SSC CHSL Tier 1 2010 Quantitative Aptitude Questions with Solutions

Explanation

The volume V of a cylinder with radius r and height h is given by V = πr²h.

For two cylinders with equal volumes V₁ and V₂, we have: πr₁²h₁ = πr₂²h₂ r₁²h₁ = r₂²h₂ h₁/h₂ = (r₂/r₁)²

Since the ratio of diameters is 3 : 2, the ratio of their radii r₁ : r₂ is also 3 : 2. Substituting r₁/r₂ = 3/2 into the height ratio equation yields: h₁/h₂ = (2/3)² = 4/9

Explanation

For two successive discounts of a% and b%, the equivalent single discount is determined using the successive percentage formula:

Equivalent Discount = a + b - (a × b)/100 %

Substituting the given successive discount rates of a = 25% and b = 10%:

Equivalent Discount = 25 + 10 - (25 × 10)/100 %

Equivalent Discount = 35 - 250/100 %

Equivalent Discount = 35 - 2.5 %

Equivalent Discount = 32.5%

Explanation

Using the successive percentage change formula for net profit or loss:

Net Change = a + b + (ab / 100)

Here, the markup percentage (a) is +40% and the discount percentage (b) is -25% because a discount represents a reduction in price.

Net Change = 40 - 25 + (40 × (-25) / 100) Net Change = 15 - 1000 / 100 Net Change = 15 - 10 Net Change = 5%

Since the net percentage change is positive, the trader makes a profit of 5%.

Explanation

Let the two numbers be 3x and 4x, where x is their highest common factor.

The LCM of 3x and 4x is the product of their non-common ratios and their highest common factor: LCM = 3 × 4 × x = 12x

Since the LCM is given as 48, we can set up the equation: 12x = 48 x = 4

The sum of the two numbers is: 3x + 4x = 7x

Substituting the value of x into the sum: 7 × 4 = 28

Explanation

To find the ratio A : B : C from the given equation 2A = 3B = 4C, we divide each term by the Least Common Multiple (LCM) of the coefficients 2, 3, and 4.

The LCM of 2, 3, and 4 is 12.

Dividing each term of the equation by 12: 2A/12 = 3B/12 = 4C/12

Simplifying each fraction gives: A/6 = B/4 = C/3

This directly represents the ratio of the variables: A : B : C = 6 : 4 : 3

Explanation

Expressing the base 4 as 2²: 4³.⁵ = (2²)³.⁵

Using the exponent rule (aᵐ)ⁿ = aᵐˣⁿ: 2²ˣ³.⁵ = 2⁷

The given ratio 4³.⁵ : 2⁵ becomes: 2⁷ : 2⁵

Dividing both sides by 2⁵: 2⁷⁻⁵ : 1 = 2² : 1 = 4 : 1

Explanation

The weighted average formula gives the average price of the mixed rice:

Average Price = (Quantity 1 × Price 1 + Quantity 2 × Price 2) / (Quantity 1 + Quantity 2)

Substituting the given values into the formula:

Average Price = (12 × 30 + 8 × 40) / (12 + 8)

Average Price = (360 + 320) / 20

Average Price = 680 / 20

Average Price = 34

Explanation

Let the present ages of the father and the son be 5x and 2x respectively.

According to the given condition, the product of their ages is 1000: 5x × 2x = 1000 10x² = 1000 x² = 100 x = 10

The father's present age is 5x = 5 × 10 = 50 years.

The father's age after 10 years will be 50 + 10 = 60 years.

Explanation

The correct average is calculated by adjusting the initial average for the error in reading the number using the relation:

Correct Average = Original Average + (Correct Value - Incorrect Value) ÷ Number of Observations

Substituting the given values into the formula:

Correct Average = 15 + (36 - 26) ÷ 10

Correct Average = 15 + 10 ÷ 10

Correct Average = 15 + 1 = 16

This direct adjustment adds the difference between the correct and incorrect values, then distributes that difference evenly across all 10 observations to find the correct mean.

Explanation

For any arithmetic progression, including successive numbers, the average of the entire sequence is always equal to the average of the first (smallest) and the last (greatest) terms.

If the eight successive numbers are represented as an arithmetic progression with the smallest term as a and the greatest term as l, the average of the entire sequence is given by the formula:

Average = (a + l) / 2

Since the average of all eight successive numbers is 6.5, the average of the smallest and the greatest numbers, which is also (a + l) / 2, must be 6.5.

Explanation

When a person travels equal distances at two different speeds, say v₁ and v₂, the average speed for the entire journey is given by the harmonic mean formula:

Average Speed = 2v₁v₂ / (v₁ + v₂)

Substituting the given values, v₁ = 36 km/h and v₂ = 45 km/h:

Average Speed = 2 × 36 × 45 / (36 + 45)

Average Speed = 3240 / 81

Average Speed = 40 km/h

Explanation

Let the average of runs after the 12th innings be A. Since the average decreased by 5 runs after the 12th innings, his average for the first 11 innings was A + 5.

The total runs after the 12th innings is the sum of the total runs of the first 11 innings and the runs scored in the 12th innings: 11 × (A + 5) + 90 = 12 × A

Expanding and solving the linear equation: 11A + 55 + 90 = 12A 11A + 145 = 12A A = 145

Explanation

Let the selling price (SP) of the article be 100.

According to the question, the cost price (CP) is 80% of the selling price: CP = 80% of 100 = 80

The profit is the difference between the selling price and the cost price: Profit = SP - CP = 100 - 80 = 20

The profit percentage is calculated using the formula: Profit % = (Profit / CP) × 100

Substituting the values into the formula: Profit % = (20 / 80) × 100 = ¼ × 100 = 25%

Explanation

The present rate per egg can be directly calculated using the price-quantity relation. When the price of an item increases by r%, the reduced quantity Q that can be purchased for a fixed amount A is related to the present price per unit by the formula:

Present Price per unit = (A × r) / (100 × Q)

Substituting the given values: Amount A = Rs. 24 Percentage increase r = 20 Reduction in quantity Q = 2

Present Price per egg = (24 × 20) / (100 × 2) = 4.8 / 2 = Rs. 2.40

Since there are 12 eggs in a dozen, the present rate of eggs per dozen is: 12 × Rs. 2.40 = Rs. 28.80

Explanation

Let the original selling price of the article be ₹100.

The cost price (CP) for Mahesh is 9/10 of this original selling price: CP = (9 / 10) × 100 = ₹90

He then sells it at 8% more than the original selling price: Selling Price (SP) = 100 + 8 = ₹108

The gain made by Mahesh is the difference between his selling price and his cost price: Gain = SP - CP = 108 - 90 = ₹18

The gain percentage is calculated as: Gain percent = (Gain / CP) × 100 = (18 / 90) × 100 = 20%

Explanation

Let the cost price of the article be CP.

The percentage of profit is given by the formula: Profit % = ((SP - CP) / CP) × 100

According to the given condition, the profit percentage when the selling price is Rs. 78 is twice the profit percentage when the selling price is Rs. 69: ((78 - CP) / CP) × 100 = 2 × ((69 - CP) / CP) × 100

Canceling out the common terms (100 and CP) on both sides of the equation simplifies the relation: 78 - CP = 2 × (69 - CP) 78 - CP = 138 - 2CP 2CP - CP = 138 - 78 CP = 60

Explanation

SP of 5 lemons = Rs. 14 SP of 1 lemon = 14 / 5 = Rs. 2.80

Since the vendor gains 40%, the relationship between cost price (CP) and selling price (SP) is: CP = SP / (1 + Gain%)

CP of 1 lemon = 2.80 / (1 + 0.40) = 2.80 / 1.40 = Rs. 2

A dozen contains 12 lemons. CP of 12 lemons = 12 × CP of 1 lemon = 12 × 2 = Rs. 24

Explanation

Since Ram and Shyam earn the same salary, their savings are directly proportional to their saving percentages.

Saving of Ram = (Ram's saving percentage ÷ Shyam's saving percentage) × Shyam's savings

Substituting the given values:

Saving of Ram = (14 ÷ 22) × 1540 Saving of Ram = 14 × 70 Saving of Ram = 980

Explanation

Let B's salary be ₹100. Since A's salary is 25% more than B's salary, A's salary is ₹125.

The difference between their salaries is ₹25. To find how much percent B's salary is less than A's salary, we calculate this difference relative to A's salary:

Percentage less = (Difference / A's salary) × 100 Percentage less = (25 / 125) × 100 = (1/5) × 100 = 20%

Alternatively, using the standard percentage relation, if one quantity is r% more than another, then the second quantity is less than the first by:

[r / (100 + r)] × 100 %

Substituting r = 25:

[25 / (100 + 25)] × 100 % = (25 / 125) × 100 % = 20%

Explanation

The area of a circle is A = πr², where r is the radius. When the radius increases by x%, the percentage change in its square is given by the successive percentage formula:

Percentage increase = 2x + x²/100 %

Substituting x = 1:

Percentage increase = 2(1) + 1²/100 % = 2 + 0.01 % = 2.01 %

Explanation

The passing marks required for the examination are the sum of the marks obtained by the student and the marks by which the student failed.

Passing Marks = 113 + 85 = 198

Since the passing marks represent 36% of the maximum marks, let the maximum marks be M.

36% of M = 198 0.36 × M = 198 M = 198 / 0.36 M = 550

Explanation

Let the two numbers be a and b. We are given their sum is 36, so a + b = 36.

The product of the two numbers is equal to the product of their H.C.F. and L.C.M.: a × b = H.C.F. × L.C.M. = 3 × 105 = 315

The sum of the reciprocals of the two numbers is: 1/a + 1/b = (a + b) / (a × b)

Substituting the known values into the expression gives: 36 / 315

Dividing the numerator and the denominator by their greatest common divisor, 9, simplifies the fraction: 36 / 315 = 4 / 35

Explanation

Let the number be N. According to the division algorithm, the number can be represented as N = 387k + 48, where k is a non-negative integer.

Since 387 is a multiple of 43 (387 = 43 × 9), the term 387k is completely divisible by 43. Therefore, the remainder when N is divided by 43 is the same as the remainder when 48 is divided by 43.

By dividing 48 by 43, we get 48 = 43 × 1 + 5. Thus, the remainder obtained is 5.

Explanation

The mixed fraction 999 98/99 can be rewritten as: 999 98/99 = 999 + 98/99 = 1000 − 1 + 98/99 = 1000 − 1/99

Multiplying this expression by 99: (1000 − 1/99) × 99 = 1000 × 99 − (1/99) × 99 = 99000 − 1 = 98999

Explanation

Using the laws of exponents, a fraction raised to a negative power is equal to its reciprocal raised to the corresponding positive power:

(a/b)⁻ⁿ = (b/a)ⁿ

Applying this property to the given expression:

(½)⁻¹⁄₂ = (2/1)¹⁄₂ = 2¹⁄₂

Since raising a base to the power of ½ is mathematically equivalent to taking its square root:

2¹⁄₂ = √2

Explanation

To find how many 1/6 together make 41⅔, let the required number be x.

First, convert the mixed fraction 41⅔ into an improper fraction: 41⅔ = (41 × 3 + 2) / 3 = 125/3

Now, set up the algebraic equation: x × (1/6) = 125/3

Multiply both sides of the equation by 6 to solve for x: x = (125 / 3) × 6 x = 125 × 2 x = 250

Explanation

The term 0.9 with a bar over 9 represents the repeating decimal 0.999...

Let x = 0.999... Multiplying both sides by 10 gives: 10x = 9.999...

Subtracting the first equation from the second: 9x = 9 x = 1

Thus, the repeating decimal 0.999... is mathematically equal to 1. Since 1 is greater than 0.9, 0.1 (which is the value of 0.0999...), and 0.0909..., it is the greatest value.

Explanation

Let the required fraction be n/30. The fraction lies between 5/8 and 7/11, which gives the inequality:

5/8 < n/30 < 7/11

Converting the boundary fractions into decimal equivalents:

5/8 = 0.625 7/11 ≈ 0.6364

Substituting these values into the inequality:

0.625 < n/30 < 0.6364

Multiplying all parts of the inequality by 30 to isolate n:

18.75 < n < 19.09

The only integer value of n that lies within this interval is 19, which yields the fraction 19/30.

Explanation

Let a = √7 and b = √5. The given expression is of the form:

(a - b)/(a + b) + (a + b)/(a - b)

Taking the common denominator, the expression becomes:

((a - b)² + (a + b)²) / ((a + b)(a - b))

Expanding the numerator using algebraic identities yields (a² - 2ab + b²) + (a² + 2ab + b²) = 2(a² + b²). The denominator is the difference of squares, which is a² - b².

Substituting these simplifications back into the expression gives:

2(a² + b²) / (a² - b²)

Since a = √7 and b = √5, we have a² = 7 and b² = 5. Substituting these values:

2(7 + 5) / (7 - 5)

= 2(12) / 2

= 12

Explanation

To find the least perfect square divisible by 16, 20, and 24, we first find the Least Common Multiple (LCM) of these numbers.

The prime factorizations are: 16 = 2⁴ 20 = 2² × 5 24 = 2³ × 3

The LCM is the product of the highest powers of all prime factors present: LCM = 2⁴ × 3¹ × 5¹ = 240

To make 240 a perfect square, each prime factor in its factorization must have an even exponent. The prime factors 3 and 5 both have odd powers of 1. To make their exponents even, we multiply 240 by 3 × 5: Least perfect square = 240 × 3 × 5 = 3600

Explanation

From the given equation: x + 9/x = 6

Multiply both sides by x to form a quadratic equation: x² - 6x + 9 = 0

This is a perfect square: (x - 3)² = 0 x = 3

Substitute the value of x = 3 in the required expression: x² + 9/x² = 3² + 9/3² = 9 + 9/9 = 9 + 1 = 10

Explanation

The sequence is formed by adding successive perfect squares to each term.

52 - 51 = 1 = 1² 56 - 52 = 4 = 2² 65 - 56 = 9 = 3²

Following this pattern, the next difference in the sequence must be 4² = 16. Adding 16 to the last term, 65, yields the next number:

65 + 16 = 81

Explanation

The sequence follows the pattern where each subsequent term is generated by multiplying the preceding term by 2 and adding 1, which can be represented by the relation aₙ = 2aₙ₋₁ + 1.

Calculating the terms using this relation: The second term is 2 × 4 + 1 = 9. The third term is 2 × 9 + 1 = 19. The fourth term is 2 × 19 + 1 = 39. The fifth term is 2 × 39 + 1 = 79. The sixth term should be 2 × 79 + 1 = 159. The seventh term is 2 × 159 + 1 = 319.

Since the sixth term in the given sequence is written as 169 instead of 159, it is the incorrect number.

Explanation

Let the daily work done by a man be M and by a woman be W.

According to the question, the work done by 2 men and 3 women together in 20 days is equal to the work done by 4 men in 20 days. Therefore:

(2M + 3W) × 20 = 4M × 20 2M + 3W = 4M 3W = 2M

This relation gives us the efficiency ratio between men and women. The total work in terms of man-days is:

Total Work = 4M × 20 = 80M

We need to find the time taken by 3 men and 3 women to complete the same work. The combined daily work of 3 men and 3 women is:

3M + 3W = 3M + 2M = 5M

Thus, the number of days required by 3 men and 3 women to complete the total work is:

80M ÷ 5M = 16 days

Explanation

To cross the standing train, the moving train must cover a total distance equal to the sum of the lengths of both trains.

Total distance = 110 m + 170 m = 280 m

Convert the speed of the moving train from km/hr to m/s: Speed = 60 km/hr = 60 × 5/18 m/s = 50/3 m/s

Using the formula for time: Time = Distance / Speed Time = 280 / (50/3) Time = (280 × 3) / 50 Time = 840 / 50 Time = 16.8 seconds

Explanation

For a constant distance, the time taken is inversely proportional to speed:

Time ∝ 1/Speed

Given that the ratio of the speeds of the three persons is 4 : 3 : 5, the ratio of the times taken by them to cover the same distance will be:

1/4 : 1/3 : 1/5

To express this in whole numbers, we multiply each term of the ratio by the least common multiple (LCM) of the denominators 4, 3, and 5, which is 60:

(1/4 × 60) : (1/3 × 60) : (1/5 × 60) = 15 : 20 : 12

Explanation

The total amount (A) after 2 years is the sum of the principal (P) and the compound interest (CI): A = P + CI A = 1500 + 449.40 = 1949.40

Using the compound interest formula: A = P(1 + r/100)ⁿ

Substituting the given values into the formula: 1949.40 = 1500(1 + r/100)² (1 + r/100)² = 1949.40 / 1500 (1 + r/100)² = 1.2996

Taking the square root on both sides: 1 + r/100 = √1.2996 1 + r/100 = 1.14 r/100 = 0.14 r = 14%

Explanation

The difference (D) between the compound interest and simple interest on a principal amount P for 2 years at an annual interest rate R% is given by the direct formula:

D = P × (R / 100)²

Given that the difference D is Rs. 129.60 and the rate R is 9%, substituting these values into the formula gives:

129.60 = P × (9 / 100)²

129.60 = P × (81 / 10000)

P = (129.60 × 10000) / 81

P = 1296000 / 81

P = 16000

The sum of money is Rs. 16000.

Explanation

Let the principal sum of money be P, the rate of interest be R, and the time period T be 2 years.

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

When the rate of interest is increased by 1%, the increase in interest is given by: Increase in SI = (P × Increase in Rate × T) / 100

Given that the increase in rate is 1% and the increase in interest is Rs. 24, substituting these values gives: 24 = (P × 1 × 2) / 100

Simplifying the equation: 24 = 2P / 100 P = (24 × 100) / 2 P = 1200

Explanation

The given expression can be simplified by separating each mixed fraction into its whole number and fractional components:

99 1/7 + 99 2/7 + 99 3/7 + 99 4/7 + 99 5/7 + 99 6/7

= (99 + 1/7) + (99 + 2/7) + (99 + 3/7) + (99 + 4/7) + (99 + 5/7) + (99 + 6/7)

Grouping the whole numbers and fractions together:

= (99 × 6) + (1/7 + 2/7 + 3/7 + 4/7 + 5/7 + 6/7)

Calculating the product of the whole numbers: 99 × 6 = 594

Calculating the sum of the fractions: (1 + 2 + 3 + 4 + 5 + 6) / 7 = 21 / 7 = 3

Adding the two results together: 594 + 3 = 597

Explanation

According to the properties of modular arithmetic, the remainder of the sum of two numbers when divided by a divisor is equal to the remainder of the sum of their individual remainders when divided by the same divisor. This relation is expressed as:

(A + B) mod d = (A mod d + B mod d) mod d

Given that the divisor d is 33, and the individual remainders are 21 and 28, substituting these values into the relation gives:

(21 + 28) mod 33 = 49 mod 33

When 49 is divided by 33, we get:

49 = 33 × 1 + 16

The remainder is 16.

Explanation

Let the two numbers be 12a and 12b, where a and b are co-prime positive integers.

According to the question, the sum of the two numbers is 84: 12a + 12b = 84 12(a + b) = 84 a + b = 7

Since 7 is a prime number, any pair of positive integers (a, b) with a ≤ b that sums to 7 will automatically be co-prime. The number of such pairs is given by the integer division 7/2, which is 3.

These three co-prime pairs (a, b) are: (1, 6), which gives the numbers 12 and 72. (2, 5), which gives the numbers 24 and 60. (3, 4), which gives the numbers 36 and 48.

Explanation

To find the value of the expression, rationalize the denominator by multiplying both the numerator and the denominator by the conjugate (3 - √2):

7 / (3 + √2) = [7 × (3 - √2)] / [(3 + √2) × (3 - √2)]

Using the algebraic identity (a + b)(a - b) = a² - b² in the denominator:

7(3 - √2) / (3² - (√2)²) = 7(3 - √2) / (9 - 2) = 7(3 - √2) / 7 = 3 - √2

Substituting the given value of √2 = 1.4142:

3 - 1.4142 = 1.5858

Rounding 1.5858 to two decimal places yields 1.59 because the digit in the third decimal place is 5, which rounds the digit in the second decimal place (8) up by 1.

Explanation

Let the required number be x. According to the question:

x × (√3 + √2) = √12 + √18

The terms √12 and √18 can be simplified as: √12 = √(4 × 3) = 2√3 √18 = √(9 × 2) = 3√2

Substituting these simplified terms back into the equation: x × (√3 + √2) = 2√3 + 3√2

Solving for x: x = (2√3 + 3√2) / (√3 + √2)

To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator, (√3 - √2): x = [(2√3 + 3√2) × (√3 - √2)] / [(√3 + √2) × (√3 - √2)]

The denominator simplifies using the algebraic identity (a + b)(a - b) = a² - b²: (√3)² - (√2)² = 3 - 2 = 1

The numerator expands as: (2√3 × √3) - (2√3 × √2) + (3√2 × √3) - (3√2 × √2) = 6 - 2√6 + 3√6 - 6 = √6

Therefore, x = √6.

Explanation

We can use the standard algebraic identity: (x + y)² + (x - y)² = 2(x² + y²)

By substituting the given values, x - y = 2 and x² + y² = 20, we get: (x + y)² + 2² = 2(20) (x + y)² + 4 = 40 (x + y)² = 40 - 4 (x + y)² = 36

Explanation

Let x = √(3√(3√(3...))

By squaring both sides of this equation, the outermost square root is removed: x² = 3√(3√(3...))

Since the infinite nested radical term is equal to x, we can substitute x back into the equation: x² = 3x

Dividing both sides by x, since x is greater than 0: x = 3

Explanation

The given expression can be simplified using the algebraic identity: x⁴ + y⁴ - 2x²y² = (x² - y²)²

Using the difference of squares identity, we have: x² - y² = (x + y)(x - y)

Now, calculate the values of (x + y) and (x - y): x + y = (a + 1/a) + (a - 1/a) = 2a x - y = (a + 1/a) - (a - 1/a) = 2/a

Substituting these values into the expression for x² - y²: x² - y² = (2a) × (2/a) = 4

Thus, the value of the original expression is: (x² - y²)² = 4² = 16

Explanation

Each term of the expression can be simplified by rationalizing the denominators.

For the first term, we simplify 2 / (√6 + 2): 2 / (√6 + 2) = 2(√6 - 2) / ((√6 + 2)(√6 - 2)) = 2(√6 - 2) / (6 - 4) = 2(√6 - 2) / 2 = √6 - 2

For the second term, we rationalize 1 / (√7 + √6): 1 / (√7 + √6) = (√7 - √6) / ((√7 + √6)(√7 - √6)) = (√7 - √6) / (7 - 6) = √7 - √6

For the third term, we rationalize 1 / (√8 - √7): 1 / (√8 - √7) = (√8 + √7) / ((√8 - √7)(√8 + √7)) = (√8 + √7) / (8 - 7) = √8 + √7 = 2√2 + √7

Substituting these simplified terms back into the original expression gives: (√6 - 2) + (√7 - √6) + (2√2 + √7) + 2 - 2√2

Grouping and combining the like terms: (√6 - √6) + (√7 + √7) + (2√2 - 2√2) + (-2 + 2) = 0 + 2√7 + 0 + 0 = 2√7

Explanation

Let the two numbers be x and y.

The sum of the two numbers is x + y = 8, and their product is xy = 15.

The sum of their reciprocals is given by the algebraic relation: 1/x + 1/y = (x + y) / xy

By substituting the known values of the sum and the product into this relation: 1/x + 1/y = 8/15

This directly yields the result 8/15.

Explanation

The area of a square can be calculated directly from the length of its diagonal using the formula:

Area = d² / 2

where d is the length of the diagonal.

Given that the diagonal d = 15√2 cm:

Area = (15√2)² / 2 Area = (225 × 2) / 2 Area = 225 cm²