SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

Expand the parentheses by multiplying through by 1/3: ⅓ · ¹²ˣ⁄₅ - ⅓ · ½ + ⁶⁄₅ = ⁷⁄₆ ⁴ˣ⁄₅ - ⅙ + ⁶⁄₅ = ⁷⁄₆

Group the numerical terms on the right side of the equation to isolate the x term: ⁴ˣ⁄₅ = ⁷⁄₆ + ⅙ - ⁶⁄₅ ⁴ˣ⁄₅ = ⁸⁄₆ - ⁶⁄₅

Simplify the right side by reducing 8/6 to 4/3 and finding a common denominator of 15: ⁴ˣ⁄₅ = ⁴⁄₃ - ⁶⁄₅ ⁴ˣ⁄₅ = (20 - 18)/(15) ⁴ˣ⁄₅ = ²⁄₁₅

Isolate x by multiplying both sides of the equation by 5/4: x = ²⁄₁₅ · ⁵⁄₄ x = ¹⁰⁄₆₀ x = ⅙

Explanation

The algebraic identity for the difference of two cubes is a³ - b³ = (a - b)³ + 3ab(a - b). Substituting the given values a - b = 10 and ab = -21 into this formula, we get a³ - b³ = (10)³ + 3(-21)(10). Evaluating the terms, 10³ = 1000 and 3 × (-21) × 10 = -630. Therefore, a³ - b³ = 1000 - 630 = 370.

Explanation

Let the fraction be x. According to the given condition, the sum of twice the fraction and five times its reciprocal is 7. This is represented by the equation 2x + ⁵⁄ₓ = 7.

To solve for x, multiply the entire equation by x to remove the fraction: 2x² + 5 = 7x 2x² - 7x + 5 = 0

This quadratic equation can be solved by factoring: 2x² - 2x - 5x + 5 = 0 2x(x - 1) - 5(x - 1) = 0 (2x - 5)(x - 1) = 0

Solving for x gives x = 5/2 or x = 1. Substituting x = 5/2 into the expression 2x + 5/x confirms the result: 2(5/2) + 5(2/5) = 5 + 2 = 7.

Explanation

In an arithmetic progression, the nᵗʰ term is defined by the formula aₙ = a + (n-1)d, where a is the first term and d is the common difference. The difference between any two terms aₘ and aₙ is given by (m-n)d. Using the given 3rd and 6th terms:

a₆ - a₃ = (6 - 3)d 37 - 19 = 3d 18 = 3d d = 6

To find the 13th term, we can use the 6th term and the common difference. The 13th term is 7 positions ahead of the 6th term:

a₁₃ = a₆ + (13 - 6)d a₁₃ = 37 + 7(6) a₁₃ = 37 + 42 a₁₃ = 79

Explanation

When a point (x, y) is reflected in a horizontal line y = k, the x-coordinate of the reflected point remains the same because the reflection is vertical. The y-coordinate of the reflected point (x', y') is determined by the fact that the line of reflection y = k is the perpendicular bisector of the segment joining the original point and its reflection. This means k is the midpoint of y and y', giving the relation y' = 2k - y.

For the point (5, -2) and the line of reflection y = -1, we identify x = 5, y = -2, and k = -1. Applying the transformation: x' = 5 y' = 2(-1) - (-2) = -2 + 2 = 0

The coordinates of the reflected point are (5, 0).

Explanation

The coordinates of the centroid (x_G, y_G) of a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) are given by the average of the coordinates of the vertices: x_G = (x₁ + x₂ + x₃)/(3) and y_G = (y₁ + y₂ + y₃)/(3)

Given the centroid is (-1, 4), A is (-3, -1), and B is (3, 5), we let the coordinates of vertex C be (x, y).

Substituting the x-values into the formula: -1 = (-3 + 3 + x)/(3) -1 = ˣ⁄₃ x = -3

Substituting the y-values into the formula: 4 = (-1 + 5 + y)/(3) 4 = (4 + y)/(3) 12 = 4 + y y = 8

The coordinates of vertex C are found to be (-3, 8).

Explanation

The slope m of a linear equation in the form Ax + By = C is calculated using the formula m = -A/B. In the given equation ax + 3y = 6, the coefficients are A = a and B = 3.

Substituting these into the slope formula: m = -ᵃ⁄₃

The problem states that the slope is -⅔. By setting the two expressions for the slope equal to each other: -ᵃ⁄₃ = -⅔

Multiplying both sides by -3 yields: a = 2

Explanation

In Δ ABC, since DE is parallel to BC, the smaller triangle Δ ADE is similar to the larger triangle Δ ABC by AA similarity. The ratio of their corresponding sides is given by (AD)/(AB). We are given AD:DB = 2:3, which implies AB = AD + DB = 2 + 3 = 5 units. Thus, the side ratio is (AD)/(AB) = ⅖.

The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides: (Area(Δ ADE))/(Area(Δ ABC)) = ((AD)/(AB))² = (⅖)² = ⁴⁄₂₅

Substituting the given value Area(Δ ABC) = 100 sq cm: Area(Δ ADE) = ⁴⁄₂₅ × 100 = 16 sq cm

The area of the quadrilateral BDEC is calculated by subtracting the area of the smaller triangle from the area of the larger triangle: Area(BDEC) = Area(Δ ABC) - Area(Δ ADE) = 100 - 16 = 84 sq cm

Explanation

Using the standard trigonometric values for 30°, we have sin 30° = ½ and cos 30° = √3{2}. Substituting these values into the expression sin 30° + 2 cos 30° yields:

½ + 2 × ( √3{2} )

The factor of 2 in the second term cancels with the denominator, simplifying the expression to:

½ + √3

To express this as a single fraction, we find a common denominator of 2:

½ + 2√3{2} = 1 + 2√3{2}

Explanation

In right-angled triangle Δ XYZ, since ∠ Y = 90°, the side ZX is the hypotenuse and YZ is the base relative to ∠ Z. The length of the base can be calculated using the cosine trigonometric ratio:

cos(Z) = (YZ)/(ZX)

Given that ∠ Z = 60° and the hypotenuse ZX = 3√3 cm, substitute these values into the equation:

cos(60°) = (YZ)/(3√(3))

Since the value of cos(60°) is ½, the equation becomes:

½ = (YZ)/(3√(3))

Multiplying both sides by 3√3 to isolate YZ:

YZ = 3√3{2} cm

Explanation

In a right-angled triangle, the secant function is defined as sec θ = (Hypotenuse)/(Base). Given sec θ = ¹³⁄₁₂, we can represent the Hypotenuse (H) as 13 and the Base (B) as 12. According to the Pythagorean theorem, the Perpendicular (P) is calculated as:

P = √(H² - B²) P = √(13² - 12²) P = √(169 - 144) P = √(25) = 5

This corresponds to the well-known Pythagorean triple (5, 12, 13). The cotangent function is defined as the ratio of the Base to the Perpendicular, cot θ = (Base)/(Perpendicular). Substituting the values, we find cot θ = ¹²⁄₅.

Explanation

Based on the data provided in the table, the number of work visas issued for each year is calculated by multiplying the total number of issued visas by the percentage of work visas issued.

For 2011: 4000 × 60% = 2400 For 2012: 3500 × 65% = 2275 For 2013: 2000 × 70% = 1400 For 2014: 3000 × 55% = 1650 For 2015: 4500 × 32% = 1440 For 2016: 3000 × 60% = 1800

To determine the number of years where work visas issued were more than the previous year, we compare consecutive years: In 2012, 2275 is less than 2400 in 2011. In 2013, 1400 is less than 2275 in 2012. In 2014, 1650 is more than 1400 in 2013. In 2015, 1440 is less than 1650 in 2014. In 2016, 1800 is more than 1440 in 2015.

The number of work visas issued was greater than the previous year in exactly 2 years, which are 2014 and 2016.

Explanation

According to the data provided for the years 2015 and 2016: Total number of visas issued in 2015 = 4500 Percentage of work visas issued in 2015 = 32% Total number of visas issued in 2016 = 3000 Percentage of work visas issued in 2016 = 60%

First, calculate the number of work visas for each year: Number of work visas in 2015 = 32% of 4500 = ³²⁄₁₀₀ × 4500 = 1440 Number of work visas in 2016 = 60% of 3000 = ⁶⁰⁄₁₀₀ × 3000 = 1800

The difference between the number of work visas in 2016 and 2015 is 1800 - 1440 = 360. To find the percentage relative to the previous year (2015): Percentage Difference = ³⁶⁰⁄₁₄₄₀ × 100 = ¼ × 100 = 25%

Explanation

Based on the data for work visas issued between 2011 and 2016, the annual figures are 3000, 3500, 3000, 3500, 4500, and 3500 respectively. The total number of visas issued over these six years is calculated by summing these individual values: 3000 + 3500 + 3000 + 3500 + 4500 + 3500 = 21000

The problem states that 10% of the people who were issued visas did not use them. This implies that 90% of the total visas issued were actually used. To find the number of people who used the visas, the following calculation is applied: Total Users = Total Issued × (100% - 10%) Total Users = 21000 × 0.90 = 18900

By calculating 90% of the cumulative total of 21,000, we find that 18,900 people used their work visas during the period from 2011 to 2016.

Explanation

To find the least number that must be subtracted from 518 to make it exactly divisible by 13, we calculate the remainder when 518 is divided by 13. The division process is as follows:

518 ÷ 13

First, divide 51 by 13: 13 × 3 = 39 51 - 39 = 12

Next, bring down the 8 to make it 128, then divide 128 by 13: 13 × 9 = 117 128 - 117 = 11

The remainder obtained is 11. In modular arithmetic, this is expressed as 518 ≡ 11 13. The remainder represents the specific value that exceeds the nearest lower multiple of 13. By subtracting this remainder from the original number, we arrive at 518 - 11 = 507. Since 507 = 13 × 39, the resulting sum is completely divisible by 13. Thus, 11 is the required least number.

Explanation

The time taken by B to complete the job alone can be found using the formula for combined work rates. If T_A₊_B is the time taken by A and B together and T_A is the time taken by A alone, the time T_B for B working alone is given by:

T_B = T_A × T_A₊_B{T_A - T_A₊_B}

Substituting the given values T_A₊_B = 12 and T_A = 20:

T_B = (20 × 12)/(20 - 12)

T_B = ²⁴⁰⁄₈ = 30

Alternatively, using the efficiency method, assume the total work is the LCM of 12 and 20, which is 60 units. The combined efficiency of A and B is 60 / 12 = 5 units per day, and the efficiency of A is 60 / 20 = 3 units per day. The efficiency of B is therefore 5 - 3 = 2 units per day. To complete 60 units of work, B takes 60 / 2 = 30 days.

Explanation

The area A of a regular hexagon with side length s is given by the formula A = 3√3{2} s². This formula is derived from the fact that a regular hexagon consists of six equilateral triangles, each having an area of √3{4} s².

Given the side length s = 9 cm, the area is calculated as follows: A = 3√3{2} × (9)² A = 3√3{2} × 81 A = 243√3{2}

The calculation yields an area of 243√3{2} square centimeters.

Explanation

In a buy 4 get 1 free offer, the total number of shirts a customer receives is 5, consisting of 4 purchased shirts and 1 free shirt. The effective discount percentage is calculated by dividing the number of free items by the total number of items received and multiplying by 100. Using the formula:

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

By substituting the given values, we get:

Discount % = (1)/(4 + 1) × 100 = ⅕ × 100 = 20%

Explanation

The ratio of the present ages of P and Q is 7:9, and the ratio 10 years ago was 5:7. Since the age difference between two people remains constant over time, we check the difference between the terms of the ratios. For both ratios, the difference is 2 (9 - 7 = 2 and 7 - 5 = 2), so the scales are already consistent. The change in ratio units for both P and Q over the 10-year period is 2 units (7 - 5 = 2 or 9 - 7 = 2). This means 2 units correspond to 10 years, which implies 1 unit is equal to 5 years. Q's present age is represented by 9 units in the present age ratio. Therefore, Q's present age is 9 × 5 = 45 years.

Explanation

The correct average is determined by calculating the net difference caused by the entry error and adjusting the initial average accordingly. The formula for the corrected average is:

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

Substituting the given values into the formula:

Correct Average = 34 + (26 - 62)/(40)

Correct Average = 34 - ³⁶⁄₄₀

Correct Average = 34 - 0.9 = 33.1

The total marks were overestimated by 36 because 62 was entered instead of 26. When this excess of 36 marks is removed and the deficit is distributed across the 40 students, the average decreases by exactly 0.9.

Explanation

The final price paid by the customer is the result of successive profit percentages applied to the initial cost price of the wholesaler. Let CP be the cost price of the good for the wholesaler.

The wholesaler sells it at a 5% profit, so the selling price to the retailer is CP × (1 + ⁵⁄₁₀₀) = CP × 1.05.

The retailer then sells it to the customer at a 10% profit on their cost price. Thus, the customer pays CP × 1.05 × (1 + ¹⁰⁄₁₀₀) = CP × 1.05 × 1.10.

Given that the customer pays Rs. 2,000, we set up the equation: CP × 1.155 = 2000

Solving for CP: CP = (2000)/(1.155) CP = ²⁰⁰⁰⁰⁰⁰⁄₁₁₅₅ ≈ 1731.6017

Rounding to one decimal place, the cost to the wholesaler was Rs. 1731.6.

Explanation

Let the original number be x, which represents 100% of itself. When the number is increased by 69, it becomes 103% of itself. This means the numerical increase of 69 corresponds to the percentage increase of 3% (since 103% - 100% = 3%).

The relationship can be expressed as: 3% of x = 69 ³⁄₁₀₀ × x = 69

To solve for x, divide 69 by 3 and multiply by 100: x = (69 × 100)/(3) x = 23 × 100 x = 2300

By calculating 103% of 2300, we get 1.03 × 2300 = 2369, which is exactly 2300 + 69.

Explanation

When a body covers the same distance at two different speeds, v₁ and v₂, the average speed for the entire journey is calculated using the harmonic mean of the speeds. The formula is:

Average Speed = (2 × v₁ × v₂)/(v₁ + v₂)

Given that v₁ = 40 km/hr and v₂ = 24 km/hr, substitute these values into the formula:

Average Speed = (2 × 40 × 24)/(40 + 24)

Average Speed = ¹⁹²⁰⁄₆₄

Average Speed = 30 km/hr

Alternatively, using the LCM method, assume the distance is the least common multiple of 40 and 24, which is 120 km. The time taken for the forward journey is 120 / 40 = 3 hours, and the time taken for the return journey is 120 / 24 = 5 hours. The total distance covered is 120 + 120 = 240 km, and the total time taken is 3 + 5 = 8 hours. Dividing total distance by total time gives 240 / 8 = 30 km/hr.

Explanation

The simple interest is calculated using the formula SI = (P × R × T)/(100), where P is the principal sum, R is the rate of interest per annum, and T is the time in years. In this case, the total simple interest accrued over 4 years at a rate of 12.5% is Rs. 5,400.

The effective interest rate for the entire duration is the product of the annual rate and the time: 12.5% × 4 = 50%. This implies that the total interest fetched is exactly half of the principal sum. By substituting the values into the equation 5400 = (P × 50)/(100), we get 5400 = (P)/(2). Solving for P gives P = 5400 × 2 = 10800.

Explanation

To find the value of x, start by distributing the fraction ⅔ into the parenthesis: ⅔ · ⁶ˣ⁄₅ - ⅔ · ¼ + ⅓ = ⁹ˣ⁄₅

Simplify the products: ⁴ˣ⁄₅ - ⅙ + ⅓ = ⁹ˣ⁄₅

Combine the constant terms -⅙ and ⅓ by using a common denominator of 6: ⅓ - ⅙ = ²⁄₆ - ⅙ = ⅙

Substitute this back into the equation: ⁴ˣ⁄₅ + ⅙ = ⁹ˣ⁄₅

To isolate x, subtract ⁴ˣ⁄₅ from both sides of the equation: ⅙ = ⁹ˣ⁄₅ - ⁴ˣ⁄₅ ⅙ = ⁵ˣ⁄₅ ⅙ = x

Explanation

Using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b), let s = a + b. Substituting the given values a³ + b³ = 341 and ab = 30 into the formula yields the cubic equation 341 = s³ - 3(30)s. This simplifies to s³ - 90s - 341 = 0. By testing the value s = 11, we calculate 11³ - 90(11) - 341 = 1331 - 990 - 341 = 0.

Alternatively, the result can be found by inspecting the factors of ab = 30. The pairs of integers (a, b) that satisfy the product are (1, 30), (2, 15), (3, 10), and (5, 6). Testing the pair (5, 6), we find that 5³ + 6³ = 125 + 216 = 341. The sum of these numbers is a + b = 5 + 6 = 11.

Explanation

Let the fraction be x. The problem defines the relationship x + 3/x = 31/6. By substituting x = 9/2 into the equation, we get 9/2 + 3(2/9) = 9/2 + 2/3. Using a common denominator of 6, the sum is (27 + 4)/6 = 31/6, which satisfies the given condition. This result can also be found by rearranging the original equation into the quadratic form 6x² - 31x + 18 = 0. Applying the quadratic formula x = -b ± √(b²⁾⁻⁴ᵃᶜ{2a} gives x = 31 ± √(31²⁾⁻⁴⁽⁶⁾⁽¹⁸⁾{2(6)} = 31 ± √(961 - 432){12} = (31 ± 23)/(12). Solving for the positive roots provides x = 54/12 = 9/2 or x = 8/12 = 2/3.

Explanation

The general term of an arithmetic progression is given by aₙ = a + (n - 1)d, where a is the first term and d is the common difference. Using the given values, we have two equations: a₅ = a + 4d = 1 a₈ = a + 7d = -17

Subtracting the first equation from the second gives 3d = -18, which simplifies to d = -6. Substituting this value back into the first equation, we find a + 4(-6) = 1, which gives a = 25.

The sum of the first n terms of an arithmetic progression is calculated using the formula Sₙ = ⁿ⁄₂ [2a + (n - 1)d]. For n = 13: S₁₃ = ¹³⁄₂ [2(25) + (13 - 1)(-6)] S₁₃ = ¹³⁄₂ [50 + 12(-6)] S₁₃ = ¹³⁄₂ [50 - 72] S₁₃ = ¹³⁄₂ [-22] S₁₃ = 13 × (-11) = -143

Alternatively, since 13 is an odd number, the sum is equal to the number of terms multiplied by the middle term, a₇. We can find a₇ by adding 2d to a₅: 1 + 2(-6) = -11. Thus, S₁₃ = 13 × (-11) = -143.

Explanation

The reflection of a point P(x, y) across a vertical line x = k results in a point P'(x', y') where the y-coordinate remains unchanged and the x-coordinate is transformed such that k is the midpoint of x and x'. The formula for the new x-coordinate is x' = 2k - x.

Given the point (-4, 3) and the line of reflection x = -2, we have x = -4, y = 3, and k = -2. Applying the formula:

x' = 2(-2) - (-4) = -4 + 4 = 0 y' = 3

The reflected point is (0, 3).

Explanation

The coordinates of point A(x, y) dividing the line segment BC with endpoints B(x₁, y₁) and C(x₂, y₂) in the ratio m:n are determined by the section formula:

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

Substituting the given values B(6, -4), C(0, 8), and the ratio m:n = 5:1:

x = (5(0) + 1(6))/(5 + 1) = (0 + 6)/(6) = ⁶⁄₆ = 1

y = (5(8) + 1(-4))/(5 + 1) = (40 - 4)/(6) = ³⁶⁄₆ = 6

The coordinates of point A are (1, 6).

Explanation

The slope m of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated using the formula m = (y₂ - y₁)/(x₂ - x₁). Substituting the given points (3, -4) and (-2, 5) into this formula, we get m = (5 - (-4))/(-2 - 3). This simplifies to m = ⁹⁄₋₅, which is equal to -9/5. In coordinate geometry, parallel lines have identical slopes. Therefore, the slope of any line parallel to the one passing through the given points is also -9/5.

Explanation

For two similar triangles, the ratio of their corresponding sides is equal to the ratio of their perimeters. Given that Δ XYZ ∼ Δ PQR, the relationship between the corresponding sides XY and PQ and their perimeters is expressed as:

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

Substituting the given values into the formula:

(XY)/(27) = ⁴⁄₉

To find the length of XY, rearrange the equation:

XY = ⁴⁄₉ × 27

XY = 4 × 3

XY = 12 cm

Explanation

The trigonometric values for the given angles are cot 45° = 1 and csc 60° = (2)/(√(3)). Substituting these values into the expression cot 45° + ⅓ csc 60°, we get 1 + ⅓ ( (2)/(√(3)) ). This simplifies to 1 + (2)/(3√(3)). To rationalize the denominator of the fractional term, we multiply the numerator and denominator by √3, which gives 1 + 2√3{3 × 3} = 1 + 2√3{9}. By taking 9 as the common denominator, the expression becomes 9 + 2√3{9}.

Explanation

In the given right-angled triangle △ DEF where ∠ E = 90° and ∠ F = 45°, the expression to evaluate is the product of the sine and tangent of the angle F. According to standard trigonometric ratios for a 45° angle, the value of sin 45° is (1)/(√(2)) and the value of tan 45° is 1. By substituting these specific values into the product sin F × tan F, we get: sin 45° × tan 45° = (1)/(√(2)) × 1 = (1)/(√(2))

Explanation

In a right-angled triangle, the trigonometric ratio cot θ is defined as the ratio of the base (B) to the perpendicular (P). Given cot θ = ²¹⁄₂₀, we can represent the base as B = 21 and the perpendicular as P = 20.

To find the value of sec θ, which is the ratio of the hypotenuse (H) to the base (B), we must first determine the length of the hypotenuse using the Pythagorean theorem: H = √(B²⁾⁺^P^^^². Substituting the known values into the formula, we have H = √(21²⁾⁺²⁰^^^² = √(441 + 400) = √(841) = 29.

Once the hypotenuse is identified as 29, the value of sec θ is calculated using the definition sec θ = (H)/(B). Substituting H = 29 and B = 21 yields the ratio ²⁹⁄₂₁. This calculation relies on the properties of the Pythagorean triple (20, 21, 29), where the sum of the squares of the two shorter sides equals the square of the longest side.

Explanation

A number is divisible by both 5 and 7 if and only if it is divisible by their least common multiple, LCM(5, 7) = 35. To find how many such numbers exist between 300 and 650, we identify the multiples of 35 within this range.

The first multiple of 35 in this range is 315 (since 35 × 8 = 280 and 35 × 9 = 315). The last multiple of 35 in this range is 630 (since 35 × 18 = 630 and 35 × 19 = 665).

These multiples form an arithmetic progression where the first term a = 315, the last term l = 630, and the common difference d = 35. The number of terms n is determined by the formula:

n = (l - a)/(d) + 1

Substituting the values into the formula:

n = (630 - 315)/(35) + 1 = ³¹⁵⁄₃₅ + 1 = 9 + 1 = 10

Explanation

Wages for a piece of work are distributed among workers in the ratio of their efficiencies. Efficiency is inversely proportional to the time taken to complete the work. Given the time taken by Raman, Jatin, and Sachin are 5, 7, and 9 days respectively, their efficiency ratio is:

Efficiency Ratio = ⅕ : ⅐ : ⅑

To simplify this ratio, multiply each term by the least common multiple of 5, 7, and 9, which is 5 × 7 × 9 = 315:

Raman : Jatin : Sachin = 63 : 45 : 35

The sum of the ratio parts is 63 + 45 + 35 = 143. The total amount paid is Rs 2860. Raman's share is calculated as:

Raman's Share = ⁶³⁄₁₄₃ × 2860

Since 143 × 20 = 2860, the calculation becomes:

Raman's Share = 63 × 20 = 1260

Explanation

The surface area of a sphere is calculated using the formula 4π r², where r is the radius of the sphere. Since the diameter is given as 14 cm, the radius is r = ¹⁴⁄₂ = 7 cm. Substituting these values into the formula:

Surface Area = 4 × ²²⁄₇ × 7 × 7 Surface Area = 4 × 22 × 7 Surface Area = 88 × 7 = 616 cm²

Explanation

The relationship between the marked price (MP) and the selling price (SP) after successive discounts is given by the formula SP = MP × (1 - d₁/100) × (1 - d₂/100). Substituting the given values, we have 50700 = MP × (1 - 20/100) × (1 - 35/100). This simplifies to 50700 = MP × (80/100) × (65/100). Expressing the percentages as simplified fractions, the equation becomes 50700 = MP × (4/5) × (13/20), which reduces to 50700 = MP × (52/100). To find the marked price, we rearrange the terms to get MP = (50700 × 100) / 52. Dividing both 50700 and 52 by 13, we get MP = (3900 × 100) / 4. Further dividing 3900 by 4 gives 975, resulting in MP = 975 × 100 = 97500.

Explanation

The total amount of Rs 3200 is distributed according to the ratio 3 : 5 : 8. The sum of the ratio units is 3 + 5 + 8 = 16 units. Since 16 units represent the total amount of Rs 3200, the value of 1 unit is ³²⁰⁰⁄₁₆ = 200. The difference between the shares of B and C in terms of ratio units is 8 - 5 = 3 units. By multiplying the value of a single unit by this difference, we calculate the required amount as 3 × 200 = 600.

Explanation

The current total age of the 5-member family is calculated as 5 × 24 = 120 years. To determine the total age of the family at the time of the birth of the youngest member, we must subtract the 8 years that have passed for each of the 5 members. The total age 8 years ago was 120 - (5 × 8) = 120 - 40 = 80 years. At the time of birth, the youngest member's age was 0, meaning the sum of ages of the remaining 4 family members was 80 years. The average age of the family at that time is found by dividing this sum by the 4 members who were present prior to the birth, which is 80 / 4 = 20 years.

Explanation

The total expenditure remains constant at Rs 100. When the price of the pens decreases by 20 percent, the saving on the total amount is 20% of 100, which equals Rs 20. According to the problem, this saving of Rs 20 allows the man to purchase 10 additional pens. Therefore, the new price of these 10 pens is Rs 20. To find the new price of each pen, divide the total saving by the number of extra pens: 20 / 10 = 2.

Explanation

Let the initial number be x. The problem describes two successive percentage deductions. After the first deduction of 60%, the remaining value is 40% of the original number. The second deduction of 15% is applied to this remainder, meaning 85% of that intermediate value is left.

The relationship can be expressed using the formula for successive percentage changes: x × (1 - ⁶⁰⁄₁₀₀) × (1 - ¹⁵⁄₁₀₀) = 1428

Substituting the values: x × 0.40 × 0.85 = 1428 x × 0.34 = 1428

To find the initial number, divide the final result by the product of the remaining percentages: x = (1428)/(0.34) x = ¹⁴²⁸⁰⁰⁄₃₄

Dividing 1428 by 34 yields 42. Thus: x = 4200

Explanation

Let the speed of the car be v km/hr. Since the train travels 40% faster than the car, its speed is 1.4v km/hr. Both cover a distance of 140 km. The car takes 140/v hours to reach the destination. The train spends 25 minutes (which is 25/60 = 5/12 hours) at stations, so its total time is the sum of its travel time and stoppage time: 140/(1.4v) + 5/12 hours.

Because both reach the destination at the same time, their total travel durations are equal: 140/v = 140/(1.4v) + 5/12

To solve for the speed of the train Sₜ = 1.4v, we can rewrite the equation in terms of Sₜ: 140 / (Sₜ/1.4) - 140 / Sₜ = 5/12 (140 × 1.4) / Sₜ - 140 / Sₜ = 5/12 (196 - 140) / Sₜ = 5/12 56 / Sₜ = 5/12 Sₜ = (56 × 12) / 5 Sₜ = 672 / 5 = 134.4 km/hr.

Explanation

For simple interest, the relationship between the number of times a sum grows and the time taken is given by the formula (n₁ - 1)/(T₁) = (n₂ - 1)/(T₂), where n represents the multiplier of the principal and T represents the time.

Given that the sum triples (n₁ = 3) in 5 years (T₁ = 5), the interest earned is 3P - P = 2P. To find the time (T₂) it takes for the sum to become five times (n₂ = 5), the required interest is 5P - P = 4P.

Substituting the values into the relation: (3 - 1)/(5) = (5 - 1)/(T₂) ⅖ = (4)/(T₂) 2 · T₂ = 20 T₂ = 10

Since simple interest is constant over equal periods, if it takes 5 years to gain 2P interest, it will take exactly double that time, 10 years, to gain 4P interest.

Explanation

From the equation x + (1/x) = 2, we can derive that (x - 1)² = 0, which means x = 1. Substituting x = 1 into the expression x⁶⁴ + x¹²¹ gives 1⁶⁴ + 1¹²¹ = 1 + 1 = 2.

Explanation

Given x = 6 + 2√6, we first find the value of x - 1: x - 1 = 6 + 2√6 - 1 = 5 + 2√6

To find the square root of 5 + 2√6, we look for two numbers whose sum is 5 and product is 6. These numbers are 3 and 2. Thus, we can write 5 + 2√6 as (√3)² + (√2)² + 2(√3)(√2), which is the expansion of (√3 + √2)². Taking the square root, we get: √(x - 1) = √3 + √2

Now, find the reciprocal: (1)/(√(x - 1)) = (1)/(√(3) + 2) Rationalizing the denominator by multiplying the numerator and denominator by √3 - √2: 1(√3 - √2){(√3 + √2)(√3 - √2)} = √3 - √2{3 - 2} = √3 - √2

Finally, add the two expressions: √(x - 1) + (1)/(√(x - 1)) = (√3 + √2) + (√3 - √2) = 2√3

Explanation

The algebraic identity states that for any three variables x, y, and z, the expression x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx). Consequently, if x + y + z = 0, the entire expression evaluates to 0. In this problem, let x = a - 7, y = b - 9, and z = c - 11. Adding these three terms results in x + y + z = (a - 7) + (b - 9) + (c - 11) = a + b + c - (7 + 9 + 11) = a + b + c - 27. Given the condition a + b + c = 27, substituting this value yields x + y + z = 27 - 27 = 0. Since the sum of the terms is zero, the value of the given cubic expression must be 0.

Explanation

Let the base of the isosceles triangle be b and the equal sides be a. According to the problem, a = ⅚b. The perimeter of the triangle is the sum of its sides: 2a + b = 32. Substituting the value of a, we get 2(⅚b) + b = 32, which simplifies to ⁵⁄₃b + b = 32. This leads to ⁸⁄₃b = 32, so b = (32 × 3)/(8) = 12 cm.

With the base b = 12 cm, the equal sides are a = ⅚ × 12 = 10 cm. In an isosceles triangle, the altitude h drawn to the base bisects the base into two equal parts of 6 cm each. Using the Pythagorean theorem in the resulting right-angled triangle, the height is h = √(a² - (ᵇ⁄₂)²) = √(10² - 6²) = √(100 - 36) = √(64) = 8 cm. The area of the triangle is given by the formula Area = ½ × base × height = ½ × 12 × 8 = 48 cm².

Explanation

In a rhombus PQRS, all sides are equal in length, thus PQ = QR = RS = SP = 8 cm. The sum of any two adjacent angles in a rhombus is 180°. Given ∠ PQR = 120°, the adjacent angle ∠ SPQ is calculated as 180° - 120° = 60°.

Consider the triangle PQS formed by sides PQ, PS, and the diagonal QS. In this triangle, PQ = PS = 8 cm, making it an isosceles triangle. Since the angle between these two equal sides is ∠ SPQ = 60°, the remaining two angles must also be 60° because (180° - 60°) / 2 = 60°. Therefore, △ PQS is an equilateral triangle. In an equilateral triangle, all sides are equal, so the length of the diagonal QS must be equal to the side length PQ, which is 8 cm. Using the Law of Cosines for verification: QS² = PQ² + PS² - 2(PQ)(PS) cos(60°) = 8² + 8² - 2(8)(8)(0.5) = 64 + 64 - 64 = 64. Taking the square root gives QS = 8 cm.