SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The ratio of the number of respondents in the High IQ category to those in the Low IQ category is equal to the ratio of their respective percentages, since both percentages are calculated from the same total number of respondents (208).

Percentage of respondents in High IQ = 12.6% Percentage of respondents in Low IQ = 13.0%

Required Ratio = 12.6 : 13.0

Multiplying both terms by 10 to eliminate the decimal gives: Required Ratio = 126 : 130

Dividing both terms by their greatest common divisor, which is 2, simplifies the ratio to: Required Ratio = 63 : 65

Explanation

The double-angle identity for sine is: sin 2θ = 2 sin θ cos θ

By substituting the given values of sin θ = 6/10 and cos θ = 8/10: sin 2θ = 2 × (6/10) × (8/10) sin 2θ = 96/100

Simplifying the fraction by dividing both the numerator and the denominator by 4 yields: sin 2θ = 24/25

Explanation

The number of male workers in the company is obtained by subtracting the number of female workers from the total number of workers: Number of male workers = 960 - 420 = 540

The ratio of the total number of male workers to the total number of female workers is: Ratio = 540 : 420

Dividing both terms by their greatest common divisor, which is 60, simplifies the ratio: 540 ÷ 60 = 9 420 ÷ 60 = 7

This yields the simplified ratio of 9:7.

Explanation

According to the Pythagorean theorem, the hypotenuse c of a right-angled triangle with adjacent sides a and b is given by:

c = √(a² + b²)

Substituting the given values, a = 1.6 cm and b = 6.3 cm:

c = √(1.6² + 6.3²) c = √(2.56 + 39.69) c = √42.25 c = 6.5 cm

Explanation

Average speed is defined as the total distance divided by the total time taken.

Total time taken for the entire journey = Total distance ÷ Average speed = 750 ÷ 50 = 15 hours.

Time taken for the first 200 km of the journey = Distance ÷ Speed = 200 ÷ 40 = 5 hours.

Remaining time for the second part of the journey = Total time - Time taken for the first part = 15 - 5 = 10 hours.

Speed required for the remaining 550 km = Remaining distance ÷ Remaining time = 550 ÷ 10 = 55 kmph.

Explanation

The given equation is x² + y² + z² = 2(x + z - 1).

Rearranging all terms to the left-hand side: x² - 2x + y² + z² - 2z + 2 = 0

Grouping the terms to complete the squares: (x² - 2x + 1) + y² + (z² - 2z + 1) = 0 (x - 1)² + y² + (z - 1)² = 0

Since the sum of squares of real numbers must be zero, each individual squared term must be equal to zero: (x - 1)² = 0 → x = 1 y² = 0 → y = 0 (z - 1)² = 0 → z = 1

Substituting these values into the required expression: x³ + y³ + z³ = 1³ + 0³ + 1³ = 1 + 0 + 1 = 2

Explanation

Correct Answer: C

The area of a sector of a circle is given by the formula: A = (θ / 360°) × π × r²

where A is the area of the sector, θ is the angle subtended at the centre, and r is the radius of the circle.

Substituting the given values into the formula: 346.5 = (22.5 / 360) × (22 / 7) × r²

Simplifying the angle fraction: 22.5 / 360 = 45 / 720 = 1 / 16

Substituting this back into the equation: 346.5 = (1 / 16) × (22 / 7) × r² 346.5 = (11 / 56) × r²

Solving for r²: r² = (346.5 × 56) / 11 r² = 31.5 × 56 r² = 1764

Taking the square root of both sides to find the radius r: r = √1764 r = 42 cm

Explanation

The length of the transverse common tangent between two circles is given by the formula:

T = √(d² - (r₁ + r₂)²)

where T is the length of the transverse common tangent, d is the distance between the centres of the two circles, and r₁ and r₂ are the radii of the two circles.

Given: r₁ = 7 cm r₂ = 4 cm T = 13 cm

Substituting these values into the formula:

13 = √(d² - (7 + 4)²) 13 = √(d² - 11²)

Squaring both sides of the equation to eliminate the square root:

169 = d² - 121 d² = 169 + 121 d² = 290 d = √290 cm

Explanation

To find the time taken, use the formula:

Time = Distance / Speed

First, convert the speed from kilometers per hour (km/h) to meters per minute (m/min). Since 1 km = 1000 m and 1 hour = 60 minutes:

Speed = 24 × 1000 / 60 = 400 m/min

Now, substitute the distance of 600 m and the speed of 400 m/min into the formula:

Time = 600 / 400 = 1.5 minutes

Explanation

The successive discount formula determines the final selling price when multiple discounts are applied one after another:

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

Here, the marked price is ₹1,000, the trade discount d₁ is 10%, and the cash discount d₂ is 10%.

Substituting these values into the formula:

Selling Price = 1000 × (1 - 10/100) × (1 - 10/100) Selling Price = 1000 × 0.90 × 0.90 Selling Price = 1000 × 0.81 Selling Price = 810

Applying successive discounts of 10% and 10% reduces the original price to 81%, yielding a final selling price of ₹810.

Explanation

For a right-angled triangle ABC with the right angle at B: tan A = opposite/adjacent = 12/5

By the Pythagorean theorem, the hypotenuse is: hypotenuse = √(opposite² + adjacent²) = √(12² + 5²) = √169 = 13

Using the definitions of cosec A and sec A: cosec A = hypotenuse/opposite = 13/12 sec A = hypotenuse/adjacent = 13/5

Explanation

To find the month with the lowest production of all types of bikes, the total sum of bike types P, Q, R, and S is calculated for each of the months given in the options:

For September: 56 + 45 + 76 + 65 = 242 For October: 15 + 26 + 80 + 37 = 158 For November: 45 + 36 + 54 + 21 = 156 For December: 33 + 40 + 41 + 44 = 158

Comparing the total production of each month, the minimum production is 156, which occurred in November.

Explanation

Using the algebraic identity (a - b)² = a² + b² - 2ab, let a = 2x and b = y:

(2x - y)² = (2x)² + y² - 2(2x)(y) (2x - y)² = 4x² + y² - 4xy

Substitute the given values 4x² + y² = 40 and xy = 6 into the equation:

(2x - y)² = 40 - 4(6) (2x - y)² = 40 - 24 (2x - y)² = 16

Taking the square root on both sides:

2x - y = ±√16 = ±4

Since 4 is the positive value given in the options, the value of 2x - y is 4.

Explanation

From the given equation: cos θ - sin θ = 0 cos θ = sin θ tan θ = 1

For the acute angle θ: θ = 45°

Substituting θ = 45° into the expression sin⁸ θ + cos⁸ θ: sin θ = 1/√2 cos θ = 1/√2

Therefore: (sin θ)⁸ + (cos θ)⁸ = (1/√2)⁸ + (1/√2)⁸ = (½)⁴ + (½)⁴ = 1/16 + 1/16 = 2/16 = 1/8

Explanation

Using the compound growth formula, the population P after n years is given by:

P = P₀ × (1 + r/100)ⁿ

Where P is the present population, P₀ is the population 2 years ago, r is the annual growth rate, and n is the number of years.

Substituting the given values into the formula:

1,33,100 = P₀ × (1 + 10/100)²

1,33,100 = P₀ × (1.1)²

1,33,100 = P₀ × 1.21

P₀ = 1,33,100 / 1.21

P₀ = 1,10,000

Explanation

To find the time taken when working together, we calculate the sum of their individual daily rates: 1/4 + 1/5 + 1/10 = 11/20 of the work per day. Taking the reciprocal gives the total time required, which is 20/11 or 1 and 9/11 days.

Explanation

Let the cost price of the item for A be x.

Using the successive transactions relation, the price paid by C is related to the cost price of A by the formula: Price paid by C = x × (1 + gain percent of A / 100) × (1 - loss percent of B / 100)

Substituting the given values: 50,520 = x × (1 + 0.25) × (1 - 0.10) 50,520 = x × 1.25 × 0.90 50,520 = x × 1.125

Solving for x: x = 50,520 ÷ 1.125 x = 50,520 × 8/9 x = 404,160 ÷ 9 x = 44,906.67

Explanation

Given the equations: p = tan θ + sin θ q = tan θ - sin θ

Adding and subtracting these equations gives: p + q = 2 tan θ p - q = 2 sin θ

Multiplying these two results yields: p² - q² = (p + q)(p - q) = 4 tan θ sin θ

Now, multiplying p and q directly: pq = (tan θ + sin θ)(tan θ - sin θ) pq = tan² θ - sin² θ pq = (sin² θ / cos² θ) - sin² θ pq = sin² θ (1/cos² θ - 1) pq = sin² θ tan² θ

Taking the square root of both sides: √(pq) = sin θ tan θ

Multiplying by 4: 4√(pq) = 4 sin θ tan θ

Equating the two derived expressions: p² - q² = 4√(pq)

Explanation

Total Selling Price (SP) is the sum of Cost Price (CP) and profit: Total SP = CP + Profit Total SP = 1200 + 396 = ₹1596

The number of bananas sold is calculated by dividing the total SP by the selling price per banana: Bananas sold = Total SP ÷ SP per banana Bananas sold = 1596 ÷ 3.5 = 456

The number of rotten bananas thrown away is: Bananas thrown away = Total bananas bought - Bananas sold Bananas thrown away = 480 - 456 = 24

The percentage of bananas thrown away is: Percentage = (Bananas thrown away ÷ Total bananas bought) × 100 Percentage = (24 ÷ 480) × 100 = 5%

Explanation

The single equivalent discount for successive discounts of d₁% and d₂% is given by the formula d₁ + d₂ - (d₁ × d₂)/100.

For Ravi, the successive discounts are 36% and 5%. Equivalent discount for Ravi = 36 + 5 - (36 × 5)/100 = 41 - 1.8 = 39.2%

For Prabhat, the single discount is 40%.

Since Prabhat received a higher discount than Ravi, Ravi paid more for the item.

The difference in discount percentages is 40% - 39.2% = 0.8%.

The difference in the amount paid is 0.8% of the marked price: Difference = (0.8/100) × 500 = 0.8 × 5 = ₹4.

Thus, Ravi paid ₹4 more than Prabhat.

Explanation

The simple interest SI on a principal P at an annual rate R% for T years is given by the formula SI = P × R × T / 100.

Let the principal sum of money be P.

The simple interest accrued in 9 years at 18% per annum is: SI₂ = P × 18 × 9 / 100 = 162P / 100

The simple interest accrued in 6 years at 15% per annum is: SI₁ = P × 15 × 6 / 100 = 90P / 100

The difference between these two interests is given as ₹1,440: SI₂ - SI₁ = 1440 162P / 100 - 90P / 100 = 1440 72P / 100 = 1440 P = 1440 × 100 / 72 P = 20 × 100 P = 2000

Explanation

The remainder of any term 9ᵏ when divided by 6 can be determined using modular arithmetic. Since 9 ≡ 3 (mod 6), any power of 9 satisfies 9ᵏ ≡ 3ᵏ (mod 6).

Evaluating the powers of 3 modulo 6 shows a constant remainder: 3¹ = 3 ≡ 3 (mod 6) 3² = 9 ≡ 3 (mod 6) 3³ = 27 ≡ 3 (mod 6)

For all positive integers k, 3ᵏ ≡ 3 (mod 6). Consequently, 9ᵏ ≡ 3 (mod 6) for each term in the series.

The given expression has 2n + 1 terms: S = 9 + 9² + ... + 9²ⁿ⁺¹

Applying the modulo 6 operation to the entire sum: S ≡ 3 + 3 + ... + 3 (mod 6) [for 2n + 1 terms] S ≡ 3 × (2n + 1) (mod 6) S ≡ 6n + 3 (mod 6)

Since 6n is a multiple of 6, it is completely divisible by 6, leaving a remainder of 0. Therefore: S ≡ 3 (mod 6)

Explanation

We use the standard algebraic identity: (a - b)² = a² + b² - 2ab

Substituting the given values, a - b = 9 and ab = 20, into the equation: 9² = a² + b² - 2(20)

Calculating the terms: 81 = a² + b² - 40

Rearranging the terms to solve for a² + b²: a² + b² = 81 + 40 a² + b² = 121

Explanation

The volume of the metallic sphere is recast into N smaller cones, which means the total volume remains conserved.

Volume of the sphere = 4/3 × π × R³ Volume of one cone = 1/3 × π × r² × h

Here, the radius of the sphere R = 21 cm, the radius of each cone r = 7 cm, and the height of each cone h = 3 cm.

The number of cones obtained, N, is given by the formula: N = Volume of the sphere / Volume of one cone N = (4/3 × π × R³) / (1/3 × π × r² × h) N = (4 × R³) / (r² × h)

Substituting the given values: N = (4 × 21 × 21 × 21) / (7 × 7 × 3) N = 4 × (21/7) × (21/7) × (21/3) N = 4 × 3 × 3 × 7 N = 252

Explanation

The relation between the Selling Price (SP), Marked Price (MP), and successive discounts d₁ and d₂ is given by the formula:

SP = MP × (1 - d₁) × (1 - d₂)

Converting the percentage discounts into fractions: First discount of 12.5% is equivalent to 1/8, leaving a multiplying factor of 7/8. Second discount of 8% is equivalent to 2/25, leaving a multiplying factor of 23/25.

Substituting the given values into the formula: 5474 = MP × (7/8) × (23/25) 5474 = MP × 161 / 200 MP = (5474 × 200) / 161

Dividing 5474 by 161: 5474 ÷ 7 = 782 782 ÷ 23 = 34

MP = 34 × 200 = 6800

Explanation

The total volume of solution taken from jar A is 7 liters, where the ratio of liquids P and Q is 2:19. The quantity of liquid P from jar A is 7 × 2/(2 + 19) = 7 × 2/21 = 2/3 liters. The quantity of liquid Q from jar A is 7 × 19/21 = 19/3 liters.

The total volume of solution taken from jar B is 4 liters, where the ratio of liquids P and Q is 1:11. The quantity of liquid P from jar B is 4 × 1/(1 + 11) = 4 × 1/12 = 1/3 liters. The quantity of liquid Q from jar B is 4 × 11/12 = 11/3 liters.

In the new mixture: Total quantity of liquid P = 2/3 + 1/3 = 3/3 = 1 liter. Total quantity of liquid Q = 19/3 + 11/3 = 30/3 = 10 liters.

The ratio of liquid P to liquid Q in the new mixture is 1:10.

Explanation

Since the line segment ST is parallel to MN, triangle PST is similar to triangle PMN.

For similar triangles, the ratio of the corresponding sides is equal: ST/MN = PS/PM

The point S divides the side PM in the ratio 5 : 7, which means: PS/SM = 5/7 PM = PS + SM = 5 + 7 = 12 units

Therefore, the ratio of PS to PM is: PS/PM = 5/12

Using this ratio to find the length of ST with MN = 24 cm: ST/24 = 5/12 ST = 24 × (5/12) ST = 10 cm

Explanation

Given the equation: x + 1/x = 2

Multiplying both sides by x: x² + 1 = 2x x² - 2x + 1 = 0 (x - 1)² = 0 x = 1

Substituting x = 1 into the expression: x¹¹ + 1/x²⁰ = 1¹¹ + 1/1²⁰ = 1 + 1 = 2

Explanation

The total enrollment for Basketball in 2020 is calculated by adding the number of boys and girls enrolled in that year:

Total Enrollment = 43 + 40 = 83

This sum of 83 is the highest overall enrollment among all games across all the years listed in the data.

Explanation

By the properties of a circle, the perpendicular drawn from the centre to a chord bisects the chord, forming a right-angled triangle where the hypotenuse is the radius, one leg is the perpendicular distance from the centre, and the other leg is half the length of the chord.

Let the radius r = 10 cm and the perpendicular distance d = 6 cm. Using Pythagoras' theorem:

r² = d² + (L/2)²

Substituting the given values:

10² = 6² + (L/2)² 100 = 36 + (L/2)² (L/2)² = 64 L/2 = 8

Multiplying by 2 gives the total length of the chord:

L = 2 × 8 = 16 cm

Explanation

Let the share of C be x. Since the share of B is thrice the share of C, B's share is 3x. The share of A is twice the share of B, which makes A's share 2 × 3x = 6x.

The ratio of their investments, and consequently their profit shares, is A : B : C = 6 : 3 : 1.

The total of these ratio parts is 6 + 3 + 1 = 10 parts.

The total profit earned is ₹6,000, which corresponds to these 10 parts. Therefore, 1 part equals 6000 / 10 = ₹600.

Since A's share is 6 parts, the money A received is 6 × ₹600 = ₹3,600.

Explanation

To find the value of x, we can eliminate y from the given system of equations.

The given equations are: ax + by = 1 (Equation 1) bx + ay = 2ab / (a² + b²) (Equation 2)

Multiply Equation 1 by a: a²x + aby = a (Equation 3)

Multiply Equation 2 by b: b²x + aby = 2ab² / (a² + b²) (Equation 4)

Subtract Equation 4 from Equation 3 to eliminate y: (a² - b²)x = a - 2ab² / (a² + b²)

To simplify the right side, take a common denominator: a - 2ab² / (a² + b²) = [a(a² + b²) - 2ab²] / (a² + b²) = [a³ + ab² - 2ab²] / (a² + b²) = [a³ - ab²] / (a² + b²) = a(a² - b²) / (a² + b²)

Now substitute this back: (a² - b²)x = a(a² - b²) / (a² + b²)

Dividing both sides by (a² - b²): x = a / (a² + b²)

Explanation

The marked price (MP) is obtained by dividing the selling price (SP) by the remaining fraction after the discount. A discount of 12.5% is equivalent to 1/8, meaning the selling price is 7/8 of the marked price.

SP = MP × 7/8

2625 = MP × 7/8

MP = 2625 × 8/7

MP = 375 × 8

MP = 3000

Explanation

The relationship between the number of mechanics (M), days (D), and work done (W) is given by the formula:

(M₁ × D₁) ÷ W₁ = (M₂ × D₂) ÷ W₂

Given: M₁ = 18 mechanics D₁ = 10 days W₁ = 90 cars D₂ = 12 days W₂ = 150 cars

Substituting these values into the formula:

(18 × 10) ÷ 90 = (M₂ × 12) ÷ 150

180 ÷ 90 = 12M₂ ÷ 150

2 = 12M₂ ÷ 150

12M₂ = 2 × 150

12M₂ = 300

M₂ = 300 ÷ 12

M₂ = 25

Explanation

At the moment Savitha starts running, Parnika already has a lead consisting of her 396 m head start plus the distance she covers during the 9 seconds before Savitha starts.

Parnika's distance covered in those 9 seconds = 6 m/s × 9 s = 54 m. Total initial lead of Parnika when Savitha starts = 396 m + 54 m = 450 m.

Savitha catches up with Parnika after running for 250 seconds. During this same 250-second period, Parnika covers additional distance.

Distance covered by Parnika in 250 seconds = 6 m/s × 250 s = 1500 m.

The total distance from the start line to the point where Savitha catches Parnika is the sum of Parnika's total lead and her subsequent distance covered: Total distance = 450 m + 1500 m = 1950 m.

Since Savitha covers this distance of 1950 m in 250 seconds, her speed is: Savitha's speed = 1950 m ÷ 250 s = 7.8 m/s.

Explanation

For two externally touching circles with radii r₁ and r₂, the distance between their centres d is equal to r₁ + r₂.

The length of the direct common tangent PQ is given by the formula: PQ = √(d² - (r₁ - r₂)²)

Substituting d = r₁ + r₂ into the formula: PQ = √((r₁ + r₂)² - (r₁ - r₂)²) PQ = √(4 × r₁ × r₂) PQ = 2√(r₁ × r₂)

Given the radii r₁ = 9 cm and r₂ = 4 cm: PQ = 2√(9 × 4) PQ = 2√36 PQ = 2 × 6 = 12 cm

Explanation

In a circle with centre O, the line segments OA, OB, and OC are all radii of the circle, which means OA = OB = OC.

In triangle OAB, since OA = OB, it is an isosceles triangle. This means the angles opposite to the equal sides are equal, so angle ABO = angle BAO = 36°.

In triangle OBC, since OB = OC, it is also an isosceles triangle, which gives angle CBO = angle BCO = 48°.

Because the chords AB and BC lie on either side of the centre O, the radius OB lies between OA and OC. Therefore, the angle ABC is the sum of angle ABO and angle CBO. angle ABC = angle ABO + angle CBO = 36° + 48° = 84°

According to the circle theorem, the angle subtended by an arc at the centre of a circle is twice the angle subtended by it at any point on the boundary. Thus, the angle subtended by the minor arc AC at the centre O is angle AOC. angle AOC = 2 × angle ABC = 2 × 84° = 168°

Explanation

In ΔXYZ, L and M are the midpoints of the sides XY and XZ respectively. According to the Midpoint Theorem, the segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of its length. Thus, LM = ½ YZ, which means YZ = 2 LM.

The point N lies on LM such that LN:NM = 1:2. Since LN = 5 cm, NM is twice LN, which gives: NM = 2 × 5 = 10 cm

The total length of LM is the sum of LN and NM: LM = LN + NM = 5 + 10 = 15 cm

Using the midpoint relation: YZ = 2 × LM = 2 × 15 = 30 cm

Explanation

Using the algebraic relation where if x + 1/x = 2, then x = 1, we can let x = tan θ. Since cot θ = 1/tan θ, the given equation tan θ + cot θ = 2 simplifies to tan θ + 1/tan θ = 2, which gives tan θ = 1.

Since θ is an acute angle, tan θ = 1 implies θ = 45° and cot θ = 1.

Substituting tan θ = 1 and cot θ = 1 into the target expression: 2 tan² θ + cot² θ + tan⁴ θ cot⁴ θ = 2(1)² + 1² + (1)⁴ × (1)⁴ = 2 + 1 + 1 = 4.

Explanation

The fundamental trigonometric identity states: cosec² A - cot² A = 1

Factoring this difference of squares gives: (cosec A + cot A)(cosec A - cot A) = 1

Given that cosec A + cot A = 7, we substitute this into the identity: 7 × (cosec A - cot A) = 1 cosec A - cot A = 1/7

Adding the given equation and this result: (cosec A + cot A) + (cosec A - cot A) = 7 + 1/7 2 cosec A = 50/7 cosec A = 25/7

Explanation

Using the trigonometric identity for the triple angle: sin(3A) = 3 sin A - 4 sin³ A

Substituting A = 30° into the identity: 3 sin(30°) - 4 sin³(30°) = sin(3 × 30°) = sin(90°) = 1

Alternatively, substituting the standard value sin(30°) = ½: 3 sin(30°) - 4 sin³(30°) = 3(½) - 4(½)³ = 3/2 - 4(1/8) = 3/2 - ½ = 1

Explanation

Correct Answer: B

Let the amount of money lent at the rate of 8% be x. Then, the remaining amount lent at the rate of 9% is 30,000 - x.

The total annual simple interest is the sum of the interests from both parts: Interest = (Principal × Rate × Time) / 100

Using this relation for both parts with Time = 1 year: (x × 8 × 1)/100 + ((30,000 - x) × 9 × 1)/100 = 2,650

Multiplying the entire equation by 100 to clear the denominators: 8x + 9(30,000 - x) = 265,000

Expanding and simplifying the equation: 8x + 270,000 - 9x = 265,000 -x = 265,000 - 270,000 -x = -5,000 x = 5,000

The amount of money lent at the rate of 8% is ₹5,000.

Explanation

To find the time taken by A, B, and C working together, we use the rate of work method.

Let the total work be the Least Common Multiple (LCM) of the individual days taken by A, B, and C. LCM(9, 12, 18) = 36 units

Now, calculate the individual daily work efficiencies: Efficiency of A = 36 / 9 = 4 units/day Efficiency of B = 36 / 12 = 3 units/day Efficiency of C = 36 / 18 = 2 units/day

Combined efficiency of A, B, and C working together: Total efficiency = 4 + 3 + 2 = 9 units/day

The number of days required to complete the total work of 36 units together: Time = Total work / Total efficiency = 36 / 9 = 4 days

Explanation

Let the marked price (MP) of 1 kg of rice be ₹100.

The MP of 50 kg of rice is 50 × 100 = ₹5000. Since Arun bought this at a 15% discount, the amount he paid is: Amount paid = 5000 × (1 - 0.15) = ₹4250

The MP of 1 kg of wheat is 60% of the MP of 1 kg of rice: MP of 1 kg of wheat = 60% of 100 = ₹60 The MP of the 5 kg of wheat given for free is 5 × 60 = ₹300.

Assuming the cost price (CP) of these goods to the shopkeeper is equal to the transaction price paid by Arun, we have: CP = ₹4250

If the shopkeeper had sold both items at their marked prices to other customers, the selling price (SP) would be the sum of their marked prices: Total SP = MP of 50 kg rice + MP of 5 kg wheat = 5000 + 300 = ₹5300

The profit percentage is calculated as: Profit percentage = ((Total SP - CP) / CP) × 100 Profit percentage = ((5300 - 4250) / 4250) × 100 = (1050 / 4250) × 100 ≈ 24.7%

Explanation

Let the total income of Roma be 100%.

Roma spends 25% on food and 15% on children's books. Total percentage spent on food and books = 25% + 15% = 40%

The remaining income after these expenditures is: 100% - 40% = 60%

She spends 75% of this remaining income on uniforms, which means she is left with 25% of the remaining income: 100% - 75% = 25%

To find the percentage of the total income she is left with, calculate 25% of the remaining 60%: Leftover percentage = 25% of 60% = 0.25 × 60% = 15%

Explanation

The selling price of an item sold at a discount is calculated using the formula:

Selling Price = Marked Price × (1 - Discount Percentage / 100)

Given that the Marked Price is 250 and the discount is 12%, substituting these values into the formula gives:

Selling Price = 250 × (1 - 12 / 100) Selling Price = 250 × (88 / 100) Selling Price = 250 × 22 / 25 Selling Price = 10 × 22 Selling Price = 220

Explanation

The ratio of TVs to rice cookers in the showroom is 2:3. This means that for every 2 TVs, there are 3 rice cookers, making the total number of parts in the ratio equal to 2 + 3 = 5.

To find the percentage of rice cookers, divide the number of parts representing rice cookers by the total number of parts, and then multiply by 100:

Percentage of rice cookers = (3 / 5) × 100% = 60%

Explanation

Using the algebraic cube identity (a + b)³ = a³ + b³ + 3ab(a + b), we substitute a = x and b = 3y:

(x + 3y)³ = x³ + (3y)³ + 3(x)(3y)(x + 3y)

Simplifying the terms:

(x + 3y)³ = x³ + 27y³ + 9xy(x + 3y)

Given that x + 3y = 6, we substitute this value into the relation:

6³ = x³ + 27y³ + 9xy(6)

216 = x³ + 27y³ + 54xy

Explanation

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

Let a = 381 and b = 119.

Substituting these values into the given expression: ((381 + 119)² + (381 - 119)²) / (381² + 119²) = 2(381² + 119²) / (381² + 119²)

Canceling the common term (381² + 119²) from the numerator and the denominator yields: 2

Explanation

When two objects move in the same direction along a circular track, the time taken for them to meet for the first time is given by the formula:

t = L / (v₁ - v₂)

where L is the length of the track, and v₁ and v₂ are the speeds of the two objects with v₁ > v₂.

First, convert the speeds from km/h to m/s: Speed of Sohan (v₁) = 90 × 5/18 = 25 m/s Speed of Raju (v₂) = 72 × 5/18 = 20 m/s

The relative speed of Sohan with respect to Raju is: v_rel = v₁ - v₂ = 25 - 20 = 5 m/s

Using the track length L = 1725 m, the time required to meet for the first time is: t = 1725 / 5 = 345 seconds