SSC CHSL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The time taken by two individuals A and B working together to complete a task is given by the formula:

Time taken together = (a × b) / (a + b)

where a and b are the individual times taken by A and B respectively.

Substituting the given values, a = 10 and b = 40:

Time taken together = (10 × 40) / (10 + 40) = 400 / 50 = 8 days

Explanation

Using the formula for average speed:

Average Speed = Total Distance ÷ Total Time

The total distance of the journey is the sum of both segments: Total Distance = 60 km + 90 km = 150 km

The time taken for the first segment is: t₁ = 60 km / 40 km/h = 1.5 hours

The time taken for the second segment is: t₂ = 90 km / 45 km/h = 2 hours

The total time for the journey is: Total Time = 1.5 hours + 2 hours = 3.5 hours = 7/2 hours

Now, calculate the overall average speed: Average Speed = 150 / (7/2) = 300/7 = 42 6/7 km/h

Explanation

The volume V of a sphere is directly proportional to the cube of its diameter d, so V ∝ d³.

Let the diameter of the smaller sphere be d₁ and that of the larger sphere be d₂. Given that d₁:d₂ = 1:4, the ratio of their volumes is: V₁/V₂ = (d₁/d₂)³ = (1/4)³ = 1/64

This gives the volume of the larger sphere as V₂ = 64 V₁.

The larger sphere is melted to make 125 identical spheres. The volume of each of these 125 spheres is: vₗarge = V₂ / 125 = 64 V₁ / 125

The smaller sphere is melted to make 27 identical spheres. The volume of each of these 27 spheres is: vₛmall = V₁ / 27

The ratio of the volume of each of the 125 spheres to each of the 27 spheres is given as 1:m: vₗarge / vₛmall = 1 / m (64 V₁ / 125) / (V₁ / 27) = 1 / m (64 / 125) × (27 / 1) = 1 / m 1728 / 125 = 1 / m m = 125 / 1728

Explanation

The coordinates of the common point of intersection are found by solving the system of simultaneous linear equations:

x + y = 5 x - y = 3

Adding the two equations: (x + y) + (x - y) = 5 + 3 2x = 8 x = 4

Substituting x = 4 into the first equation: 4 + y = 5 y = 1

The coordinates of the common point are (4, 1).

Explanation

Let the total salary of the man be S.

The man spends 60% of his salary on household expenses and 20% on rent, making the total expenditure 80% of his salary.

The remaining salary after these expenses is: Remaining = S × (1 - 0.80) = 0.20S

He donates 80% of this remaining amount to a trust, which leaves him with 20% of the remaining salary: Final savings = 0.20 × 0.20S = 0.04S

Given that the final savings are ₹20,000, we set up the equation: 0.04S = 20,000 S = 20,000 / 0.04 S = 5,00,000

Explanation

The perpendicular from the center of a circle to a chord bisects the chord. Let the radius of the circle be R = 5√13 cm.

For the chord of length 20 cm, the half-length is 10 cm. The perpendicular distance d₁ from the center to this chord is given by the Pythagorean theorem: d₁² + 10² = R² d₁² + 100 = (5√13)² d₁² + 100 = 325 d₁² = 225 d₁ = 15 cm

For the chord of length 12 cm, the half-length is 6 cm. The perpendicular distance d₂ from the center to this chord is: d₂² + 6² = R² d₂² + 36 = 325 d₂² = 289 d₂ = 17 cm

Since both chords lie on the same side of the center, the distance between them is the difference between their perpendicular distances from the center: d = d₂ - d₁ = 17 - 15 = 2 cm

Explanation

The angle between two tangents drawn from an external point to a circle and the angle subtended by the line segment joining the points of contact at the center are supplementary. This is because the radii OQ and OR are perpendicular to the tangents PQ and PR respectively, making ∠OQP = 90° and ∠ORP = 90°.

In quadrilateral OQPR, the sum of all interior angles is 360°: ∠QPR + ∠OQP + ∠QOR + ∠ORP = 360°

Substituting the known values: ∠QPR + 90° + 150° + 90° = 360° ∠QPR + 330° = 360° ∠QPR = 360° - 330° = 30°

Explanation

The formula for simple interest is SI = P × R × T / 100, where P is the principal, R is the rate of interest per annum, and T is the time in years.

We are given that the simple interest is equal to the principal, which means SI = P.

The rate of interest is R = 2 6/7% = 20/7% per annum.

Substituting these values into the formula: P = P × (20/7) × T / 100

Dividing both sides by P: 1 = 20 × T / (7 × 100) 1 = T / 35 T = 35 years

Explanation

When a quantity is successively changed by x% and y%, the net percentage change is calculated using the formula:

Net change% = x + y + xy/100

In this case, the salary is first decreased by 50% (x = -50) and then increased by 50% (y = +50). Substituting these values into the formula:

Net change% = -50 + 50 + (-50 × 50)/100 Net change% = 0 - 2500/100 Net change% = -25%

The negative sign indicates a net decrease, meaning Arun lost 25% of his salary.

Explanation

Let the cost price of 1000 grams of oil be ₹1000.

The marked price of the oil after a 40% markup is: Marked Price = 1000 × 1.40 = ₹1400

With a discount of 32% offered on the marked price, the selling price becomes: Selling Price = 1400 × (1 - 0.32) = 1400 × 0.68 = ₹952

Since the faulty machine measures 15% less, the quantity actually delivered is: Delivered Quantity = 1000 × (1 - 0.15) = 850 grams

The actual cost price of this delivered quantity to the shopkeeper is: Actual Cost Price = ₹850

Now, the profit percentage is calculated using the formula: Profit % = ((Selling Price - Actual Cost Price) ÷ Actual Cost Price) × 100 Profit % = ((952 - 850) ÷ 850) × 100 Profit % = (102 ÷ 850) × 100 = 12% Profit

Explanation

Let the number of ₹5, ₹10, and ₹20 coins received by Radhika be 3x, 5x, and 7x respectively.

The total value of these coins is the sum of the products of the number of coins and their respective values: (3x × 5) + (5x × 10) + (7x × 20) = 1845

Simplifying the equation: 15x + 50x + 140x = 1845 205x = 1845 x = 1845 ÷ 205 x = 9

The smallest coin in denomination is the ₹5 coin. The total worth of the ₹5 coins is: 3x × 5 = 15x

Substituting the value of x: 15 × 9 = 135

Explanation

In any triangle ABC, the sum of all interior angles is 180°. Since ∠B = 90°, the sum of the remaining acute angles A and C is: A + C = 180° - 90° = 90°

Using the trigonometric addition formula for sine: sin(A + C) = sinA cosC + cosA sinC

Substituting the value of A + C = 90° into the identity: sin(90°) = sinA cosC + cosA sinC

Since sin(90°) = 1, the value of the given expression is 1.

Explanation

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

Area(ΔABC) / Area(ΔDEF) = (Perimeter of ΔABC / Perimeter of ΔDEF)²

Substituting the given perimeter values into the formula:

Area(ΔABC) / Area(ΔDEF) = (24 / 9)²

Simplifying the fraction inside the parentheses by dividing both the numerator and the denominator by their greatest common divisor, 3:

Area(ΔABC) / Area(ΔDEF) = (8 / 3)² = 64 / 9

This gives the ratio of the areas as 64 : 9.

Explanation

Using the formula Distance = Speed × Time, since the distance covered is the same in both cases, the relationship between speeds and times is S₁ × T₁ = S₂ × T₂.

First, convert the times into hours: Walking time T₁ = 3 hours 45 minutes = 3 + 45/60 = 15/4 hours Cycling time T₂ = 2 hours 10 minutes = 2 + 10/60 = 13/6 hours

Given the walking speed S₁ = 5 km/h, substitute these values into the formula to find the cycling speed S₂: 5 × 15/4 = S₂ × 13/6 75/4 = S₂ × 13/6 S₂ = (75/4) × (6/13) S₂ = 225/26 S₂ = 8 17/26 km/h

Explanation

To find the percentage of marks obtained by Mohan, first calculate the individual marks obtained in each subject based on the given percentages and maximum marks:

Physics: 80% of 300 = 240 marks Maths: 60% of 300 = 180 marks Music: 70% of 100 = 70 marks Hindi: 95% of 100 = 95 marks Botany: 80% of 200 = 160 marks

Total marks obtained by Mohan = 240 + 180 + 70 + 95 + 160 = 745 marks

Total maximum marks for all subjects = 300 + 300 + 100 + 100 + 200 = 1000 marks

Percentage of marks obtained = (Total marks obtained ÷ Total maximum marks) × 100 Percentage of marks obtained = (745 ÷ 1000) × 100 = 74.5%

Explanation

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

For the 5-digit number 627x5, the sum of the digits is: 6 + 2 + 7 + x + 5 = 20 + x

The smallest multiple of 9 greater than or equal to 20 is 27. Setting the sum of the digits to this multiple: 20 + x = 27 x = 7

Explanation

According to the BODMAS rule, operations must be performed in the order of Division and Multiplication (from left to right), followed by Addition and Subtraction (from left to right).

The given expression is: 3 + 28 ÷ 35 × 15 + 4 - 3 × 4

First, perform the division: 28 ÷ 35 = 28/35 = 4/5

Next, perform the multiplications: (4/5) × 15 = 12 3 × 4 = 12

Substitute these values back into the expression: 3 + 12 + 4 - 12

Finally, perform addition and subtraction from left to right: 15 + 4 - 12 = 19 - 12 = 7

Explanation

Let the original marked price of the article be 100.

The shopkeeper raises the marked price by 40%, making the new marked price: 100 + 40 = 140

To sell the article at the original marked price of 100, the shopkeeper must offer a discount of: 140 - 100 = 40

The discount percentage is calculated on the new marked price of 140: Discount percentage = (Discount / New Marked Price) × 100 Discount percentage = (40 / 140) × 100 = (2 / 7) × 100 = 200/7% = 28 4/7%

Explanation

When two circles touch each other, either externally or internally, they share a unique common tangent line at their point of contact. According to the properties of circles, the radius drawn from the center of a circle to the point of tangency is perpendicular to the tangent line at that point. Since both circles share the same tangent line at their point of contact, the line segment connecting the center of the first circle to the point of contact and the line segment connecting the center of the second circle to the point of contact are both perpendicular to this common tangent line at the same point. This makes the two centers and the point of contact collinear. Therefore, the point of contact always lies on the line joining the centers of the two circles.

Explanation

Let the sale amount two years ago be S. The sale increases by 12% in the first year and then by 15% in the second year. The formula for successive percentage increase is: Present Sale = S × (1 + r₁/100) × (1 + r₂/100)

Substituting the given values: 1,28,800 = S × (1 + 12/100) × (1 + 15/100) 1,28,800 = S × 1.12 × 1.15 1,28,800 = S × 1.288 S = 1,28,800 / 1.288 S = 1,00,000

Explanation

Since PQ is parallel to SR in trapezium PQRS, the alternate interior angles are equal: ∠OPQ = ∠ORS and ∠OQP = ∠OSR. Thus, triangle POQ is similar to triangle ROS.

By the property of similar triangles, the ratio of their corresponding sides is equal: PO / OR = OQ / OS

Substitute the given values into the equation: (3x - 15) / (x - 5) = (x + 9) / 5

Cross-multiply to solve the equation: 5 × (3x - 15) = (x - 5) × (x + 9) 15x - 75 = x² + 4x - 45 x² - 11x + 30 = 0

Factor the quadratic equation: (x - 6)(x - 5) = 0

This gives two values for x: x₁ = 6 x₂ = 5

Now, calculate the value of (x₁² - x₂²): x₁² - x₂² = 6² - 5² = 36 - 25 = 11

Explanation

The list price of the article is £100.

After successive discounts of 10% and 20%, the purchase price of the article is calculated as: Purchase Price = 100 × (1 - 0.10) × (1 - 0.20) Purchase Price = 100 × 0.90 × 0.80 = 72

The dealer spends 10% of this purchase price on transportation: Transportation Cost = 72 × 0.10 = 7.2

The total cost price (CP) to the dealer is the sum of the purchase price and transportation cost: Total CP = 72 + 7.2 = 79.2

To earn a profit of 15%, the selling price (SP) is calculated using the formula: SP = Total CP × (1 + Profit%) SP = 79.2 × 1.15 = 91.08

Explanation

The perimeter of a square is given by the formula P = 4s, where s is the side length of the square. Given that the perimeter is 64 cm, the side length is:

s = 64 / 4 = 16 cm

The area of a square is given by the formula A = s². Substituting the side length of 16 cm into this formula yields:

A = 16² = 256 cm²

Explanation

The volume V of a right circular cone is given by the formula V = ⅓πr²h, where r is the radius of the base and h is the height.

Given: Radius r = 10.5 cm = 21/2 cm Height h = 24 cm π = 22/7

Substituting these values into the formula:

V = ⅓ × 22/7 × (21/2)² × 24

V = ⅓ × 22/7 × 441/4 × 24

V = 22/21 × 441/4 × 24

V = 22 × 21/4 × 24

V = 22 × 21 × 6

V = 132 × 21

V = 2772 cm³

Explanation

To find the company with the minimum average of manufacturing computers over the five-year period from 2010 to 2014, we calculate the total production for each company, as the number of years is constant (5 years) for all.

The total production for Company A is: 240 + 300 + 1250 + 470 + 320 = 2580

The average production for Company A is: 2580 ÷ 5 = 516 thousand computers

Comparing this with the total production of the other companies: Company B: 120 + 530 + 650 + 320 + 1070 = 2690 Company C: 270 + 490 + 430 + 1180 + 550 = 2920 Company D: 280 + 650 + 710 + 890 + 480 = 3010

Company A has the lowest total production of 2580 thousand, which gives it the minimum average production of 516 thousand computers. This corresponds to Company A.

Explanation

The given trigonometric equation is: 3 cos²θ - 2 cos θ = sin θ - 3 sin²θ

Rearranging the terms to group the quadratic and linear terms on separate sides: 3 cos²θ + 3 sin²θ = 2 cos θ + sin θ

Factor out 3 on the left side: 3(cos²θ + sin²θ) = 2 cos θ + sin θ

Using the fundamental trigonometric identity cos²θ + sin²θ = 1: 3 = 2 cos θ + sin θ

To express this relation in terms of cotangent, divide both sides of the equation by sin θ: 3/sin θ = 2 cos θ/sin θ + sin θ/sin θ 3 cosec θ = 2 cot θ + 1

Squaring both sides to eliminate the cosecant term: (3 cosec θ)² = (2 cot θ + 1)² 9 cosec²θ = 4 cot²θ + 4 cot θ + 1

Substitute the identity cosec²θ = 1 + cot²θ into the equation: 9(1 + cot²θ) = 4 cot²θ + 4 cot θ + 1 9 + 9 cot²θ = 4 cot²θ + 4 cot θ + 1 5 cot²θ - 4 cot θ + 8 = 0

This quadratic equation in terms of cot θ directly relates the given equation to cot²θ.

Explanation

In a circle, the tangents AB and AC drawn from an external point A to the points of contact B and C are perpendicular to the radii XB and XC respectively. This means angle ABX = 90° and angle ACX = 90°.

Since the sum of the interior angles of quadrilateral ABXC is 360°, the angles BAC and BXC are supplementary:

angle BAC + angle BXC = 180°

Substituting the given value of angle BXC = 125° into the equation:

angle BAC + 125° = 180° angle BAC = 180° - 125° = 55°

Explanation

Let the income of Q be 100.

Since the income of P is 40% less than the income of Q: Income of P = 100 - 40 = 60

The combined income of P and Q is: Combined Income = 60 + 100 = 160

To find how much percent the combined income is more than the income of P, we calculate the difference relative to the income of P: Percentage excess = ((Combined Income - Income of P) / Income of P) × 100 Percentage excess = ((160 - 60) / 60) × 100 Percentage excess = (100 / 60) × 100 Percentage excess = (5 / 3) × 100 Percentage excess = 500 / 3 % Percentage excess = 166 2/3 %

Explanation

To convert a percentage to a fraction, divide the percentage value by 100 and simplify:

Fraction = Percentage / 100

Substituting 12.75% into the formula: 12.75% = 12.75 / 100

Multiply both the numerator and the denominator by 100 to eliminate the decimal: 12.75 / 100 = 1275 / 10000

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 25: 1275 ÷ 25 = 51 10000 ÷ 25 = 400

This yields the simplified fraction of 51 / 400.

Explanation

The simple interest earned is the difference between the final amount and the principal amount: Simple Interest = ₹72,000 - ₹50,000 = ₹22,000

The formula for simple interest is: Simple Interest = (Principal × Rate × Time) / 100

Rearranging this formula to find the rate of interest: Rate = (Simple Interest × 100) / (Principal × Time)

Substituting the given values: Rate = (22,000 × 100) / (50,000 × 4) Rate = 2,200,000 / 200,000 Rate = 11% per annum

Explanation

The number of valid votes obtained by the other candidate is calculated directly using the formula:

Other candidate's votes = Total votes × (100% - Invalid votes%) × (100% - Winner's share of valid votes)

Substituting the given values into the formula:

Other candidate's votes = 8400 × (1 - 0.15) × (1 - 0.60) Other candidate's votes = 8400 × 0.85 × 0.40 Other candidate's votes = 7140 × 0.40 = 2856

Explanation

The total price of the laptop is ₹48,000. To match a total discount of 9.5% on this amount, the total discount value required is:

Total discount = 9.5% of 48,000 = (9.5 / 100) × 48,000 = 4,560

Rohan has already offered two discounts: First discount of 12% on ₹28,000 = (12 / 100) × 28,000 = 3,360 Second discount of 8% on ₹12,000 = (8 / 100) × 12,000 = 960

The sum of these two discounts is: 3,360 + 960 = 4,320

The remaining discount to be offered on the final ₹8,000 is the difference between the required total discount and the sum of the discounts already given: Remaining discount = 4,560 - 4,320 = 240

Explanation

The volume of each cube is 216 cm³. Let the side length of each cube be a.

Using the formula for the volume of a cube: a³ = 216 a = 6 cm

When four such cubes are joined end to end, they form a cuboid. The dimensions of this new cuboid are: Length, l = 4 × 6 = 24 cm Width, w = 6 cm Height, h = 6 cm

The formula for the total surface area of a cuboid is 2(lw + wh + hl). Substituting these values: Surface area = 2 × (24 × 6 + 6 × 6 + 6 × 24) Surface area = 2 × (144 + 36 + 144) Surface area = 2 × 324 = 648 cm²

Alternatively, joining four cubes end to end creates three joints, hiding two adjacent faces at each joint. Out of the 24 total faces of the four independent cubes, 6 faces are hidden. The surface area of the remaining 18 exposed faces is: 18 × a² = 18 × 6² = 18 × 36 = 648 cm²

Explanation

Correct Answer: D

When two objects move in opposite directions from the same point, their relative speed is the sum of their individual speeds.

Relative Speed = Speed of Shyam + Speed of Ram

Given: Speed of Shyam = 5¼ km/hr = 21/4 km/hr Speed of Ram = 3 km/hr

Relative Speed = 21/4 + 3 = 33/4 km/hr

The required distance between them is 44 km.

Time = Distance ÷ Relative Speed Time = 44 ÷ (33/4) = 44 × 4/33 = 16/3 hours

Converting 16/3 hours into hours and minutes: 16/3 hours = 5⅓ hours ⅓ hour = ⅓ × 60 minutes = 20 minutes

This gives a total time of 5 hours 20 minutes.

Explanation

When x is added to four numbers a, b, c, and d to make them in proportion, the value of x can be directly calculated using the formula:

x = (ad - bc) / ((b + c) - (a + d))

Given the numbers are a = 7, b = 11, c = 18, and d = 23, substituting these values yields:

x = (7 × 23 - 11 × 18) / ((11 + 18) - (7 + 23)) x = (161 - 198) / (29 - 30) x = -37 / -1 = 37

Using x = 37, the two expressions become:

x - 1 = 37 - 1 = 36 2x - 10 = 2 × 37 - 10 = 64

The mean proportional between two numbers A and B is defined by the formula √(A × B).

Mean proportional = √(36 × 64) = 6 × 8 = 48

Explanation

Let the number of correct answers be c. Since there are 160 questions in total, the number of wrong or unattempted questions is 160 - c.

Each correct answer earns 4 marks, and each wrong or unattempted question deducts 1 mark. The equation for the total score is:

4c - 1 × (160 - c) = 400

Simplifying the equation:

4c - 160 + c = 400

5c - 160 = 400

5c = 560

c = 112

Therefore, the candidate answered 112 questions correctly.

Explanation

Volume of the cylindrical rod with radius R and height H = 12R is: V_cylinder = π R² H = π R² (12R) = 12 π R³

Volume of a spherical ball with radius r = R/3 is: V_ball = 4/3 π r³ = 4/3 π (R/3)³ = 4/81 π R³

The number of spherical balls N is the ratio of the volume of the cylinder to the volume of one spherical ball: N = V_cylinder / V_ball N = (12 π R³) / (4/81 π R³) N = (12 × 81) / 4 N = 3 × 81 = 243

Explanation

The length of the direct common tangent to two circles touching each other externally is given by the formula:

Length = 2 × √(R × r)

where R and r are the radii of the two circles.

Given: Radius of the first circle, R = 6 cm Radius of the second circle, r = 3 cm

Substituting these values into the formula:

Length = 2 × √(6 × 3) Length = 2 × √18 Length = 2 × 3√2 Length = 6√2 cm

Explanation

Let the total work be the LCM of 6 and 8, which is 24 units.

The daily work efficiency of Riya is 24 ÷ 6 = 4 units per day, and the daily work efficiency of Sangeeta is 24 ÷ 8 = 3 units per day.

Riya worked alone for the first 3 days, completing 3 × 4 = 12 units of work.

The remaining work to be completed is 24 - 12 = 12 units.

When Sangeeta joined Riya, their combined daily efficiency became 4 + 3 = 7 units per day. The time taken by both to finish the remaining work is 12 ÷ 7 = 1 5/7 days.

The total duration of the work is the sum of the days Riya worked alone and the days they worked together, which is 3 + 1 5/7 = 4 5/7 days.

Explanation

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their perimeters.

Since the ratio of the perimeters is 4 : 7, the ratio of their areas is (4/7)² = 16 : 49.

Let the areas of the two triangles be 16x and 49x.

The sum of their areas is given as: 16x + 49x = 195 65x = 195 x = 3

The difference between their areas is: 49x - 16x = 33x

We need to find 1/3 of the difference between the areas: 1/3 × 33x = 11x

Substituting x = 3 into the expression: 11 × 3 = 33 cm²

Explanation

Using the formula for simple interest: SI = (P × R × T) / 100

Let the principal sum be P, the rate of interest per annum be R, and the time T be 8 years. The simple interest SI is 4/5 of the sum P: SI = (4/5) × P

Substitute these values into the formula: (4/5) × P = (P × R × 8) / 100

Cancel P from both sides: 4/5 = 8R / 100

Rearrange the equation to solve for R: R = (4 × 100) / (5 × 8) R = 400 / 40 R = 10%

Explanation

To simplify the expression, we use the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction).

First, convert all the mixed fractions into improper fractions: 2¼ = 9/4 3⅕ = 16/5 5⅙ = 31/6 2 4/31 = 66/31 3⅔ = 11/3 4⅐ = 29/7 1 6/29 = 35/29

Substituting these fractions into the original expression gives: (9/4) × (16/5) + (31/6) of (66/31) ÷ (11/3) + (29/7) of (35/29)

According to BODMAS, we first evaluate the "of" terms: (31/6) of (66/31) = (31/6) × (66/31) = 11 (29/7) of (35/29) = (29/7) × (35/29) = 5

Substitute these values back into the expression: (9/4) × (16/5) + 11 ÷ (11/3) + 5

Next, perform the division operation: 11 ÷ (11/3) = 11 × (3/11) = 3

Substitute this value back: (9/4) × (16/5) + 3 + 5

Next, perform the multiplication operation: (9/4) × (16/5) = 36/5

Finally, perform the addition: 36/5 + 3 + 5 = 36/5 + 8 = (36 + 40)/5 = 76/5 = 15⅕

Explanation

In a circle, equal chords are equidistant from the center. Since the two parallel chords are of equal length (8 units each) and are 6 units apart, the center of the circle lies exactly midway between them. Therefore, the perpendicular distance from the center to each chord is d = 6 / 2 = 3 units.

A perpendicular from the center to a chord bisects the chord. The length of the half-chord is a = 8 / 2 = 4 units.

The radius r of the circle, the perpendicular distance d from the center to the chord, and the half-chord length a form a right-angled triangle. Applying the Pythagorean theorem:

r² = d² + a²

r² = 3² + 4²

r² = 9 + 16

r² = 25

r = √25 = 5 units.

Explanation

The decrease in the selling price of the item is: 5,000 - 4,680 = 320

Since loss percentage is calculated on the Cost Price (CP), this decrease of 320 corresponds to the 4% increase in the loss percentage.

4% of CP = 320 CP = (320 × 100) / 4 = 8,000

To make a profit of 4%, the selling price must be 104% of the CP: Required Selling Price = 8,000 × 1.04 = 8,320

Explanation

Using the trigonometric identity for the tangent of the sum of two angles: tan(α + β) = (tan α + tan β) / (1 - tan α tan β)

Given that α + β = 45° and tan(45°) = 1, we substitute these values: (tan α + tan β) / (1 - tan α tan β) = 1 tan α + tan β = 1 - tan α tan β tan α + tan β + tan α tan β = 1

Now, expanding the given product: (tan α + 1)(tan β + 1) = tan α tan β + tan α + tan β + 1

Substituting the relation tan α + tan β + tan α tan β = 1 into the expanded expression: (tan α + 1)(tan β + 1) = 1 + 1 = 2

Since the given equation is (tan α + 1)(tan β + 1) = 2x, we have: 2x = 2 x = 1

Explanation

Total percentage of income spent on various expenses = 10% + 10% + 20% + 15% = 55%

Percentage of income saved = 100% - 55% = 45%

Monthly income = ₹30,000

Savings = 45% of ₹30,000 = (45/100) × 30,000 = 45 × 300 = ₹13,500

Explanation

Let CP be the cost price and SP be the selling price of the article.

According to the given condition: 70% of CP = 40% of SP

70/100 × CP = 40/100 × SP 7 × CP = 4 × SP SP/CP = 7/4

The formula for profit percentage is: Profit % = (SP - CP)/CP × 100

Substituting the ratio values: Profit % = (7 - 4)/4 × 100 Profit % = 3/4 × 100 Profit % = 75%

Explanation

A number is divisible by 22 if it is divisible by both 2 and 11.

To be divisible by 2, the number must end in an even digit. The number 893002 ends in 2, which is even, so it is divisible by 2.

To be divisible by 11, the difference between the sum of the digits at the odd positions and the sum of the digits at the even positions must be 0 or a multiple of 11.

For the number 893002: Sum of digits at odd positions from the left (1st, 3rd, and 5th digits) = 8 + 3 + 0 = 11. Sum of digits at even positions from the left (2nd, 4th, and 6th digits) = 9 + 0 + 2 = 11. Difference = 11 - 11 = 0.

Since the difference is 0, the number 893002 is divisible by 11. Therefore, it is divisible by 22.

Explanation

To convert a percentage into a ratio, express the percentage value as a fraction with a denominator of 100 and then simplify it to its lowest terms.

28% = 28/100

Dividing both the numerator and the denominator by their greatest common divisor, which is 4, yields:

28 ÷ 4 = 7 100 ÷ 4 = 25

This gives the simplified fraction 7/25, which is expressed in ratio form as 7:25.

Explanation

To find the number nearest to 7658 that is exactly divisible by 45, we divide 7658 by 45 to determine the remainder: 7658 ÷ 45 = 170 with a remainder of 8.

The multiple of 45 just below 7658 is: 7658 - 8 = 7650

The multiple of 45 just above 7658 is: 7658 + (45 - 8) = 7695

Comparing the absolute differences from 7658: 7658 - 7650 = 8 7695 - 7658 = 37

Since 8 is smaller than 37, 7650 is the nearest number to 7658 that is exactly divisible by 45.