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Shortcut Trick

Total area of the circle = πr² = (22/7) × 14 × 14 = 616 cm².

Sector area as a fraction of the total area = 154 / 616 = 1/4.

Angle of the sector = Fraction of area × 360°.

Angle = (1/4) × 360° = 90°.

∴ The correct answer is 90°.

Alternate Method

Given:

Radius (r) = 14 cm

Area of the sector = 154 cm²

Formula Used:

Area of a sector = (θ / 360°) × πr²

O

r = 14 cm r = 14 cm

θ = ? Area = 154

Calculations:

⇒ 154 = (θ / 360°) × (22 / 7) × (14)²

⇒ 154 = (θ / 360°) × 22 × 2 × 14

⇒ 154 = (θ / 360°) × 616

⇒ θ / 360° = 154 / 616

⇒ θ / 360° = 1/4

⇒ θ = 360° / 4

⇒ θ = 90°

∴ The correct answer is 90°.

Additional Information

Length of an Arc

The length of an arc (l) of a circle can be calculated as: l = (θ / 360°) × 2πr.

Area of Sector using Arc Length

If the arc length is already known, the area of the sector can be found directly using the formula: Area = (1 / 2) × l × r.

Area of a Segment

The area of a segment of a circle is defined as: Area of the corresponding sector − Area of the corresponding triangle formed by the chord and the bounding radii.

Shortcut Trick

Total Cost = (Percentage Painted) × (Area of Circle) × Rate

⇒ Total Cost = 90% × (π × r²) × 60

⇒ Total Cost = 0.9 × 3.14 × 2² × 60

⇒ Total Cost = 0.9 × 12.56 × 60 = 678.24

⇒ Total Cost ≈ ₹678.2

∴ The correct answer is ₹678.2.

Alternate Method

90% Painted 10% r = 2 m Total Cost = Painted Area × Cost per m²

Given:

Radius of the circular signboard (r) = 2 m

Cost of painting = ₹60 per m²

Unpainted portion = 10%

Formula Used:

Area of a circle = πr²

Calculation:

⇒ Total area of the board = 3.14 × 2² = 12.56 m²

⇒ Painted portion = 100% − 10% = 90%

⇒ Area to be painted = 90% of 12.56 = 0.9 × 12.56 = 11.304 m²

⇒ Total cost of painting = Painted area × Cost per m²

⇒ Total cost = 11.304 × 60 = ₹678.24

⇒ Total cost ≈ ₹678.2

∴ The correct answer is ₹678.2.

Additional Information

Area of a Sector

A sector represents a fraction of the circle. The formula is Area = (θ ÷ 360°) × πr², where θ is the central angle.

Circumference of a Circle

The total length of the boundary of a circle is given by C = 2πr.

Percentage Application in Geometry

Instead of finding the excluded area and subtracting, directly calculate the effective percentage (e.g., 100% − 10% = 90%) and multiply it with the total area.

Shortcut Trick

For any linear equation in the standard form Ax + By = C, the y-intercept can be calculated directly by dividing the constant term by the y-coefficient.

Formula: y-intercept = C ÷ B

Here, the given equation is 4x + 5y = 20, where C = 20 and B = 5.

y-intercept = 20 ÷ 5 = 4

∴ The correct answer is 4.

Alternate Method

Given:

The linear equation of the line is 4x + 5y = 20.

Formula Used:

The y-intercept is the exact point where the graph of the line crosses the y-axis. At this specific point, the x-coordinate is always 0.

X Y O

(0, 4) y-intercept (5, 0) x-intercept

4x + 5y = 20

Calculations:

⇒ To find the y-intercept, substitute x = 0 in the given equation.

⇒ 4(0) + 5y = 20

⇒ 0 + 5y = 20

⇒ 5y = 20

⇒ y = 20 ÷ 5

⇒ y = 4

∴ The correct answer is 4.

Additional Information

Intercept Form of a Line

The equation of a straight line can be expressed as x/a + y/b = 1, where a represents the x-intercept and b represents the y-intercept.

Slope-Intercept Form

A line can be represented as y = mx + c, where m signifies the slope of the line and c represents the y-intercept.

Finding the x-intercept

The x-intercept is the exact point where the graph of the line intersects the x-axis. It is determined by setting y = 0 in the equation.

Shortcut Trick

For any given algebraic expression where x + 1/x = k, the value of x³ + 1/x³ is directly given by k³ − 3k.

Here, the given value is k = 4.

Substituting the value in the shortcut formula, we get: 4³ − 3 × 4

Value = 64 − 12 = 52

∴ The correct answer is 52.

Alternate Method

Given:

x + 1/x = 4

Formula Used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculation:

⇒ Cubing both sides of the given equation, we get:

⇒ (x + 1/x)³ = 4³

⇒ x³ + 1/x³ + 3 × x × (1/x) × (x + 1/x) = 64

⇒ x³ + 1/x³ + 3 × 1 × 4 = 64

⇒ x³ + 1/x³ + 12 = 64

⇒ x³ + 1/x³ = 64 − 12

⇒ x³ + 1/x³ = 52

∴ The correct answer is 52.

Additional Information

Algebraic Identity 1

If x − 1/x = k, then the direct result is x³ − 1/x³ = k³ + 3k.

Algebraic Identity 2

If x + 1/x = k, then the square values relate as x² + 1/x² = k² − 2.

Algebraic Identity 3

If x − 1/x = k, then the square values relate as x² + 1/x² = k² + 2.

Shortcut Trick

When adding decimals with a single 1 in different place values, just position 1 in each corresponding decimal place.

0.1 has 1 in the tenths place.

0.01 has 1 in the hundredths place.

0.001 has 1 in the thousandths place.

⇒ Combining them directly gives 0.111

∴ The correct answer is 0.111.

Alternate Method

Given:

Expression: 0.1 + 0.01 + 0.001

Method Used:

Convert the given decimals into fractions with a common denominator of 1000, and then add them together.

Calculation:

⇒ 0.1 = 1 ÷ 10 = 100 ÷ 1000

⇒ 0.01 = 1 ÷ 100 = 10 ÷ 1000

⇒ 0.001 = 1 ÷ 1000

⇒ Sum = (100 ÷ 1000) + (10 ÷ 1000) + (1 ÷ 1000)

⇒ Sum = (100 + 10 + 1) ÷ 1000

⇒ Sum = 111 ÷ 1000

⇒ Sum = 0.111

∴ The correct answer is 0.111.

Additional Information

BODMAS Rule

It determines the correct sequence of operations in any mathematical expression: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Decimal Fractions

Fractions in which denominators are powers of 10 (like 10, 100, 1000) are known as decimal fractions. Example: 1 ÷ 10 = 0.1

Like Decimals

Decimals having the exact same number of decimal places are called like decimals. Adding trailing zeros converts unlike decimals to like decimals, making addition simpler (e.g., 0.100 + 0.010 + 0.001).

Shortcut Trick

To find the compound ratio N : P directly, we can multiply the given intermediate ratios N : O and O : P.

N / P = (N / O) × (O / P)

N / P = (6 / 7) × (9 / 11)

N / P = 54 / 77

∴ The correct answer is 54 : 77.

Alternate Method

Given:

M : N = 4 : 5

N : O = 6 : 7

O : P = 9 : 11

N O P

6 7 → 7

9 ← 9 11

54 63 77

Calculations:

⇒ To find the combined ratio, make the common term (O) equal in both ratios by taking the LCM of 7 and 9, which is 63.

⇒ Multiply the N : O ratio by 9 ⇒ N : O = (6 × 9) : (7 × 9) = 54 : 63

⇒ Multiply the O : P ratio by 7 ⇒ O : P = (9 × 7) : (11 × 7) = 63 : 77

⇒ Now, the combined ratio N : O : P = 54 : 63 : 77

⇒ Extracting the required values N and P from the combined ratio, we get N : P = 54 : 77

(Note: M : N is extraneous information and is not required for solving the problem.)

∴ The correct answer is 54 : 77.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the resulting ratio is called a compound ratio. For a : b and c : d, the compound ratio is ac : bd.

Combining Ratios

If A : B = x : y and B : C = p : q, you can combine them by equating the common term 'B'. Multiply the first ratio by p and the second by y to get A : B : C = xp : yp : yq.

Extraneous Data in Exams

Questions often contain additional data (like M : N in this problem) that is not required to find the final answer. Reading the question carefully to identify the core requirement saves valuable exam time.

Shortcut Trick

Convert the mixed fraction directly to a decimal: 2¹/₄ = 2.25

Dividing any number by 0.5 is mathematically equivalent to multiplying that number by 2.

⇒ 2.25 × 2 = 4.5

∴ The correct answer is 4.5.

Alternate Method

Given:

Expression: 2¹/₄ ÷ 0.5

Calculations:

⇒ Convert the mixed fraction into an improper fraction:

⇒ 2¹/₄ = (2 × 4 + 1) / 4 = 9/4

⇒ Convert the decimal 0.5 into a fraction:

⇒ 0.5 = 5/10 = 1/2

⇒ Substitute the fractions into the given expression:

⇒ 9/4 ÷ 1/2

⇒ To divide by a fraction, multiply by its reciprocal:

⇒ 9/4 × 2/1

⇒ Multiply the numerators and denominators:

⇒ 18/4

⇒ Simplify the resulting fraction:

⇒ 9/2 = 4.5

∴ The correct answer is 4.5.

Additional Information

Fraction Division Rule

To divide by a fraction, multiply the first number by the reciprocal of the second fraction: (a/b) ÷ (c/d) = (a/b) × (d/c).

Mixed Fraction to Improper Fraction

Multiply the whole number part by the denominator, add the numerator, and place the result over the original denominator.

Decimal to Fraction Conversion

Decimals can be easily converted to fractions by placing the number over its place value (e.g., 0.5 = 5/10 = 1/2, or 0.25 = 25/100 = 1/4).

Shortcut Trick

Let the given numbers be a = 10, b = 12, c = 13, and d = 16.

Direct formula for the number to be subtracted: x = (a × d − b × c) ÷ ((a + d) − (b + c))

⇒ x = (10 × 16 − 12 × 13) ÷ ((10 + 16) − (12 + 13))

⇒ x = (160 − 156) ÷ (26 − 25)

⇒ x = 4 ÷ 1 = 4

∴ The correct answer is 4.

Alternate Method

Given:

The numbers are 10, 12, 13, and 16.

Let the number to be subtracted from each term be x.

Formula Used:

If four numbers A, B, C, and D are in proportion, then A : B :: C : D or A ÷ B = C ÷ D.

10 − x

12 − x

=

13 − x

16 − x

Proportion: (10 − x) : (12 − x) :: (13 − x) : (16 − x)

Calculations:

⇒ (10 − x) ÷ (12 − x) = (13 − x) ÷ (16 − x)

⇒ Cross-multiplying both the sides:

⇒ (10 − x) × (16 − x) = (13 − x) × (12 − x)

⇒ 160 − 10x − 16x + x² = 156 − 13x − 12x + x²

⇒ 160 − 26x = 156 − 25x

⇒ 26x − 25x = 160 − 156

⇒ x = 4

∴ The correct answer is 4.

Additional Information

Fourth Proportional

If a, b, c, and d are in proportion (a : b :: c : d), then d is called the fourth proportional. It is calculated as d = (b × c) ÷ a.

Third Proportional

For two numbers a and b, the third proportional c is defined such that a : b :: b : c. It is calculated as c = b² ÷ a.

Mean Proportional

The mean proportional between two positive numbers a and b is given by √(a × b). It represents the middle term in a continuous proportion.

Componendo and Dividendo Rule

If a ÷ b = c ÷ d, then according to this mathematical rule: (a + b) ÷ (a − b) = (c + d) ÷ (c − d).

Shortcut Trick

Difference in percentage = 45% − 30% = 15%

Value of 15% of the number = 0.9

1% of the number = 0.9 ÷ 15 = 0.06

100% of the number = 0.06 × 100 = 6

∴ The correct answer is 6.

Alternate Method

Given:

45% of a number is 0.9 more than 30% of it.

30% 15%

0.9 Difference

55%

Total = 100% = x

Calculations:

Let the number be x.

⇒ 45% of x − 30% of x = 0.9

⇒ (45 − 30)% of x = 0.9

⇒ 15% of x = 0.9

⇒ (15 ÷ 100) × x = 0.9

⇒ x = 0.9 × (100 ÷ 15)

⇒ x = 90 ÷ 15

⇒ x = 6

∴ The correct answer is 6.

Additional Information

Percentage Difference

When comparing two percentages of the same base value, the difference between their absolute values is equal to the percentage of the difference applied to the base. Formula: a% of x − b% of x = (a − b)% of x.

Base of Percentage

A percentage is always calculated on a base value (considered as 100%). Finding the unknown base given a percentage value involves multiplying the value by 100 and dividing by the given percentage.

Percentage to Fraction Conversion

To convert a percentage into a fraction, divide the value by 100. For example, 15% is equivalent to 15 ÷ 100 or 3 ÷ 20 in simplified fractional form.

Shortcut Trick

In partnership, Profit Ratio = Ratio of (Investment × Time).

Working in terms of thousands (k) to simplify calculations:

Sohan's effective capital = (90 × 8) + (60 × 4) = 720 + 240 = 960

Rohan's effective capital = 150 × 7 = 1050

Profit Ratio = 960 : 1050 = 96 : 105 = 32 : 35

∴ The correct answer is 32:35.

Alternate Method

Given:

Sohan's initial investment = ₹90,000

Time for which Sohan invested his initial amount = 5 + 3 = 8 months

Amount withdrawn by Sohan after 8 months = ₹30,000

Remaining time for Sohan's adjusted investment = 12 − 8 = 4 months

Rohan's investment = ₹1,50,000

Rohan joined after 5 months, so his investment time = 12 − 5 = 7 months

Formula Used:

Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

0m 5m 8m 12m

Sohan

₹90k × 8m

₹60k × 4m

Rohan

₹150k × 7m

Effective Capital = Investment × Time

Calculations:

⇒ Sohan's remaining investment after withdrawal = 90,000 − 30,000 = ₹60,000

⇒ Sohan's effective capital = (90,000 × 8) + (60,000 × 4)

⇒ Sohan's effective capital = 7,20,000 + 2,40,000 = 9,60,000

⇒ Rohan's effective capital = 1,50,000 × 7 = 10,50,000

⇒ Profit Ratio (Sohan : Rohan) = 9,60,000 : 10,50,000

⇒ Ratio = 96 : 105

Dividing both sides by 3:

⇒ Ratio = 32 : 35

∴ The correct answer is 32:35.

Additional Information

Partnership Profit Sharing

Profit is shared among partners strictly based on their effective capital. The effective capital is calculated as the product of the capital invested and the time duration of the investment.

Compound Partnership

When partners invest for varying time intervals, or they alter (add/withdraw) their invested capital mid-way through the business term, it represents a compound partnership. The total effective capital is the sum of (capital × time) for each sub-period.

Working vs Sleeping Partner

A working partner actively manages the business and frequently takes a specified salary or percentage of the profit before the remainder is divided. A sleeping partner simply invests funds without engaging in day-to-day operations and receives only their proportionate share.