SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

The marked price (MP) is determined by increasing the cost price x by 120%, which gives MP = x × (1 + 1.20) = 2.2x. A discount of 20% is then applied to this marked price to find the selling price (SP). The relation is SP = MP × (1 - 0.20) = 2.2x × 0.8.

Given the selling price is Rs 8800, we can solve for x using the equation: 8800 = 1.76x x = (8800)/(1.76) x = ⁸⁸⁰⁰⁰⁰⁄₁₇₆ x = 5000

Explanation

Let the number of 5-rupee, 50-paise, and 10-paise coins be x, 5x, and 11x respectively, following the given ratio 1:5:11. The total value of the bag is calculated by multiplying the quantity of each coin by its denomination value in rupees.

The value equation is: 5(x) + 0.50(5x) + 0.10(11x) = 43

Simplifying the terms: 5x + 2.5x + 1.1x = 43 8.6x = 43

Solving for x: x = (43)/(8.6) = 5

The total number of 50-paise coins is represented by the term 5x. Substituting the value of x: Number of 50-paise coins = 5 × 5 = 25

Explanation

To find the runs already scored, multiply the number of overs completed by the current run rate: 45 × 5.8 = 261 runs. The target score is 295 runs, so the number of runs remaining to be scored is 295 - 261 = 34 runs. Since there are 50 - 45 = 5 overs left in the innings, the required run rate is calculated by dividing the remaining runs by the remaining overs: 34 / 5 = 6.8 runs per over.

Explanation

Let the initial quantity of milk purchased be 5 litres. The cost price of milk is Rs 24 per litre, so the total cost price CP = 5 × 24 = Rs 120.

The milkman adds 1/5 of water to the milk, which is 1/5 × 5 = 1 litre of water. The total volume of the mixture becomes 5 + 1 = 6 litres. Since water is free, the cost price remains the same.

The mixture is sold at Rs 32 per litre. Therefore, the total selling price SP = 6 × 32 = Rs 192.

The gain is calculated as SP - CP = 192 - 120 = Rs 72.

The gain percentage is given by the formula: Gain % = (Gain)/(CP) × 100 Gain % = ⁷²⁄₁₂₀ × 100 Gain % = ⁶⁄₁₀ × 100 = 60%

Explanation

Let the original number be x. The problem states that when the number is increased by 105, it becomes 135% of its original value. Since the original value is 100%, the increase of 105 represents the additional 35% (135% - 100%).

This relationship can be represented by the equation: 35% of x = 105

Expressing the percentage as a fraction: ³⁵⁄₁₀₀ × x = 105

To solve for x, rearrange the equation: x = (105 × 100)/(35)

Dividing 105 by 35 gives 3: x = 3 × 100 = 300

Explanation

When two objects move towards each other, their relative speed is calculated by adding their individual speeds. In this case, the relative speed of cyclists P and Q is 20 km/hr + 16 km/hr = 36 km/hr. The time taken to meet is given as 40 minutes, which needs to be converted into hours to match the speed units: 40 / 60 = 2/3 hours. The initial distance between the two cyclists is the product of their relative speed and the time elapsed until they meet. Applying the formula Distance = Relative Speed × Time, we get 36 × (2/3) = 24 km.

Explanation

The amount A in compound interest is calculated using the formula A = P(1 + ʳ⁄₁₀₀)ⁿ, where P is the principal, r is the annual interest rate, and n is the time in years. Given that A = 19965, r = 10, and n = 3, the equation is:

19965 = P(1 + ¹⁰⁄₁₀₀)³ 19965 = P(¹¹⁄₁₀)³ 19965 = P × ¹³³¹⁄₁₀₀₀

To isolate P, we rearrange the formula: P = (19965 × 1000)/(1331)

Dividing 19965 by 1331 gives 15. Multiplying this result by 1000 yields the principal amount: P = 15 × 1000 = 15000

Explanation

The given equation is ⁸ˣ⁄₃ + (7(5 - 2x)/(3)){2} = ½. To solve for x, multiply the entire equation by the least common multiple of the denominators 3 and 2, which is 6, to clear the fractions: 6 · (⁸ˣ⁄₃) + 6 · [(7(5 - 2x)/(3)){2}] = 6 · (½) 2(8x) + 3 · 7(5 - ²ˣ⁄₃) = 3 16x + 21(5 - ²ˣ⁄₃) = 3

Now, distribute 21 into the parentheses: 16x + 105 - ⁴²ˣ⁄₃ = 3 16x + 105 - 14x = 3

Combine the terms containing x: 2x + 105 = 3

Isolate the variable x by subtracting 105 from both sides: 2x = 3 - 105 2x = -102

Divide both sides by 2: x = ⁻¹⁰²⁄₂ x = -51

Explanation

The value of the expression a³ - b³ is found using the algebraic identity a³ - b³ = (a - b)³ + 3ab(a - b). By substituting the given values a - b = -1 and ab = 6 into this formula, the calculation is as follows:

a³ - b³ = (-1)³ + 3(6)(-1) a³ - b³ = -1 + 18(-1) a³ - b³ = -1 - 18 a³ - b³ = -19

Alternatively, using the identity a³ - b³ = (a - b)(a² + ab + b²), we first determine a² + b² using (a - b)² = a² + b² - 2ab, which gives a² + b² = (-1)² + 2(6) = 13. Then, a³ - b³ = (-1)(13 + 6) = -19.

Explanation

Let the fraction be x. According to the given condition, the sum of the fraction and 3 times its reciprocal is ³¹⁄₆. This relationship can be expressed by the equation x + 3 ( ¹⁄ₓ ) = ³¹⁄₆. By substituting the value x = ⁹⁄₂ into the equation, we calculate the sum as ⁹⁄₂ + 3 × ²⁄₉. Simplifying the second term gives ⅔, so the expression becomes ⁹⁄₂ + ⅔. To add these fractions, we use a common denominator of 6, resulting in (27 + 4)/(6) = ³¹⁄₆. This matches the sum provided in the question. Alternatively, the equation x + ³⁄ₓ = ³¹⁄₆ can be rewritten as 6x² - 31x + 18 = 0. Solving this quadratic equation using the factorization method, we find 6x² - 27x - 4x + 18 = 0, which simplifies to (3x - 2)(2x - 9) = 0. This yields two possible values for x, which are ⅔ and ⁹⁄₂. Of these, ⁹⁄₂ is the value provided in the options.

Explanation

The n-th term of an arithmetic progression is defined by the formula aₙ = a + (n-1)d, where a is the first term and d is the common difference.

The common difference d can be calculated using the values and positions of any two terms: d = (a₈ - a₂)/(8 - 2) d = (-1 - 17)/(6) = ⁻¹⁸⁄₆ = -3

To find the 14th term, we can start from the 8th term and add the common difference for the number of steps between the 8th and 14th positions: a₁₄ = a₈ + (14 - 8)d a₁₄ = -1 + 6(-3) a₁₄ = -1 - 18 = -19

Explanation

When reflecting a point (x, y) across a horizontal line y = k, the x-coordinate remains unchanged while the y-coordinate is transformed. The line y = k acts as the midpoint between the original y-coordinate and the reflected y-coordinate y'. This relationship is expressed by the formula y' = 2k - y.

Given the point (2, 3) and the line y = 4, we have x = 2, y = 3, and k = 4. The x-coordinate of the reflected point remains 2. Applying the formula for the y-coordinate:

y' = 2(4) - 3 y' = 8 - 3 y' = 5

The coordinates of the reflected point are (2, 5).

Explanation

The midpoint M(xₘ, yₘ) of a line segment joining two points A(x₁, y₁) and B(x₂, y₂) is calculated using the formula xₘ = (x₁ + x₂)/(2) and yₘ = (y₁ + y₂)/(2).

In this problem, point P(-2, 5) is the midpoint of segment AB, where A is (-5, y) and B is (x, 3). To find the value of x, we only need to use the formula for the x-coordinate of the midpoint:

-2 = (-5 + x)/(2)

Multiplying both sides by 2 gives:

-4 = -5 + x

Rearranging the equation to solve for x:

x = -4 + 5 x = 1

Explanation

The equation of a line in slope-intercept form is defined as y = mx + c, where m represents the slope and c represents the y-intercept. Substituting the given values m = -1/3 and c = 6 into this formula results in the following equation:

y = -⅓x + 6

To eliminate the fraction and convert the equation into standard form, multiply all terms by 3:

3y = -x + 18

Adding x to both sides of the equation to bring the variables to one side yields the linear equation:

x + 3y = 18

Explanation

In a right-angled triangle ABC where the right angle is at B and BD is the altitude to the hypotenuse AC, the property of similar triangles states that the square of a leg is equal to the product of its adjacent segment on the hypotenuse and the entire hypotenuse. Thus, AB² = AD × AC.

First, calculate the total length of the hypotenuse AC by adding the segments AD and DC: AC = AD + DC = 4 + 12 = 16 cm

Substitute the values of AD and AC into the formula: AB² = 4 × 16 = 64 AB = √64 = 8 cm

This relationship is derived from the fact that △ ABD is similar to △ ABC. Since both triangles share ∠A and contain a right angle (∠ADB and ∠ABC respectively), they are similar by the Angle-Angle similarity criterion. The proportionality of their sides leads to the ratio AB/AC = AD/AB, which confirms AB² = AD × AC. Substituting the given values AD = 4 and AC = 16 directly yields the result.

Explanation

To find the value of the expression sin 60° + ½ csc 45°, we substitute the standard values of the trigonometric functions. The value of sin 60° is √3{2}. The value of csc 45° is the reciprocal of sin 45°, which is (1)/(1/√(2)) = √2. Substituting these into the given expression, we have:

sin 60° + ½ csc 45° = √3{2} + ½(√2)

sin 60° + ½ csc 45° = √3{2} + √2{2}

By combining the fractions over a common denominator of 2, the result is:

√3 + √2{2}

Explanation

In a right-angled triangle Δ LMN where ∠ M = 90°, the side NL is the hypotenuse. Since ∠ N = 45°, the third angle ∠ L must also be 45° because the sum of angles in a triangle is 180°. This identifies Δ LMN as an isosceles right triangle.

In any 45°-45°-90° triangle, the sides are in the ratio 1 : 1 : √2. The relationship between a leg and the hypotenuse is defined by: Leg = (Hypotenuse)/(√(2))

Substituting the given value of the hypotenuse NL = 6√2 cm: MN = 6√2{√2} = 6 cm

Alternatively, using the trigonometric ratio for cosine: cos(N) = (Base)/(Hypotenuse) = (MN)/(NL) cos(45°) = (MN)/(6√(2)) (1)/(√(2)) = (MN)/(6√(2)) MN = 6√2{√2} = 6 cm

Explanation

Given cos θ = ³⁵⁄₃₇, we identify the base b = 35 and the hypotenuse h = 37 of a right-angled triangle. According to the Pythagorean theorem, the perpendicular p is calculated as:

p = √(h² - b²) p = √(37² - 35²) p = √((37 - 35)(37 + 35)) p = √(2 × 72) = √(144) = 12

The cosecant function is defined as the ratio of the hypotenuse to the perpendicular:

csc θ = ʰ⁄ₚ csc θ = ³⁷⁄₁₂

Explanation

Based on the line graph for the net exports (in million USD) of the country from 2011 to 2016, the values are as follows:

2011: 10 2012: 20 2013: -10 2014: -20 2015: 30 2016: 30

To find the number of years where net exports were more than the previous year, we compare each year's value with its predecessor:

In 2012, the net exports (20) were more than in 2011 (10). In 2013, the net exports (-10) were less than in 2012 (20). In 2014, the net exports (-20) were less than in 2013 (-10). In 2015, the net exports (30) were more than in 2014 (-20). In 2016, the net exports (30) were equal to those in 2015 (30), not more.

The net exports increased compared to the previous year in only 2 years: 2012 and 2015.

Explanation

The relationship between imports and exports is defined by the following ratio: Ratio = (Import)/(Export)

From the data provided for the year 2013, the ratio of imports to exports is approximately 1.11. Given that the exports in 2013 were 90 million USD, the value of imports is calculated by rearranging the formula: Import = Ratio × Export Import = 1.11 × 90 = 99.9 million USD

Rounding this value to the nearest integer, the imports for the year 2013 were 100 million USD.

Explanation

Net export is defined as the difference between total exports and total imports for a specific year. Based on the data provided for the years 2016 and 2012:

Net Exports in 2016 = Exports₂₀₁₆ - Imports₂₀₁₆ = 140 - 110 = 30 Net Exports in 2012 = Exports₂₀₁₂ - Imports₂₀₁₂ = 110 - 90 = 20

The ratio of net exports in 2016 to that in 2012 is calculated by dividing the net export values of the respective years:

Ratio = ³⁰⁄₂₀ = ³⁄₂ = 3 : 2

Explanation

Cumulative net exports are calculated by summing the net export values for each year within the specified period. Based on the data for the years 2012 to 2015, the net export values (in million USD) are:

2012: 20 2013: -10 2014: -20 2015: 30

The cumulative net exports from the beginning of 2012 to the end of 2015 is the sum of these annual values: Cumulative Net Exports = 20 + (-10) + (-20) + 30 Cumulative Net Exports = 20 - 10 - 20 + 30 Cumulative Net Exports = 10 - 20 + 30 Cumulative Net Exports = -10 + 30 = 20 million USD

Explanation

To determine the least number that must be subtracted from 3401 to make the result divisible by 11, we calculate the remainder when 3401 is divided by 11. Using the division algorithm, we find that 3401 = (11 × 309) + 2. Alternatively, the divisibility rule for 11 states that the difference between the sum of digits at odd positions and the sum of digits at even positions identifies the remainder. For 3401, the sum of digits at odd positions (1st and 3rd) is 1 + 4 = 5, and the sum of digits at even positions (2nd and 4th) is 0 + 3 = 3. The difference is 5 - 3 = 2. Since the remainder is 2, subtracting 2 from 3401 yields 3399, which is perfectly divisible by 11.

Explanation

The efficiency of a workman is inversely proportional to the time taken to complete a task. Given that M is thrice as good as N, their efficiency ratio is M:N = 3:1. This means that for every 1 unit of work N completes, M completes 3 units in the same amount of time.

To find the total work, we multiply their combined efficiency by the time they take to finish the work together. Let the efficiency of N be 1 unit per day and the efficiency of M be 3 units per day. Their combined efficiency is 3 + 1 = 4 units per day. Since they finish the work in 30 days, the total work is 4 × 30 = 120 units.

The time taken by M to finish the work alone is the total work divided by M's efficiency. Time = ¹²⁰⁄₃ = 40 days.

Explanation

A regular hexagon can be divided into six identical equilateral triangles. The area of one equilateral triangle with side s is √3{4}s². Therefore, the area of a regular hexagon is given by the formula A = 6 × √3{4}s², which simplifies to A = 3√3{2}s².

Given the side length s = 14 cm, we substitute this value into the formula: A = 3√3{2} × (14)² A = 3√3{2} × 196 A = 3√3 × 98 A = 294√3

The total area of the regular hexagon is 294√3 sq cm.

Explanation

The effective discount percentage is calculated by dividing the number of free items by the total number of items included in the offer.

Effective Discount % = (Number of Free Items)/(Total Number of Items) × 100

In this scenario, 2 T-shirts are offered free on a purchase of 5 T-shirts. This implies that the total quantity involved in the transaction is 5 T-shirts, out of which 2 are provided free of cost. Consequently, the customer pays for only 5 - 2 = 3 T-shirts but receives the full set of 5.

Given: Number of Free Items = 2 Total Number of Items = 5

Substituting these values into the formula: Discount % = ⅖ × 100 Discount % = 0.4 × 100 = 40%

This value represents the actual percentage reduction in price for each T-shirt received as part of the deal.

Explanation

Let the present ages of R and S be 11x and 17x respectively. According to the problem, 11 years ago, the ratio of their ages was 11:20. This relationship can be written as:

(11x - 11)/(17x - 11) = ¹¹⁄₂₀

Factoring out 11 from the numerator of the left side:

(11(x - 1))/(17x - 11) = ¹¹⁄₂₀

Dividing both sides by 11 simplifies the equation to:

(x - 1)/(17x - 11) = ¹⁄₂₀

Cross-multiplying the terms:

20(x - 1) = 17x - 11 20x - 20 = 17x - 11 20x - 17x = 20 - 11 3x = 9 x = 3

R's present age is defined as 11x. Substituting x = 3:

R = 11 × 3 = 33 years

Explanation

To find the correct average, calculate the net change in the total marks and divide it by the number of students. The formula for the correct average is:

Correct Average = Original Average + (Correct Value - Incorrect Value)/(Total Students)

Substituting the given values: Correct Average = 25 + (37 - 73)/(40) Correct Average = 25 - ³⁶⁄₄₀ Correct Average = 25 - 0.9 = 24.1

The error of entering 73 instead of 37 resulted in the total sum being 36 marks higher than it should have been. When this excess of 36 is distributed across the 40 students, the average decreases by exactly 36 / 40, which is 0.9. Subtracting this correction from the initial average of 25 yields the accurate value.

Explanation

The cost price to the wholesaler can be calculated using the successive profit formula: SP = CP × (1 + (r₁)/(100)) × (1 + (r₂)/(100))

Given the final price SP = 4158, the wholesaler profit r₁ = 5%, and the retailer profit r₂ = 10%, the equation is: 4158 = CP × ¹⁰⁵⁄₁₀₀ × ¹¹⁰⁄₁₀₀ 4158 = CP × ²¹⁄₂₀ × ¹¹⁄₁₀ 4158 = CP × ²³¹⁄₂₀₀

Rearranging to solve for CP: CP = (4158 × 200)/(231) CP = 18 × 200 CP = 3600

The calculation is simplified by first dividing 4158 by 11 to get 378, then dividing 378 by 21 to get 18, and finally multiplying by the denominator product of 200. This confirms the initial cost to the wholesaler was Rs. 3600.

Explanation

Let the original number be 100%. When the number is increased by 84, it becomes 107% of itself. This indicates that the numerical increase of 84 is equivalent to a percentage increase of 7% (calculated as 107% - 100%).

Using the unitary method: 7% of the number = 84 1% of the number = ⁸⁴⁄₇ = 12 100% of the number = 12 × 100 = 1200

Algebraically, let the number be x. The condition states x + 84 = 1.07x. Rearranging the equation to solve for x gives 1.07x - x = 84, which simplifies to 0.07x = 84. Thus, x = (84)/(0.07) = ⁸⁴⁰⁰⁄₇ = 1200.

Explanation

When a body covers the same distance at two different speeds, say x km/hr and y km/hr, the average speed for the entire journey is given by the harmonic mean formula:

Average Speed = (2xy)/(x + y)

Given the speed while going is x = 30 km/hr and the speed while returning is y = 60 km/hr, substitute these values into the formula:

Average Speed = (2 · 30 · 60)/(30 + 60)

Average Speed = ³⁶⁰⁰⁄₉₀

Average Speed = 40 km/hr

This formula is derived from the basic definition of average speed as total distance divided by total time. If the distance for one way is d, the total distance is 2d, and the total time is d/30 + d/60, which simplifies to the harmonic mean of the two speeds.

Explanation

Simple Interest (SI) is calculated using the formula SI = (P × R × T)/(100), where P represents the principal amount, R is the annual interest rate, and T is the time in years. Given that R = 6.25% and T = 4 years, the cumulative interest percentage over the period is 6.25 × 4 = 25%. Since the total simple interest fetched is Rs. 3200, the equation is:

3200 = (P × 25)/(100)

Simplifying the fraction, 25/100 equals 1/4. The equation becomes 3200 = (P)/(4). Solving for P, we get P = 3200 × 4 = 12800.

Explanation

The given linear equation is x/2 - [4(15/2 - x/3)]/3 = -x/18. To solve for x, we first simplify the term inside the square brackets. The expression 15/2 - x/3 is equivalent to (45 - 2x)/6. Multiplying this by 4 and then dividing by 3 results in [4(45 - 2x)/6] / 3, which simplifies to (90 - 4x)/9. Substituting this back into the original equation, we get x/2 - (90 - 4x)/9 = -x/18. To eliminate the denominators, we multiply the entire equation by the least common multiple of 2, 9, and 18, which is 18. This gives 9x - 2(90 - 4x) = -x. Expanding the bracket, we have 9x - 180 + 8x = -x. Combining the x terms on the left side gives 17x - 180 = -x. Adding x to both sides and adding 180 to both sides results in 18x = 180. Dividing by 18, we obtain the value x = 10.

Explanation

The value of ab is determined using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Substituting the provided values a³ + b³ = 152 and a + b = 8 into the equation results in 152 = 8³ - 3ab(8). Calculating the cube of 8 gives 512, so the equation becomes 152 = 512 - 24ab. Rearranging the equation to isolate the term with ab gives 24ab = 512 - 152, which simplifies to 24ab = 360. Solving for ab by dividing 360 by 24 results in ab = 15.

Explanation

Let the fraction be x. According to the problem, the difference between the fraction and its reciprocal is 9/20, which gives the equation:

x - ¹⁄ₓ = ⁹⁄₂₀

To find the value of x efficiently, substitute the value 5/4 into the equation. The reciprocal of 5/4 is 4/5. Evaluating the difference:

⁵⁄₄ - ⅘ = (25 - 16)/(20) = ⁹⁄₂₀

The calculation confirms that 5/4 satisfies the condition. Algebraically, multiplying the original equation by 20x results in the quadratic equation 20x² - 9x - 20 = 0. Factoring this as (4x - 5)(5x + 4) = 0 provides the positive root x = 5/4.

Explanation

The sum of the first n terms of an arithmetic progression is calculated using the formula Sₙ = ⁿ⁄₂(a + l), where n is the number of terms, a is the first term, and l is the last term. Given n = 9, a = 7, and l = 55, the sum is:

S₉ = ⁹⁄₂(7 + 55)

S₉ = ⁹⁄₂(62)

S₉ = 9 × 31 = 279

Explanation

The reflection of a point (x, y) in a vertical line x = k is given by the formula (2k - x, y). In this case, the point is (5, -2) and the line of reflection is x = -1, meaning k = -1. Substituting these values into the formula, the x-coordinate of the reflected point is 2(-1) - 5, which simplifies to -2 - 5 = -7. The y-coordinate remains unchanged at -2. Consequently, the coordinates of the reflected point are (-7, -2).

Explanation

The coordinates of point A, which divides the line segment joining B (x₁, y₁) = (6, 1) and C (x₂, y₂) = (⁷⁄₂, 6) in the ratio m:n = 4:1, are determined using the section formula:

x = (mx₂ + nx₁)/(m + n) y = (my₂ + ny₁)/(m + n)

Substituting the given values into the formula for x: x = (4 × (7)/(2)) + 1 × 6{4 + 1} = (14 + 6)/(5) = ²⁰⁄₅ = 4

Substituting the given values into the formula for y: y = (4 × 6 + 1 × 1)/(4 + 1) = (24 + 1)/(5) = ²⁵⁄₅ = 5

The calculation yields the coordinates (4, 5) for point A.

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). Using the given coordinates (5, -1) and (4, -4), the calculation is m = (-4 - (-1))/(4 - 5) = (-4 + 1)/(-1) = ⁻³⁄₋₁ = 3. By definition, parallel lines have equal slopes, meaning any line parallel to the line passing through these points must also have a slope of 3.

Explanation

The trigonometric values for the angles in the expression are sec 30° = (2)/(√(3)) and tan 60° = √3. Substituting these values into the given expression ½ sec 30° + √2 tan 60° results in:

½ ( (2)/(√(3)) ) + √2 ( √3 )

By canceling the common factor of 2 in the first term, the expression simplifies to (1)/(√(3)) + √6. To combine these terms into a single fraction, a common denominator of √3 is used. Multiplying √6 by √3{√3} gives:

(1)/(√(3)) + √6 · √3{√3} = (1)/(√(3)) + √(18){√3}

Since √(18) can be simplified as √(9 · 2) = 3√2, the final simplified form of the expression is:

1 + 3√2{√3}

Explanation

In a right-angled triangle Δ DEF where ∠ E = 90° and ∠ D = 45°, the third angle ∠ F is determined by the angle sum property: ∠ D + ∠ E + ∠ F = 180°. Substituting the known values gives 45° + 90° + ∠ F = 180°, which simplifies to ∠ F = 45°.

The objective is to find the value of csc F × cot D, which is csc 45° × cot 45°. According to trigonometric ratios for standard angles, csc 45° = √2 and cot 45° = 1. Multiplying these values together results in √2 × 1 = √2.

Explanation

The secant function is defined as sec θ = Hypotenuse/Base. Given sec θ = 25/24, we can identify the Hypotenuse (H) as 25 and the Base (B) as 24. To find the value of sin θ, which is the ratio of the Perpendicular (P) to the Hypotenuse, we use the Pythagorean theorem:

P = √(H² - B²) P = √(25² - 24²) P = √(625 - 576) P = √49 = 7

With the Perpendicular determined to be 7, we apply the definition of the sine function:

sin θ = P/H = 7/25

This result is consistent with the properties of the Pythagorean triplet (7, 24, 25), where 25 is the longest side.

Explanation

The divisibility rule for 9 states that a number is divisible by 9 if the sum of its digits is divisible by 9. For the number 213, the sum of the digits is 2 + 1 + 3 = 6. The smallest multiple of 9 that is greater than 6 is 9 itself. The difference between this target sum and the current sum is 9 - 6 = 3. Therefore, 3 is the least number that must be added to 213 to make the sum of the digits equal to 9, ensuring the resulting number 216 is divisible by 9. Alternatively, dividing 213 by 9 gives a remainder of 6. The number to be added to reach the next multiple is the divisor minus the remainder, which is 9 - 6 = 3.

Explanation

The individual efficiencies of A and B are ¹⁄₁₂ and ¹⁄₂₄ of the work per day, respectively. Their combined efficiency when working together is:

Combined Efficiency = ¹⁄₁₂ + ¹⁄₂₄ = (2 + 1)/(24) = ³⁄₂₄ = ⅛

In 4 days, the amount of work completed by A and B together is:

Work Done = 4 × ⅛ = ½

The fraction of work remaining is calculated by subtracting the work done from the total work:

Work Left = 1 - ½ = ½

Explanation

The area A of an equilateral triangle with side length s is calculated using the formula A = √3{4}s². Given that the side length s is 6 cm, the calculation is A = √3{4} × 6². Substituting 36 for 6², the expression becomes A = √3{4} × 36. Dividing 36 by 4 gives 9, resulting in an area of 9√3 sq cm.

Explanation

For two successive discounts of x% and y%, the effective discount D is given by the formula D = x + y - ˣʸ⁄₁₀₀.

Substituting the given values x = 20 and y = 25 into the formula:

D = 20 + 25 - (20 × 25)/(100)

D = 45 - ⁵⁰⁰⁄₁₀₀

D = 45 - 5 = 40%

Alternatively, if the initial price is 100, a 20% discount reduces the price to 80. Applying a 25% discount on the remaining 80 results in a further reduction of 0.25 × 80 = 20. The final price is 80 - 20 = 60, which represents a total discount of 40 from the original price of 100.

Explanation

The total profit of Rs 12,400 is shared among partners A, B, and C in the ratio 5:7:8. To find the specific amount received by partner B, we first calculate the sum of the ratios: 5 + 7 + 8 = 20.

The share of any partner is calculated as: Partner's Share = (Ratio of the Partner)/(Sum of Ratios) × Total Profit

Substituting the given values for partner B: B's Share = ⁷⁄₂₀ × 12,400

Simplifying the expression: B's Share = 7 × (12,400)/(20) B's Share = 7 × 620 B's Share = 4,340

Explanation

The weight of Q is calculated by finding the total sum of weights and using the sums of the given pairs. According to the average formula, Sum = Average × Number of observations.

Sum of weights of P, Q, and R = 47 × 3 = 141 kg Sum of weights of P and Q = 32.5 × 2 = 65 kg Sum of weights of Q and R = 48.5 × 2 = 97 kg

To find Q, we add the sums of the two pairs and subtract the total sum of all three individuals. This is because adding the pairs (P + Q) and (Q + R) counts Q twice: Q = (P + Q) + (Q + R) - (P + Q + R) Q = 65 + 97 - 141 Q = 162 - 141 = 21 kg

Explanation

To find the profit percentage, let the selling price (SP) of one item be x. The total selling price for 5 items is 5x. According to the problem, the profit earned is equal to the selling price of 1 item, so Profit = x.

The cost price (CP) of the 5 items is calculated using the formula CP = SP - Profit. Substituting the values, we get CP = 5x - x = 4x.

The profit percentage is determined by the formula Profit % = ((Profit)/(CP)) × 100. Substituting the values into this formula gives Profit % = (ˣ⁄₄ₓ) × 100 = ¼ × 100 = 25%.

Explanation

To find the value, we convert the percentages into their fractional or decimal equivalents and multiply them by the base number. The term percent means per hundred, so the expression is calculated as follows:

Value = (20/100) × (500/100) × 50

By simplifying the fractions:

20/100 = 0.2 500/100 = 5

Now, substitute these values back into the equation:

Value = 0.2 × 5 × 50

First, multiply 0.2 and 5:

0.2 × 5 = 1.0

Finally, multiply by the remaining factor:

1 × 50 = 50