the percentage increase required to restore the price of an article to its original value after a reduction of r%, we use the standard percentage formula:

Percentage increase = (r / (100 - r)) × 100%

Given that the price of the article is cut by r = 42%, we substitute this value into the formula:

Percentage increase = (42 / (100 - 42)) × 100% Percentage increase = (42 / 58) × 100% Percentage increase = (21 / 29) × 100% Percentage increase = 2100 / 29% Percentage increase ≈ 72.41%

Explanation

The difference between compound interest and simple interest on a principal sum P for a period of 2 years at an annual interest rate R is given by the formula:

D = P × (R / 100)²

Here, the difference D is Rs 8 and the rate of interest R is 4% per annum. Substituting these values into the formula gives:

8 = P × (4 / 100)²

8 = P × (1 / 25)²

8 = P × (1 / 625)

P = 8 × 625

P = 5000

Explanation

Total cost price (CP) of 16 quintals of rice = Rs 5632

Since 20% of the rice is lost in transportation, the quantity of rice remaining for sale is: Remaining rice = 16 × (1 - 0.20) = 12.8 quintals

To earn a profit of 25%, the desired total selling price (SP) is: Total SP = CP × (1 + Profit%) = 5632 × 1.25 = Rs 7040

The selling rate per quintal is calculated by dividing the desired total selling price by the remaining quantity of rice: Selling rate = Total SP ÷ Remaining rice = 7040 ÷ 12.8 = Rs 550 per quintal

Explanation

Correct Answer: D

The first inequality is: 2 + 4x < 5 - x/2

Multiplying the entire inequality by 2 to clear the fraction: 4 + 8x < 10 - x 9x < 6 x < 6/9 x < 2/3

The second inequality is: 3x + 3 > -5 - 3x

Solving for x: 6x > -8 x > -8/6 x > -4/3

Combining both inequalities gives the range for x: -4/3 < x < 2/3

In decimal form, this range is approximately: -1.33 < x < 0.67

The value -1 satisfies both conditions because -1.33 < -1 < 0.67.

Explanation

Let the tens digit of the two-digit number be t and the units digit be u. The value of the number is 10t + u.

According to the first condition, the tens digit is greater than the units digit by 7: t - u = 7

For the number 81, the tens digit is 8 and the units digit is 1. The difference is 8 - 1 = 7, which satisfies this condition.

When 63 is subtracted from 81: 81 - 63 = 18

The number 18 is formed by interchanging the tens and units digits of 81, which satisfies the second condition.

Explanation

According to the properties of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Area(Δ DEF) / Area(Δ GHI) = (DE / GH)²

Given that DE = 9 cm and GH = 16 cm, substituting these values:

Ratio of areas = (9 / 16)² = 81 / 256

Explanation

The total work required to build one cupboard is represented as 1.

The carpenter can build the cupboard in 48 hours, which means his rate of work is 1/48 of the cupboard per hour.

In 12 hours, the fraction of the cupboard built by the carpenter is: 12 / 48 = 1/4 = 0.25

The fraction of the cupboard that is yet to be built is: 1 - 0.25 = 0.75

Explanation

To find the revenue of the 4th year, we use the sum of the revenues.

The total revenue for all 7 years is: 7 × 83 = 581 lakhs

The sum of the revenues of the first 4 years is: 4 × 78 = 312 lakhs

The sum of the revenues of the last 4 years is: 4 × 90 = 360 lakhs

Since the 4th year is included in both the first 4 years and the last 4 years, we can find its revenue by adding these two sums and subtracting the total revenue of all 7 years: Revenue of the 4th year = (312 + 360) - 581 = 672 - 581 = 91 lakhs

Explanation

The slope m of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Substituting the given points (7, -2) and (x, 1), and the slope m = -3/10 into the formula:

-3/10 = (1 - (-2)) / (x - 7)

-3/10 = 3 / (x - 7)

By cross-multiplying:

-3(x - 7) = 30

Dividing both sides by -3:

x - 7 = -10

x = -10 + 7

x = -3

Explanation

To find the percentage of jewellery in the total exports, we first calculate the total value of all exports listed in the table:

Total Exports = 1000 + 725 + 700 + 700 + 700 = 3825 million USNext, we use the percentage formula:

Percentage of Jewellery = (Value of Jewellery / Total Exports) × 100

Percentage of Jewellery = (1000 / 3825) × 100 ≈ 26.14%

Explanation

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

Effective Discount = d₁ + d₂ - (d₁ × d₂) / 100

Here, the initial discount is d₁ = 10%, and the cashback of 5% on the transaction acts as a successive discount of d₂ = 5%.

Substituting these values into the formula:

Effective Discount = 10 + 5 - (10 × 5) / 100 Effective Discount = 15 - 50 / 100 Effective Discount = 15 - 0.5 = 14.5%

Alternatively, if the marked price of the item is assumed to be 100: Price after the 10% discount = 100 - 10 = 90 Cashback of 5% is received on this price = 5% of 90 = 4.5 Total savings = 10 + 4.5 = 14.5 This represents an effective discount of 14.5% on the original price of 100.

Explanation

The sum of the measures of the interior angles of any polygon with n sides is given by the formula (n - 2) × 180°.

Given that the polygon has 12 sides, we substitute n = 12 into the formula:

Sum = (12 - 2) × 180° Sum = 10 × 180° Sum = 1800°

Explanation

The diagonals of a rhombus are perpendicular bisectors of each other. If the diagonals are d₁ and d₂, then the relationship between the side length a and the diagonals is given by the formula: a² = (d₁/2)² + (d₂/2)²

Let the shorter diagonal d₁ = x. Since one diagonal is half the other, the longer diagonal d₂ = 2x.

Given that the side length a = 20 cm, substituting these values into the formula gives: 20² = (x/2)² + (2x/2)² 400 = x²/4 + x² 400 = 5x²/4 5x² = 1600 x² = 320

The area of a rhombus is calculated using the formula: Area = ½ × d₁ × d₂ Area = ½ × x × 2x Area = x²

Substituting the value of x² gives the area of the rhombus as 320 sq cm.

Explanation

The relationship between the volume (V) and the curved surface area (CSA) of a hemisphere is given by the formula:

V / CSA = ((2/3)πr³) / (2πr²) = r / 3

Rearranging this formula to solve directly for the radius (r) gives:

r = 3 × V / CSA

Substituting the given values, V = 19.404 cm³ and CSA = 27.72 cm²:

r = 3 × 19.404 / 27.72 r = 58.212 / 27.72 r = 2.1 cm

Explanation

For any relation of the form xA = yB = zC, the ratio A : B : C can be directly calculated using the formula: A : B : C = yz : xz : xy

In the given equation, x = 6, y = 11, and z = 7. Substituting these values into the formula gives: A : B : C = (11 × 7) : (6 × 7) : (6 × 11) A : B : C = 77 : 42 : 66

This gives the direct ratio in its simplest integer form.

Explanation

The cosine function is an even function, meaning cos(-θ) = cos(θ). Applying this property: cos(-4π/3) = cos(4π/3)

Using the quadrant identity cos(π + θ) = -cos(θ), we can express 4π/3 as π + π/3: cos(4π/3) = cos(π + π/3) = -cos(π/3)

Since the standard value of cos(π/3) is ½: -cos(π/3) = -½

Therefore, x = -½.

Explanation

By the definition of the cotangent function, cot(A + B) can be expressed in terms of sine and cosine as: cot(A + B) = cos(A + B) / sin(A + B)

Using the trigonometric sum formulas for sine and cosine, we have: cos(A + B) = cosA cosB - sinA sinB sin(A + B) = sinA cosB + cosA sinB

Substituting these formulas into the quotient gives: cot(A + B) = (cosA cosB - sinA sinB) / (sinA cosB + cosA sinB)

To express the right-hand side in terms of cotangent, we divide every term in both the numerator and the denominator by sinA sinB: cot(A + B) = ((cosA cosB / sinA sinB) - (sinA sinB / sinA sinB)) / ((sinA cosB / sinA sinB) + (cosA sinB / sinA sinB))

Applying the identity cotθ = cosθ / sinθ to simplify each term: cot(A + B) = (cotA cotB - 1) / (cotB + cotA)

Comparing this mathematical identity with the given equation: x = (cotAcotB - 1) / (cotB + cotA)

This directly yields: x = cot(A + B)