SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

To find the weight of Kangra Valley per kilogram in the new brand, we mix equal parts of both barrels, say 0.5 kg of each to make 1 kg of the new brand. Since Barrel 1 contains no Kangra Valley, and Barrel 2 contains Kangra Valley in the ratio of 3 out of 12 parts (or 1/4th of its weight), the weight of Kangra Valley in 0.5 kg of Barrel 2 is 0.5 kg * (1/4) = 1/8 kg, which equals 125 gm. Thus, 1 kg of the new brand contains 125 gm of Kangra Valley.

Explanation

The height of Vanaja is 117 cm and the height of Uma is 133 cm. To find the number of students whose height is between that of Vanaja and Uma, we identify the students with heights strictly greater than 117 cm and less than 133 cm.

From the given table, the students falling within this range (117 cm < height < 133 cm) are Sailaja with a height of 125 cm, Amala with a height of 132 cm, and Leela with a height of 121 cm. This gives a total of three students.

Explanation

Using the simple interest formula: Amount = Principal + Simple Interest

Since the sum of money becomes five times its initial value, the final amount is 5P, where P is the principal. This gives the Simple Interest: SI = 5P - P = 4P

The formula for simple interest is: SI = (P × R × T) / 100

Substituting the values SI = 4P and T = 18 years into the formula: 4P = (P × R × 18) / 100 4 = 18R / 100 R = 400 / 18 R = 22.22%

Rounding 22.22% to the nearest integer yields 22%.

Explanation

If one diagonal of a rhombus is equal to its side, the rhombus is made of two equilateral triangles joined base-to-base, making the half-diagonals a/2 and (a√3)/2. Thus, the ratio of the diagonals is d1 : d2 = a : a√3, which simplifies to 1 : √3.

Explanation

Let x = cos θ. Since sec θ = 1/cos θ, the given relation is: x + 1/x = √3

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

Substituting the given value of x + 1/x: (√3)³ = x³ + 1/x³ + 3(√3) 3√3 = x³ + 1/x³ + 3√3

Subtracting 3√3 from both sides gives: x³ + 1/x³ = 0

Thus, cos³θ + sec³θ = 0.

Explanation

Tanvi secured the following percentage of marks in each of the six subjects: Maths: 70% Chemistry: 75% Physics: 65% Geography: 85% History: 40% Computer Science: 60%

Since a student is considered to have received a distinction if they secure 80% or more in a subject, Tanvi achieved a distinction only in Geography, which counts as exactly one subject.

Explanation

To find the number nearest to 13051 that is divisible by 9, we apply the divisibility rule for 9, which states that a number is divisible by 9 if the sum of its digits is divisible by 9.

Checking the sum of the digits for 13059: 1 + 3 + 0 + 5 + 9 = 18

Since 18 is divisible by 9, the number 13059 is divisible by 9. It is the only number among the options that satisfies this divisibility condition.

Explanation

The slant height l, height h, and radius r of a cone are related by the formula l² = r² + h².

Substituting the given values: 29² = r² + 21² 841 = r² + 441 r² = 400 r = 20 cm

The volume V of a cone is given by the formula V = (1/3) × π × r² × h.

Substituting the values into the formula: V = (1/3) × (22/7) × 400 × 21 V = 22 × 400 V = 8800 cm³

Explanation

To find the total amount paid, we first calculate the number of days for which the interest is charged from 26th March 2022 to 7th June 2022.

Number of days in March = 31 − 26 = 5 days Number of days in April = 30 days Number of days in May = 31 days Number of days in June = 7 days Total duration = 5 + 30 + 31 + 7 = 73 days

Since 2022 is a non-leap year, the time period in years is T = 73/365 = 1/5 year.

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

Substituting the given values: SI = (75000 × 9 × 1/5) / 100 SI = 150 × 9 SI = 1350

The total amount paid to clear the account is: Amount = Principal + Simple Interest Amount = 75000 + 1350 = 76350

Explanation

Using the trigonometric identity cos(θ) = sin(90° - θ), the equation cos(3α) = sin(α - 22°) can be rewritten as:

sin(90° - 3α) = sin(α - 22°)

Since both angles are in the first quadrant, we can equate the arguments directly:

90° - 3α = α - 22°

Rearranging the terms to solve for α:

90° + 22° = α + 3α

112° = 4α

α = 112° ÷ 4

α = 28°

This value satisfies the condition 3α < 90° since 3 × 28° = 84° < 90°.

Explanation

Let the total number of voters be V.

With 80% of the voters casting their votes: Votes cast = 0.80 × V

Since 2% of the cast votes were invalid, the valid votes constitute 98% of the cast votes: Valid votes = 0.98 × 0.80 × V = 0.784 × V

Candidate A received 60% of the valid votes: Votes secured by A = 0.60 × 0.784 × V = 0.4704 × V

Given that A got 9,408 votes: 0.4704 × V = 9,408 V = 9,408 ÷ 0.4704 = 20,000

Explanation

Using the algebraic identity (x + y)² = x² + y² + 2xy, the given equation can be simplified and squared to find the required value.

First, simplify the given equation by adding 2 to both sides: 5a + 4/a = 15

Next, square both sides of this equation: (5a + 4/a)² = 15²

Expand the left side using the identity (x + y)² = x² + y² + 2xy: 25a² + 16/a² + 2 × 5a × (4/a) = 225

Simplify the middle term by canceling a: 25a² + 16/a² + 40 = 225

Subtract 40 from both sides to find the target value: 25a² + 16/a² = 225 - 40 25a² + 16/a² = 185

Explanation

A number is divisible by 6 if it is divisible by both 2 and 3.

For the number 3426x to be divisible by 2, its last digit x must be an even number, meaning x must be 0, 2, 4, 6, or 8.

For the number 3426x to be divisible by 3, the sum of its digits must be divisible by 3. The sum of the digits is 3 + 4 + 2 + 6 + x = 15 + x. Since 15 is already divisible by 3, the value of x must also be a multiple of 3, meaning x must be 0, 3, 6, or 9.

To satisfy both divisibility conditions, x must be an even multiple of 3, which limits the possible values of x to 0 or 6.

Additionally, the number is not divisible by 5, which means its units digit x cannot be 0 or 5. Excluding 0 from the possible values leaves x = 6 as the only valid solution.

Explanation

The weighted average formula for a mixture of three components is used to find the price of the third variety:

Average Price = (w₁p₁ + w₂p₂ + w₃p₃) / (w₁ + w₂ + w₃)

Here, the ratios of the three varieties are w₁ = 1, w₂ = 1, and w₃ = 2. The prices of the first two varieties are p₁ = 504 and p₂ = 540. The average price of the mixture is 612. Let the price of the third variety be p₃.

Substituting the values into the formula:

612 = (1 × 504 + 1 × 540 + 2 × p₃) / (1 + 1 + 2)

612 = (1044 + 2p₃) / 4

Multiplying both sides by 4:

2448 = 1044 + 2p₃

2p₃ = 2448 - 1044

2p₃ = 1404

p₃ = 702

Explanation

The weight of the new player can be calculated directly using the replacement formula:

Weight of new player = Weight of replaced player + (Total number of players × Increase in average weight)

Substituting the given values into the formula:

Weight of new player = 63 + 12 × 1.5 Weight of new player = 63 + 18 Weight of new player = 81 kg

Explanation

Using the algebraic identity for the difference of cubes: x³ - y³ = (x - y)(x² + xy + y²)

Substitute this identity into the given expression: ((x³ - y³)(x + y)) / (x² + xy + y²) = ((x - y)(x² + xy + y²)(x + y)) / (x² + xy + y²)

Cancel the common term (x² + xy + y²) from the numerator and the denominator: (x - y)(x + y)

Applying the difference of squares formula: (x - y)(x + y) = x² - y²

Explanation

When two circles of different radii r₁ and r₂ touch each other externally, the distance d between their centers is exactly equal to the sum of their radii, so d = r₁ + r₂.

For this configuration, we can draw two types of common tangents: Two direct common tangents, which touch both circles on the same side and do not cross the line segment connecting the centers of the two circles. One transverse common tangent, which passes through the point of contact where the two circles touch externally and crosses the line segment connecting the centers.

Adding these together, the maximum number of common tangents that can be drawn to the two externally touching circles is 2 + 1 = 3.

Explanation

Using the basic trigonometric identity, tan A is equal to sin A / cos A. Substituting this into the given expression simplifies the term tan² A cos² A:

tan² A cos² A = (sin² A / cos² A) × cos² A = sin² A

Now, substituting this back into the original expression:

2(tan² A cos² A + cos² A) = 2(sin² A + cos² A)

Since sin² A + cos² A = 1 for any angle A, the expression simplifies directly to:

2(1) = 2

Explanation

The question contains a typographical error and intends to ask for the value of A + B + C instead of A + B - C. From the given trigonometric equations, we have A + B - C = 30 degrees, B + C - A = 60 degrees, and C + A - B = 45 degrees. Adding these three equations yields A + B + C = 30 + 60 + 45 = 135 degrees, which corresponds to Option D.

Explanation

To find the ratio of the average number of female athletes from England to France over all the years, we first calculate the total number of female athletes for each country because the ratio of the averages is equal to the ratio of their sums over the same number of years.

Total number of female athletes from England over the years 2016 to 2019 is: 480 + 420 + 130 + 250 = 1280

Average number of female athletes from England is: 1280 ÷ 4 = 320

Total number of female athletes from France over the years 2016 to 2019 is: 280 + 120 + 160 + 600 = 1160

Average number of female athletes from France is: 1160 ÷ 4 = 290

The ratio of these averages is: 320 : 290 = 32 : 29

Explanation

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

By setting x = a and y = 5b, the relation becomes: (a + 5b)² = a² + 2(a)(5b) + (5b)² (a + 5b)² = a² + 25b² + 10ab

Substituting the given values, a + 5b = 13 and ab = 9, into this identity: 13² = a² + 25b² + 10(9) 169 = a² + 25b² + 90

Subtracting 90 from both sides gives the required value: a² + 25b² = 169 - 90 a² + 25b² = 79

Explanation

The sum of angles in a triangle is 180°, which gives the relation: ∠A + ∠B + ∠C = 180°

We are given the relation: 8 ∠A = 6 ∠B = 3 ∠C

Let this relation be equal to a constant k: 8 ∠A = 6 ∠B = 3 ∠C = k

This allows us to express each angle in terms of k: ∠A = k/8 ∠B = k/6 ∠C = k/3

Substituting these expressions into the angle sum equation: k/8 + k/6 + k/3 = 180°

Using the least common multiple of 8, 6, and 3, which is 24, we combine the fractions: (3k + 4k + 8k) / 24 = 180° 15k / 24 = 180° 5k / 8 = 180° k = (180° × 8) / 5 k = 288°

Now, we calculate the individual angle measures: ∠A = 288° / 8 = 36° ∠B = 288° / 6 = 48° ∠C = 288° / 3 = 96°

Explanation

When a constant P is subtracted from four numbers a, b, c, and d such that the resulting numbers are in proportion, the value of P is given by the formula:

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

Substituting the values a = 8, b = 6, c = 2, and d = 9 into the formula:

P = (8 × 9 - 6 × 2) / ((8 + 9) - (6 + 2)) P = (72 - 12) / (17 - 8) P = 60 / 9 = 20/3

Now, substitute the value of P into the two expressions:

3P - 6 = 3 × (20/3) - 6 = 20 - 6 = 14 9P - 4 = 9 × (20/3) - 4 = 60 - 4 = 56

The mean proportional between two numbers x and y is given by √(x × y).

Mean proportional = √(14 × 56) Mean proportional = √(14 × 14 × 4) Mean proportional = 14 × 2 = 28

Explanation

The time taken by Pankaj to complete the 140 m race at a speed of 5/2 m/s is calculated using the formula:

Time = Distance / Speed

Time taken by Pankaj = 140 / (5/2) = 140 × 2/5 = 56 seconds.

Arun has a start of 20 m, which means he only needs to cover a distance of:

Distance covered by Arun = 140 - 20 = 120 m.

Since Pankaj defeats Arun by 4 seconds, Arun takes 4 seconds more than Pankaj to cover his distance.

Time taken by Arun = 56 + 4 = 60 seconds.

The speed of Arun is:

Speed = Distance / Time = 120 / 60 = 2 m/s.

Explanation

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

Total items received by the customer = 13 + 7 = 20 Number of free items = 7

Discount Percentage = (7 / 20) × 100 = 35%

Explanation

The formula for compound interest is: A = P(1 + R/100)ⁿ

Here, Principal (P) = ₹3,20,000, Amount (A) = ₹4,05,000, and Rate of interest (R) = 12.5% per annum.

Substituting these values into the formula: 4,05,000 = 3,20,000 × (1 + 12.5/100)ⁿ

Since 12.5/100 = 1/8, we can simplify the equation: 4,05,000 / 3,20,000 = (1 + 1/8)ⁿ 405 / 320 = (9/8)ⁿ

Dividing the numerator and the denominator on the left side by 5: 81 / 64 = (9/8)ⁿ

Expressing 81/64 as a power of 9/8: (9/8)² = (9/8)ⁿ

Comparing the exponents on both sides gives n = 2 years.

Explanation

Let the total work be 120 units, which is the least common multiple of 24 and 30.

The daily work efficiency of A is 120 ÷ 24 = 5 units per day. The daily work efficiency of B is 120 ÷ 30 = 4 units per day.

Their combined daily efficiency is 5 + 4 = 9 units per day.

In 11 days, the total work completed by A and B together is 11 × 9 = 99 units.

The remaining work is 120 - 99 = 21 units.

The percentage of the work still to be finished is (21 ÷ 120) × 100 = 17.5%.

Explanation

In similar triangles, corresponding angles are equal and corresponding sides are in proportion. Given ΔLMN ~ ΔPQR:

First, find the third angle of ΔLMN. The sum of angles in a triangle is 180°: Angle L + Angle M + Angle N = 180° 40° + Angle M + 80° = 180° Angle M = 180° - 120° = 60°

Since ΔLMN ~ ΔPQR, corresponding angles are equal: Angle Q = Angle M = 60°

Next, use the ratio of corresponding sides to find PR: LM/PQ = LN/PR

Substitute the given values into the ratio: 6/7.5 = 8/PR PR = (8 × 7.5)/6 PR = 60/6 PR = 10 cm

Explanation

The length of a direct common tangent to two circles with radii r₁ and r₂ and distance d between their centres is given by the formula:

L = √(d² - (r₂ - r₁)²)

Given: r₁ = 2.4 cm r₂ = 4.0 cm d = 6.5 cm

Substitute these values into the formula:

L = √(6.5² - (4.0 - 2.4)²) L = √(6.5² - 1.6²)

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

L = √((6.5 - 1.6) × (6.5 + 1.6)) L = √(4.9 × 8.1) L = √(39.69) L = 6.3 cm

Explanation

We can simplify the given expression by writing sec θ in terms of cos θ using the relation sec θ = 1/cos θ.

The expression is: ((1 + sec θ) / sec θ) (1 - cos θ)

We can rewrite the first term as: (1/sec θ + 1) (1 - cos θ)

Since 1/sec θ = cos θ, this becomes: (cos θ + 1) (1 - cos θ)

Rearranging the terms: (1 + cos θ) (1 - cos θ)

Applying the algebraic identity (a + b)(a - b) = a² - b²: 1 - cos² θ

Using the fundamental Pythagorean identity sin² θ + cos² θ = 1, we can write: 1 - cos² θ = sin² θ

Explanation

To find the second-highest scorer, we compare the total scores of all the batsmen given in the table:

E: 1380 H: 1365 A: 1356 B: 1301 F: 1281 C: 1278 D: 1156 G: 1129

Arranging these scores in descending order: 1380 > 1365 > 1356 > 1301 > 1281 > 1278 > 1156 > 1129

The highest score is 1380, belonging to batsman E. The second-highest score is 1365, which belongs to batsman H.

Explanation

Let P be the principal amount. The increase in simple interest due to an increase in the interest rate is given by the formula:

Increase in Interest = P × Increase in Rate × Time / 100

Substituting the given values:

600 = P × 6 × 4 / 100 600 = 24P / 100 P = 600 × 100 / 24 P = 2500

The question asks for the value of 2.5 times this principal amount:

2.5 × P = 2.5 × 2500 = 6250

Explanation

The net gain percentage when an article is marked up by m% and then sold at a discount of d% can be calculated using the successive percentage formula:

Net Gain % = m - d - (m × d)/100

Substituting the given values, where the markup percentage m = 90% and the discount percentage d = 40%:

Net Gain % = 90 - 40 - (90 × 40)/100 Net Gain % = 50 - 3600/100 Net Gain % = 50 - 36 Net Gain % = 14%

Alternatively, assuming the cost price is ₹100, a 90% markup makes the marked price ₹190. Offering a 40% discount on this marked price results in a selling price of:

190 × (1 - 0.40) = 190 × 0.60 = ₹114

The gain is the difference between the selling price and the cost price, which is:

114 - 100 = 14%

Explanation

The cost price for A (CP) is calculated using the relation CP = SP / (1 - Loss Rate). Since A sells the television for ₹4,860 at a 19% loss: CP of A = 4860 / (1 - 0.19) = 4860 / 0.81 = ₹6,000

B buys the television at this selling price, so the cost price for B is ₹4,860.

B then sells the television to C at a price that would have given A a 17% profit on A's cost price: Selling price of B = CP of A × (1 + 0.17) = 6000 × 1.17 = ₹7,020

The gain of B is the difference between B's selling price and B's cost price: Gain of B = 7020 - 4860 = ₹2,160

Explanation

In triangle AOB, OA and OB are radii of the circle, so OA = OB = 12 cm. The chord AB is the base of the triangle with a length of 20 cm.

Let OD be the perpendicular drawn from the centre O to the chord AB. Since the perpendicular from the centre of a circle to a chord bisects the chord, D is the midpoint of AB. Therefore, AD = DB = 10 cm.

In the right-angled triangle OAD, we can apply the Pythagorean theorem: OA² = OD² + AD² 12² = OD² + 10² 144 = OD² + 100 OD² = 44 OD = √44 = 2√11 cm

The area of triangle AOB is given by the formula: Area = ½ × base × height Area = ½ × AB × OD Area = ½ × 20 × 2√11 Area = 20√11 cm²

Explanation

The circumference of the base of the hemispherical dome is given by the formula C = 2πr, where r is the radius of the base.

2 × (22/7) × r = 154 r = (154 × 7) / 44 r = 24.5 cm

The curved surface area (CSA) of a hemispherical dome is given by the formula 2πr².

CSA = 2 × (22/7) × 24.5 × 24.5 CSA = 2 × (22/7) × (49/2) × (49/2) CSA = 11 × 7 × 49 CSA = 3773 cm²

The cost of painting is ₹4 per 100 cm².

Total Cost = CSA × (4 / 100) Total Cost = 3773 × 0.04 Total Cost = ₹150.92

Explanation

Let b be the cost of one bread and j be the cost of one packet of juice. Based on the given conditions, we can set up the following system of linear equations:

3b + j = 323.50 5b + j = 433.50

Subtracting the first equation from the second equation to eliminate j:

(5b + j) - (3b + j) = 433.50 - 323.50 2b = 110.00 b = 55.00

Now, substituting b = 55.00 back into the first equation:

3(55.00) + j = 323.50 165.00 + j = 323.50 j = 323.50 - 165.00 j = 158.50

To find the cost of 2 breads and 2 packets of juice, we calculate:

2b + 2j = 2(55.00) + 2(158.50) 2b + 2j = 110.00 + 317.00 2b + 2j = 427.00

Explanation

Correct Answer: B

The compound interest formula is given by A = P(1 + r/100)ⁿ, where A is the amount, P is the principal, r is the annual interest rate, and n is the time in years.

Substituting the given values into the formula: 46656 = 40000(1 + 8/100)ⁿ 46656 / 40000 = (1.08)ⁿ 1.1664 = (1.08)ⁿ

To find the value of n, we can calculate the powers of 1.08: (1.08)² = 1.08 × 1.08 = 1.1664

Since (1.08)² matches 1.1664, the value of n is 2.

Explanation

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Area of larger triangle / Area of smaller triangle = (Side of larger triangle / Side of smaller triangle)²

Substituting the given values into the formula:

243 / Area of smaller triangle = (9 / 4)²

243 / Area of smaller triangle = 81 / 16

Area of smaller triangle = 243 × 16 / 81

Area of smaller triangle = 3 × 16 = 48 sq.cm

Explanation

From the given equation, we have a(a - √6) = 1, which can be rewritten as a - 1/a = √6. Using the algebraic identity (a + 1/a)² = (a - 1/a)² + 4, we get (a + 1/a)² = 6 + 4 = 10. Since a > 0, we take the positive square root to get a + 1/a = √(10).

Explanation

The net profit or loss percentage when an item is marked up by x% and then sold at a discount of y% can be directly calculated using the successive percentage formula:

Net Percentage = x - y - (x × y) / 100

Here, the markup percentage x is 30% and the discount percentage y is 12%. Substituting these values into the formula:

Net Percentage = 30 - 12 - (30 × 12) / 100 Net Percentage = 18 - 3.6 Net Percentage = 14.4%

Since the net percentage change is positive, it represents a profit of 14.4%.

Explanation

Given the equation: x = 1 / (x - 5)

Multiplying both sides by x - 5 gives: x(x - 5) = 1 x² - 5x = 1

Dividing both sides by x (since x > 0): x - 1/x = 5

To find the value of x + 1/x, we use the standard algebraic identity: (x + 1/x)² = (x - 1/x)² + 4

Substituting the value of x - 1/x: (x + 1/x)² = 5² + 4 (x + 1/x)² = 25 + 4 = 29

Taking the positive square root on both sides because x > 0: x + 1/x = √29

Explanation

The profit percentage can be calculated directly using the successive percentage change formula:

Net percentage change = A + B + (A × B)/100

Here, the markup percentage A is +40% and the discount percentage B is -10%.

Substituting these values into the formula:

Net percentage change = 40 - 10 + (40 × (-10))/100 Net percentage change = 30 - 400/100 Net percentage change = 30 - 4 = 26%

Since the net percentage change is positive, the shopkeeper makes a profit of 26%. The selling price of ₹3,790 is redundant information.

Explanation

The total amount obtained after a period of time under simple interest is given by the formula:

Amount = Principal + Simple Interest Simple Interest = Principal × Rate × Time / 100

For the first investment: Principal = ₹4,400 Rate = 10% Time = 12 years Simple Interest = 4400 × 10 × 12 / 100 = ₹5,280

For the second investment: Principal = ₹7,500 Rate = 8% Time = 12 years Simple Interest = 7500 × 8 × 12 / 100 = ₹7,200

Total Simple Interest = 5280 + 7200 = ₹12,480 Total Principal = 4400 + 7500 = ₹11,900

Total Amount = Total Principal + Total Simple Interest = 11900 + 12480 = ₹24,380

Explanation

The time difference between the starts of the two men is from 8:30 A.M. to 8:54 A.M., which is 24 minutes or 24/60 = 0.4 hours.

During this time, the first man travels a distance of: Distance = Speed × Time = 30 km/h × 0.4 h = 12 km

The remaining distance between them at 8:54 A.M. is: 181 km - 12 km = 169 km

Since they are moving towards each other, their relative speed is the sum of their individual speeds: Relative Speed = 30 km/h + 35 km/h = 65 km/h

The time taken to cover the remaining distance after 8:54 A.M. is: Time = Distance / Relative Speed = 169 km / 65 km/h = 2.6 hours

Converting 2.6 hours to hours and minutes: 2.6 hours = 2 hours + 0.6 × 60 minutes = 2 hours and 36 minutes

Adding 2 hours and 36 minutes to 8:54 A.M.: 8:54 A.M. + 2 hours = 10:54 A.M. 10:54 A.M. + 36 minutes = 11:30 A.M.

Explanation

According to the circle theorem, the angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circumference.

Given that the angle subtended at the circumference is 55°, the angle subtended at the center is calculated as:

Angle at the center = 2 × Angle at the circumference Angle at the center = 2 × 55° = 110°

Explanation

The percentage increment in the production of D-type cars is calculated using the formula:

Percentage increment = ((Production in 2022 - Production in 2021) / Production in 2021) × 100

From the given table, the production of D-type cars is: Production in 2021 = 1975 Production in 2022 = 2100

Substituting these values into the formula: Percentage increment = ((2100 - 1975) / 1975) × 100 Percentage increment = (125 / 1975) × 100 Percentage increment = (5 / 79) × 100 Percentage increment ≈ 6.33%

Explanation

Let the present age of the father and the son be 5x and 3x, respectively.

According to the given condition, the product of their ages is: 5x × 3x = 960 15x² = 960 x² = 64 x = 8

Thus, the present age of the father is 5 × 8 = 40 years, and the present age of the son is 3 × 8 = 24 years.

After 6 years: Father's age = 40 + 6 = 46 years Son's age = 24 + 6 = 30 years

The ratio of their ages after 6 years is: 46:30 = 23:15

Explanation

Let the marked price (MP) of the article be 100.

The cost price (CP) is 72% of the marked price, so CP = 72.

The selling price (SP) after a discount of 6% on the marked price is SP = 100 - 6 = 94.

Profit = SP - CP = 94 - 72 = 22

Profit Percentage = (Profit / CP) × 100 Profit Percentage = (22 / 72) × 100 = 30.555... %

Rounding off to two decimal places, the profit percentage is 30.56%.

Explanation

Using the difference of squares formula, x² - y² = (x - y)(x + y), the numerator can be rewritten as: 36a² - 49b² = (6a)² - (7b)² = (6a - 7b)(6a + 7b)

Substituting this back into the original expression: ((6a - 7b)(6a + 7b)) / (6a + 7b) = 6a - 7b