SSC CGL Tier 1 2010 Quantitative Aptitude Questions with Solutions

Explanation

The relationship between men, days, and work is defined by the formula (M₁ × D₁)/(W₁) = (M₂ × D₂)/(W₂). Given M₁ = 7 men, D₁ = 12 days, and W₁ = 1 unit of work, we need to find the total men M₂ required for W₂ = 2 units of work in D₂ = 8 days.

(7 × 12)/(1) = (M₂ × 8)/(2)

84 = 4 × M₂

M₂ = 21

The total number of men required is 21. To find the additional men needed, subtract the original 7 men from the total: 21 - 7 = 14.

Explanation

Let the rate of the slower pipe be x. Since the faster pipe fills the tank three times faster, its rate is 3x. When both pipes work together, their combined rate of filling is x + 3x = 4x.

The total capacity of the tank can be expressed as the product of the combined rate and the time taken: Capacity = 4x × 36 = 144x

The time taken by the slower pipe alone to fill the tank is the total capacity divided by its individual rate: Time = ¹⁴⁴ˣ⁄ₓ = 144 minutes

To convert 144 minutes into hours and minutes, we divide by 60: 144 minutes = 120 minutes + 24 minutes = 2 hours and 24 minutes

Explanation

Let the number of correct answers be x. Since the student attempted all 200 questions, the number of incorrect answers is 200 - x. Based on the scoring rules of 4 marks for a correct answer and a penalty of 1 mark for a wrong answer, the total marks can be represented by the equation:

4x - 1(200 - x) = 200

Expanding the equation gives: 4x - 200 + x = 200 5x = 400 x = 80

Alternatively, using the rule of alligation, if all 200 questions were correct, the student would score 800 marks. If all 200 were wrong, the student would score -200 marks. The actual score is 200. The ratio of correct to wrong answers is (200 - (-200)) : (800 - 200), which is 400 : 600 or 2 : 3. Dividing the 200 questions in the ratio 2 : 3 gives 200 × (2 / 5) = 80 correct answers.

Explanation

The odd numbers up to 100 consist of the sequence 1, 3, 5, …, 99. This sequence is an arithmetic progression where the first term a = 1 and the last term l = 99. The average of any arithmetic progression can be calculated using the formula (a + l)/(2). Substituting the values into this formula, we get (1 + 99)/(2) = ¹⁰⁰⁄₂ = 50.

Alternatively, the number of terms n in this sequence is 50. Using the mathematical property that the sum of the first n odd natural numbers is n², the average is calculated as (Sum)/(n) = n²{n} = n. Since there are exactly 50 odd numbers between 1 and 100, the average is 50.

Explanation

Let B's income be 100 units. Since A's income is 25% less than B's, A's income is 100 - 25 = 75 units. To determine by how much percent B's income is more than A's, the comparison must be made relative to A's income. The absolute difference between their incomes is 100 - 75 = 25 units. The percentage by which B's income exceeds A's is calculated as (Difference)/(A's income) × 100, which results in ²⁵⁄₇₅ × 100. Simplifying the fraction ²⁵⁄₇₅ gives ⅓, and ⅓ × 100 = ¹⁰⁰⁄₃ %.

This can also be solved directly using the percentage change formula for inverse relations: if A is r% less than B, then B is [(r)/(100 - r) × 100]% more than A. Substituting r = 25 into the formula, we get [(25)/(100 - 25) × 100]% = ²⁵⁄₇₅ × 100 = ¹⁰⁰⁄₃ %.

Explanation

The sum of the squares of the first n natural numbers is calculated using the formula S = ⁿ⁽ⁿ⁺¹⁾⁽²ⁿ⁺¹⁾⁄₆. Here, the series consists of the squares of numbers from 1 to 10, meaning n = 10. Substituting n = 10 into the formula:

S = (10(10 + 1)(2 × 10 + 1))/(6)

S = (10 × 11 × 21)/(6)

Dividing 21 and 6 by 3, and 10 and 2 by 2:

S = 5 × 11 × 7

S = 385

Explanation

The given sequence is 2, 6, 11, 17. The pattern is determined by examining the differences between consecutive terms: 6 - 2 = 4 11 - 6 = 5 17 - 11 = 6

The difference between terms increases by 1 at each step. Following this progression, the difference between the 4th and 5th terms is 7, and the difference between the 5th and 6th terms is 8. Using this arithmetic logic: 5th term = 17 + 7 = 24 6th term = 24 + 8 = 32

Explanation

Let the two numbers be 7x and 11x. According to the problem, when 7 is added to both numbers, the ratio becomes 2:3. This relationship can be expressed with the following equation:

(7x + 7)/(11x + 7) = ⅔

To solve for x, cross-multiply the terms:

3(7x + 7) = 2(11x + 7) 21x + 21 = 22x + 14

Rearrange the equation to isolate x:

22x - 21x = 21 - 14 x = 7

The smaller number in the original ratio is 7x. By substituting the value of x, we calculate the smaller number:

Smaller number = 7 × 7 = 49

Alternatively, using the ratio balancing method, the initial ratio is 7:11 (difference of 4 units) and the final ratio is 2:3 (difference of 1 unit). To equate the differences, multiply the final ratio by 4 to get 8:12. The increase from 7 to 8 units (or 11 to 12 units) is 1 unit, which corresponds to the 7 added. Since 1 unit equals 7, the smaller number of 7 units equals 49.

Explanation

The given expression is a telescoping product where each term can be simplified by performing the subtraction within the parentheses: (1 - 1/3) = 2/3 (1 - 1/4) = 3/4 (1 - 1/5) = 4/5 Continuing this pattern up to the final term: (1 - 1/25) = 24/25

The product of these terms is: ⅔ × ¾ × ⅘ × … × ²⁴⁄₂₅

In this sequence, the denominator of each fraction cancels out with the numerator of the subsequent fraction. For instance, the 3 in the denominator of the first term cancels with the 3 in the numerator of the second term, the 4 in the denominator of the second term cancels with the 4 in the numerator of the third term, and so on. This cancellation continues until the numerator of the final term (24) is cancelled by the denominator of the preceding term.

After all cancellations, only the numerator of the first fraction and the denominator of the last fraction remain: ²⁄₂₅

Explanation

A number N can be expressed as N = Dq + R, where D is the divisor, q is the quotient, and R is the remainder. In this case, the number is N = 136k + 36. When this number is divided by 17, we evaluate the expression modulo 17: N 17 = (136k + 36) 17.

Since 136 is exactly divisible by 17 (17 × 8 = 136), the term 136k leaves a remainder of 0. The problem then simplifies to finding the remainder when the original remainder, 36, is divided by 17. Performing the division 36 = (17 × 2) + 2, we find that the remainder is 2.

Explanation

The simplification of this expression follows the power of a power rule, which states that (xᵃ)ᵇ = x^(a·b). The radical sign ⁵√() represents an exponent of 1/5. By identifying all the exponents applied to the base x in the given structure, we find the sequence of powers to be -3/5, 1/5, -5/3, and 5. To find the simplified form, these exponents are multiplied together:

Exponent = ⁻³⁄₅ · ⅕ · ⁻⁵⁄₃ · 5

In this multiplication, the product of ⁻³⁄₅ and ⁻⁵⁄₃ is 1, and the product of ⅕ and 5 is also 1. Multiplying these intermediate results together gives a final exponent of 1 · 1 = 1. Therefore, the entire expression simplifies to x¹, which is simply x.

Explanation

Let a two-digit number be represented as 10x + y, where x and y are the digits. When this number is repeated to form a four-digit number, it takes the form xyxy. Using decimal expansion based on place values, this four-digit number can be written as 1000x + 100y + 10x + y. By rearranging the terms, we get (1000x + 10x) + (100y + y), which simplifies to 1010x + 101y. Factoring out the common term 101, the expression becomes 101 × (10x + y). This demonstrates that the resulting four-digit number is always the product of the original two-digit number and 101. Consequently, any number of this form is exactly divisible by 101.

Explanation

The expression can be simplified by expressing the decimals as powers of 10. We have 0.1 = 10⁻¹, 0.01 = 10⁻², and 0.001 = 10⁻³. Substituting these values into the given expression, we get:

10⁻¹ × 10⁻² × 10⁻³ × 10⁷

Using the law of exponents, aᵐ × aⁿ = aᵐ⁺ⁿ, the powers of 10 are added together:

10⁽⁻¹⁾⁺⁽⁻²⁾⁺⁽⁻³⁾⁺⁷

10⁻⁶⁺⁷

10¹ = 10

Explanation

The given equation is 2p + ¹⁄ₚ = 4. To find the value of p³ + (1)/(8p³), we first simplify the given expression by dividing both sides by 2: p + ¹⁄₂ₚ = 2

We recognize that p³ + (1)/(8p³) is in the form of a³ + b³, where a = p and b = ¹⁄₂ₚ. Using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b), we substitute the values: p³ + (1)/((2p)³) = (p + ¹⁄₂ₚ)³ - 3(p)(¹⁄₂ₚ)(p + ¹⁄₂ₚ)

Substituting p + ¹⁄₂ₚ = 2 into the identity: p³ + (1)/(8p³) = (2)³ - 3(½)(2) p³ + (1)/(8p³) = 8 - 3 p³ + (1)/(8p³) = 5

Explanation

The expression 5p9 + 327 + 2q8 = 1114 can be evaluated by expanding each term according to its decimal place value. This gives (500 + 10p + 9) + 327 + (200 + 10q + 8) = 1114. Combining the constant terms results in 1044 + 10p + 10q = 1114. Subtracting 1044 from both sides of the equation leads to 10(p + q) = 70, which simplifies to the linear relation p + q = 7 after dividing by 10. Because p and q are defined as digits, their values must be non-negative integers between 0 and 9. To obtain the maximum possible value for q, the value of p must be minimized. The minimum possible value for the digit p is 0. Substituting p = 0 into the relation p + q = 7 yields q = 7.

Explanation

For the fractions 15/16, 19/20, 24/25, and 34/35, the difference between the denominator and the numerator is a constant value of 1. In such cases involving proper fractions where the difference (D - N) is the same, the fraction with the smallest numerator is the smallest fraction. This can be demonstrated by expressing each fraction in the form 1 - 1/n. The given fractions are 1 - 1/16, 1 - 1/20, 1 - 1/25, and 1 - 1/35. To find the least value, we look for the largest quantity being subtracted from 1. Among the values 1/16, 1/20, 1/25, and 1/35, the term 1/16 is the largest because it has the smallest denominator. Consequently, 1 - 1/16 yields the smallest result.

Explanation

To convert the recurring decimal 1.27 into the form (p)/(q), we use the standard formula where the numerator is the difference between the complete number and the non-repeating part, and the denominator contains a 9 for every repeating digit.

x = 1.27

x = (127 - 1)/(99)

x = ¹²⁶⁄₉₉

To simplify the fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 9:

x = (126 ÷ 9)/(99 ÷ 9)

x = ¹⁴⁄₁₁

Alternatively, this can be solved by setting x = 1.2727... and multiplying by 100 to get 100x = 127.2727.... Subtracting the first equation from the second gives 99x = 126, which simplifies to ¹⁴⁄₁₁.

Explanation

The expression is simplified by using the distributive property of multiplication over subtraction in the numerator. By factoring out the common term 3.20, the expression becomes:

(3.20 × (3.25 - 3.05))/(0.064)

Subtracting the terms inside the parentheses gives 3.25 - 3.05 = 0.20. Substituting this back into the expression results in:

(3.20 × 0.20)/(0.064)

Calculating the product in the numerator:

3.20 × 0.20 = 0.6400 = 0.64

The expression then simplifies to the division of the numerator by the denominator:

(0.64)/(0.064) = ⁶⁴⁰⁄₆₄ = 10

Explanation

Let the six consecutive natural numbers be represented as n, n+1, n+2, n+3, n+4, and n+5. The sum of the first three numbers is given as n + (n+1) + (n+2) = 27. Simplifying this equation gives 3n + 3 = 27, which results in 3n = 24 and n = 8. The first three numbers are therefore 8, 9, and 10. The following three consecutive numbers are 11, 12, and 13. Their sum is 11 + 12 + 13 = 36.

Alternatively, using a property of consecutive integers, if the sum of k consecutive integers is S, the sum of the next k consecutive integers is S + k². In this problem, k = 3 and S = 27. Applying the formula, the sum of the other three numbers is 27 + 3² = 27 + 9 = 36.

Explanation

The relationship between two numbers and their mathematical properties is defined by the formula: First Number × Second Number = H.C.F × L.C.M

Given the values from the problem: First Number = 84 H.C.F = 12 L.C.M = 336

Let the second number be x. Substituting the known values into the formula gives: 84 × x = 12 × 336

To solve for x, rearrange the equation: x = (12 × 336)/(84)

Simplifying the fraction by dividing 84 by 12, we get: x = ³³⁶⁄₇

Performing the final division: x = 48

This calculation shows that the product of 84 and 48 equals 4032, which is identical to the product of 12 and 336.

Explanation

For any two numbers a and b, the relationship between the numbers, their Highest Common Factor (H.C.F.), and their Least Common Multiple (L.C.M.) is given by the product formula a × b = H.C.F. × L.C.M..

Given: a + b = 36 H.C.F. = 3 L.C.M. = 105

Using the product formula: a × b = 3 × 105 = 315

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

Substituting the known values: ¹⁄ₐ + (1)/(b) = ³⁶⁄₃₁₅

Dividing both the numerator and the denominator by their greatest common divisor, which is 9: (36 ÷ 9)/(315 ÷ 9) = ⁴⁄₃₅

Explanation

The expression n³ - n can be factorized as n(n² - 1), which further simplifies to (n - 1)n(n + 1). This result represents the product of three consecutive integers. According to the property of consecutive integers, the product of k consecutive integers is always divisible by k! (k factorial). For three consecutive integers, the product is always divisible by 3! = 3 × 2 × 1 = 6.

Alternatively, evaluating the expression for the smallest natural numbers confirms this. For n = 1, 1³ - 1 = 0. For n = 2, 2³ - 2 = 6. For n = 3, 3³ - 3 = 24. For n = 4, 4³ - 4 = 60. The greatest common divisor of 6, 24, 60, … is 6. Since at least one of any three consecutive integers must be even (divisible by 2) and exactly one must be a multiple of 3, their product must be divisible by 2 × 3 = 6.

Explanation

To find the number of perfect squares between 120 and 300, we determine the range of integers n such that 120 < n² < 300. We first find the smallest integer whose square exceeds 120. Since 10² = 100 and 11² = 121, the first integer is 11. Next, we find the largest integer whose square is less than 300. Since 17² = 289 and 18² = 324, the last integer is 17.

The set of integers whose squares lie within the specified range is {11, 12, 13, 14, 15, 16, 17}. The number of terms in this sequence is calculated as 17 - 11 + 1 = 7. The specific perfect squares are 121, 144, 169, 196, 225, 256, and 289.

Explanation

The expression is evaluated by calculating the squared terms, performing the division, and then the addition. The terms (0.1)² and (0.01)² are calculated as follows: (0.1)² = 0.1 × 0.1 = 0.01 (0.01)² = 0.01 × 0.01 = 0.0001

The expression is interpreted as the difference of the squares divided by the denominator, followed by addition. Substituting the squared values into the fraction: (0.01 - 0.0001)/(0.0001) + 1

Subtracting the values in the numerator: 0.01 - 0.0001 = 0.0099

Dividing by the denominator: (0.0099)/(0.0001) = 99

Adding the final term: 99 + 1 = 100

Alternatively, by distributing the denominator: ((0.1)²)/(0.0001) - ((0.01)²)/(0.0001) + 1 = (0.01)/(0.0001) - (0.0001)/(0.0001) + 1 = 100 - 1 + 1 = 100

Explanation

Let the cost price (CP) of the article be 100 units. A 20% profit on the cost price implies that the profit amount is 20 units. The selling price (SP) is determined by adding the profit to the cost price: SP = CP + Profit = 100 + 20 = 120 The percentage of profit calculated on the selling price is determined by the ratio of the profit to the selling price, multiplied by 100: Profit % on SP = ( (Profit)/(SP) ) × 100 Substituting the values: Profit % on SP = ( ²⁰⁄₁₂₀ ) × 100 = ⅙ × 100 = ⁵⁰⁄₃%

Explanation

Let the cost price of one book be x. The cost price of 15 books is 15x. As given, the selling price of 20 books is equal to the cost price of 15 books, so the selling price of 20 books is 15x. This means the selling price of one book is ¹⁵ˣ⁄₂₀ = ³ˣ⁄₄.

Loss is calculated as the difference between cost price and selling price: Loss = x - ³ˣ⁄₄ = ˣ⁄₄. The loss percentage is calculated relative to the cost price: Loss Percent = (Loss)/(Cost Price) × 100 = (x)/(4){x} × 100 = ¹⁰⁰⁄₄ = 25.

Alternatively, when the cost price of M items equals the selling price of N items, the loss percent (when N > M) is given by the formula (N - M)/(N) × 100. Substituting the values M = 15 and N = 20 yields (20 - 15)/(20) × 100 = ⁵⁄₂₀ × 100 = 25.

Explanation

Let the cost price (CP) of the article be 100. Profit is given as 200% of the cost price.

Profit = 200% of 100 = 200

The selling price (SP) is calculated using the relation: SP = CP + Profit SP = 100 + 200 = 300

The ratio of the cost price to the selling price is: CP : SP = 100 : 300

Dividing both terms by 100, the ratio simplifies to 1 : 3.

Using the general formula for selling price with profit percentage: SP = CP × (1 + (Profit %)/(100)) SP = CP × (1 + ²⁰⁰⁄₁₀₀) SP = CP × (1 + 2) = 3CP

Therefore, the ratio (CP)/(SP) = (CP)/(3CP) = ⅓.

Explanation

The equivalent single discount for two successive discounts of x% and y% is calculated using the formula x + y - ˣʸ⁄₁₀₀. Substituting x = 20 and y = 10, the effective discount is 20 + 10 - (20 × 10)/(100) = 28%. The difference between the single discount of 30% and the effective successive discount of 28% is 2%. Since the difference in selling prices is directly proportional to the difference in discount rates applied to the marked price, we have 2% of Marked Price = Rs. 72. This yields the equation ²⁄₁₀₀ × Marked Price = 72. Solving for the Marked Price gives (72 × 100)/(2) = 3,600.

Explanation

Let the total amount of the electricity bill be x. The problem states that a 4% reduction on the bill amount equals Rs. 13. This relationship is expressed by the formula:

4% of x = 13

Converting the percentage to a decimal gives:

0.04x = 13

To solve for x, divide the reduction value by the percentage rate:

x = (13)/(0.04)

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

x = ¹³⁰⁰⁄₄

x = 325

Explanation

To find the single equivalent discount for successive discounts of d₁, d₂, and d₃, we can use the multiplier method. Let the initial price be 100. After the first discount of 10%, the price becomes 100 × (1 - 0.10) = 90. After the second discount of 20% is applied to this new price, it becomes 90 × (1 - 0.20) = 72. Finally, applying the third discount of 30% results in a final price of 72 × (1 - 0.30) = 50.4. The total discount offered is the difference between the initial price and the final price, which is 100 - 50.4 = 49.6.

Alternatively, using the formula for equivalent discount D = [1 - (1 - d₁)(1 - d₂)(1 - d₃)] × 100: D = [1 - (0.9 × 0.8 × 0.7)] × 100 D = [1 - 0.504] × 100 = 0.496 × 100 = 49.6%

Explanation

Let the original price of the article be P. When a price is increased successively by x% and y%, the final price is calculated using the formula P × (1 + ˣ⁄₁₀₀) × (1 + ʸ⁄₁₀₀).

Substituting the given values into the equation: P × (1 + ¹⁰⁄₁₀₀) × (1 + ²⁰⁄₁₀₀) = 33 P × ¹¹⁰⁄₁₀₀ × ¹²⁰⁄₁₀₀ = 33 P × 1.1 × 1.2 = 33 1.32P = 33

To find the original price P: P = (33)/(1.32) P = ³³⁰⁰⁄₁₃₂ Dividing both numerator and denominator by 11, we get: P = ³⁰⁰⁄₁₂ = 25

Alternatively, the effective percentage increase can be found using the successive change formula a + b + ᵃᵇ⁄₁₀₀. Here, 10 + 20 + (10 × 20)/(100) = 32%. This means the final price of Rs. 33 is 132% of the original price. Original Price = ³³⁄₁₃₂ × 100 = ¼ × 100 = 25

Explanation

The area of a square is determined by the product of its two dimensions, Area = side × side. When each dimension of a figure is increased, the total percentage increase in the area can be calculated using the successive percentage change formula: x + y + ˣʸ⁄₁₀₀.

Given that each side of the square is increased by 10%, we set x = 10 and y = 10. Substituting these values into the formula:

Percentage increase in area = 10 + 10 + (10 × 10)/(100) Percentage increase in area = 20 + ¹⁰⁰⁄₁₀₀ Percentage increase in area = 20 + 1 = 21%

This can also be verified by considering the side as s. The original area is s². After a 10% increase, the new side becomes 1.1s. and the new area is (1.1s)² = 1.21s². The fractional increase is 1.21 - 1 = 0.21, which is equivalent to 21%.

Explanation

The quantity of milk relative to water is determined by calculating the ratio of milk to water for each container. For the third container, the ratio 3:2 yields a value of 3 ÷ 2 = 1.5. Comparing the ratios for all containers—1.66 (5/3), 2.0 (2/1), 1.5 (3/2), and 1.75 (7/4)—the value 1.5 is the lowest. This indicates that the third container has the lowest proportion of milk for every unit of water. Alternatively, expressing the milk as a fraction of the total mixture for the third container gives 3/(3+2) = 0.60, which is less than the concentrations of 0.625, 0.667, and 0.636 found in the other mixtures. Since 1.5 (or 0.60) is the minimum value, the third container satisfies the condition.

Explanation

Let the two numbers be x and 3x according to the ratio 1:3. The problem states that their sum is 240, which gives the equation x + 3x = 240. Combining the terms, we get 4x = 240, and dividing by 4 results in x = 60. The difference between the two numbers is represented by 3x - x = 2x. Substituting the value of x into this expression, the difference is 2 × 60 = 120.

Explanation

The ratio of income to expenditure is 11:10. Since Savings = Income - Expenditure, the savings ratio part is 11 - 10 = 1. Given that the person saves Rs. 9,000 per annum, this 1 part represents the annual savings of Rs. 9,000.

Annual Income = 11 parts = 11 × 9,000 = Rs. 99,000

To find the monthly income, divide the annual income by 12:

Monthly Income = (99,000)/(12) = Rs. 8,250

Explanation

To find the ratio W₂:W₃, we can express both W₂ and W₃ in terms of the common variable W₁. From the first ratio, W₁:W₂ = 2:3, we can write W₂ = ³⁄₂W₁. From the second ratio, W₁:W₃ = 1:2, we can write W₃ = 2W₁.

The ratio W₂:W₃ is therefore calculated as follows: (W₂)/(W₃) = (3)/(2)W₁{2W₁} (W₂)/(W₃) = ³⁄₂ × ½ (W₂)/(W₃) = ¾

Alternatively, by equating the value of W₁ in both ratios, we have W₁:W₂ = 2:3 and W₁:W₃ = 2:4 (by multiplying both parts of the second ratio by 2). Combining these gives the compound ratio W₁:W₂:W₃ = 2:3:4. Taking the last two terms, we get W₂:W₃ = 3:4.

Explanation

To find the radius of the sphere, we equate the volume of the cylindrical copper wire to the volume of the sphere, as the total volume remains constant during melting. First, we convert all dimensions to centimeters. The length of the wire L is 36 m = 3600 cm. The diameter of the wire is 2 mm = 0.2 cm, so the radius of the wire r is 0.1 cm.

The volume of the cylindrical wire is given by V_wᵢᵣₑ = π r² L. Substituting the values: V_wᵢᵣₑ = π × (0.1)² × 3600 = π × 0.01 × 3600 = 36π cm³

The volume of the sphere is given by Vₛₚₕₑᵣₑ = ⁴⁄₃ π R³, where R is the radius of the sphere. Setting the volumes equal: ⁴⁄₃ π R³ = 36π

Dividing both sides by π and solving for R³: ⁴⁄₃ R³ = 36 R³ = 36 × ¾ R³ = 9 × 3 R³ = 27

Taking the cube root of both sides: R = ³√(27) = 3 cm

Explanation

The circumference C of a wheel is calculated using the formula C = 2π r, where r represents the radius of the wheel. Since 2 and π are constants, the circumference is directly proportional to the radius (C ∝ r).

To find the ratio of the circumferences of two wheels with radii r₁ and r₂, we set up the ratio of their respective formulas: C₁ / C₂ = (2π r₁) / (2π r₂)

By cancelling the common factor of 2π from both the numerator and the denominator, the ratio of the circumferences simplifies directly to the ratio of the radii: C₁ / C₂ = r₁ / r₂

Given that the ratio of the radii r₁ : r₂ is 3 : 4, the ratio of their circumferences C₁ : C₂ is also 3 : 4.

Explanation

The net percentage change in the area of a rectangle when its dimensions are modified can be determined using the successive percentage change formula x + y + ˣʸ⁄₁₀₀, where x and y are the percentage changes in length and breadth. For an increase of 10% in length, x = +10, and for a decrease of 10% in breadth, y = -10. Substituting these values into the formula yields 10 - 10 + (10 × (-10))/(100) = 0 - ¹⁰⁰⁄₁₀₀ = -1%. The resulting negative value indicates a decrease in the overall area by 1%.

Explanation

For a sum of money to double itself under simple interest, the interest earned must be equal to the principal amount itself (SI = P). Using the standard simple interest formula SI = (P × r × t)/(100), where P is the principal, r is the annual interest rate, and t is the time in years, we can substitute the given values to find t.

Given that the rate r = ²⁵⁄₄ percent per annum, the equation becomes P = (P × 25)/(4) × t{100}. Dividing both sides by P gives 1 = (25 × t)/(4 × 100). Simplifying the expression, we get 1 = (25 × t)/(400), which leads to t = ⁴⁰⁰⁄₂₅. Dividing 400 by 25 results in t = 16.

Explanation

In compound interest, if a sum of money becomes k times in t years, it becomes kⁿ times in n × t years. The problem states that the sum doubles (k=2) in 5 years (t=5). To find the amount after 20 years, we calculate the number of 5-year cycles: n = 20 / 5 = 4. Consequently, the principal will grow by a factor of 2⁴ = 16 times. Multiplying the initial principal by this factor, we get 12,000 × 16 = 1,92,000.

Alternatively, using the standard compound interest formula A = P(1 + r/100)ᵗ, we are given 2P = P(1 + r/100)⁵, which simplifies to (1 + r/100)⁵ = 2. After 20 years, the amount is A = P(1 + r/100)²⁰. This can be rewritten as A = P[(1 + r/100)⁵]⁴. Substituting the values, A = 12,000 × (2)⁴ = 12,000 × 16 = 1,92,000.