SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

To find the value of x, begin by simplifying the given linear equation -3/2 + (2/3)(3x + 9) = x/2. First, distribute the factor 2/3 into the parentheses: (2/3) · 3x = 2x and (2/3) · 9 = 6. The equation is rewritten as -3/2 + 2x + 6 = x/2.

To simplify further and eliminate the denominators, multiply every term in the equation by 2. This results in 2(-3/2) + 2(2x) + 2(6) = 2(x/2), which simplifies to -3 + 4x + 12 = x. Combining the constants on the left side gives 4x + 9 = x.

Next, isolate the variable x by moving all x terms to one side and the constant terms to the other. Subtracting x from both sides gives 3x + 9 = 0. Subtracting 9 from both sides yields 3x = -9. Finally, dividing both sides by 3 results in x = -3.

Explanation

The value of the expression a³ - b³ can be calculated using the algebraic identity a³ - b³ = (a - b)³ + 3ab(a - b).

Given the values a - b = 2 and ab = 8, substitute them into the formula:

a³ - b³ = (2)³ + 3(8)(2)

a³ - b³ = 8 + 48

a³ - b³ = 56

Alternatively, by observing the given conditions, the integers a = 4 and b = 2 satisfy both a - b = 2 and ab = 8. Substituting these into the target expression yields 4³ - 2³ = 64 - 8 = 56.

Explanation

According to the divisibility rule for 11, a number is divisible by 11 if the difference between the sum of the digits at odd positions and the sum of the digits at even positions is either 0 or a multiple of 11. For the four-digit number 27N4, we identify the digits based on their positions.

Sum of digits at odd positions (1st and 3rd) = 2 + N Sum of digits at even positions (2nd and 4th) = 7 + 4 = 11

For the number to be divisible by 11, the difference between these sums must be a multiple of 11, including zero. We set the equation (2 + N) - 11 = 0, which simplifies to N - 9 = 0. Solving this gives N = 9. Since N must be a single-digit integer between 0 and 9, this is the only valid solution. Any other multiple of 11, such as (2 + N) - 11 = 11 or (2 + N) - 11 = -11, would result in N = 20 or N = -2, which are not possible for a single digit. Therefore, the value of N that makes 2794 divisible by 11 is 9.

Explanation

Let the non-zero number be x. The problem states that the number exceeds 7 times its reciprocal by 9.3, which is expressed by the algebraic equation x - ⁷⁄ₓ = 9.3. Testing x = 10 in this equation, we get 10 - ⁷⁄₁₀, which simplifies to 10 - 0.7 = 9.3.

Explanation

The sum of the first n terms of an arithmetic progression is calculated using the formula Sₙ = ⁿ⁄₂(a + l), where n represents the number of terms, a is the first term, and l is the last term. Given that n = 12, a = 3, and l = 47, substituting these values into the formula yields S₁₂ = ¹²⁄₂(3 + 47). This simplifies to 6 × 50, which results in a total sum of 300.

Explanation

Reflection of a point (x, y) in the origin results in the point (-x, -y). Therefore, if point P is (a, b), its reflection in the origin P₁ is (-a, -b).

Reflection of a point (x, y) in the Y-axis results in the point (-x, y). Applying this transformation to P₁(-a, -b), the new coordinates are (-(-a), -b), which simplifies to (a, -b).

According to the problem, this final position is (4, -3). By equating the coordinates: a = 4 -b = -3 ⟹ b = 3

Substituting these values back into the original point P(a, b) yields the coordinates (4, 3).

Explanation

The Y-axis divides the line segment joining two points (x₁, y₁) and (x₂, y₂) in the ratio -x₁ : x₂. For the given points A(-1, 3) and B(2, -4), the x-coordinates are x₁ = -1 and x₂ = 2. Substituting these values into the ratio formula gives -(-1) : 2, which simplifies to 1 : 2.

This can also be derived using the section formula. Any point on the Y-axis has an x-coordinate of 0. If the Y-axis divides the segment AB in the ratio m : n, the x-coordinate of the point of division is given by:

x = (m x₂ + n x₁)/(m + n)

Setting x = 0 and substituting x₁ = -1 and x₂ = 2:

0 = (m(2) + n(-1))/(m + n)

2m - n = 0

2m = n

ᵐ⁄ₙ = ½

This confirms the ratio is 1 : 2.

Explanation

The slope m of a line expressed in the general form Ax + By + C = 0 is calculated using the formula m = -(A)/(B). For the given equation 2x + 3y = 12, the coefficient of x is A = 2 and the coefficient of y is B = 3. Substituting these values into the formula gives m = -⅔.

This can also be determined by converting the equation into the slope-intercept form y = mx + c. Starting with 2x + 3y = 12, we isolate the y term by subtracting 2x from both sides to get 3y = -2x + 12. Dividing every term by 3 results in y = -⅔x + 4. In this form, the coefficient of x represents the slope of the line.

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 side lengths. The mathematical relationship is: (Area(Δ XYZ))/(Area(Δ PQR)) = ((XY)/(PQ))²

Given that Area(Δ XYZ) = 100 cm², Area(Δ PQR) = 25 cm², and PQ = 4 cm, these values are substituted into the formula: ¹⁰⁰⁄₂₅ = ((XY)/(4))²

Simplifying the ratio of the areas: 4 = (XY²)/(16)

Multiplying both sides by 16 to isolate XY²: XY² = 64

Taking the square root of both sides to find the length: XY = √(64) = 8

Explanation

To find the value of the expression sec 30° + tan 60°, we substitute the standard trigonometric ratios for the given angles. The value of cos 30° is √3{2}, which means its reciprocal sec 30° is (2)/(√(3)). The value of tan 60° is √3.

Substituting these values into the expression: sec 30° + tan 60° = (2)/(√(3)) + √3

To simplify, we find a common denominator by expressing √3 as a fraction with √3 in the denominator: (2)/(√(3)) + √3 × √3{√3} = (2)/(√(3)) + (3)/(√(3))

Adding the fractions together yields: (2 + 3)/(√(3)) = (5)/(√(3))

Explanation

In a right-angled triangle Δ PQR where the right angle is at Q, the sum of the angles is 180°. Given ∠ Q = 90° and ∠ R = 30°, the measure of ∠ P is calculated as:

∠ P = 180° - (∠ Q + ∠ R) ∠ P = 180° - (90° + 30°) ∠ P = 60°

To find the value of cot P, we substitute the value of the angle into the trigonometric function:

cot P = cot 60°

According to the standard trigonometric table, the value of cot 60° is (1)/(√(3)).

Explanation

In a right-angled triangle, the secant function is defined as the ratio of the hypotenuse to the base. Given sec θ = ⁵⁄₃, we have the hypotenuse H = 5 and the base B = 3.

Using the Pythagorean theorem, H² = P² + B², where P is the perpendicular: 5² = P² + 3² 25 = P² + 9 P² = 16 P = 4

The cosecant function is defined as the ratio of the hypotenuse to the perpendicular: csc θ = (H)/(P) = ⁵⁄₄

Converting the fraction to a decimal: ⁵⁄₄ = 1.25

Explanation

The price of the watch after the first discount of 25% is calculated by reducing the marked price by one-fourth: Price₁ = 8400 × (1 - 0.25) = 8400 × 0.75 = 6300

The final selling price after the second discount is Rs 5040. The amount of the second discount is the difference between the intermediate price and the final selling price: Discount₂ = 6300 - 5040 = 1260

The second discount percentage d₂ is calculated relative to the intermediate price: d₂ = ¹²⁶⁰⁄₆₃₀₀ = ¹²⁶⁄₆₃₀

Simplifying the fraction by dividing both the numerator and denominator by 63: d₂ = ²⁄₁₀ = 0.2

Explanation

The sum of the shares of E and B is equal to the sum of the shares of A, C, D, and F. In a standard distribution representing 100% of the data, where the combined share of E and B is 50%, the combined share of the remaining entities A, C, D, and F is calculated as 100% - 50% = 50%.

Ratio = (S_E + S_B) : (S_A + S_C + S_D + S_F)

Substituting the respective values:

Ratio = 50 : 50 = 1 : 1

This equality indicates that the two groups of entities hold identical portions of the total allocation, resulting in a ratio of 1 : 1.

Explanation

The company earns a total profit of Rs 20,00,000. The problem states that the company decides to distribute half of this profit to its shareholders. Therefore, the total amount available for distribution is calculated as: Amount for distribution = 20,00,000 / 2 = 10,00,000

According to the shareholding pattern associated with this data set, shareholder D holds a 15 percent stake in the company. To determine the specific amount received by D, we calculate 15 percent of the total amount being distributed: Amount received by D = 15/100 × 10,00,000 = 1,50,000

Thus, the share of profit that D receives is Rs 1,50,000.

Explanation

Given the ratio P : Q : R = 2 : 3 : 5, we can represent the variables as P = 2k, Q = 3k, and R = 5k, where k is a non-zero constant. To find the ratio (P + Q) : (Q + R) : (R + P), we perform the addition for each part of the expression using these proportional values.

For the first part of the ratio, P + Q = 2k + 3k = 5k. For the second part of the ratio, Q + R = 3k + 5k = 8k. For the third part of the ratio, R + P = 5k + 2k = 7k.

The required ratio is then expressed as 5k : 8k : 7k. Since k is a common factor in all three terms of the ratio, it can be cancelled out by dividing each term by k, simplifying the expression to 5 : 8 : 7. Direct substitution of the ratio parts 2, 3, and 5 into the expression works here because the terms P+Q, Q+R, and R+P are all linear combinations of the original variables, ensuring the homogeneity of the resulting ratio.

Explanation

Total Age = Average Age × Number of Students

Initial total age of the 20 students = 20 × 21 = 420 years. Final total age of the group (retaining the base of 20 students) = 20 × 20 = 400 years.

The reduction in the total age of the group is: 420 - 400 = 20 years.

This net reduction is the difference between the sum of the ages of the students who left and the age of the student who joined: (Age_Lₑ_fₜ₁ + Age_Lₑ_fₜ₂) - Age_Jₒᵢₙₑ_d = 20 (26 + Age_Lₑ_fₜ₂) - 20 = 20 6 + Age_Lₑ_fₜ₂ = 20 Age_Lₑ_fₜ₂ = 20 - 6 = 14 years.

Explanation

To find the least number that must be subtracted from 210 to make it divisible by 11, we divide 210 by 11 to find the remainder. Using the division algorithm N = dq + R, where N is the dividend, d is the divisor, q is the quotient, and R is the remainder:

210 = (11 × 19) + 1

The remainder R is 1. For a number to be perfectly divisible by a divisor, the remainder must be zero. By subtracting the remainder from the original number, we obtain 210 - 1 = 209, which is exactly divisible by 11 (11 × 19 = 209). Therefore, the least number that must be subtracted is 1.

Explanation

The work rate of an individual is the reciprocal of the time taken to complete the task. Given that A takes 25 days and B takes 10 days, their individual work rates are 1/25 and 1/10 respectively. When A, B, and C work together, they complete the job in 6.25 days. Since 6.25 = 25/4, the combined work rate of A, B, and C is 4/25 per day.

The work rate of C alone is found by subtracting the rates of A and B from the combined rate: Rate_C = Rate_A_B_C - (Rate_A + Rate_B) Rate_C = ⁴⁄₂₅ - (¹⁄₂₅ + ⅒) Rate_C = ³⁄₂₅ - ⅒

Using the common denominator of 50: Rate_C = ⁶⁄₅₀ - ⁵⁄₅₀ = ¹⁄₅₀

Since C paints 1/50 of the house per day, the total time required for C to paint the house alone is 50 days.

Explanation

The effective discount in a Buy X Get Y Free offer is calculated based on the total number of items received by the customer. In this case, for every 3 sarees purchased, 1 saree is provided free of cost, making the total number of sarees received equal to 3 + 1 = 4.

The discount percentage is the ratio of the free items to the total items received, expressed as a percentage. Using the formula:

Discount % = (Free Items)/(Total Items) × 100

Substituting the given values:

Discount % = ¼ × 100 = 25%

The customer effectively pays for 3 units but receives 4 units, resulting in a 25% discount on the total transaction.

Explanation

Let the present ages of R and S be 6x and 7x respectively. Based on the information provided, twelve years ago, the ratio of their ages was 9:11. This relationship can be expressed using the following algebraic equation:

(6x - 12)/(7x - 12) = ⁹⁄₁₁

To solve for x, cross-multiply the terms of the equation: 11(6x - 12) = 9(7x - 12) 66x - 132 = 63x - 108

Subtract 63x from both sides and add 132 to both sides to isolate the variable: 66x - 63x = 132 - 108 3x = 24 x = 8

R's present age is defined as 6x. Substituting the calculated value of x: 6 × 8 = 48

The same result can be achieved by equalizing the ratio differences. The present ratio is 6:7 (difference of 1) and the past ratio is 9:11 (difference of 2). Multiplying the present ratio by 2 gives 12:14, making the differences equal. The change from 12 to 9 units represents the 12-year period, meaning 3 units equal 12 years, or 1 unit equals 4 years. R's present age is 12 × 4 = 48 years.

Explanation

The correct average can be calculated using the formula for adjusting the mean when a value is incorrectly recorded: Correct Average = Old Average + (Correct Value - Wrong Value)/(n). In this case, the initial average is 25, the number of students (n) is 10, the correct mark is 43, and the wrongly entered mark is 34.

First, determine the net change in the total marks, which is 43 - 34 = 9. Since the correct mark is higher than the wrong mark, the total sum of marks must increase by 9. To find the increase in the average, divide this net change by the total number of students: ⁹⁄₁₀ = 0.9. Adding this adjustment to the original average gives the correct average: 25 + 0.9 = 25.9.

Explanation

Let the cost price for the wholesaler be x. The wholesaler sells the jacket to the retailer at a profit of 25%, which means the selling price to the retailer is x × (1 + 25/100) = 1.25x. The retailer then sells it to the customer at a profit of 20%, so the final price paid by the customer is 1.25x × (1 + 20/100) = 1.25x × 1.20.

According to the problem, the customer pays Rs. 2,400. This gives the following equation: x × 1.25 × 1.20 = 2400

Using fractions for faster calculation: x × (5/4) × (6/5) = 2400 x × (6/4) = 2400 x × 1.5 = 2400 x = 2400 / 1.5 x = 1600

Explanation

The problem states that when a number is increased by 24, it becomes 104% of itself. Since any number is 100% of itself, the increase of 24 corresponds to the percentage increase of 104% - 100% = 4%.

Let the number be x. We can represent the relationship as: 4% of x = 24 ⁴⁄₁₀₀ × x = 24

To find the value of x, we rearrange the equation: x = (24 × 100)/(4) x = 6 × 100 x = 600

Explanation

When a journey involves covering the same distance twice at two different speeds, v₁ and v₂, the average speed for the entire trip is the harmonic mean of the speeds. The formula for average speed in this scenario is Vₐᵥ_g = (2v₁v₂)/(v₁ + v₂). Substituting the given values v₁ = 30 km/h and v₂ = 20 km/h into the formula yields Vₐᵥ_g = (2 × 30 × 20)/(30 + 20) = ¹²⁰⁰⁄₅₀. Dividing the terms, we find the average speed to be 24 km/h. This formula is derived from the fundamental definition of average speed as total distance divided by total time, where the distances for each leg of the journey are equal.

Explanation

The Simple Interest (SI) formula is SI = (P × R × T)/(100). Substituting the given values where SI = 1200, R = 7.5, and T = 4, we have 1200 = (P × 7.5 × 4)/(100). Since 7.5 × 4 = 30, the equation simplifies to 1200 = (30P)/(100), or 1200 = 0.3P. Calculating for the principal sum, P = (1200)/(0.3) = 4000.

Explanation

The given equation is -3[1 - ˣ⁄₂] + ⁵ˣ⁄₃ = ⅙. Expanding the first term results in -3 + ³ˣ⁄₂ + ⁵ˣ⁄₃ = ⅙. Adding 3 to both sides of the equation gives ³ˣ⁄₂ + ⁵ˣ⁄₃ = ⅙ + 3. To combine the terms on the left side, we find the common denominator of 6, resulting in (9x + 10x)/(6) = (1 + 18)/(6). This simplifies the equation to ¹⁹ˣ⁄₆ = ¹⁹⁄₆. By multiplying both sides by 6, we get 19x = 19, and dividing by 19 yields x = 1.

Explanation

The value of a³ + b³ is determined using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). By substituting the given values a + b = 3 and ab = -4 into this formula, the expression becomes (3)³ - 3(-4)(3). Performing the arithmetic operations results in 27 - (-36), which simplifies to 27 + 36 = 63. Alternatively, the identity a³ + b³ = (a + b)(a² - ab + b²) can be used by first finding a² + b² = (a + b)² - 2ab = 3² - 2(-4) = 17, leading to 3(17 - (-4)) = 3(21) = 63.

Explanation

Let the non-zero number be x. According to the problem, the sum of this number and 9 times its reciprocal equals 10. This is represented by the equation x + 9(¹⁄ₓ) = 10. To solve for x, multiply the entire equation by x to eliminate the fraction, resulting in the quadratic equation x² + 9 = 10x. Rearranging the terms into standard form gives x² - 10x + 9 = 0. This quadratic can be factored into (x - 9)(x - 1) = 0. Setting each factor to zero provides the possible values for x as 9 or 1. Among the choices provided, the value 9 satisfies the condition.

Explanation

In an arithmetic progression, the n-th term is defined by the formula aₙ = a + (n - 1)d, where a is the first term and d is the common difference. Given that the first term a = -10 and the third term a₃ = -4, we can determine the common difference using the relation a₃ = a + 2d. Substituting the given values yields -4 = -10 + 2d, which simplifies to 2d = 6, resulting in d = 3. To find the 12th term, we apply the formula a₁₂ = a + 11d. Substituting the values for a and d gives a₁₂ = -10 + 11(3) = -10 + 33 = 23.

Explanation

When a point (x, y) is reflected in the y-axis, the y-coordinate remains unchanged while the sign of the x-coordinate is reversed. The transformation rule for reflection across the y-axis is defined as (x, y) → (-x, y). Given the point (2, -3.5), the x-coordinate is 2 and the y-coordinate is -3.5. Applying the rule, the x-coordinate changes from 2 to -2, and the y-coordinate stays -3.5. Therefore, the reflected point is (-2, -3.5).

Explanation

The coordinates of a point A (x, y) that divides a line segment joining B(x₁, y₁) and C(x₂, y₂) in the ratio m:n are given by the section formula: x = (mx₂ + nx₁)/(m+n) and y = (my₂ + ny₁)/(m+n). Here, the coordinates of B are (4, -4), the coordinates of C are (0, 6), and the ratio m:n is 1:3.

To find the x-coordinate of A: x = (1(0) + 3(4))/(1 + 3) = ¹²⁄₄ = 3

To find the y-coordinate of A: y = (1(6) + 3(-4))/(1 + 3) = (6 - 12)/(4) = ⁻⁶⁄₄ = -1.5

The resulting coordinates of point A are (3, -1.5).

Explanation

The slope m of a line passing through two points (x₁, y₁) and (x₂, y₂) is determined by the formula m = (y₂ - y₁)/(x₂ - x₁). Substituting the given coordinates (-2, -1) and (4, -3) into this equation:

m = (-3 - (-1))/(4 - (-2)) m = (-3 + 1)/(4 + 2) m = ⁻²⁄₆ m = -⅓

According to the properties of coordinate geometry, parallel lines must have the same slope. Consequently, the slope of any line parallel to the line passing through the points (-2, -1) and (4, -3) is exactly -⅓.

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides. Since Δ XYZ ∼ Δ PQR, we have the relationship:

(Perimeter(Δ XYZ))/(Perimeter(Δ PQR)) = (XY)/(PQ)

Substituting the given values into the formula:

³⁄₂ = (XY)/(6)

To find XY, we rearrange the equation:

XY = ³⁄₂ × 6

XY = 3 × 3

XY = 9 cm

Explanation

The trigonometric values for the given angles are sec 45° = √2 and tan 30° = (1)/(√(3)). Substituting these values into the expression sec 45° + tan 30° gives:

√2 + (1)/(√(3))

To combine these terms, we find a common denominator of √3. Multiplying the first term by √3{√3} yields:

√2 · √3{√3} + (1)/(√(3)) = √6 + 1{√3}

Explanation

In the right-angled triangle Δ DEF, the sum of all interior angles is 180°. Given that ∠ E = 90° and ∠ D = 45°, the measure of the third angle ∠ F is determined by the equation ∠ D + ∠ E + ∠ F = 180°. Substituting the known values, we get 45° + 90° + ∠ F = 180°, which simplifies to ∠ F = 45°. To find csc F, we evaluate csc 45°. Using the trigonometric identity csc θ = (1)/(sin θ) and the standard value sin 45° = (1)/(√(2)), we find that csc 45° = (1)/(1/√(2)) = √2.

Explanation

In a right-angled triangle, cos θ is defined as the ratio of the adjacent side to the hypotenuse. Given cos θ = ⁵⁄₁₃, we can represent the adjacent side as 5 units and the hypotenuse as 13 units. Using the Pythagorean theorem, the length of the opposite side is √(13² - 5²) = √(169 - 25) = √(144) = 12. This corresponds to the well-known Pythagorean triplet (5, 12, 13). The cosecant function, csc θ, is the reciprocal of the sine function and is defined as the ratio of the hypotenuse to the opposite side. Substituting the known values, we get csc θ = ¹³⁄₁₂.

Explanation

Based on the data provided in the bar graph for the given companies, the number of employees in Company A and Company D are as follows:

Number of employees in Company A = 150 Number of employees in Company D = 250

The difference in the number of employees between Company D and Company A is calculated by subtracting the smaller value from the larger value:

Difference = Employees in Company D - Employees in Company A Difference = 250 - 150 = 100

Explanation

Based on the data provided in the bar graph for the number of employees in different companies, the number of employees in companies B, D, and F are as follows: Number of employees in company B = 150 Number of employees in company D = 180 Number of employees in company F = 420

If an insurance agent sells a policy to every employee in these three companies, the total number of policies sold is the sum of the employees in companies B, D, and F. Total policies = Employees(B) + Employees(D) + Employees(F) Total policies = 150 + 180 + 420 = 750

Explanation

Based on the statistical data for the specified organizations, the number of employees in company B is 650 and the number of employees in company E is 130. To determine the percentage by which the staff count in company E is lower than that in company B, the following formula is applied:

Percentage Lower = (Employees in B - Employees in E)/(Employees in B) × 100

The absolute difference in the number of employees between the two companies is 650 - 130 = 520. Substituting the values into the percentage formula gives:

Percentage = ⁵²⁰⁄₆₅₀ × 100

The fraction ⁵²⁰⁄₆₅₀ simplifies to ⅘ by dividing both the numerator and the denominator by 130. Multiplying this fraction by 100 yields the final value:

⅘ × 100 = 80%

Explanation

Based on the data provided for this shift, Company B has 250 employees and Company F has 400 employees. The number of employees being transferred from Company B to Company F is calculated as 20% of Company B's total strength.

Number of employees transferred = 20% of 250 = ²⁰⁄₁₀₀ × 250 = 50

To find the new employee count for Company F, we add these transferred employees to Company F's existing workforce:

New total for Company F = Original employees in F + Employees from B New total = 400 + 50 = 450

Explanation

To find the Least Common Multiple (LCM) of 120 and 450, we use the prime factorization method. The prime factorization of 120 is 2³ × 3¹ × 5¹. The prime factorization of 450 is 2¹ × 3² × 5². The LCM is determined by taking the highest power of each prime factor present in either number:

LCM(120, 450) = 2ᵐᵃˣ⁽³^,¹⁾ × 3ᵐᵃˣ⁽¹^,²⁾ × 5ᵐᵃˣ⁽¹^,²⁾ LCM = 2³ × 3² × 5² LCM = 8 × 9 × 25 LCM = 72 × 25 = 1800

Alternatively, using the common division method, we divide both numbers by their common factors. Dividing 120 and 450 by 10 gives 12 and 45. Dividing these results by 3 gives 4 and 15. Since 4 and 15 are co-prime, the LCM is calculated as the product of the divisors and the remaining quotients: 10 × 3 × 4 × 15 = 1800.

Explanation

Let the total work be the Least Common Multiple (LCM) of the individual days taken by A, B, and C. The LCM of 6, 9, and 18 is 18 units.

The efficiency (work per day) of each individual is calculated as follows: Efficiency of A = 18 / 6 = 3 units/day Efficiency of B = 18 / 9 = 2 units/day Efficiency of C = 18 / 18 = 1 unit/day

The combined efficiency of A, B, and C working together is 3 + 2 + 1 = 6 units/day. Since they work together for 1 day, the work completed is 6 × 1 = 6 units.

The remaining work to be done is 18 - 6 = 12 units. After A and B quit, C must finish this remaining work alone. The time taken by C is the remaining work divided by C's efficiency: Time = 12 / 1 = 12 days.

Explanation

The surface area S of a sphere is calculated using the formula S = 4π r², where r is the radius. Given the surface area is 6.16 sq cm and using π = ²²⁄₇, we set up the equation:

4 × ²²⁄₇ × r² = 6.16

⁸⁸⁄₇ r² = 6.16

To find r², rearrange the terms:

r² = (6.16 × 7)/(88)

r² = (43.12)/(88)

r² = 0.49

Taking the square root of both sides gives r = √(0.49) = 0.7 cm. The diameter d of a sphere is defined as twice the radius (d = 2r). Therefore, d = 2 × 0.7 = 1.4 cm.

Explanation

The total marked price for the items purchased is calculated by summing the cost of 2 shirts and 3 pairs of trousers: 2 × 500 + 3 × 1000 = 1000 + 3000 = 4000 rupees. The total discount amount is the sum of the discounts applied to each category: 20% of 1000 for the shirts and 40% of 3000 for the trousers, which results in 200 + 1200 = 1400 rupees. To find the effective discount percentage, divide the total discount by the total marked price and multiply by 100: (1400 / 4000) × 100 = 35%.

Explanation

To find the ratio of C's wealth to A's wealth, express both A and C in terms of B's wealth. Given that A = ⁴⁄₉B and C = ⁷⁄₆B, the ratio C:A is calculated as:

(C)/(A) = (7)/(6)B{⁴⁄₉B}

Canceling the common term B and multiplying by the reciprocal of the denominator:

(C)/(A) = ⁷⁄₆ × ⁹⁄₄

Reducing the terms by dividing 9 and 6 by their common factor 3:

(C)/(A) = (7 × 3)/(2 × 4) = ²¹⁄₈

This results in the ratio 21:8.

Explanation

The average of an even number of consecutive terms in an arithmetic progression is the arithmetic mean of the two middle terms. Since the four consecutive odd numbers have an average of 40, the two middle numbers must be 39 and 41. Because the numbers are consecutive odd integers, the common difference is 2. Therefore, the sequence of four numbers is 37, 39, 41, and 43. The largest number in this sequence is 43.

Alternatively, let the four consecutive odd numbers be represented as x - 3, x - 1, x + 1, and x + 3. The sum of these four numbers is (x - 3) + (x - 1) + (x + 1) + (x + 3) = 4x. The average is the sum divided by the count, which is 4x / 4 = x. Given that the average is 40, we have x = 40. The largest number in the set is x + 3. Substituting the value of x, we get 40 + 3 = 43.

Explanation

The cost price CP of the coffee tin can be determined from the initial selling price and the loss percentage using the relation CP = (SP)/(1 - Loss %). Substituting the given values, CP = (528)/(1 - 0.12) = (528)/(0.88) = 600.

When the wholesaler sells the tin at the new selling price of Rs 636, the profit earned is the difference between the new selling price and the cost price, which is 636 - 600 = 36. The profit percentage is calculated by dividing the profit by the cost price and multiplying by 100, which gives ³⁶⁄₆₀₀ × 100 = 6%.

Using a direct ratio approach, the ratio of the selling price to the corresponding percentage of the cost price remains constant: (528)/(100 - 12) = (636)/(100 + P). This simplifies to ⁵²⁸⁄₈₈ = (636)/(100 + P). Since 528 ÷ 88 = 6, we have 6 = (636)/(100 + P), which leads to 100 + P = 106, confirming a profit of 6%.

Explanation

Let the total wealth be represented as 100 units. The wife is willed 40% of the wealth, which amounts to 40 units. The children receive the remainder of the wealth, which is calculated as 100 - 40 = 60 units. To determine what percent of the wife's share the children receive, we calculate the ratio of the children's share to the wife's share and multiply by 100:

Percentage = (Children's Share)/(Wife's Share) × 100

Substituting the values into the formula:

Percentage = ⁶⁰⁄₄₀ × 100 = ³⁄₂ × 100 = 1.5 × 100 = 150%

Explanation

The time taken for both speeds is the same. Let the actual distance traveled at 10 km/hr be D. According to the problem, walking at 12 km/hr results in a distance of D + 1 in the same time t. Using the relation Time = (Distance)/(Speed), we can equate the time for both cases:

(D)/(10) = (D + 1)/(12)

Cross-multiplying gives: 12D = 10(D + 1) 12D = 10D + 10 2D = 10 D = 5

Alternatively, the difference in speed is 12 - 10 = 2 km/hr. This extra speed accounts for the additional distance of 1 km covered in time t. Therefore, t = (Distance Difference)/(Speed Difference) = ½ hour. The actual distance traveled at the original speed is 10 × ½ = 5 km.