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Shortcut Trick

Difference between the digits = Difference in numbers ÷ 9 = 27 ÷ 9 = 3.

Sum of digits is given as 11.

Smaller digit (Tens place) = (Sum − Difference) ÷ 2 = (11 − 3) ÷ 2 = 4.

Larger digit (Units place) = (Sum + Difference) ÷ 2 = (11 + 3) ÷ 2 = 7.

Original Number = 47 and Reversed Number = 74.

The required ratio = 47 : 74.

∴ The correct answer is 47:74.

Alternate Method

Given:

Sum of digits of a two-digit number = 11

Original Number + 27 = Reversed Number

⇒ Let the two-digit number be 10x + y, where x is the tens digit and y is the units digit.

⇒ According to the first condition: x + y = 11 → (Equation 1)

⇒ According to the second condition: (10x + y) + 27 = 10y + x

⇒ 10y − y + x − 10x = 27

⇒ 9y − 9x = 27

⇒ y − x = 3 → (Equation 2)

⇒ Adding (1) and (2): (x + y) + (y − x) = 11 + 3 ⇒ 2y = 14 ⇒ y = 7

⇒ Substituting y = 7 in (1): x + 7 = 11 ⇒ x = 4

⇒ Original number = 10(4) + 7 = 47

⇒ Number on reversing = 10(7) + 4 = 74

⇒ Ratio = 47 : 74

∴ The correct answer is 47:74.

Additional Information

Two-Digit Number Properties

Any two-digit number can be represented as 10x + y. The difference between the number and its reverse is always 9 × (x − y).

Divisibility of the Sum

The sum of a two-digit number and its reverse is always divisible by 11 and is equal to 11 × (x + y).

Finding Digits

If the sum (S) and difference (D) of two digits are known, the digits are (S + D) ÷ 2 and (S − D) ÷ 2.

Shortcut Trick

In a trapezium (where AB || DC), the distance between the midpoints of the diagonals is half the difference of the parallel sides.

Length of EF = 1/2 × |DC − AB|

⇒ EF = 1/2 × |92 − 24|

⇒ EF = 1/2 × 68 = 34 cm

∴ The correct answer is 34 cm.

Alternate Method

Given:

Quadrilateral ABCD where AB || DC (Trapezium).

Parallel sides: AB = 24 cm, DC = 92 cm.

Non-parallel sides: AD = 95 cm, BC = 62 cm.

E and F are the midpoints of diagonals AC and BD respectively.

Formula Used:

The length of the line segment joining the midpoints of the diagonals of a trapezium is given by:

EF = (Length of longer parallel side − Length of shorter parallel side) ÷ 2

Calculations:

⇒ AB = 24 cm (Shorter parallel side)

⇒ DC = 92 cm (Longer parallel side)

⇒ Length of EF = (DC − AB) ÷ 2

⇒ EF = (92 − 24) ÷ 2

⇒ EF = 68 ÷ 2

⇒ EF = 34 cm

∴ The correct answer is 34 cm.

Additional Information

Median of a Trapezium

The segment joining the midpoints of the non-parallel sides is parallel to the bases and its length is the average of the bases: MN = (a + b) / 2.

Distance between Midpoints of Diagonals

For any trapezium with parallel sides a and b, the segment connecting the diagonal midpoints lies on the median and has length |a − b| / 2.

Area of a Trapezium

The area is calculated as Area = 1/2 × (Sum of parallel sides) × Height = 1/2 × (a + b) × h.

Shortcut Trick

The ratio of the first two terms is 27 : 81 → 1 : 3.

The third proportional (length) follows the same ratio: 81 × 3 = 243 units.

Area of rectangle = Length × Width = 243 × 4 = 972 units.

∴ The correct answer is 972 units.

Alternate Method

Given:

First term (a) = 27

Second term (b) = 81

Width of rectangle (w) = 4 units

Formula Used:

For a continued proportion a : b :: b : c, the third proportional is c = b² ÷ a.

Area of Rectangle = Length (l) × Width (w)

Calculations:

⇒ Third proportional (Length) = (81 × 81) ÷ 27

⇒ Length = 3 × 81 = 243 units

⇒ Area of rectangle = 243 × 4

⇒ Area = 972 units

∴ The correct answer is 972 units.

Additional Information

Third Proportional

For two numbers a and b, the third proportional c is such that a/b = b/c, resulting in c = b²/a.

Mean Proportional

The mean proportional between two numbers a and b is given by √(a × b).

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is such that a/b = c/d, resulting in d = (b × c) / a.

Shortcut Trick

Total Distance = 150 + 120 + 180 = 450 km

Total Time = (150 ÷ 60) + (120 ÷ 80) + (180 ÷ 90) = 2.5 + 1.5 + 2 = 6 hours

Average Speed = Total Distance ÷ Total Time = 450 ÷ 6 = 75 km/hr

∴ The correct answer is 75 km/hr.

Alternate Method

Given:

Distances (d₁, d₂, d₃) = 150 km, 120 km, 180 km

Speeds (s₁, s₂, s₃) = 60 km/hr, 80 km/hr, 90 km/hr

Formula Used:

Time = Distance ÷ Speed

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Time taken for first part (t₁) = 150 ÷ 60 = 2.5 hours

⇒ Time taken for second part (t₂) = 120 ÷ 80 = 1.5 hours

⇒ Time taken for third part (t₃) = 180 ÷ 90 = 2.0 hours

⇒ Total Time (T) = 2.5 + 1.5 + 2.0 = 6 hours

⇒ Total Distance (D) = 150 + 120 + 180 = 450 km

⇒ Average Speed = 450 ÷ 6 = 75 km/hr

∴ The correct answer is 75 km/hr.

Additional Information

Average Speed for Equal Distances

If a journey covers two equal distances at speeds x and y, the average speed is 2xy ÷ (x + y). For three equal distances at speeds x, y, and z, it is 3xyz ÷ (xy + yz + zx).

Unit Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert from m/s to km/hr, multiply by 18/5.

Relative Speed

If two objects move in the same direction, relative speed is s₁ − s₂. If they move in opposite directions, relative speed is s₁ + s₂.

Shortcut Trick

Recognize that the expression (1 − tan²x) / (1 + tan²x) is equal to cos(2x).

We know cos(2x) = cos²x − sin²x.

This expands as (cos x − sin x)(cos x + sin x).

Given cos x − sin x = a and cos x + sin x = b, we simply multiply them.

Required value = a × b = ab.

∴ The correct answer is ab.

Alternate Method

Given:

⇒ cos x − sin x = a

⇒ cos x + sin x = b

Formula Used:

⇒ (1 − tan²x) / (1 + tan²x) = cos(2x)

⇒ cos(2x) = cos²x − sin²x

⇒ cos²x − sin²x = (cos x − sin x)(cos x + sin x)

Calculations:

⇒ (1 − tan²x) / (1 + tan²x) = (1 − sin²x/cos²x) / (1 + sin²x/cos²x)

⇒ ((cos²x − sin²x)/cos²x) / ((cos²x + sin²x)/cos²x)

⇒ (cos²x − sin²x) / (cos²x + sin²x)

⇒ Since cos²x + sin²x = 1, the expression simplifies to cos²x − sin²x.

⇒ cos²x − sin²x = (cos x − sin x)(cos x + sin x)

⇒ Substituting the given values: a × b = ab.

∴ The correct answer is ab.

Additional Information

Double Angle Identities

cos(2x) = cos²x − sin²x = 2 cos²x − 1 = 1 − 2 sin²x = (1 − tan²x) / (1 + tan²x).

Pythagorean Identity

sin²x + cos²x = 1 is the fundamental identity used to convert between sine and cosine squares.

Tangent Double Angle

sin(2x) = 2 tan x / (1 + tan²x) and tan(2x) = 2 tan x / (1 − tan²x).

Given:

(14/5)¹¹ × (5/14)¹⁷ × (14/5)¹⁴ = (5/14)^{3m + 7}

Formula used:

1. Reciprocal rule: (a/b)ⁿ = (b/a)⁻ⁿ
2. Law of exponents: a^x × a^y × a^z = a^{x + y + z}

Calculations:

(14/5)¹¹ × (5/14)¹⁷ × (14/5)¹⁴ = (5/14)^{3m + 7}

Convert all terms to the same base (5/14) using the reciprocal rule:

⇒ (5/14)⁻¹¹ × (5/14)¹⁷ × (5/14)⁻¹⁴ = (5/14)^{3m + 7}

Applying the law of exponents for multiplication to add the powers:

⇒ (5/14)^{(−11 + 17 − 14)} = (5/14)^{3m + 7}

⇒ (5/14)⁻⁸ = (5/14)^{3m + 7}

Since the bases are identical, equate the exponents:

⇒ −8 = 3m + 7

⇒ 3m = −8 − 7

⇒ 3m = −15

⇒ m = −15 ÷ 3

⇒ m = −5

The value of m that satisfies the equation is −5.

∴ The correct answer is option (4).

Shortcut Trick

Apply the divisibility rule of 11: The difference between the sum of digits at odd positions and even positions must be 0 or a multiple of 11.

Sum of digits at odd positions (4, 8, 3) = 15

Sum of digits at even positions (5, 7) = 12

Current difference = 15 − 12 = 3

To make the difference 11 (the next multiple), we need to add 8 to the odd positions sum (3 + 8 = 11).

∴ The correct answer is 8.

Alternate Method

Given: Number = 45873, Divisor = 11

Formula Used: Smallest number to be added = Divisor − Remainder

⇒ 45873 ÷ 11

⇒ 11 × 4170 = 45870

⇒ Remainder = 45873 − 45870 = 3

⇒ Required number to add = 11 − 3

⇒ Required number to add = 8

∴ The correct answer is 8.

Additional Information

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.

Smallest Number to Subtract

To find the smallest number to subtract so a number becomes divisible, the answer is simply the Remainder itself (in this case, 3).

Divisibility Rule of 9

A number is divisible by 9 if the sum of all its digits is divisible by 9.

Given:

The ratios are 11:15, 23:55, 32:33, 4:11, and 26:35.

Formula used:

To arrange ratios in ascending order, convert each into its decimal equivalent.

Calculations:

Converting ratios to decimal values:

4:11 = 4 ÷ 11 ≈ 0.3636

23:55 = 23 ÷ 55 ≈ 0.4181

11:15 = 11 ÷ 15 ≈ 0.7333

26:35 = 26 ÷ 35 ≈ 0.7428

32:33 = 32 ÷ 33 ≈ 0.9696

Comparison of decimals:

0.3636 < 0.4181 < 0.7333 < 0.7428 < 0.9696

⇒ 4:11 < 23:55 < 11:15 < 26:35 < 32:33

∴ The correct answer is option (2).

Shortcut Trick

The total simple interest accumulated over 5 years is 25% of the Principal.

Since Simple Interest is calculated linearly every year, the Rate of Interest per annum is the total percentage divided by the number of years.

⇒ Rate of Interest = 25% ÷ 5

⇒ Rate of Interest = 5%

∴ The correct answer is 5%.

Alternate Method

Given:

Let the Principal (P) be 100.

Simple Interest (SI) = 25% of 100 = 25.

Time (T) = 5 years.

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Calculations:

⇒ 25 = (100 × R × 5) ÷ 100

⇒ 25 = R × 5

⇒ R = 25 ÷ 5

⇒ R = 5%

∴ The correct answer is 5%.

Additional Information

Simple Interest (SI) Formula

The basic formula is SI = (P × R × T) ÷ 100, where P is Principal, R is Rate per annum, and T is Time in years.

Amount in Simple Interest

The total Amount (A) after T years is calculated as A = P + SI or A = P × (1 + RT ÷ 100).

Rate of Interest Definition

The Rate of Interest is the interest earned on a Principal of 100 units for a period of 1 year.

Linear Growth

Unlike Compound Interest, Simple Interest remains constant every year because it is always calculated on the original Principal.

Shortcut Trick

Required percentage Reduction in Consumption = [r ÷ (100 + r)] × 100%

Given Price Increase (r) = 3%

Reduction = [3 ÷ (100 + 3)] × 100 = [3 ÷ 103] × 100

Reduction ≈ 2.912% ≈ 2.9% (rounded to one decimal place)

∴ The correct answer is 2.9%.

Alternate Method

Given: The Price of Sugar is increased by 3%.

Formula Used: Expenditure = Price × Consumption

⇒ Let the Initial Price of sugar be ₹100 and the Initial Consumption be 100 kg. ⇒ Initial Expenditure = 100 × 100 = ₹10,000. ⇒ New Price = 100 + 3% of 100 = ₹103. ⇒ Since Expenditure must remain constant, the New Expenditure = ₹10,000. ⇒ New Consumption = Total Expenditure ÷ New Price = 10,000 ÷ 103. ⇒ New Consumption ≈ 97.087 kg. ⇒ Reduction in Consumption = 100 − 97.087 = 2.913 kg. ⇒ Percentage Reduction = (2.913 ÷ 100) × 100 = 2.913% ≈ 2.9%.

∴ The correct answer is 2.9%.

Additional Information

Inverse Variation

When the product of two quantities is constant, an increase in one requires a proportional decrease in the other (Price ∝ 1/Consumption).

Percentage Decrease Formula

If a value increases by r%, the percentage decrease required to return to the original value is [r ÷ (100 + r)] × 100%.

Successive Percentage Changes

The net effect of two successive percentage changes a% and b% is given by the formula [a + b + (a × b ÷ 100)]%.

∴ The correct answer is 2.9%.