SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The given expression is x = [cosA/(1+sinA)]².

Squaring the terms in the numerator and denominator, we get: x = cos²A / (1 + sinA)²

Using the trigonometric identity cos²A = 1 - sin²A, we can rewrite the numerator as a difference of squares: x = (1 - sin²A) / (1 + sinA)² x = (1 - sinA)(1 + sinA) / (1 + sinA)²

Canceling out the common term (1 + sinA) from both the numerator and the denominator simplifies the expression to: x = (1 - sinA) / (1 + sinA)

To express this in terms of cosecA, we divide both the numerator and the denominator by sinA: x = (1/sinA - 1) / (1/sinA + 1)

Since cosecA = 1/sinA, substituting this relationship gives: x = (cosecA - 1) / (cosecA + 1)

Explanation

The relationship between Selling Price (SP) and Marked Price (MP) given a discount percentage (D) is expressed by the formula:

SP = MP × (100 - D)/100

Using the ratio of two different selling prices for the same marked price:

SP₂ / SP₁ = (100 - D₂)/(100 - D₁)

Substituting the given values, where SP₁ = 24000, D₁ = 19%, and D₂ = 32.5%:

SP₂ / 24000 = (100 - 32.5)/(100 - 19)

SP₂ / 24000 = 67.5 / 81

Simplifying the fraction 67.5/81 by dividing both the numerator and the denominator by 13.5 gives 5/6:

SP₂ = 24000 × 5/6

SP₂ = 4000 × 5 = 20000

Explanation

The volume of a sphere is given by the formula V = (4/3) π r³.

Let R be the radius of the larger ball and r be the radius of each smaller ball. The diameter of the larger ball is 80 cm, which gives a radius R = 80 / 2 = 40 cm. The radius of the smaller ball r is 10 cm.

The number of smaller balls n that can be made by melting the bigger ball is equal to the ratio of their volumes:

n = ((4/3) π R³) / ((4/3) π r³)

n = R³ / r³

n = (R / r)³

Substituting the values:

n = (40 / 10)³ = 4³ = 64

Explanation

By the division algorithm, the relationship between the dividend, divisor, quotient, and remainder is given by the formula:

Dividend = (Divisor × Quotient) + Remainder

Substituting the given values into the formula:

Dividend = (54 × 86) + 13

Multiplying 54 by 86:

54 × 86 = 4644

Adding the remainder of 13:

4644 + 13 = 4657

Explanation

The total runs scored in the first 44 overs are calculated by multiplying the number of overs by the initial run rate: 44 × 5.1 = 224.4 runs

The remaining runs needed to reach the target of 294 runs are: 294 - 224.4 = 69.6 runs

To find the required run rate in the remaining 6 overs, divide the remaining runs by the remaining overs: 69.6 / 6 = 11.6 runs per over

This matches the value of approximately 11.59 runs per over.

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides.

Perimeter of ΔGHI / Perimeter of ΔKLM = GH / KL

Substituting the given values into this relation:

4/9 = 8 / KL

Solving for KL:

KL = (8 × 9) / 4

KL = 18 cm

Explanation

In a square ABCD, the diagonals AC and BD are perpendicular to each other and bisect each other at their midpoint.

First, calculate the slope of diagonal AC: Slope of AC = (7 - (-3)) / (0 - 4) = 10 / -4 = -5/2

Since diagonal BD is perpendicular to diagonal AC, the product of their slopes is -1. Therefore, the slope of BD (m) is: m = -1 / (-5/2) = 2/5

Next, find the midpoint of diagonal AC, which is also the midpoint of diagonal BD: Midpoint M = ((4 + 0)/2, (-3 + 7)/2) = (2, 2)

Using the point-slope form, the equation of diagonal BD passing through M(2, 2) with slope 2/5 is: y - 2 = (2/5) × (x - 2) 5(y - 2) = 2(x - 2) 5y - 10 = 2x - 4 2x - 5y = -6

Explanation

The sum of the exterior angles of any convex polygon is always 360°. For a regular polygon with n sides, all exterior angles are equal in measure. The measure of each exterior angle is given by the formula:

Exterior angle = 360° ÷ n

A regular nonagon has 9 sides, so n = 9. Substituting this value into the formula:

Exterior angle = 360° ÷ 9 = 40°

Explanation

The circumference of a circle is given by the formula C = π × d, where d is the diameter of the circle and π is approximately 22/7.

Substituting the given circumference of 88 cm into the formula: 88 = (22/7) × d

Solving for the diameter d: d = 88 × 7/22 d = 4 × 7 d = 28 cm

Explanation

To find the ratio A : B : C from the equation 10A = 3B = 7C, assume that 10A = 3B = 7C = k, where k is a constant.

This yields: A = k/10 B = k/3 C = k/7

Therefore, the ratio is: A : B : C = 1/10 : 1/3 : 1/7

To simplify the fractions, multiply each term by the least common multiple of the denominators 10, 3, and 7, which is 210.

A : B : C = (1/10 × 210) : (1/3 × 210) : (1/7 × 210) A : B : C = 21 : 70 : 30

Explanation

To find the coefficient of x² in the product (x + 5)(3 - 2x)(4x - 2), we first multiply the first two factors:

(x + 5)(3 - 2x) = 3x - 2x² + 15 - 10x = -2x² - 7x + 15

Next, we multiply this intermediate expression by the third factor (4x - 2):

(-2x² - 7x + 15)(4x - 2)

To find the coefficient of x², we only need to calculate the terms in the expansion that result in x²:

(-2x²) × (-2) = 4x² (-7x) × (4x) = -28x²

Summing these terms together:

4x² - 28x² = -24x²

The coefficient of x² is -24.

Explanation

Simple interest is earned equally each year. Since the simple interest for two years is Rs 2600, the simple interest for the first year is: 2600 / 2 = Rs 1300

The difference between the compound interest and simple interest for two years is the interest earned on the first year's interest. This difference is: 2769 - 2600 = Rs 169

This Rs 169 is the interest on Rs 1300 for one year at the rate of interest R%. Using the relation: Rate = (Difference / Simple Interest for 1st year) × 100 R = (169 / 1300) × 100 R = 169 / 13 R = 13%

Explanation

Using the standard algebraic identity: (x + y)² = x² + y² + 2xy

Substitute the given values into the identity: 14² = 100 + 2xy

Simplify the terms: 196 = 100 + 2xy 2xy = 196 - 100 2xy = 96 xy = 48

Explanation

Let the cost price of the oven be O and the cost price of the refrigerator be R.

Given that the total cost of both items is Rs 12000: O + R = 12000

The oven is sold at a profit of 20%, so its selling price is: Selling Price = 1.20 × O

The refrigerator is sold at a loss of 28%, so its selling price is: Selling Price = 0.72 × R

Since the selling prices of both items are equal, we can equate the two expressions: 1.20 × O = 0.72 × R

To find the ratio of their cost prices: O / R = 0.72 / 1.20 O / R = 72 / 120 O / R = 3 / 5

The ratio of the cost price of the oven to the refrigerator is 3 : 5. This gives a total of 8 parts (3 + 5) representing the total cost.

Value of 1 part = 12000 / 8 = Rs 1500

The cheaper item is the oven, which corresponds to 3 parts: Cost price of the cheaper item = 3 × 1500 = Rs 4500

Explanation

A completes 75% (or 3/4) of the work in 45 days. This means A can complete the entire work in 45 × 4/3 = 60 days. Therefore, A's daily work rate is 1/60.

The remaining 25% (or 1/4) of the work is completed by A and B together in 5 days. This means A and B together can complete the entire work in 5 × 4 = 20 days. Therefore, their joint daily work rate is 1/20.

To find B's daily work rate, subtract A's daily work rate from their joint rate: 1/20 - 1/60 = 3/60 - 1/60 = 2/60 = 1/30

This rate of 1/30 of the work per day means B alone takes 30 days to complete the entire work.

Explanation

The distance between two cities when the speeds and the difference in travel time are known can be calculated using the relation:

Distance = (Speed₁ × Speed₂ / (Speed₂ - Speed₁)) × Difference in Time

Substituting the given speeds of 48 km/hr and 60 km/hr, and the time difference of 3.5 hours:

Distance = (48 × 60 / (60 - 48)) × 3.5

Distance = (2880 / 12) × 3.5

Distance = 240 × 3.5

Distance = 840 km

Explanation

Let the tens digit of the two-digit number be x and the units digit be y. The number is represented as 10x + y.

The sum of the digits is 12: x + y = 12

When 54 is added to the number, its digits are reversed to form 10y + x: 10x + y + 54 = 10y + x 9y - 9x = 54 y - x = 6

Adding these two equations: (x + y) + (y - x) = 12 + 6 2y = 18 y = 9

Substituting y = 9 into the first equation: x + 9 = 12 x = 3

The original two-digit number is 10(3) + 9 = 39.

Explanation

The given inequality is 2x + 2(1 - x) > 3x - 2 > 2x - 5.

First, simplifying the leftmost part of the inequality: 2x + 2 - 2x > 3x - 2 2 > 3x - 2 4 > 3x x < 4/3

Next, solving the second part of the inequality: 3x - 2 > 2x - 5 3x - 2x > -5 + 2 x > -3

Combining both results gives the solution interval for x: -3 < x < 4/3

The value -2 falls within this interval since -3 < -2 < 4/3.

Explanation

The area of a circle is A = πr², meaning the area is proportional to the square of its radius. When a quantity depends on the square of a variable, a percentage increase of x% in that variable results in a percentage increase in the quantity given by the successive percentage formula 2x + x²/100.

Substituting the radius increase of x = 12 into the formula:

Percentage increase in area = 2(12) + 12²/100 = 24 + 144/100 = 24 + 1.44 = 25.44%.

Explanation

The total value of the stock is calculated by summing the products of the quantity and the average cost for each item category.

Total value of Mobile Phones = 89 × 12000 = 1,068,000 Rs Total value of Cameras = 69 × 19000 = 1,311,000 Rs Total value of TVs = 96 × 32000 = 3,072,000 Rs Total value of Refrigerators = 40 × 42000 = 1,680,000 Rs Total value of ACs = 94 × 25000 = 2,350,000 Rs

Summing these values gives the total stock value: Total Stock Value = 1,068,000 + 1,311,000 + 3,072,000 + 1,680,000 + 2,350,000 = 9,481,000 Rs

To express this value in lakh rupees, divide by 100,000 (since 1 lakh = 100,000): 9,481,000 ÷ 100,000 = 94.81 lakh rupees

Explanation

In 2012, the ratio of Import/Export is 0.8. Import in 2012 = 900 crores. Export in 2012 = Import / Ratio = 900 / 0.8 = 1125 crores.

The total exports in 2012 and 2013 together is 4000 crores. Export in 2013 = Total exports - Export in 2012 = 4000 - 1125 = 2875 crores.

In 2013, the ratio of Import/Export is 1.3. Import in 2013 = Export in 2013 × Ratio = 2875 × 1.3 = 3737.5 crores.

Rounding to the nearest integer, the imports in 2013 is 3738 crores.

Explanation

According to the given data table: Height of the child on the 10th Birthday = 135 cms Height of the child on the 14th Birthday = 155 cms

The increase in height is calculated as: Increase in height = Height on the 14th Birthday - Height on the 10th Birthday Increase in height = 155 - 135 = 20 cms

Explanation

The total time duration between 10:00 PM and 6:00 AM is 8 hours.

The values given in the table represent the percentage of time spent in each state, which sum up to 100 (10 + 20 + 15 + 25 + 30 = 100).

The percentage of time the user spent in Light sleep is 15% and in Extremely light sleep is 25%. The combined percentage of these two states is: 15% + 25% = 40%

To find the duration in hours, we calculate 40% of the total 8 hours: Duration = 0.40 × 8 = 3.2 hours

Explanation

Using the standard trigonometric values for specific angles: tan 45° = 1 csc 30° = 1/sin 30° = 1/½ = 2

Substituting these values into the given equation: x = tan 45° + csc 30° x = 1 + 2 = 3

Explanation

Let C be the cost price of the watch for the wholesaler.

The wholesaler sells the watch to the retailer at a gain of 20%, so the cost price for the retailer is C × (1 + 0.20) = 1.2C.

The retailer then sells the watch to the customer at a loss of 10%, which makes the final selling price equal to 1.2C × (1 - 0.10) = 1.2C × 0.90 = 1.08C.

Since the customer pays Rs 874.8: 1.08C = 874.8

Solving for C: C = 874.8 / 1.08 C = 810

Explanation

Using the double-angle identity for cosine: cos(2θ) = 2 cos²(θ) - 1

Rearranging the terms gives: 1 + cos(2θ) = 2 cos²(θ)

Dividing both sides by 2: (1 + cos(2θ))/2 = cos²(θ)

Taking the square root of both sides: √((1 + cos(2θ))/2) = cos(θ)

Substituting 2θ = A, which means θ = A/2, we get: √((1 + cos A)/2) = cos(A/2)

Since the given expression is x = √((1 + cos A)/2), it follows that x = cos(A/2).

Explanation

The time taken by P to complete the project is 150 days, which means P's rate of work is 1/150 of the project per day. The time taken by Q to complete the project is 100 days, which means Q's rate of work is 1/100 of the project per day.

Working together, their combined rate of work per day is: 1/150 + 1/100 = (2 + 3) / 300 = 5 / 300 = 1/60 of the project per day.

This means P and Q working together can complete the entire project in 60 days.

To find the time required to complete 60% of the project, we multiply the total time required for the whole project by 60%: 60 days × 60% = 60 × 0.60 = 36 days.

Explanation

Using basic trigonometric identities, we have: sec A = 1/cos A tan A = sin A/cos A

Substituting these into the given expression: x = sec A + tan A x = (1/cos A) + (sin A/cos A) x = (1 + sin A)/cos A

Using the identity cos² A = 1 - sin² A, we can write cos A as √(1 - sin² A): x = (1 + sin A) / √(1 - sin² A)

Since 1 - sin² A can be factored as (1 - sin A)(1 + sin A), we get: x = (1 + sin A) / √((1 - sin A)(1 + sin A))

We can rewrite the numerator 1 + sin A as √(1 + sin A) × √(1 + sin A): x = (√(1 + sin A) × √(1 + sin A)) / (√(1 - sin A) × √(1 + sin A))

Canceling the common term √(1 + sin A) from the numerator and denominator: x = √(1 + sin A) / √(1 - sin A) x = √((1 + sin A)/(1 - sin A))

Explanation

The volume of the solid is the sum of the volume of the cone and the volume of the hemisphere.

The radius of both the hemisphere and the cone is half of the diameter, which gives r = 5 cm. The height of the cone is h = 12 cm.

The formula for the volume of a cone is ⅓ × π × r² × h, and the formula for the volume of a hemisphere is ⅔ × π × r³.

Total Volume = ⅓ × π × r² × h + ⅔ × π × r³ = ⅓ × π × r² × (h + 2r)

Substituting the given values into the formula: Total Volume = ⅓ × π × 5² × (12 + 2 × 5) Total Volume = ⅓ × π × 25 × 22 Total Volume = 550/3 × π

Using the approximation π = 22/7: Total Volume = 550/3 × 22/7 = 12100/21 = 576.19 cubic cm

Explanation

Let the diagonal of the rectangle be d and the two adjacent sides be a and b. According to the Pythagorean theorem, the diagonal and the two sides of a rectangle form a right-angled triangle, where:

d² = a² + b²

Substituting the given values of the diagonal d = 26 cm and one side a = 10 cm into the equation:

26² = 10² + b² 676 = 100 + b² b² = 576 b = 24 cm

The area of a rectangle is the product of its length and breadth:

Area = a × b Area = 10 × 24 = 240 sq cm.

Explanation

To find the smallest number to be added to 2401 so that the sum is completely divisible by 14, we first divide 2401 by 14 to determine the remainder.

Dividing 2401 by 14: 2401 ÷ 14 = 171 with a remainder of 7

The smallest number that needs to be added is the difference between the divisor and the remainder: 14 - 7 = 7

Adding 7 to 2401 results in 2408, which is perfectly divisible by 14.

Explanation

Let the cost price be CP and the list price be L.

In the first case, a 25% discount on the list price results in a 5% loss. This gives the relation: 0.75 L = 0.95 CP L = (0.95 / 0.75) CP = (19 / 15) CP

In the second case, the discount offered is 10%, which means the new selling price (SP) is 90% of the list price: SP = 0.90 L SP = 0.90 × (19 / 15) CP SP = (9 / 10) × (19 / 15) CP SP = (3 / 10) × (19 / 5) CP = 57 / 50 CP = 1.14 CP

Since the selling price is 1.14 times the cost price, it represents a profit of 14%.

Explanation

The diagonals of a rhombus are perpendicular bisectors of each other. Thus, the diagonal QS is the perpendicular bisector of the diagonal PR.

First, find the midpoint M of PR, through which QS must pass: M = ((3 + (-7))/2, (1 + 5)/2) = (-2, 3)

Next, find the slope of PR (m₁): m₁ = (5 - 1)/(-7 - 3) = 4/(-10) = -2/5

Since QS is perpendicular to PR, the slope of QS (m₂) is the negative reciprocal of m₁: m₂ = -1/(-2/5) = 5/2

Using the point-slope form, the equation of diagonal QS passing through M(-2, 3) with slope 5/2 is: y - 3 = (5/2)(x - (-2)) 2(y - 3) = 5(x + 2) 2y - 6 = 5x + 10 5x - 2y = -16

Explanation

To find the ratio A : B : C from the equation 5A = 13B = 7C, we can express each variable as a fraction of a constant k. Let 5A = 13B = 7C = k. This gives A = k/5, B = k/13, and C = k/7.

The ratio A : B : C is therefore 1/5 : 1/13 : 1/7.

To convert this fractional ratio into an integer ratio, we multiply each term by the product of the denominators, which is 5 × 13 × 7 = 455:

A : B : C = (13 × 7) : (5 × 7) : (5 × 13) A : B : C = 91 : 35 : 65

Explanation

Let the time taken by the Rajdhani train to cover the distance of 708 km be T hours.

Since the Express train takes 6 hours more than the Rajdhani train, its initial time is T + 6 hours.

When the speed of the Express train is doubled, the time taken by it becomes half of its initial time, which is (T + 6) / 2 hours. According to the problem, this new time is 3 hours less than the Rajdhani train's time.

We can set up the equation: (T + 6) / 2 = T - 3

Multiplying both sides by 2: T + 6 = 2T - 6 T = 12 hours

Now, using the speed formula: Speed of Rajdhani train = Distance / Time = 708 / 12 = 59 km/hr

Explanation

The length of an arc of a circle is given by the formula L = (θ / 360°) × 2πr, where L is the arc length, r is the radius of the circle, and θ is the central angle in degrees.

Substituting the given values L = 22 cm, r = 28 cm, and π = 22/7:

22 = (θ / 360°) × 2 × (22 / 7) × 28

22 = (θ / 360°) × 176

1 = (θ / 360°) × 8

θ = 360° / 8

θ = 45°

Explanation

The formula for the area of an equilateral triangle with side length a is Area = (√3 / 4) × a².

Given that the area of the triangle is 9√3 sq cm, we set up the equation: (√3 / 4) × a² = 9√3

Dividing both sides by √3: ¼ × a² = 9

Multiplying both sides by 4: a² = 36

Taking the square root of both sides gives the side length: a = 6 cm

The perimeter of an equilateral triangle is the sum of its three equal sides: Perimeter = 3 × a

Substituting the side length into the perimeter formula: Perimeter = 3 × 6 = 18 cm

Explanation

First, we solve the linear equation for x:

3x - 10 = -(2x + 5) 3x - 10 = -2x - 5 3x + 2x = 10 - 5 5x = 5 x = 1

Now, we substitute the value of x into the given expression (x - 4)³:

(1 - 4)³ = (-3)³ = -27

Explanation

Revenue of the 7th year is calculated using the relation: Revenue of 7th year = (Sum of first 7 years) + (Sum of last 7 years) - (Sum of all 13 years)

Substituting the given average values: Sum of 13 years = 13 × 82 = 1066 lakhs Sum of first 7 years = 7 × 77 = 539 lakhs Sum of last 7 years = 7 × 89 = 623 lakhs

Revenue of 7th year = 539 + 623 - 1066 = 1162 - 1066 = 96 lakhs.

Explanation

Using the algebraic identity: (x + y)² = x² + y² + 2xy

Substitute the given values x² + y² = 61 and xy = -30 into the identity: (x + y)² = 61 + 2(-30) (x + y)² = 61 - 60 (x + y)² = 1

Taking the square root of both sides gives: x + y = ±1

Since 1 is among the given choices, the value of (x + y) is 1.

Explanation

The total amount accumulated at the end of the year is the sum of the accumulated values of both deposits. Since the interest is 10% per half-year:

For the first deposit of ₹3600 made on 1st January, interest is compounded for 2 half-years: Amount 1 = 3600 × (1 + 10/100)² = 3600 × (1.1)² = 3600 × 1.21 = 4356

For the second deposit of ₹3600 made on 1st July, interest is compounded for 1 half-year: Amount 2 = 3600 × (1 + 10/100)¹ = 3600 × 1.1 = 3960

Total amount at the end of the year: Total Amount = 4356 + 3960 = 8316

Total principal deposited: Total Principal = 3600 + 3600 = 7200

The total interest gained: Interest = Total Amount - Total Principal Interest = 8316 - 7200 = 1116

Explanation

Let the required number be x.

According to the given condition, the number is greater than five times its reciprocal by 19/2. This can be represented mathematically as:

x - 5/x = 19/2

To solve for x, multiply the entire equation by 2x to eliminate the denominators:

2x² - 10 = 19x

Rearranging the terms into a standard quadratic equation:

2x² - 19x - 10 = 0

Splitting the middle term:

2x² - 20x + x - 10 = 0

2x(x - 10) + 1(x - 10) = 0

(2x + 1)(x - 10) = 0

This gives two possible values for x:

x = 10 or x = -½

Explanation

The passing marks required for the student to pass is the sum of the marks obtained and the marks by which he failed:

Passing Marks = Marks Obtained + Failing Marks Passing Marks = 53 + 37 = 90

According to the problem, the passing marks represent 36% of the maximum marks. Let M be the maximum marks:

36% of M = 90 0.36 × M = 90 M = 90 ÷ 0.36 = 250

Explanation

To find the value of x, the compound inequality is split into two separate inequalities:

2x + 4(x - 3) < -2 - x -2 - x < 2x - 1

Solving the first inequality: 2x + 4x - 12 < -2 - x 6x - 12 < -2 - x 7x < 10 x < 10/7

Solving the second inequality: -2 - x < 2x - 1 -1 < 3x x > -1/3

Combining both parts gives the range for x: -1/3 < x < 10/7

Evaluating x = 1 shows that it lies within this range because -0.33 < 1 < 1.43. Substituting x = 1 back into the original inequality confirms it: 2(1) + 4(1 - 3) < -2 - 1 < 2(1) - 1 -6 < -3 < 1

Explanation

To find the least weight to height ratio, we calculate the ratio of weight in kilograms to height in meters for each person:

Anju: 68 / 1.74 ≈ 39.08 Ankita: 72 / 1.54 ≈ 46.75 Anupama: 69 / 1.77 ≈ 38.98 Anuradha: 68 / 1.71 ≈ 39.77

Comparing the calculated values, Anupama has the lowest weight to height ratio of approximately 38.98.

Explanation

The percentage profit is the least in the year 2015, which is 10%.

The relation between Revenue, Expenditure, and Profit percentage is given by the formula: Revenue = Expenditure × (1 + Profit % / 100)

Given that the Expenditure in this year is ₹475 crore and the Profit is 10%, substituting these values into the formula: Revenue = 475 × (1 + 10 / 100) Revenue = 475 × 1.1 Revenue = 522.5 crore

Explanation

The average number of children per house is calculated using the formula:

Average = ∑(x × f) ÷ ∑f

where x represents the number of children and f represents the number of houses.

First, calculate the total number of houses, which is the sum of the frequencies:

∑f = 6 + 17 + 17 + 3 = 43

Next, calculate the total number of children by multiplying the number of children by the number of houses for each category and summing them up:

∑(x × f) = (0 × 6) + (1 × 17) + (2 × 17) + (3 × 3) = 0 + 17 + 34 + 9 = 60

Now, divide the total number of children by the total number of houses:

Average = 60 ÷ 43 ≈ 1.395 ≈ 1.4

Explanation

The company has issued a total of 6,00,000 shares. According to the given table, Dhiraj initially holds 15% of the total shares.

Initial shares held by Dhiraj = 15% of 6,00,000 = (15 ÷ 100) × 6,00,000 = 90,000 shares.

If Anand sells 15,000 of his shares to Dhiraj, Dhiraj's total number of shares increases by this amount.

Dhiraj's new total number of shares = 90,000 + 15,000 = 1,05,000 shares.

Explanation

Using the algebraic identity for the difference of two squares, x² - y² = (x - y)(x + y), the numerator can be expressed as:

4a² - 9b² = (2a)² - (3b)² = (2a - 3b)(2a + 3b)

Substituting this factored form back into the original expression:

(4a² - 9b²) / (2a + 3b) = (2a - 3b)(2a + 3b) / (2a + 3b)

The term (2a + 3b) in the numerator and the denominator cancels out, simplifying the expression to 2a - 3b.

Explanation

Let the marked price of the item be 100.

After offering a 5% discount, the selling price of the item becomes: 100 - 5 = 95

The customer then wins a 20% cashback on the amount paid (95): Cashback = 20% of 95 = (20/100) × 95 = 19

The effective price paid by the customer after the cashback is: 95 - 19 = 76

The total savings or effective discount on the original price of 100 is: 100 - 76 = 24, which is equivalent to 24%.

Alternatively, this can be solved using the successive percentage change formula: Effective Discount = d₁ + d₂ - (d₁ × d₂)/100

Substituting d₁ = 5 and d₂ = 20: Effective Discount = 5 + 20 - (5 × 20)/100 = 25 - 1 = 24%