SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The time taken by Abraham and Andrew working together is calculated using the formula (T1 × T2) / (T1 + T2). Substituting the given values: (9 × 6) / (9 + 6) = 54 / 15 = 18 / 5 = 3.6 days, which is equal to 3 3/5 days.

Explanation

The average speed is calculated as total distance divided by total time. Total distance is 45 km (20 km + 25 km) and total time is 270 minutes (4.5 hours), resulting in an average speed of 10 km/h.

Explanation

The expression x⁸ - 1/x⁸ can be factored using the difference of squares identity, a² - b² = (a + b)(a - b), as follows: x⁸ - 1/x⁸ = (x⁴ + 1/x⁴)(x⁴ - 1/x⁴) = (x⁴ + 1/x⁴)(x² + 1/x²)(x² - 1/x²) = (x⁴ + 1/x⁴)(x² + 1/x²)(x + 1/x)(x - 1/x)

Given x + 1/x = √6, square both sides to find x² + 1/x²: (x + 1/x)² = (√6)² ⟹ x² + 1/x² + 2 = 6 ⟹ x² + 1/x² = 4

Square this result to find x⁴ + 1/x⁴: (x² + 1/x²)² = 4² ⟹ x⁴ + 1/x⁴ + 2 = 16 ⟹ x⁴ + 1/x⁴ = 14

Next, find x - 1/x using the relation (x - 1/x)² = (x + 1/x)² - 4: (x - 1/x)² = (√6)² - 4 = 6 - 4 = 2 Since x > 1, the value of x - 1/x is positive, so x - 1/x = √2.

Substitute these values into the factored expression: x⁸ - 1/x⁸ = 14 × 4 × √6 × √2 x⁸ - 1/x⁸ = 56 × √(12) x⁸ - 1/x⁸ = 56 × 2√3 = 112√3

Explanation

Let the monthly income be I. The total pocket money is 0.08I, and the elder son receives 85% of that, which is 0.85 * 0.08I = 0.068I. Since he spends 90%, he saves 10%, meaning 0.10 * 0.068I = 17, which simplifies to 0.0068I = 17, resulting in I = 2500.

Explanation

The total surface area of a cubical box is determined using the geometric formula 6a², where a represents the length of one side of the cube. Since a cube is composed of six congruent square faces, the total area is six times the area of a single face. Given that the side length a is 12 cm, we substitute this value into the formula to obtain 6 × (12)². Squaring the side length, 12 × 12, gives 144. Multiplying this area by the six faces, 6 × 144, results in a total surface area of 864 cm².

Explanation

The given expression follows the algebraic identity (x + y)(x² - xy + y²) = x³ + y³. By setting x = (5)/(b) and y = 5b, we can verify that x² = (25)/(b²), y² = 25b², and the middle term xy = ((5)/(b))(5b) = 25. Therefore, the entire expression simplifies to ((5)/(b))³ + (5b)³.

Given b = 5, we substitute this value into the simplified cubic form: (⁵⁄₅)³ + (5 × 5)³ = 1³ + 25³

Calculating the powers: 1³ = 1 25³ = 25 × 25 × 25 = 625 × 25 = 15625

Adding the results: 1 + 15625 = 15626

Explanation

Since the efficiencies of A, B, and C are all equal (A = B and C = A), the amount of work done each day is the same regardless of who is working. If the work is completed in 15 days in the order A, B, C, it will also take exactly 15 days in the order C, A, B because the daily productivity remains constant throughout the period.

Explanation

In a linear race, when P beats Q by 120 metres or 30 seconds, it signifies that Q takes 30 seconds to cover the remaining 120 metres of the 1000-metre race distance.

The speed of Q is calculated as: Speed of Q = (Distance)/(Time) = (120 m)/(30 sec) = 4 m/s

The total time taken by Q to complete the 1 km (1000 m) race is: Time taken by Q = (Total Distance)/(Speed of Q) = (1000 m)/(4 m/s) = 250 sec

Since P finishes the race 30 seconds before Q, the time taken by P is: Time taken by P = 250 sec - 30 sec = 220 sec

Explanation

The expression is solved using the BODMAS rule, which dictates the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction. Starting with the innermost parentheses, we calculate 7 - 7 - 2 = -2. Substituting this into the curly braces gives {15 - (-2)} = 15 + 2 = 17. Moving to the square brackets, we have [18 - 17] = 1. The expression now simplifies to 14 ÷ 7 × 1 + 3 - 1. Following the order of operations, division is performed first, resulting in 2 × 1 + 3 - 1. Next, multiplication is performed to get 2 + 3 - 1. Finally, addition and subtraction are carried out from left to right, where 2 + 3 = 5, and 5 - 1 = 4.

Explanation

The algebraic identity a³ + b³ + c³ - 3abc = (a + b + c)((a + b + c)² - 3(ab + bc + ca)) provides the most direct path to the solution. Substituting the given values a³ + b³ + c³ - 3abc = 98 and a + b + c = 14 into this formula:

98 = 14(14² - 3(ab + bc + ca))

Dividing both sides by 14 simplifies the equation to: 7 = 196 - 3(ab + bc + ca)

Rearranging the terms to solve for the expression (ab + bc + ca): 3(ab + bc + ca) = 196 - 7 3(ab + bc + ca) = 189 ab + bc + ca = 189 / 3 = 63

Explanation

Given that A = 22.5°, we first find the value of 2A, which is 2 × 22.5° = 45°. We use the standard trigonometric values for 45°, where sin 45° = (1)/(√(2)), cos 45° = (1)/(√(2)), and tan 45° = 1.

Substituting these values into the given expression: (10√2sin 2A - 7√2cos 2A + 9 tan 2A) = 10√2((1)/(√(2))) - 7√2((1)/(√(2))) + 9(1) = 10 - 7 + 9 = 3 + 9 = 12

Explanation

The relationship between the sum of two numbers, the sum of their cubes, and their product is given by the algebraic identity a³ + b³ = (a + b)((a + b)² - 3ab). Given a³ + b³ = 9728 and a + b = 32, we substitute these values into the formula:

9728 = 32 × (32² - 3ab)

Dividing both sides by 32 gives:

304 = 1024 - 3ab

Rearranging the equation to solve for 3ab:

3ab = 1024 - 304 3ab = 720

Dividing by 3 yields the product of the two numbers:

ab = ⁷²⁰⁄₃ = 240

Explanation

The force F required to keep the car from skidding is determined by the joint variation formula F = k · W · v²{r}, where W is the weight, v is the speed, r is the radius, and k is the constant of variation.

Using the initial values provided to find k: 6125 = k · 2750 · 35²{550} 6125 = (k · 2750 · 1225)/(550) 6125 = k · 5 · 1225 6125 = k · 6125 k = 1

Now, applying this constant to the second set of conditions to find the required force F: F = 1 · 3600 · 50²{750} F = (3600 · 2500)/(750) F = (3600 · 10)/(3) F = 1200 · 10 F = 12000

Explanation

The provided data represents a cumulative frequency distribution of the less-than type. To determine the number of students who scored more than 40 marks, we identify the number of students in the marks range of 40 to 50. The cumulative frequency for students scoring less than 50 is 10, and the cumulative frequency for those scoring less than 40 is 8. The count of students scoring more than 40 is the difference between these two values: 10 - 8 = 2.

Explanation

In simple interest, the interest earned remains constant every year. The interest earned in one year is the difference between the amounts at the end of the fourth year and the third year.

Interest for 1 year = A₄ - A₃ = 1350 - 1200 = 150

The interest for 3 years is 150 × 3 = 450.

The principal sum (P) can be calculated by subtracting the interest earned over 3 years from the total amount after 3 years:

P = A₃ - I₃ = 1200 - 450 = 750

Explanation

For three numbers a, b, and c to be in continued proportion, where c is the third proportional to a and b, the relationship is defined as (a)/(b) = (b)/(c), which simplifies to b² = ac. Let the number to be subtracted be x. The two numbers then become 2 - x and 3 - x, with 4 being the third proportional.

Substituting these values into the formula b² = ac: (3 - x)² = 4(2 - x) 9 - 6x + x² = 8 - 4x x² - 2x + 1 = 0

This is a perfect square trinomial that can be factored as: (x - 1)² = 0 x - 1 = 0 x = 1

By subtracting 1 from both numbers, they become 1 and 2. The third proportional to 1 and 2 is indeed 4 because 1/2 = 2/4.

Explanation

Let the original price of the cell phone be ₹100. Ram bought the phone at 30% less than this price, which means his Cost Price (CP) is: CP = 100 - (30% of 100) = 100 - 30 = 70

Ram sold the phone at a profit of 50% on the price he paid. To find his Selling Price (SP), we apply the profit percentage to his CP: SP = CP + (50% of 70) = 70 + 35 = 105

The profit earned by Ram relative to the original price is the difference between his selling price and the original price: Profit over original price = SP - Original Price = 105 - 100 = 5

The percentage of profit earned on the original price is calculated as: Profit % = (Profit)/(Original Price) × 100 = ⁵⁄₁₀₀ × 100 = 5%

Explanation

The overall average marks per paper is calculated using the weighted average formula:

Average marks = (Σ (nᵢ × aᵢ))/(Σ nᵢ)

Where nᵢ is the number of papers in each subject and aᵢ is the average marks in that subject.

Total marks in Mathematics = 3 × 45 = 135 Total marks in English = 2 × 55 = 110 Total marks in Hindi = 2 × 60 = 120

Sum of total marks across all 7 papers = 135 + 110 + 120 = 365

Average marks per paper = ³⁶⁵⁄₇ = 52.1428...

Rounding to one decimal place, the average marks per paper is 52.1.

Explanation

The expression for tan(θ/2) is derived using the standard trigonometric identities for double angles. The identity for sine is sin θ = 2 sin(θ/2) cos(θ/2), while the identity derived from the cosine double-angle formula is 1 + cos θ = 2 cos²(θ/2). By forming a ratio of these two identities, we can simplify the expression as follows:

sin θ / (1 + cos θ) = (2 sin(θ/2) cos(θ/2)) / (2 cos²(θ/2))

After cancelling the common factor 2 cos(θ/2) from both the numerator and the denominator, the expression reduces to:

sin(θ/2) / cos(θ/2) = tan(θ/2)

This confirms that the function tan(θ/2) is equivalent to the ratio of the sine of the angle to the sum of one and the cosine of the angle.

Explanation

The volume of a cube is determined by the formula V = s³, where s is the side length of the cube. Given that the volume is 4096 cm³, the side length is found by calculating the cube root of the volume:

s = ³√(4096)

To solve for s, we can use prime factorization or recognize powers of 2. The number 4096 is 2¹² because 2¹⁰ = 1024 and 1024 × 4 = 4096. Applying the cube root to this power:

s = ³√(2¹²⁾ = 2¹²^/³ = 2⁴ = 16

Alternatively, we can verify the result through direct multiplication. Calculating 16 × 16 gives 256. Multiplying 256 by 16 results in 4096. This confirms that a side length of 16 cm corresponds to a volume of 4096 cm³.

Explanation

To evaluate the expression sin 600° cos 750° + sin 150° cos 240°, we simplify each trigonometric term using the periodic properties and quadrant rules. Since the period of sine and cosine is 360°, we can reduce large angles by subtracting multiples of 360°.

For the first term, sin 600° = sin(360° + 240°) = sin 240°. Applying the quadrant rule, sin 240° = sin(180° + 60°) = -sin 60° = -√3{2}.

For the second term, cos 750° = cos(2 × 360° + 30°) = cos 30° = √3{2}.

For the third term, sin 150° = sin(180° - 30°) = sin 30° = ½.

For the fourth term, cos 240° = cos(180° + 60°) = -cos 60° = -½.

Substituting these specific values back into the expression: (-√3{2})(√3{2}) + (½)(-½) = -¾ - ¼ = -⁴⁄₄ = -1

Explanation

To expand and simplify the expression (x - 5)² + (x + 3)² + 4x, use the algebraic identities (a - b)² = a² - 2ab + b² and (a + b)² = a² + 2ab + b².

Expanding the first term: (x - 5)² = x² - 2(x)(5) + 5² = x² - 10x + 25

Expanding the second term: (x + 3)² = x² + 2(x)(3) + 3² = x² + 6x + 9

Now, combine all parts of the expression: (x² - 10x + 25) + (x² + 6x + 9) + 4x

Group the like terms together: (x² + x²) + (-10x + 6x + 4x) + (25 + 9) 2x² + 0x + 34 2x² + 34

By factoring out the common factor of 2, the expression becomes 2(x² + 17).

Explanation

The total work is taken as the Least Common Multiple (LCM) of the time taken by each group, which is LCM(10, 12) = 60 units. The efficiency (work done per day) of the 15 men is ⁶⁰⁄₁₀ = 6 units per day, and the efficiency of the 15 women is ⁶⁰⁄₁₂ = 5 units per day. When both groups work together, their combined efficiency is 6 + 5 = 11 units per day. The total time required to complete the work is the total work divided by the combined efficiency: Time = ⁶⁰⁄₁₁ = 5⁵⁄₁₁ days

Explanation

The final selling price after successive discounts is calculated using the formula SP = MP × (1 - d/100)ⁿ, where MP is the marked price, d is the discount percentage, and n is the number of successive discounts. Given MP = ₹55,000, d = 10%, and n = 3, we substitute these values into the formula.

SP = 55,000 × (1 - 10/100)³ SP = 55,000 × (0.9)³ SP = 55,000 × 0.729

To find the final result, multiply 55,000 by 0.729: 55,000 × 0.729 = 55 × 729 = 40,095

Alternatively, applying a 10% discount three times is equivalent to multiplying the price by 0.9 each time. The first discount reduces the price to ₹49,500, the second to ₹44,550, and the third to ₹40,095.

Explanation

The time taken by Somesh alone can be calculated using the work rate formula 1/Tₜₒₜₐₗ = 1/T_Rₐₘₑₛₕ + 1/T_Sₒₘₑₛₕ. Given that Ramesh takes 24 days and together they take 8 days, we substitute these values into the equation: 1/8 = 1/24 + 1/T_Sₒₘₑₛₕ. Rearranging the formula to solve for Somesh's rate gives 1/T_Sₒₘₑₛₕ = 1/8 - 1/24.

By finding a common denominator, we get 1/T_Sₒₘₑₛₕ = 3/24 - 1/24 = 2/24. Simplifying the fraction results in 1/T_Sₒₘₑₛₕ = 1/12. Therefore, T_Sₒₘₑₛₕ = 12 days.

Alternatively, using the efficiency method, let the total work be the LCM of 24 and 8, which is 24 units. Ramesh's efficiency is 24 / 24 = 1 unit per day. The combined efficiency of Ramesh and Somesh is 24 / 8 = 3 units per day. Somesh's individual efficiency is the difference between the combined efficiency and Ramesh's efficiency, which is 3 - 1 = 2 units per day. The time taken by Somesh to complete the work alone is 24 / 2 = 12 days.

Explanation

The expression is solved using the BODMAS rule, which dictates the order of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, solve the innermost parentheses and curly braces: 8 - 11 = -3 4 × 2 - (-3) = 8 + 3 = 11

Next, simplify the expression inside the square brackets: 11 × 4 ÷ 4 of 11 According to BODMAS, the of operation is performed before multiplication and division: 4 of 11 = 44 The expression becomes 11 × 4 ÷ 44. Following the order of operations, division is performed before multiplication: 4 ÷ 44 = 1/11 11 × 1/11 = 1

Now substitute this back into the main expression: 120 ÷ 15 of 4 + 1 Perform the of operation first: 15 of 4 = 60 120 ÷ 60 + 1 Finally, perform division then addition: 2 + 1 = 3

Explanation

Loss is calculated as the difference between the Cost Price (CP) and the Selling Price (SP). In this case, the CP is ₹3,200 and the SP is ₹2,700.

Loss = CP - SP Loss = 3,200 - 2,700 = 500

The formula for loss percentage is: Loss Percentage = (Loss)/(CP) × 100

Substituting the known values into the formula: Loss Percentage = (500)/(3,200) × 100 Loss Percentage = ⁵⁰⁰⁄₃₂

To find the decimal value, divide 500 by 32: 500 ÷ 32 = 15.625

When expressed to two decimal places as per the provided options, the value is 15.62%.

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds.

Relative Speed = 7.5 km/h + 9 km/h = 16.5 km/h

To find the meeting time in minutes, convert the relative speed from km/h to m/min: Relative Speed in m/min = 16.5 × ¹⁰⁰⁰⁄₆₀ = ¹⁶⁵⁰⁰⁄₆₀ = 275 m/min

The time taken for them to meet for the first time is calculated by dividing the total distance (circumference) by the relative speed: Time = (Distance)/(Relative Speed) Time = ¹⁴⁵²⁄₂₇₅ minutes

Dividing both numerator and denominator by 11: Time = ¹³²⁄₂₅ = 5.28 minutes

Explanation

Let the number of boys be 11x and the number of girls be 15x based on the given ratio 11:15. The problem states that there are 200 more girls than boys, which leads to the equation 15x - 11x = 200. Simplifying this gives 4x = 200, which results in x = 50. To find the number of boys, we multiply the boy's ratio part by x: 11 × 50 = 550.

Explanation

To determine the required ratio, calculate the sum of sweets manufactured by factories P, Q, and R in the year 1998 and the sum of sweets manufactured by factories S, T, and U in the same year. According to the data table, the production values for the year 1998 are P = 178, Q = 240, and R = 150. Summing these values:

Sum(P, Q, R) = 178 + 240 + 150 = 568

Next, the production values for factories S, T, and U in 1998 are S = 130, T = 177, and U = 192. Summing these values:

Sum(S, T, U) = 130 + 177 + 192 = 499

The respective ratio of the total sweets manufactured by the first group to the second group is expressed as:

Ratio = 568 : 499

Explanation

Let the monthly income be 800 and the expenditure be 500 based on the given 8:5 ratio. Since Savings = Income - Expenditure, the initial savings are 800 - 500 = 300.

The change in income is 20% of 800, which is 0.20 × 800 = 160. The change in expenditure is 30% of 500, which is 0.30 × 500 = 150. Using the relation for changes Δ Savings = Δ Income - Δ Expenditure, the change in savings is 160 - 150 = 10.

A positive change indicates an increase. To find the percentage increase in savings, use the formula: Percentage Increase = (Δ Savings)/(Initial Savings) × 100 Substituting the values: ¹⁰⁄₃₀₀ × 100 = ¹⁰⁄₃% = 3⅓%

Explanation

The expression is simplified using the BODMAS rule, which establishes the order of operations as Brackets, Orders (Exponents), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

First, evaluate the first bracket (2 × 7 - 5 + 9 ÷ 3). Inside this bracket, prioritize multiplication and division: 2 × 7 = 14 9 ÷ 3 = 3 The expression inside the bracket becomes 14 - 5 + 3. Proceeding from left to right, 14 - 5 = 9, and 9 + 3 = 12.

Next, evaluate the second bracket and its exponent: (4 + 2)² = 6² = 36

Substitute these results back into the original expression: 12 ÷ 36 × 2

Now, perform division and multiplication from left to right: 12 ÷ 36 = ¹²⁄₃₆ = ⅓ ⅓ × 2 = ⅔

Explanation

Let the two natural numbers be x and y. We are given x³ - y³ = 6272 and x - y = 8. Using the algebraic identity x³ - y³ = (x - y)(x² + xy + y²), we substitute the known values: 6272 = 8(x² + xy + y²), which simplifies to x² + xy + y² = 784.

We also know (x - y)² = x² - 2xy + y² = 8² = 64. Subtracting this equation from x² + xy + y² = 784 gives 3xy = 720, which results in xy = 240. To find the individual numbers, we look for two factors of 240 that differ by 8. Since 20 × 12 = 240 and 20 - 12 = 8, the numbers are determined to be 20 and 12.

The sum of the cubes of these two numbers is calculated as x³ + y³ = 20³ + 12³. Evaluating the terms, we get 8000 + 1728 = 9728.

Explanation

The relationship between expenditure, price, and consumption is defined by the formula Expenditure = Price × Consumption. When the total expenditure remains constant, price and consumption are inversely proportional. If the price increases by a percentage r, the percentage decrease in consumption required to maintain the same expenditure is calculated using the following relation:

Percentage Decrease = ( (r)/(100 + r) ) × 100

Given that the price of petrol increases by r = 81, substitute this value into the formula:

Percentage Decrease = ( (81)/(100 + 81) ) × 100

Percentage Decrease = ⁸¹⁰⁰⁄₁₈₁

To calculate the value correct to two decimal places, divide 8100 by 181:

8100 ÷ 181 ≈ 44.75138...

Rounding this result to two decimal places gives 44.75%. This reduction in consumption compensates exactly for the 81% increase in price, ensuring the total money spent on petrol remains unchanged.

Explanation

The expression 8x³ + 27y³ is in the form a³ + b³, where a = 2x and b = 3y. Using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b), we substitute the given terms:

8x³ + 27y³ = (2x + 3y)³ - 3(2x)(3y)(2x + 3y)

Simplify the term 3(2x)(3y) to 18xy. The equation becomes:

8x³ + 27y³ = (2x + 3y)³ - 18xy(2x + 3y)

Substitute the given values 2x + 3y = 16 and xy = 9 into the formula:

8x³ + 27y³ = (16)³ - 18(9)(16)

Calculating the cubes and products: 16³ = 4096 18 × 9 × 16 = 162 × 16 = 2592

Perform the subtraction: 4096 - 2592 = 1504

Explanation

The value of the expression x³ + y³ is found using the algebraic identity x³ + y³ = (x + y)³ - 3xy(x + y). Given the values x + y = 25 and xy = 20, substitute them directly into the formula:

x³ + y³ = (25)³ - 3(20)(25)

Calculating the power and the product: 25³ = 15625 3 × 20 × 25 = 60 × 25 = 1500

Performing the subtraction: 15625 - 1500 = 14125

Explanation

To find the time taken for Sohan and Amit to meet for the first time, we first convert their speeds from km/h to m/s. Sohan's speed is 36 × ⁵⁄₁₈ = 10 m/s and Amit's speed is 54 × ⁵⁄₁₈ = 15 m/s. When two objects move in the same direction, their relative speed is the difference between their individual speeds. Here, the relative speed is 15 - 10 = 5 m/s. On a circular track, the time taken for them to meet for the first time at any point is given by the formula Time = (Track Length)/(Relative Speed). Substituting the given values, we get Time = ¹¹²⁵⁄₅ = 225 seconds.

Explanation

The smallest 5-digit number is 10000. To find the smallest 5-digit number exactly divisible by 526, we first divide 10000 by 526 to determine the remainder.

10000 = (526 × 19) + 6

The remainder is 6. The smallest 5-digit multiple is calculated by adding the difference between the divisor and the remainder to 10000 using the formula: Smallest multiple = 10000 + (526 - 6) Smallest multiple = 10000 + 520 = 10520

Alternatively, since 10000 divided by 526 results in a quotient of approximately 19.01, the smallest 5-digit number divisible by 526 must be the 20ᵗʰ multiple: 526 × 20 = 10520

The previous multiple, 526 × 19 = 9994, is a 4-digit number, confirming 10520 is the first 5-digit number in the sequence of multiples of 526.

Explanation

The distance D covered by two individuals traveling at different speeds v₁ and v₂ with a known time difference Δ t is given by the formula D = (v₁ × v₂)/(|v₁ - v₂|) × Δ t. In this problem, the speeds are v₁ = 4 km/h and v₂ = 6 km/h, and the time difference is half an hour, which is 0.5 hours.

Substituting these values into the formula: D = (4 × 6)/(6 - 4) × 0.5 D = ²⁴⁄₂ × 0.5 D = 12 × 0.5 D = 6 km

This can also be understood through the relationship between speed and time. Since the distance is constant, speed is inversely proportional to time. The ratio of the speeds is 4:6, or 2:3. Consequently, the ratio of the time taken by the two motorists is 3:2. The difference in the time units is 3 - 2 = 1 unit, which is given as 0.5 hours. The time taken by the faster motorist (at 6 km/h) is 2 units, which equals 2 × 0.5 = 1 hour. Thus, the distance is speed multiplied by time, 6 km/h × 1 hour = 6 km.

Explanation

Given that A + B = 90°, we have B = 90° - A. Applying the trigonometric complement identity, tan B = tan(90° - A) = cot A. Since sin A = ⅗, we can consider a right-angled triangle where the perpendicular is 3 and the hypotenuse is 5. Using the Pythagorean triplet (3, 4, 5) or the formula base = √(hypotenuse²⁾⁻ᵖᵉʳᵖᵉⁿᵈⁱᶜᵘˡᵃʳ^^^², the base is √(5²⁾⁻³^^^² = 4. The value of cot A is the ratio of the base to the perpendicular, which is ⁴⁄₃. Therefore, tan B = cot A = ⁴⁄₃.

Explanation

First, determine the simple interest earned in the first scenario using the formula I = A - P. The interest is 104.4 - 72 = 32.4. Using the simple interest formula I = (P · R · T)/(100), the rate of interest R is calculated as R = (32.4 · 100)/(72 · 3) = ³²⁴⁰⁄₂₁₆ = 15% per annum.

For the second scenario, apply the same rate to the new principal of 120 over 5 years. The interest is calculated as I = (120 · 15 · 5)/(100) = (120 · 75)/(100) = 90. The final amount is found by adding the interest to the principal, resulting in 120 + 90 = 210.

Explanation

The payment for a piece of work is divided in the ratio of the work done by each individual. Since Ashok, Anil, and Amar worked together for the same duration of 4 days, the total payment of 4,500 is distributed in the ratio of their efficiencies.

To find the efficiencies, let the total work be 24 units (the least common multiple of 8, 12, and 4). Ashok's efficiency = ²⁴⁄₈ = 3 units per day. Anil's efficiency = ²⁴⁄₁₂ = 2 units per day. The combined efficiency of Ashok, Anil, and Amar = ²⁴⁄₄ = 6 units per day.

The efficiency of Amar is determined by subtracting the efficiencies of Ashok and Anil from the combined efficiency: Amar's efficiency = 6 - (3 + 2) = 1 unit per day.

The ratio of their efficiencies is Ashok : Anil : Amar = 3 : 2 : 1. The sum of the ratio terms is 3 + 2 + 1 = 6. Amar's share is calculated as: Amar's share = ⅙ × 4500 = 750

Explanation

The sum of the two angles is given by the equation: ₁ + ₂ = (π)/(2)

To find the value of ₂, we rearrange the equation: ₂ = (π)/(2) - ₁

Substituting the value ₁ = (π)/(6) (which is the standard angle approximately equal to 0.5 or 1/2 in radian measure calculations): ₂ = (π)/(2) - (π)/(6)

To solve the subtraction, we convert the terms to a common denominator of 6: ₂ = (3π)/(6) - (π)/(6) ₂ = (2π)/(6) ₂ = (π)/(3)

Explanation

The simple interest is calculated using the formula SI = (P × R × T)/(100). In this problem, the principal P is 27,000, the annual interest rate R is 14⅔% = ⁴⁴⁄₃%, and the time T is 8 months. To express the time in years, we convert months to years as T = ⁸⁄₁₂ = ⅔ years.

Substituting these values into the formula: SI = (27,000 × 44)/(3) × ⅔{100} SI = (27,000 × 44 × 2)/(100 × 3 × 3) SI = (27,000 × 88)/(900) SI = 30 × 88 SI = 2,640

By simplifying the expression, the calculation yields a simple interest of 2,640.

Explanation

The BODMAS rule dictates the order of operations as Division, then Addition and Subtraction. First, perform the division: 80.40 ÷ 20 = 4.02. Substituting this value back into the expression gives 4.02 - (- 4.02) + 2.06.

The subtraction of a negative number is equivalent to addition, so - (- 4.02) becomes + 4.02. The expression now simplifies to 4.02 + 4.02 + 2.06. Adding the first two terms results in 4.02 + 4.02 = 8.04. Finally, adding the remaining term gives 8.04 + 2.06 = 10.10.

Explanation

Given the equation 3cot A = ⁷⁄₃, we can determine that cot A = ⁷⁄₉. To find the value of the expression (3cos A + 2sin A)/(3cos A - 2sin A), we divide both the numerator and the denominator by sin A. This simplifies the expression to (3cot A + 2)/(3cot A - 2) because (cos A)/(sin A) = cot A.

By substituting the value of 3cot A from the given equation into this simplified form:

(7)/(3) + 2{⁷⁄₃ - 2}

To solve this, we find a common denominator for the terms in the numerator and the denominator:

(7 + 6)/(3){(7 - 6)/(3)} = (13/3)/(1/3) = 13

Alternatively, using the ratio cot A = (cos A)/(sin A) = ⁷⁄₉, we can substitute cos A = 7k and sin A = 9k directly into the expression:

(3(7k) + 2(9k))/(3(7k) - 2(9k)) = (21k + 18k)/(21k - 18k) = ³⁹ᵏ⁄₃ₖ = 13

Explanation

To find the number of boys needed, let the total work be the Least Common Multiple of the days taken by the individuals, which is LCM(3, 5, 15) = 15 units. From this, we calculate the daily work (efficiency) of each person: Efficiency of 1 man = 15 / 3 = 5 units/day Efficiency of 1 woman = 15 / 5 = 3 units/day Efficiency of 1 boy = 15 / 15 = 1 unit/day

The job must be completed in 1/5 of a day. Using the formula Work = Efficiency × Time, the required total efficiency of the group must be: Total Efficiency = Work / Time = 15 / (1/5) = 75 units/day

Let x be the number of boys assisting 1 man and 1 woman. The sum of their efficiencies must equal the required total efficiency: 1(5) + 1(3) + x(1) = 75 8 + x = 75 x = 67

Explanation

The final selling price (SP) of an article after two successive discounts is given by the formula: SP = L × (1 - d₁/100) × (1 - d₂/100) where L is the labelled price, d₁ is the first discount percentage, and d₂ is the second discount percentage.

Substituting the given values SP = 792, d₁ = 25, and d₂ = 12: 792 = L × (1 - 25/100) × (1 - 12/100) 792 = L × (75/100) × (88/100) 792 = L × (3/4) × (22/25) 792 = L × (66/100)

To find L, rearrange the equation: L = (792 × 100) / 66 L = 12 × 100 L = 1200

Alternatively, using the concept of effective multipliers, the first discount reduces the price to 0.75 of its value, and the second discount reduces it further to 0.88 of that discounted price. The total reduction is 0.75 × 0.88 = 0.66. Dividing the final price of 792 by the total multiplier 0.66 yields the original labelled price of 1200.

Explanation

To find the smallest ratio among 7/9, 13/18, 5/6, and 11/12, we convert them to a common denominator. The least common multiple of the denominators 9, 18, 6, and 12 is 36. Multiplying the numerators accordingly, the ratios become 28/36, 26/36, 30/36, and 33/36. Comparing these values, 26/36 is the lowest, which corresponds to the ratio 13/18. Alternatively, converting to decimals shows that 13/18 is approximately 0.722, while 7/9 is 0.777, 5/6 is 0.833, and 11/12 is 0.916, confirming 13/18 as the minimum value.

Explanation

The curved surface area of a hemisphere is 2π r². Since the bowl is painted on both the inside and the outside and the thickness is negligible, the total area to be painted is 2 × 2π r² = 4π r².

Using the radius r = 7 cm, the total surface area is: A = 4 × ²²⁄₇ × 7 × 7 A = 4 × 22 × 7 = 616 cm²

The cost of painting is given as Rs. 40 per 20 cm². This corresponds to a rate of ⁴⁰⁄₂₀ = 2 Rs per cm². The total cost is calculated by multiplying the total area by the rate: Total Cost = 616 × 2 = 1232