SSC CGL Tier 1 2010 Quantitative Aptitude Questions with Solutions

Explanation

The fundamental relationship between two numbers and their Greatest Common Divisor (H.C.F.) and Least Common Multiple (L.C.M.) is given by the formula:

Product of two numbers = H.C.F. × L.C.M.

Let the two numbers be x and y. Given that x = 24, H.C.F. = 8, and L.C.M. = 48, we substitute these values into the equation:

24 × y = 8 × 48

To find the value of y, we divide both sides by 24:

y = (8 × 48)/(24)

Simplifying the fraction, we see that 48 divided by 24 equals 2:

y = 8 × 2 = 16

Explanation

To find the greatest number that can be subtracted from 5834 to yield a number divisible by 20, 28, 32, and 35, we must identify the smallest number that is exactly divisible by these four divisors. This number is their Least Common Multiple (LCM).

The prime factorization of the given numbers is: 20 = 2² × 5 28 = 2² × 7 32 = 2⁵ 35 = 5 × 7

The LCM is the product of the highest powers of all prime factors present: LCM = 2⁵ × 5¹ × 7¹ = 32 × 5 × 7 = 1120

Let the required greatest number be x. To maximize x, the result of the expression 5834 - x should be the smallest multiple of the LCM that is greater than zero. Thus, we set the result equal to 1120: 5834 - x = 1120 x = 5834 - 1120 x = 4714

Explanation

The sequence is defined by the general formula aₙ = n² - 1, where n represents the ordinal position of the term. To find the ninth term, we substitute n = 9 into the formula, which yields a₉ = 9² - 1 = 81 - 1 = 80. This pattern is consistently observed throughout the given sequence: the first term is 1² - 1 = 0, the second is 2² - 1 = 3, the third is 3² - 1 = 8, the fourth is 4² - 1 = 15, the fifth is 5² - 1 = 24, and the sixth is 6² - 1 = 35. By applying this quadratic relationship, the value of the term at any specific position can be determined directly, confirming that the ninth position corresponds to the value 80.

Explanation

The ratio y / x is the mathematical reciprocal of x / y. Given the relationship x / y = 1.25, the value of y / x is determined by the formula:

y / x = (1)/(x / y)

Substituting the given value:

y / x = (1)/(1.25)

Expressing the decimal 1.25 as a fraction gives 5/4. Therefore:

y / x = (1)/(5/4) = ⅘ = 0.8

The equation x + y = 9 is redundant for finding the ratio, as the value of y / x depends solely on the given quotient of the two variables.

Explanation

The series (1 + 0.6 + 0.06 + 0.006 + 0.0006 + …) is an infinite series where, starting from the second term, the terms form a geometric progression. The sum can be represented as 1 + (0.6 + 0.06 + 0.006 + …).

The expression in the parentheses is an infinite geometric progression with the first term a = 0.6 and the common ratio r = 0.1. The sum S of an infinite geometric progression is calculated using the formula: S = a / (1 - r) S = 0.6 / (1 - 0.1) = 0.6 / 0.9 = 6/9 = 2/3.

Adding the first term of the series to this result gives the total sum: 1 + 2/3 = 5/3 = 1 ⅔.

Alternatively, the sum of the series can be observed as the recurring decimal 1.666…, which is denoted as 1.6. Converting the recurring part to a fraction, 0.6 is equivalent to 6/9, which simplifies to 2/3. Thus, the total value is 1 ⅔.

Explanation

The given number can be expressed in the form N = 114k + 21, where k is the quotient. To find the remainder when the same number N is divided by 19, we check if the first divisor (114) is a multiple of the second divisor (19). Since 114 = 19 × 6, the term 114k is perfectly divisible by 19, leaving a remainder of 0. Therefore, the required remainder is simply the remainder obtained by dividing the original remainder (21) by 19. Calculation shows that 21 = (19 × 1) + 2. Thus, the remainder is 2.

Explanation

The square root of a decimal number can be found by converting the decimal into a fraction. The value 0.09 is equivalent to ⁹⁄₁₀₀. Applying the square root to this fraction involves taking the square root of the numerator and the denominator independently:

√(0.09) = √(⁹⁄₁₀₀) = √9{√(100)} = ³⁄₁₀

The resulting fraction ³⁄₁₀ is equal to 0.3 in decimal form. In mathematics, trailing zeros after the decimal point do not change the value of the number, so 0.3 is identical to 0.30. Verification can be performed by squaring the result: 0.3 × 0.3 = 0.09, which confirms the calculation.

Explanation

The decimal 0.121212... is a pure recurring decimal where the digits 12 repeat infinitely. To convert a pure recurring decimal into a fraction of the form p/q, the numerator is the repeating sequence and the denominator consists of as many nines as there are digits in that sequence.

Let x = 0.12. The formula for such a conversion is x = (R)/(10ⁿ - 1), where R is the repeating block and n is the number of digits in that block. Here, R = 12 and n = 2.

x = (12)/(10² - 1) x = ¹²⁄₉₉

To reduce the fraction to its simplest form, we divide both the numerator and the denominator by their greatest common divisor, which is 3.

x = (12 ÷ 3)/(99 ÷ 3) x = ⁴⁄₃₃

Explanation

According to the laws of exponents, when multiplying two terms with the same base, we add their powers: aᵐ · aⁿ = aᵐ⁺ⁿ. Applying this to the left side of the equation:

(⅗)³ · (⅗)⁻⁶ = (⅗)³⁺⁽⁻⁶⁾ = (⅗)⁻³

The equation is now (⅗)⁻³ = (⅗)²ˣ⁻¹. Since the bases on both sides are identical and not equal to 0, 1, or -1, we can equate the exponents:

-3 = 2x - 1

To solve for x, add 1 to both sides:

2x = -3 + 1 2x = -2

Dividing by 2:

x = -1

Explanation

Let the two numbers be 3x and 4x, where x represents their highest common factor (HCF). The least common multiple (LCM) of two numbers expressed as ax and bx (where a and b are coprime) is given by the formula LCM = a × b × x.

Substituting the given values into the formula: 3 × 4 × x = 84 12x = 84 x = 7

The two numbers are 3 × 7 = 21 and 4 × 7 = 28. Between these two, the greater number is 28.

Explanation

The capacity can be found by calculating the fractional difference caused by removing the kerosene. The drum's status changes from 3/4 full to 1/2 full. By using a common denominator, 1/2 is equivalent to 2/4.

The difference in the filled fraction is: ¾ - ²⁄₄ = ¼

This ¼ reduction in volume represents the 30 litres that were drawn from the drum. If C represents the total capacity of the drum, the relationship is: ¼C = 30

To find the total capacity, multiply the removed volume by the reciprocal of the fraction: C = 30 × 4 = 120

The total capacity of the drum is 120 litres.

Explanation

To find the least number by which 675 must be multiplied to obtain a perfect cube, we first perform the prime factorization of 675: 675 = 5 × 135 135 = 5 × 27 27 = 3 × 3 × 3 So, the prime factorization of 675 is 3³ × 5².

For a number to be a perfect cube, the exponent of every prime factor in its prime factorization must be a multiple of 3. In the prime factorization of 675, the prime factor 3 has an exponent of 3, which satisfies the condition. However, the prime factor 5 has an exponent of 2. To make this exponent a multiple of 3, we need to multiply the expression by one more factor of 5.

675 × 5 = (3³ × 5²) × 5 = 3³ × 5³ = (3 × 5)³ = 15³ = 3375

Since 3375 is a perfect cube, the smallest number required for multiplication is 5.

Explanation

The expression 1 ½ + 11 ½ + 111 ½ + 1111 ½ is solved by separating the integer parts from the fractional parts. Since a mixed fraction n (a)/(b) is equivalent to n + (a)/(b), the sum can be rewritten as (1 + 11 + 111 + 1111) + (½ + ½ + ½ + ½).

First, add the integer parts: 1 + 11 + 111 + 1111 = 1234. Next, add the fractional parts: ½ + ½ + ½ + ½ = ⁴⁄₂ = 2. Finally, combine the two sums to obtain 1234 + 2 = 1236. This method of grouping terms simplifies the arithmetic significantly by avoiding the large numerators that would result from converting each mixed fraction into an improper fraction.

Explanation

To convert a pure recurring decimal into a vulgar fraction, the digits in the repeating sequence are placed in the numerator, and the denominator is formed by as many nines as there are digits in the repeating block. In the context of this question, 0.001 represents a repeating decimal where the block 001 recurs infinitely.

The number of digits in the repeating sequence 001 is 3. According to the formula for a pure recurring decimal 0.pqr, the fraction is (pqr)/(999).

Applying this to the given value: Numerator = 1 Denominator = 999

The relationship can also be expressed using the formula x = (n)/(10ᵏ⁾⁻¹, where n is the repeating number and k is the number of digits in that number. Substituting n = 1 and k = 3: x = (1)/(10³⁾⁻¹ = (1)/(1000 - 1) = ¹⁄₉₉₉

Explanation

The numerator is 0.441 × 0.16 = 0.07056. The denominator is 2.1 × 1.6 × 0.21 = 3.36 × 0.21 = 0.7056. Dividing the numerator by the denominator gives 0.07056 / 0.7056 = 0.1.

Explanation

The expression a⁴ - b⁴ can be factored as (a² - b²)(a² + b²). For any odd integer x, the square x² is always of the form 8k + 1. This is derived from x = 2n + 1, which gives x² = 4n(n + 1) + 1; since the product of two consecutive integers n(n + 1) is always even, 4n(n + 1) is divisible by 8. Given that a and b are odd positive integers, we have a² = 8m + 1 and b² = 8n + 1 for some integers m and n. Substituting these into the first factor, a² - b² = (8m + 1) - (8n + 1) = 8(m - n), which proves that (a² - b²) is always divisible by 8. Because a⁴ - b⁴ is the product of (a² - b²) and (a² + b²), and (a² - b²) is a multiple of 8, the entire expression a⁴ - b⁴ is always divisible by 8.

Explanation

The expression is simplified using the algebraic identity for the difference of two cubes: a³ - b³ = (a - b)(a² + ab + b²). Substituting this into the given expression, we have:

(a² + b² + ab)/(a³ - b³) = (a² + b² + ab)/((a - b)(a² + ab + b²))

The term (a² + ab + b²) is common to both the numerator and the denominator and can be cancelled out, simplifying the entire expression to:

(1)/(a - b)

Substituting the given values a = 11 and b = 9:

(1)/(11 - 9) = ½

Explanation

The expression is in the form of the algebraic identity (a² - b²)/(k), where a = 256 and b = 144. Using the identity a² - b² = (a + b)(a - b), we calculate the components:

a + b = 256 + 144 = 400 a - b = 256 - 144 = 112

Substituting these into the original expression:

(256² - 144²)/(112) = ((256 + 144)(256 - 144))/(112)

(400 × 112)/(112) = 400

The common factor of 112 in the numerator and denominator cancels out, resulting in 400.

Explanation

The expression a² - b² = 19 can be factored using the algebraic identity a² - b² = (a - b)(a + b). Since 19 is a prime number, its only possible positive integer factors are 1 and 19. Given that a and b are positive integers, the sum a + b must be greater than the difference a - b.

Setting up the system of equations based on these factors: a + b = 19 a - b = 1

To solve for a, add the two equations together: (a + b) + (a - b) = 19 + 1 2a = 20 a = 10

Substituting a = 10 into the first equation gives 10 + b = 19, which results in b = 9. Both a and b are positive integers, satisfying the initial conditions.

Explanation

The profit percentage is calculated using the standard formula Profit % = (SP - CP)/(CP) × 100. Given the ratio of cost price to selling price is 10:11, we can substitute CP = 10 and SP = 11 directly into the relation. This yields Profit % = (11 - 10)/(10) × 100 = ⅒ × 100 = 10.

Explanation

The relationship between the Cost Price (CP) and Marked Price (MP) is determined by the formula CP = MP × (100 - Discount%)/(100 + Profit%). Substituting the given values where MP = Rs. 50, Discount = 20%, and Profit = 25%:

CP = 50 × (100 - 20)/(100 + 25)

CP = 50 × ⁸⁰⁄₁₂₅

CP = 50 × ¹⁶⁄₂₅ = 2 × 16 = 32

Alternatively, the Selling Price (SP) is calculated by applying the 20% discount on the marked price: SP = 50 × (1 - 0.20) = 40. Since there is a 25% profit, the CP is found by dividing the SP by 1.25: CP = (40)/(1.25) = 32.

Explanation

The relationship between Cost Price (CP) and Printed Price (MP) when discount and profit percentages are given is expressed by the formula:

(CP)/(MP) = (100 - Discount%)/(100 + Profit%)

Substituting the given values where Discount is 10% and Profit is 12%:

(CP)/(MP) = (100 - 10)/(100 + 12)

(CP)/(MP) = ⁹⁰⁄₁₁₂

Dividing both the numerator and the denominator by their greatest common divisor, which is 2:

(CP)/(MP) = ⁴⁵⁄₅₆

This results in the ratio 45:56.

Explanation

The selling price (SP) of an article is directly proportional to (100 + Gain%) when the cost price (CP) is constant. Using the ratio of two different selling prices:

(SP₁)/(100 + G₁) = (SP₂)/(100 + G₂)

Given SP₁ = 2850, G₁ = 14, and G₂ = 8, we can calculate the new selling price SP₂ as:

SP₂ = 2850 × (100 + 8)/(100 + 14)

SP₂ = 2850 × ¹⁰⁸⁄₁₁₄

Simplifying the fraction ¹⁰⁸⁄₁₁₄ by dividing both the numerator and the denominator by 6:

SP₂ = 2850 × ¹⁸⁄₁₉

Dividing 2850 by 19 yields 150. Therefore:

SP₂ = 150 × 18 = 2700

Explanation

Profit is defined here as 25% of the selling price (SP). If we assume the selling price is 100, the profit earned is 25. The cost price (CP) is calculated by subtracting the profit from the selling price, which gives CP = 100 - 25 = 75. The actual profit percent is determined by the ratio of profit to the cost price. Using the formula Profit % = (Profit)/(CP) × 100, we substitute the values to get ²⁵⁄₇₅ × 100. This reduces to ⅓ × 100, which results in 33.33%.

Explanation

To calculate the percentage by which B's income exceeds A's income, use the standard formula for percentage change: Percentage Increase = (x)/(100 - x) × 100, where x represents the percentage by which A's income is less than B's.

Given x = 50, substituting the value into the formula: Percentage Increase = (50)/(100 - 50) × 100 Percentage Increase = ⁵⁰⁄₅₀ × 100 = 100%

Alternatively, assume B's income is 100. Since A's income is 50% less, A's income is 50. The difference between their incomes is 50. To find how much percent B is more than A, calculate the difference relative to A's income: ⁵⁰⁄₅₀ × 100 = 100%.

Explanation

To express one value as a percentage of another, the formula is (Value)/(Total) × 100. Applying the given numbers where the value is 1.14 and the total is 1.9:

Percentage = (1.14)/(1.9) × 100

First, multiply 1.14 by 100 to simplify the numerator:

(114)/(1.9)

To remove the decimal point from the denominator, multiply both the numerator and the denominator by 10:

¹¹⁴⁰⁄₁₉

By division, 114 divided by 19 equals 6. Therefore, 1140 divided by 19 equals 60.

Explanation

Let the two natural numbers be 3x and 5x based on the given ratio 3:5. The product of these numbers is stated to be 2160, which allows us to set up the following equation:

3x × 5x = 2160 15x² = 2160

To find the value of x², divide the product by the coefficient 15: x² = ²¹⁶⁰⁄₁₅ = 144

Taking the square root of both sides determines the common factor x: x = √(144) = 12

The smaller number in the ratio is 3x. Substituting the calculated value of x: 3 × 12 = 36

Explanation

The given equation is 60% × A = 48% × B. To find the ratio A:B, we can express the percentages as decimals or fractions:

0.60 × A = 0.48 × B

By rearranging the terms to isolate the ratio (A)/(B), we get:

(A)/(B) = (0.48)/(0.60)

To simplify the expression, multiply both the numerator and the denominator by 100:

(A)/(B) = ⁴⁸⁄₆₀

Both 48 and 60 are divisible by 12. Dividing both terms by their greatest common divisor:

(A)/(B) = (48 ÷ 12)/(60 ÷ 12) = ⅘

This yields the ratio A:B = 4:5.

Explanation

The net effect of two successive percentage increases of a% and b% is calculated using the formula a + b + ᵃᵇ⁄₁₀₀. Substituting the given values a = 10 and b = 10 into the formula, the equivalent single increase is 10 + 10 + (10 × 10)/(100). This simplifies to 20 + ¹⁰⁰⁄₁₀₀ = 20 + 1 = 21%.

Alternatively, using the multiplier method, an increase of 10% is equivalent to multiplying the price by 1.10. Applying this twice results in a final multiplier of 1.10 × 1.10 = 1.21. A final value that is 1.21 times the original value represents a total increase of 0.21, or 21%.

Explanation

Applying the rule of Componendo and Dividendo to the equation √(3+x) + √(3-x){√(3+x) - √(3-x)} = ²⁄₁, we have:

(√(3+x) + √(3-x)) + (√(3+x) - √(3-x)){(√(3+x) + √(3-x)) - (√(3+x) - √(3-x))} = (2 + 1)/(2 - 1)

By simplifying the numerator and the denominator on the left side, the equation reduces to:

2√(3+x){2√(3-x)} = 3

√(3+x){√(3-x)} = 3

Squaring both sides of the equation to eliminate the square roots yields:

³⁺ˣ⁄₃₋ₓ = 9

Multiplying both sides by the denominator 3-x:

3 + x = 9(3 - x)

3 + x = 27 - 9x

Rearranging the terms to solve for x:

10x = 24

x = ²⁴⁄₁₀ = ¹²⁄₅

Explanation

To form a regular hexagon from an equilateral triangle, each side of the triangle must be trisected. If the side of the equilateral triangle is s, the side of the resulting regular hexagon is a = s/3. Given s = 6 cm, the side of the hexagon is a = 6/3 = 2 cm.

A regular hexagon is composed of six congruent equilateral triangles. The area of a regular hexagon with side a is calculated using the formula A = 3√3{2} a². Substituting the value of a = 2 cm into the formula, we get A = 3√3{2} × (2)² = 3√3{2} × 4 = 6√3 cm². Alternatively, the original triangle consists of 9 smaller equilateral triangles of side 2 cm. Removing the 3 corner triangles to form the hexagon leaves 6 such triangles, each with an area of √3{4} × 2² = √3 cm², resulting in a total area of 6√3 cm².

Explanation

The length of the longest rod that can be placed in a rectangular room is equal to the length of the space diagonal of the cuboid. For a room with dimensions l, w, and h, the length of the diagonal d is given by the formula d = √(l² + w² + h²).

Substituting the given dimensions l = 10 m, w = 10 m, and h = 5 m into the formula:

d = √(10² + 10² + 5²) d = √(100 + 100 + 25) d = √(225) d = 15 m

The calculation shows that the space diagonal of the room is exactly 15 metres, which represents the maximum possible length for a straight rod to fit inside the given dimensions.

Explanation

The total amount of Rs. 1000 is divided into 3 + 2 = 5 equal parts based on the given ratio of 3:2. To calculate the specific portion allocated to A, we use the ratio division formula:

A's share = (Ratio of A)/(Sum of Ratios) × Total Amount

Substituting the given values:

A's share = (3)/(3 + 2) × 1000

A's share = ⅗ × 1000

By simplifying the expression, we divide 1000 by 5 to get 200, then multiply by 3:

A's share = 3 × 200 = 600

A receives Rs. 600 as their portion of the total sum.

Explanation

Let the quantity to be added to each term be x. The relationship is defined by the equation:

(7 + x)/(11 + x) = ¾

Cross-multiplying the terms gives:

4(7 + x) = 3(11 + x)

28 + 4x = 33 + 3x

Rearranging the terms to solve for x:

4x - 3x = 33 - 28

x = 5

Alternatively, this can be solved by equating the differences between the terms of the ratios. The difference between 7 and 11 is 4, while the difference between 3 and 4 is 1. To make the differences equal, multiply the second ratio by 4, resulting in 12:16. Comparing the original terms to the new terms, the increase required is 12 - 7 = 5 or 16 - 11 = 5.

Explanation

In compound interest, if a sum of money becomes x times in t years, it will become xⁿ times in n × t years. Here, the sum doubles (x = 2) in t = 15 years. We want to find the time required for the sum to become eight times its original value. Since 8 = 2³, the value of n is 3. Applying the formula, the time required is 3 × 15 = 45 years. This is because the amount grows geometrically; it becomes 2 times the principal in 15 years, 2 × 2 = 4 times in 30 years, and 4 × 2 = 8 times in 45 years.

Explanation

The distance between two consecutive buses leaving the terminal is the distance covered by a bus in the given time interval, which is D = V_bᵤₛ × Tᵢₙₜₑᵣᵥₐₗ. When a man moves towards the bus terminal from the opposite direction, the relative speed between the man and the bus is V_bᵤₛ + Vₘₐₙ. The time interval t at which the man meets consecutive buses is the time taken to cover the distance D at this relative speed.

Using the relation D = (V_bᵤₛ + Vₘₐₙ) × t, we substitute the known values: 20 × 10 = (20 + Vₘₐₙ) × 8 200 = 160 + 8Vₘₐₙ 8Vₘₐₙ = 40 Vₘₐₙ = 5 km / hr

This result is derived from the principle that the distance a bus travels in 10 minutes is equal to the distance the bus and the man together cover in 8 minutes when moving towards each other.

Explanation

The circumference of a circle is C = 2π r, which implies C ∝ r. Therefore, a 50% decrease in the circumference results in an identical 50% decrease in the radius. The area of a circle is A = π r². When the linear dimension of a two-dimensional figure changes by x%, the percentage change in its area is calculated using the formula 2x + (x²)/(100).

Substituting x = -50 for the percentage decrease:

Percentage change in area = 2(-50) + ((-50)²)/(100) = -100 + ²⁵⁰⁰⁄₁₀₀ = -100 + 25 = -75%

The negative sign signifies a reduction, confirming the area decreases by 75%.

Explanation

The relationship between the amount A, principal P, rate of interest R, and time n under compound interest is given by the formula A = P(1 + (R)/(100))ⁿ. Substituting the given values into the formula:

1102.50 = 1000(1 + (R)/(100))²

(1102.50)/(1000) = (1 + (R)/(100))²

1.1025 = (1 + (R)/(100))²

To find R, take the square root of both sides. Since 105² = 11025, it follows that 1.05² = 1.1025.

1.05 = 1 + (R)/(100)

(R)/(100) = 1.05 - 1

(R)/(100) = 0.05

R = 5

Explanation

The annual installment x required to discharge a debt A due in t years at a simple interest rate r per annum is calculated using the principle that the sum of the installments and the interest earned on them until the end of the term must equal the total debt. The formula is A = tx + (rx · t(t-1))/(200).

Substituting the given values A = 6450, t = 4, and r = 5 into the formula: 6450 = 4x + (5 · x · 4(4-1))/(200) 6450 = 4x + (5 · x · 12)/(200) 6450 = 4x + ⁶⁰ˣ⁄₂₀₀ 6450 = 4x + ³ˣ⁄₁₀ 6450 = (40x + 3x)/(10) 6450 = ⁴³ˣ⁄₁₀

Solving for x: x = (6450 · 10)/(43) x = 150 · 10 x = 1500

Explanation

The first 100 positive integers form an arithmetic progression starting with 1 and ending with 100. For any sequence in an arithmetic progression, the average is simply the arithmetic mean of the first and last terms, calculated using the formula (a + l)/(2), where a is the first term and l is the last term. Substituting the values, the average is (1 + 100)/(2) = ¹⁰¹⁄₂ = 50.5. Alternatively, the sum of the first n natural numbers is defined by the formula (n(n + 1))/(2). Dividing this sum by n to find the average yields (n + 1)/(2). For n = 100, this result is (100 + 1)/(2) = 50.5.

Explanation

The total age of a group is determined by the formula Sum = Average × Number of persons.

For the father and mother, the sum of their ages is 35 × 2 = 70 years.

For the family including the son, the sum of the ages of the father, mother, and son is 27 × 3 = 81 years.

The age of the son is obtained by subtracting the sum of the parents' ages from the total sum of the three family members: 81 - 70 = 11 years.

Explanation

Daily earning of one man is calculated as 5250 / 5 = 1050 rupees. Daily earning of one woman is calculated as 5250 / 7 = 750 rupees. To find the total daily earnings of 7 men and 13 women, we multiply the individual rates by the respective number of persons and sum them: (7 × 1050) + (13 × 750) = 7350 + 9750 = 17100 rupees.

Explanation

The total work can be represented as the Least Common Multiple (LCM) of the given time periods, 15 and 20, which is 60 units. The combined efficiency of A and B is calculated by dividing the total work by their combined time, giving 60 / 15 = 4 units per day. The efficiency of B alone is 60 / 20 = 3 units per day. The efficiency of A is the difference between the combined efficiency and B's efficiency, which is 4 - 3 = 1 unit per day. To find the number of days A takes alone, divide the total work by A's efficiency: 60 / 1 = 60 days.

Using the work rate formula, the time taken by A alone is T = (xy) / (y - x), where x is the time taken by A and B together and y is the time taken by B alone. Substituting the values: T = (15 × 20) / (20 - 15) = 300 / 5 = 60 days.

Explanation

The relationship between speed and time for a constant distance is inversely proportional, expressed as S × T = D. If a man walks at 3/4 of his usual speed, the time taken will be the reciprocal of the speed ratio, which is 4/3 of his usual time.

Let the usual time be T. The new time is 4/3 T. The difference between the new time and the usual time is given as 20 minutes.

4/3 T - T = 20 1/3 T = 20 T = 60 minutes

The usual time taken by the man to reach his office is 60 minutes.

Explanation

Let the total work be the Least Common Multiple (LCM) of the individual completion times: LCM(18, 20, 30) = 180 units.

The efficiency of each person (work done per day) is calculated as: Efficiency of A (E_A) = 180 / 18 = 10 units/day Efficiency of B (E_B) = 180 / 20 = 9 units/day Efficiency of C (E_C) = 180 / 30 = 6 units/day

B and C worked together for 2 days. Their combined efficiency is E_B + E_C = 9 + 6 = 15 units/day. The work completed by B and C in 2 days is 15 × 2 = 30 units.

The remaining work is 180 - 30 = 150 units. A alone completes this remaining work at a rate of 10 units/day. The time taken by A is calculated as: Time = (Remaining Work)/(E_A) = ¹⁵⁰⁄₁₀ = 15 days

Explanation

When a train passes a man moving in the same direction, the relative speed is the difference between the speed of the train and the speed of the man. Let the speed of the train be Vₜ km/hr. The speed of the man is given as 3 km/hr. Thus, the relative speed is (Vₜ - 3) km/hr.

To find the relative speed in m/s, we use the formula Vᵣₑₗ = (Distance)/(Time). Here, the distance is the length of the train (300 m) and the time taken is 33 seconds.

Vᵣₑₗ = ³⁰⁰⁄₃₃ m/s

To convert this speed into km/hr, we multiply by ¹⁸⁄₅:

Vᵣₑₗ = ³⁰⁰⁄₃₃ × ¹⁸⁄₅ = (60 × 18)/(33) = (20 × 18)/(11) = ³⁶⁰⁄₁₁ ≈ 32.72 km/hr

Since Vᵣₑₗ = Vₜ - 3, we have:

Vₜ - 3 = 32.72 Vₜ = 32.72 + 3 = 35.72 km/hr

Rounding to one decimal place, the speed of the train is 35.7 km/hr.

Explanation

Kamal speed in meters per second is calculated by multiplying the speed in kmph by ⁵⁄₁₈. Thus, Kamal speed is 18 × ⁵⁄₁₈ = 5 m/s. The time taken by Kamal to complete the 100m race is t = (Distance)/(Speed) = ¹⁰⁰⁄₅ = 20 seconds.

Since Kamal defeats Bimal by 5 seconds, Bimal takes 5 seconds more than Kamal to cover the same distance. Therefore, Bimal time is 20 + 5 = 25 seconds. Bimal speed in meters per second is ¹⁰⁰⁄₂₅ = 4 m/s. To convert this back to kmph, multiply by ¹⁸⁄₅, resulting in 4 × ¹⁸⁄₅ = ⁷²⁄₅ = 14.4 kmph.

Explanation

To find the speed of the train, we use the concept of relative speed. When two objects move in opposite directions, their relative speed is the sum of their individual speeds. Let Vₜ be the speed of the train in kmph and Vₘ be the speed of the man, which is given as 3 kmph.

The distance covered to cross the man is equal to the length of the train, which is 240 meters. The time taken is 10 seconds. We first calculate the relative speed in meters per second:

Relative Speed (Vᵣ) = (Distance)/(Time) = ²⁴⁰⁄₁₀ = 24 m/s

To convert this relative speed into kmph, we multiply by ¹⁸⁄₅:

Vᵣ in kmph = 24 × ¹⁸⁄₅ = ⁴³²⁄₅ = 86.4 kmph

Since the man and train are moving in opposite directions:

Vᵣ = Vₜ + Vₘ 86.4 = Vₜ + 3 Vₜ = 86.4 - 3 = 83.4 kmph

Explanation

Downstream speed D is the speed along the stream. Since the boatman rows 1 km in 5 minutes, we convert the time to hours: 5 minutes = 5/60 = 1/12 hours. Thus, D = (1)/(1/12) = 12 kmph.

Upstream speed U is the speed against the stream. The boatman rows 6 km in 1 hour, so U = 6/1 = 6 kmph.

The speed of the stream v is given by the formula v = (D - U)/(2). Substituting the calculated values, v = (12 - 6)/(2) = ⁶⁄₂ = 3 kmph.

Explanation

If A completes 1/3 of the work in 5 days, the total time taken by A to complete the entire work is 5 × 3 = 15 days. If B completes 2/5 of the work in 10 days, the total time taken by B to complete the entire work is 10 × ⁵⁄₂ = 25 days. To find the time taken by both working together, we use the formula T = (xy)/(x+y), where x and y are the number of days each takes individually. Substituting the values gives T = (15 × 25)/(15 + 25) = ³⁷⁵⁄₄₀. Dividing both the numerator and denominator by 5 yields ⁷⁵⁄₈, which converts to the mixed fraction 9 ⅜ days.