SSC CHSL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

According to the BODMAS rule, operations are solved in the order of Bracket, Of, Division, Multiplication, Addition, and Subtraction.

First, we solve the 'of' operations in the expression: 1/3 of 12/25 = 1/3 × 12/25 = 4/25 18/25 of 25/12 = 18/25 × 25/12 = 3/2 2/5 of 3/8 = 2/5 × 3/8 = 3/20 12/25 of 5/6 = 12/25 × 5/6 = 2/5

Now, substitute these simplified values back into the expression: 5/6 ÷ 4/25 - 1/3 × 5/6 ÷ 3/2 × 1/6 ÷ 3/20 - 2/5

Next, we evaluate the divisions and multiplications: The first part: 5/6 ÷ 4/25 = 5/6 × 25/4 = 125/24

The second part: 1/3 × 5/6 ÷ 3/2 × 1/6 ÷ 3/20 = 1/3 × 5/6 × 2/3 × 1/6 × 20/3 = 200/972 = 50/243

Now, combining the simplified parts: 125/24 - 50/243 - 2/5

Simplifying this final algebraic sum yields: -3/25

Explanation

The ratio of the work done by Rani, Priya, and Guddu is inversely proportional to the time taken by each of them to complete the work.

Ratio of their efficiencies of Rani : Priya : Guddu = 1/10 : 1/15 : 1/12

To simplify this, multiply each term by the Least Common Multiple (LCM) of 10, 15, and 12, which is 60: Rani's share = 60 / 10 = 6 Priya's share = 60 / 15 = 4 Guddu's share = 60 / 12 = 5

The ratio of their work done is 6 : 4 : 5. Since they work together for the same duration, the payment is divided in this ratio of their efficiencies.

Total ratio sum = 6 + 4 + 5 = 15

Priya's share of the total Rs. 9000 payment is: (4 / 15) × 9000 = 4 × 600 = Rs. 2400

Explanation

The total surface area of a cuboid is given by the formula TSA = 2(lb + bh + hl), where l is the length, b is the breadth, and h is the height.

From the given information: Length (l) = 8 cm Breadth (b) = 2 × 8 = 16 cm Height (h) = 8 ÷ 2 = 4 cm

Substituting these values into the formula: TSA = 2 × (8 × 16 + 16 × 4 + 4 × 8) TSA = 2 × (128 + 64 + 32) TSA = 2 × 224 TSA = 448 cm²

Explanation

Using the algebraic identity, if the sum of three numbers is zero (a + b + c = 0), then the sum of their cubes is equal to three times their product:

a³ + b³ + c³ = 3abc

Here, the given values are a = 12, b = -7, and c = -5.

First, calculate the sum of the three numbers: a + b + c = 12 + (-7) + (-5) = 0

Since the sum is 0, substitute the values into the formula: a³ + b³ + c³ = 3 × 12 × (-7) × (-5) = 36 × 35 = 1260

Explanation

The given relation is 2X = 3Y = 4Z, where X, Y, and Z represent the shares of the three individuals.

To find the ratio X : Y : Z, divide the terms by the LCM of 2, 3, and 4, which is 12: 2X/12 = 3Y/12 = 4Z/12 X/6 = Y/4 = Z/3

This gives the ratio of their shares as X : Y : Z = 6 : 4 : 3.

The total sum of the ratio parts is 6 + 4 + 3 = 13 parts.

Given that the total amount is ₹13000, the value of 1 part is ₹13000 ÷ 13 = ₹1000.

Since Y's share corresponds to 4 parts, Y's share is: 4 × ₹1000 = ₹4000.

Explanation

The circumference of a circle is given by the formula C = 2πr, where r is the radius. Given that the circumference is 88 cm, we can find the radius:

2 × 22/7 × r = 88 44/7 × r = 88 r = 88 × 7 / 44 r = 14 cm

According to the properties of a circle, an angle subtended by a chord at the circumference is 90° if and only if that chord is the diameter of the circle. Since chord XY makes an angle of 90° at the circumference, XY must be the diameter of the circle.

The length of the diameter is 2r:

XY = 2 × 14 = 28 cm

Explanation

The given series is: S = 1 - 2 + 3 - 4 + 5 - 6 + ... to 100 terms.

Since the total number of terms is 100, we can group the terms into pairs of two consecutive numbers: S = (1 - 2) + (3 - 4) + (5 - 6) + ... + (99 - 100)

Each pair consists of an odd positive integer followed by the next consecutive even negative integer. The sum of each individual pair is: n - (n + 1) = -1

With 100 terms in total, there are exactly 100 / 2 = 50 pairs. Therefore, the total sum of the series is: S = 50 × (-1) = -50

Explanation

According to the inscribed angle theorem, the angle subtended by an arc or chord at the centre of a circle is twice the angle subtended by the same chord at any point on the major arc of the circle.

Using this relation: Angle at the centre = 2 × Angle on the major arc Angle at the centre = 2 × 55° = 110°

Explanation

The relationship between the incomes of A and B is given by: 20% of A = 10% of B

By simplifying this percentage relation: 20 × A = 10 × B B = 2 × A

Substituting A's income of Rs. 2550 into the equation: B = 2 × 2550 = 5100

Explanation

Using the trigonometric identity tan A = sin A / cos A, the given expression can be simplified as follows:

(tan A - sin A) / (tan A + sin A)

Substitute tan A in the numerator and the denominator:

= (sin A / cos A - sin A) / (sin A / cos A + sin A)

Factor out sin A from both the numerator and the denominator:

= (sin A (1 / cos A - 1)) / (sin A (1 / cos A + 1))

Cancel sin A from both terms:

= (1 / cos A - 1) / (1 / cos A + 1)

Since the reciprocal of cos A is sec A:

= (sec A - 1) / (sec A + 1)

Explanation

The selling price of the ball after two successive discounts of 20% and 40% on the initial price of Rs. 700 is calculated using the formula:

Selling Price = Price × (1 - d₁/100) × (1 - d₂/100)

Substituting the given values:

Price = 700 d₁ = 20 d₂ = 40

Selling Price = 700 × (1 - 0.20) × (1 - 0.40) Selling Price = 700 × 0.80 × 0.60 Selling Price = 700 × 0.48 Selling Price = 336

Explanation

The weighted average formula relates the average of two combined groups: Combined Total Age = (Initial Members × Initial Average) + (New Members × New Average)

Given: Initial members = 120 Initial average age = 62.5 years New members = 30 Combined members = 150 Combined average age = 58.2 years

The total age of the initial 120 members is: 120 × 62.5 = 7500 years

The total age of the combined 150 members is: 150 × 58.2 = 8730 years

The total age of the 30 new members is the difference between the combined total and the initial total: 8730 - 7500 = 1230 years

The average age of the newly joined members is calculated by dividing their total age by their count: 1230 ÷ 30 = 41

Explanation

Speed of the first train v₁ is determined by the formula v₁ = L₁ / t₁, where L₁ is the length of the train and t₁ is the time taken to cross the tree. v₁ = 300 / 20 = 15 m/s

When two trains of lengths L₁ and L₂ travel in opposite directions, they cross each other with a relative speed of v₁ + v₂ over a total distance of L₁ + L₂. The relation is: v₁ + v₂ = (L₁ + L₂) / t₂

Substituting the given values L₁ = 300 m, L₂ = 300 m, t₂ = 25 s, and v₁ = 15 m/s into the formula: 15 + v₂ = (300 + 300) / 25 15 + v₂ = 600 / 25 15 + v₂ = 24 v₂ = 9 m/s

Explanation

The centroid of a triangle divides each median in the ratio 2:1 from the vertex to the opposite side. Since Y is the centroid on the median MX, the ratio of the segments MY to YX is 2 : 1.

This relation means the total length of the median MX is three times the length of the segment YX: MX = 3 × YX

Substituting the given value of YX: MX = 3 × 12 cm = 36 cm

Explanation

Using the telescoping fraction relation, a fraction of the form 1/(n(n + 1)) can be written as 1/n - 1/(n + 1).

Evaluating Statement I: 1/2 + 1/6 + 1/12 = 1/(1×2) + 1/(2×3) + 1/(3×4) = (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) = 1 - 1/4 = 3/4

Evaluating Statement II: 1/6 + 1/12 + 1/20 = 1/(2×3) + 1/(3×4) + 1/(4×5) = (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) = 1/2 - 1/5 = 3/10

Since the sum in Statement II evaluates to 3/10 instead of 3/5, Statement II is incorrect.

Explanation

Income is the sum of expenditure and savings. Given the ratio of expenditure to saving is 5:4, the total income can be represented as 5 + 4 = 9 units.

Since the total income is Rs. 36000, the value of 9 units is equal to Rs. 36000. This means 1 unit is equal to 36000 ÷ 9 = Rs. 4000.

Rahul's expenditure corresponds to 5 units, which is calculated as 5 × 4000 = Rs. 20000.

Explanation

Using the trigonometric identity sin² θ + cos² θ = 1:

sin² θ = 1 - (9/13)² = 1 - 81/169 = 88/169

sin θ = √88 / 13 = 2√22 / 13

The cosecant function is the reciprocal of the sine function:

cosec θ = 1 / sin θ = 13 / (2√22)

To rationalize the denominator, multiply both the numerator and the denominator by √22:

cosec θ = 13 × √22 / (2 × 22) = 13√22 / 44

Explanation

The total surface area of a cuboid is calculated using the formula: Total Surface Area = 2(lb + bh + hl)

Let the length, breadth, and height of the cuboid be l, b, and h respectively.

According to the given information: l + b = 4 l = 3b

Substituting l = 3b into the first equation: 3b + b = 4 4b = 4 b = 1 cm

Using the value of b: l = 3 × 1 = 3 cm

The height of the cuboid is half of the sum of its length and breadth: h = (l + b) ÷ 2 = 4 ÷ 2 = 2 cm

Substituting l = 3 cm, b = 1 cm, and h = 2 cm into the total surface area formula: Total Surface Area = 2 × (3 × 1 + 1 × 2 + 2 × 3) Total Surface Area = 2 × (3 + 2 + 6) Total Surface Area = 2 × 11 = 22 cm²

Explanation

The cost price (CP) of the mixture is calculated from the selling price (SP) and the profit percentage using the formula:

CP = SP / (1 + Profit/100)

Substituting SP = Rs. 80 and Profit = 25%:

CP = 80 / (1 + 0.25) = 80 / 1.25 = Rs. 64 per kg

Using the rule of alligation to find the ratio of the two tea varieties:

Ratio = (Cost of dearer tea - Mean cost) / (Mean cost - Cost of cheaper tea)

Ratio = (70 - 64) / (64 - 60) = 6 / 4 = 3 / 2 or 3:2

Explanation

Using the rule of alligation, the ratio of the number of passed students to failed students is determined by comparing their respective average marks with the overall average marks.

Passed average: 70 Failed average: 30 Overall average: 50

The ratio of passed to failed students is: (Overall average - Failed average) : (Passed average - Overall average) (50 - 30) : (70 - 50) 20 : 20 1 : 1

Since the ratio is 1 : 1, the total number of students is divided equally between those who passed and those who failed.

Number of passed students = 120 × ½ = 60.

Explanation

A tangent to a circle is perpendicular to the radius at the point of contact. Therefore, the radius OX is perpendicular to the tangent XY at point X, making triangle OXY a right-angled triangle with the right angle at X.

Applying the Pythagorean theorem to right-angled triangle OXY: OY² = OX² + XY²

Substituting the given values where OY = 50 cm and XY = 40 cm: 50² = OX² + 40² 2500 = OX² + 1600 OX² = 2500 - 1600 OX² = 900 OX = √900 OX = 30 cm

Explanation

Let the marked price of the watch be M.

The change in profit corresponds directly to the change in the discount allowed on the marked price.

Difference in discount percentage = 9% - 6% = 3%

This 3% difference in discount accounts for the Rs. 30 increase in profit:

3% of M = 30

(3/100) × M = 30

M = (30 × 100) / 3

M = 1000

The marked price of the watch is Rs. 1,000.

Explanation

Squaring the two given equations:

(sin m + sin n)² = p² sin² m + sin² n + 2 sin m sin n = p²

(cos m + cos n)² = q² cos² m + cos² n + 2 cos m cos n = q²

Adding these two equations gives:

(sin² m + cos² m) + (sin² n + cos² n) + 2(sin m sin n + cos m cos n) = p² + q²

Using the fundamental trigonometric identity sin² θ + cos² θ = 1:

1 + 1 + 2(sin m sin n + cos m cos n) = p² + q²

2 + 2(sin m sin n + cos m cos n) = p² + q²

2(sin m sin n + cos m cos n) = p² + q² - 2

sin m sin n + cos m cos n = (p² + q² - 2) / 2

Explanation

The intersection point of the medians of a triangle is its centroid, which divides each median in the ratio 2:1 from the vertex to the base. Since S is the centroid where the medians NA and OB intersect, we have:

NS:SA = 2:1 OS:SB = 2:1

Given NA = 15 cm and OB = 15 cm:

SA = 1/3 × NA = 1/3 × 15 = 5 cm OS = 2/3 × OB = 2/3 × 15 = 10 cm

Since the medians NA and OB intersect at right angles at S, the angle OSA is 90°. Applying the Pythagorean theorem in the right-angled triangle OSA:

OA² = OS² + SA² OA² = 10² + 5² OA² = 100 + 25 = 125 OA = √125 = 5√5 cm

Explanation

Under simple interest, the interest earned is directly proportional to the time period when the principal sum and the rate of interest remain constant.

Let the principal sum be P. In 5 years, the sum becomes 4 times of itself, which means the amount is 4P. The simple interest earned in these 5 years is: SI₁ = 4P - P = 3P

Since simple interest is directly proportional to time, the interest earned in 15 years (SI₂) is: SI₂ = SI₁ × (15 / 5) = 3P × 3 = 9P

The total amount after 15 years is the sum of the principal and the interest earned in 15 years: Amount = P + 9P = 10P

Thus, the sum becomes 10 times of itself in 15 years.

Explanation

In any triangle PQR, the angle subtended by two vertices at the incentre O is related to the opposite vertex angle by the formula:

∠POR = 90° + ½ ∠PQR

Given that ∠POR = 120°, we can substitute this value into the formula:

120° = 90° + ½ ∠PQR

Subtracting 90° from both sides:

30° = ½ ∠PQR

Multiplying by 2:

∠PQR = 60°

This gives the measure of ∠PQR as 60°.

Explanation

Using the trigonometric identity sin² B + cos² B = 1, we can find sin B:

sin² B = 1 - cos² B

Substituting cos B = 5/9 into the equation:

sin² B = 1 - (5/9)² sin² B = 1 - 25/81 sin² B = 56/81

Taking the square root of both sides:

sin B = √56 / √81 sin B = √(4 × 14) / 9 sin B = 2√14 / 9

Since cosec B is the reciprocal of sin B:

cosec B = 1 / sin B cosec B = 9 / (2√14)

Explanation

According to the divisibility rule of 9, a number is divisible by 9 if the sum of its digits is a multiple of 9.

For the number 84pp153p, the sum of the digits is: 8 + 4 + p + p + 1 + 5 + 3 + p = 21 + 3p

For this sum to be divisible by 9, the expression 21 + 3p must be a multiple of 9. Testing the smallest non-negative integer values for the single-digit p: For p = 0, the sum is 21, which is not divisible by 9. For p = 1, the sum is 24, which is not divisible by 9. For p = 2, the sum is 21 + 3 × 2 = 27, which is divisible by 9.

Explanation

Using the standard algebraic identity for the difference of squares: m² - n² = (m - n)(m + n)

Substitute the given values m² - n² = 21 and m - n = 3 into the identity: 21 = 3 × (m + n)

Dividing both sides by 3: m + n = 21/3 = 7

Explanation

Let Q's income be 100. Since Q saves 20% of his income, Q's expenditure is 80% of his income, which is 80.

P's income is 150% of Q's income: 1.50 × 100 = 150

P's expenditure is 60% of Q's expenditure: 0.60 × 80 = 48

P's savings is the difference between P's income and P's expenditure: 150 - 48 = 102

The percentage of his own income that P saves is: (102 / 150) × 100 = 68%

Explanation

The given equations are: a + b = 24 8ab = 256

From the second equation, we can find the value of ab: ab = 256 / 8 = 32

Using the algebraic identity: a² + b² = (a + b)² - 2ab

Substitute the values of (a + b) and ab into the identity: a² + b² = 24² - 2(32) a² + b² = 576 - 64 a² + b² = 512

Now, multiply the entire expression by 3 to find the required value: 3a² + 3b² = 3(a² + b²) 3a² + 3b² = 3 × 512 3a² + 3b² = 1536

Explanation

Let the selling price of one notebook be S and the cost price of one notebook be C.

For 14 notebooks: Total Selling Price = 14S Total Cost Price = 14C

The profit made is equal to the selling price of 9 notebooks, which is 9S.

Using the profit relation: Profit = Total Selling Price - Total Cost Price 9S = 14S - 14C 14C = 5S S/C = 14/5

This gives the ratio of the selling price to the cost price as 14:5. If the cost price is 5 units, the selling price is 14 units, meaning the profit is 9 units.

Profit percent = (Profit / Cost Price) × 100 Profit percent = (9 / 5) × 100 = 180%

Explanation

Let the total work be the LCM of 30 and 15, which is 30 units.

The efficiency of A is 30 / 30 = 1 unit per day. The efficiency of B is 30 / 15 = 2 units per day.

Since B works alone for 8 days, the work completed by B is: 2 × 8 = 16 units.

The remaining work is: 30 - 16 = 14 units.

The time taken by A alone to finish this remaining work is: 14 / 1 = 14 days.

Explanation

Let 6ˣ = 3ʸ = 2ᶻ = k.

Expressing each base in terms of k: 6 = k¹/ˣ 3 = k¹/ʸ 2 = k¹/ᶻ

Using the relation 2 × 3 = 6: k¹/ᶻ × k¹/ʸ = k¹/ˣ k¹/ᶻ⁺¹/ʸ = k¹/ˣ

Equating the exponents: 1/z + 1/y = 1/x

Rearranging the terms: 1/y + 1/z - 1/x = 0

Explanation

The formula for the total amount under simple interest is: A = P × (1 + RT/100)

Here, the total amount A is Rs. 403, the rate of interest R is 10% per annum, and the time period T is 3 years. Substituting these values into the formula: 403 = P × (1 + (10 × 3)/100) 403 = P × (1 + 30/100) 403 = P × 1.3 P = 403 / 1.3 P = 310

Explanation

Using the fundamental trigonometric identity cosec²θ - cot²θ = 1, we can express the number 1 in the numerator as: 1 = cosec²θ - cot²θ = (cosecθ - cotθ)(cosecθ + cotθ)

Now, substitute this into the numerator of the expression: cotθ + cosecθ - 1 = (cotθ + cosecθ) - (cosec²θ - cot²θ) = (cosecθ + cotθ) - (cosecθ - cotθ)(cosecθ + cotθ)

Factoring out the common term (cosecθ + cotθ): = (cosecθ + cotθ) [1 - (cosecθ - cotθ)] = (cosecθ + cotθ) (1 - cosecθ + cotθ) = (cosecθ + cotθ) (cotθ - cosecθ + 1)

Substitute this factored form back into the original fraction: ((cosecθ + cotθ) (cotθ - cosecθ + 1)) / (cotθ - cosecθ + 1)

Canceling the common term (cotθ - cosecθ + 1) from both the numerator and the denominator yields: cosecθ + cotθ

Explanation

To find the value of the expression, we first simplify the given relation x⁴ + 1/x⁴ = 47.

By adding 2 to both sides of the equation: x⁴ + 1/x⁴ + 2 = 47 + 2 (x² + 1/x²)² = 49

Taking the square root of both sides gives: x² + 1/x² = 7

Adding 2 to both sides of this new relation: x² + 1/x² + 2 = 7 + 2 (x + 1/x)² = 9

Taking the square root of both sides gives: x + 1/x = 3

Using the algebraic identity for the sum of cubes: x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)

Substituting the value of x + 1/x = 3 into the identity: x³ + 1/x³ = 3³ - 3(3) x³ + 1/x³ = 27 - 9 x³ + 1/x³ = 18

Explanation

The weight of the new person is calculated directly by comparing the sum of weights before and after the new person joins.

Sum of weights of the initial 20 persons = 20 × 60 = 1200 kg

Sum of weights of the 21 persons = 21 × 59 = 1239 kg

Weight of the new person = 1239 kg - 1200 kg = 39 kg

Alternatively, using the shortcut method:

Weight of new person = New average + Initial number of persons × Change in average

Weight of new person = 59 + 20 × (59 - 60) = 59 - 20 = 39 kg

Explanation

Multiplying the numerator and the denominator inside the square root by (1 - sin A):

√((1 - sin A) / (1 + sin A)) = √((1 - sin A)(1 - sin A) / ((1 + sin A)(1 - sin A)))

Using the trigonometric identity 1 - sin² A = cos² A in the denominator:

= √((1 - sin A)² / cos² A)

Taking the square root yields:

= (1 - sin A) / cos A

Separating the terms gives:

= 1/cos A - sin A/cos A

Using the basic definitions sec A = 1/cos A and tan A = sin A/cos A:

= sec A - tan A

Explanation

The given equation is K + 1/K - 2 = 0.

Rearranging this equation gives K + 1/K = 2.

Multiplying the entire equation by K yields: K² - 2K + 1 = 0

This quadratic equation can be factored as a perfect square: (K - 1)² = 0

Solving for K gives: K = 1

Substituting K = 1 into the expression K¹⁷ + 1/K¹²: 1¹⁷ + 1/1¹² = 1 + 1 = 2.

Explanation

First, calculate the initial speed of the car using the formula: Speed = Distance / Time

The initial distance is 40 km and the time taken is 24 minutes, which is equal to 24/60 hours, or 0.4 hours. Initial speed = 40 / 0.4 = 100 km/hr

The speed is decreased by 40 km/hr, making the new speed: New speed = 100 - 40 = 60 km/hr

Now, calculate the new time taken to cover the same distance of 40 km using the formula: Time = Distance / Speed

New time = 40 / 60 hours = 2/3 hours

Convert the new time into minutes: New time in minutes = 2/3 × 60 = 40 minutes

Explanation

To find the value of L2 - L1, we first calculate the total production of cars by company J (L1) and company K (L2) over the five years.

L1 is the sum of the production by company J: L1 = 120 + 80 + 40 + 30 + 20 = 290

L2 is the sum of the production by company K: L2 = 50 + 60 + 120 + 100 + 70 = 400

Subtracting L1 from L2 gives the required difference: L2 - L1 = 400 - 290 = 110

Explanation

The given expression can be simplified by evaluating the two terms separately.

Let the first term be: T₁ = ((1 + x) / (1 - x²)) ÷ ((1 + x) / (1 - x⁴))

By converting the division into multiplication with the reciprocal: T₁ = ((1 + x) / (1 - x²)) × ((1 - x⁴) / (1 + x)) T₁ = (1 - x⁴) / (1 - x²)

Since 1 - x⁴ = (1 - x²)(1 + x²), substituting this gives: T₁ = ((1 - x²)(1 + x²)) / (1 - x²) T₁ = 1 + x²

Let the second term be: T₂ = ((1 - x⁴) / (1 - x)) × ((1 + x) / (1 - x²))

Using the same identity for 1 - x⁴: T₂ = (((1 - x²)(1 + x²)) / (1 - x)) × ((1 + x) / (1 - x²)) T₂ = ((1 + x²)(1 + x)) / (1 - x)

Now, subtracting T₂ from T₁: T₁ - T₂ = (1 + x²) - ((1 + x²)(1 + x)) / (1 - x)

Factoring out (1 + x²): T₁ - T₂ = (1 + x²) [1 - (1 + x) / (1 - x)] T₁ - T₂ = (1 + x²) [((1 - x) - (1 + x)) / (1 - x)] T₁ - T₂ = (1 + x²) [-2x / (1 - x)] T₁ - T₂ = 2x(1 + x²) / (x - 1)

Explanation

The centroid of a triangle is the geometric center of the shape, defined as the point of concurrency where the three medians of the triangle intersect. A median is a line segment drawn from any vertex of the triangle to the midpoint of its opposite side. The centroid always lies inside the triangle and divides each median in a 2:1 ratio, with the larger part being closer to the vertex.

Explanation

Using the trigonometric co-function identity cos x = sin(90° - x), we can rewrite cos 50° as:

cos 50° = sin(90° - 50°) = sin 40°

Substituting this back into the given equation:

sin(θ + 30°) = sin 40°

Equating the angles:

θ + 30° = 40° θ = 10°

Explanation

Correct Answer: B

The cost price of the first watch is Rs. 12000 and it is sold at a profit of 10%. Profit on the first watch = 12000 × 10% = Rs. 1200

The cost price of the second watch is Rs. 20000 and it is sold at a loss of 20%. Loss on the second watch = 20000 × 20% = Rs. 4000

Since the loss is greater than the profit, the transaction results in an overall loss. Overall Loss = Loss on second watch - Profit on first watch Overall Loss = 4000 - 1200 = Rs. 2800

Explanation

The stoppage time of a bus per hour is calculated using the standard relation:

Stoppage time per hour = (Speed excluding stoppages - Speed including stoppages) ÷ Speed excluding stoppages × 60 minutes

Substituting the given values:

Stoppage time per hour = (60 - 54) ÷ 60 × 60 = 6 ÷ 60 × 60 = 6 minutes

Due to stoppages, the bus covers 6 km less distance in an hour. The time required to cover this remaining distance of 6 km at the non-stop speed of 60 kmph is 6/60 hours, which is equal to 6 minutes. This represents the total stoppage time of the bus per hour.

Explanation

The salary of the employee after n years under a constant percentage increase can be calculated using the formula for compound growth:

Future Salary = Current Salary × (1 + r/100)ⁿ

Given that the current salary is Rs.14,000, the annual increase rate is 20%, and the duration is 3 years:

Future Salary = 14000 × (1 + 20/100)³ Future Salary = 14000 × (1.2)³ Future Salary = 14000 × 1.728 Future Salary = 24192

Explanation

Let the two real numbers be x and y. We are given the sum of their squares as x² + y² = 12 and their product as xy = 4.

Using the algebraic identity for the square of a difference: (x - y)² = x² + y² - 2xy

Substitute the given values into the formula: (x - y)² = 12 - 2 × 4 (x - y)² = 12 - 8 (x - y)² = 4

Taking the square root of both sides gives the absolute difference between the two numbers: |x - y| = √4 = 2

Explanation

The length of an arc of a circle with radius r and central angle θ is given by the formula:

L = (θ / 360°) × 2πr

Substituting the given values, r = 7 units and θ = 60°, and using π ≈ 22/7:

L = (60 / 360) × 2 × (22/7) × 7 L = 1/6 × 44 L = 22/3 units