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Shortcut Trick

Remaining pizza percentage = 100% − 75% = 25% = 1/4 of total pizza.

Area remaining = (1/4) × π × r²

⇒ Area = (1/4) × (22/7) × 21 × 21

⇒ Area = (1/4) × 1386 = 346.5 cm²

∴ The correct answer is 346.5 cm².

Alternate Method

Given:

Radius of the circular pizza (r) = 21 cm

Percentage of pizza eaten = 75%

Formula Used:

Total Area of a circle = πr² (where π ≈ 22/7)

Remaining Area = Total Area × (Remaining Percentage / 100)

Eaten (75%) 25% Left r = 21 cm Pizza Area Remaining = 25%

Calculation:

⇒ Total Area of the pizza = (22/7) × 21 × 21

⇒ Total Area = 22 × 3 × 21 = 1386 cm²

⇒ Remaining Percentage = 100% − 75% = 25%

⇒ Remaining Area = 25% of Total Area

⇒ Remaining Area = (25 / 100) × 1386

⇒ Remaining Area = 1386 ÷ 4 = 346.5 cm²

∴ The correct answer is 346.5 cm².

Additional Information

Area of a Circle

The total area enclosed by a circle is given by A = πr², where r is the radius of the circle.

Area of a Sector

A sector is a portion of a circle enclosed by two radii and an arc. Its area is calculated as (θ/360) × πr², where θ is the central angle.

Percentage to Fraction Conversion

Converting standard percentages to fractions simplifies calculations. For example, 25% = 1/4, 50% = 1/2, and 75% = 3/4.

Shortcut Trick

Ratio of ring area to outer circle area = (R² − r²) : R²

⇒ Ratio = (10² − 7²) : 10² = (100 − 49) : 100

⇒ Exact Ratio = 51 : 100

⇒ Approximate Ratio ≈ 50 : 100 = 1 : 2

∴ The correct answer is 1:2.

Alternate Method

Given:

Outer radius (R) = 10 cm

Inner radius (r) = 7 cm

Formula Used:

Area of a circle = πR²

Area of a ring = π(R² − r²)

R = 10 cm

r = 7 cm

Ring-Shaped Disc

Calculations:

⇒ Area of the outer circle = π × (10)² = 100π cm²

⇒ Area of the inner circle = π × (7)² = 49π cm²

⇒ Area of the ring = Area of outer circle − Area of inner circle

⇒ Area of the ring = 100π − 49π = 51π cm²

⇒ Ratio of ring's area to outer circle's area = 51π : 100π

⇒ Exact ratio = 51 : 100

⇒ Approximating 51 to 50 for simplification

⇒ Approximate ratio ≈ 50 : 100 = 1 : 2

∴ The correct answer is 1:2.

Additional Information

Area of a Ring

The region bounded by two concentric circles of outer radius R and inner radius r is called a ring. Its area is calculated using the formula π(R² − r²).

Concentric Circles

Circles having the same center but different radii are called concentric circles. The perpendicular distance between the boundaries of these circles is uniform and equal to (R − r).

Area of a Sector

The area of a sector of a circle with radius r and central angle θ (in degrees) is calculated using the formula (θ ÷ 360) × πr².

Shortcut Trick

The percentage of circumference covered depends solely on the fraction of the turn, making the radius extraneous data.

A full turn (1 complete revolution) always covers exactly 100% of the circumference.

Therefore, a quarter turn covers exactly 1/4 of a full revolution.

Percentage covered = (1 ÷ 4) × 100% = 25%.

∴ The correct answer is 25%.

Alternate Method

Given:

Radius (r) = 35 cm

π = 22/7

Formula Used:

Circumference of a circle = 2 × π × r

Distance covered in a quarter turn = Circumference ÷ 4

90°

r = 35 cm Full Turn (360°) = 100% of Circumference Quarter Turn = 1/4 of Circle = 25% 1/4 Turn = 25% of Circumference

Calculations:

⇒ Total Circumference = 2 × (22/7) × 35

⇒ Total Circumference = 2 × 22 × 5 = 220 cm

⇒ Distance covered in a quarter turn = Total Circumference ÷ 4

⇒ Distance covered = 220 ÷ 4 = 55 cm

⇒ Percentage of circumference covered = (Distance ÷ Total Circumference) × 100

⇒ Percentage = (55 ÷ 220) × 100

⇒ Percentage = (1 ÷ 4) × 100 = 25%

∴ The correct answer is 25%.

Additional Information

Extraneous Data in Math Problems

When a question asks for a ratio, fraction, or percentage of a total, exact numerical measurements (like radius = 35 cm) are often not required to solve the problem and can be bypassed to save time.

Arc Length Formula

The length of an arc is proportional to its central angle: Arc Length = (θ ÷ 360) × 2 × π × r. A quarter turn has an angle of 90°, so the arc length is exactly 1/4 of the full circumference.

Distance and Rotations

The total linear distance covered by a rolling wheel equals the number of rotations multiplied by its circumference (Distance = n × 2 × π × r).

Shortcut Trick

If a line passes through a given point, the coordinates of the point directly satisfy the equation of the line.

Substitute the point (1, 8) into the line equation y = mx + 5.

⇒ 8 = m × 1 + 5

⇒ m = 8 − 5 = 3

∴ The correct answer is 3.

Alternate Method

Given:

Equation of the straight line: y = mx + 5

The line passes through the point (1, 8).

x y O

(0, 5)

(1, 8)

Δx = 1 Δy = 3

Line: y = mx + 5 y-intercept c = 5 → point (0, 5) Slope m = Δy ÷ Δx = 3 ÷ 1 = 3

Formula Used:

If a point (x, y) lies on a line, its values must satisfy the equation y = mx + c.

Calculations:

⇒ By placing x = 1 and y = 8 in the given equation y = mx + 5:

⇒ 8 = m(1) + 5

⇒ 8 = m + 5

⇒ m = 8 − 5

⇒ m = 3

∴ The correct answer is 3.

Additional Information

Slope-Intercept Form

The equation of a straight line is commonly written as y = mx + c, where m is the slope of the line and c is its y-intercept.

Calculating Slope

The slope m of a line passing through any two distinct points (x₁, y₁) and (x₂, y₂) is evaluated as m = (y₂ − y₁) ÷ (x₂ − x₁).

Point-Slope Form

If a line has a given slope m and passes through a specific point (x₁, y₁), its exact equation is found using y − y₁ = m(x − x₁).

X and Y Intercepts

The x-intercept is the point where the line touches the x-axis (y = 0). The y-intercept is the point where it crosses the y-axis (x = 0).

Shortcut Trick

Apply the Digital Sum (DS) method to solve the equation quickly.

DS of 31³ = 4³ = 64 ⇒ 1 | DS of 18³ = 9³ = 729 ⇒ 0.

DS of 37³ = 1³ = 1 | DS of 210 = 3.

Overall Expression DS = 1 + 0 − 1 + 3 = 3.

Check options: DS of −14820 = −(1+4+8+2) = −15 ⇒ −6. Add 9 to get positive DS: −6 + 9 = 3.

∴ The correct answer is -14820.

Alternate Method

Given:

Expression: 31³ + 18³ − 37³ + 210

Formula Used:

x³ = x × x × x

Calculations:

⇒ 31³ = 31 × 31 × 31 = 29791

⇒ 18³ = 18 × 18 × 18 = 5832

⇒ 37³ = 37 × 37 × 37 = 50653

Substitute the obtained values back into the expression:

⇒ Expression = 29791 + 5832 − 50653 + 210

⇒ Expression = 35623 − 50653 + 210

⇒ Expression = −15030 + 210

⇒ Expression = −14820

∴ The correct answer is -14820.

Additional Information

Simplification Rules

Always follow the proper order of operations (BODMAS). First evaluate powers and exponents before proceeding with addition and subtraction.

Digital Sum Concept

The digital sum of a number is the single-digit sum of its digits. The digital sum of the problem expression must equal the digital sum of its exact solution, making it an excellent verification tool.

Unit Digit Concept

The unit digit of an arithmetic expression relies purely on the unit digits of the numbers involved. For example, 1³ + 8³ − 7³ + 0 yields a unit digit calculation of 1 + 2 − 3 + 0 = 0.

Shortcut Trick Square both values to compare them directly without estimating decimal roots. x2 = (√3 + √5)2 = 8 + 2√15 y2 = (√8 + √2)2 = 10 + 2√16 = 18 Since √15 < 4, the expression 8 + 2√15 is strictly less than 16. Because 18 > 16, it follows that y2 > x2, which means y > x. ∴ The correct answer is y > x. Alternate Method Given: x = √3 + √5 y = √8 + √2 Formula Used: (a + b)2 = a2 + b2 + 2ab Calculations: ⇒ Square the value of x: ⇒ x2 = (√3 + √5)2 ⇒ x2 = (√3)2 + (√5)2 + 2 × √3 × √5 ⇒ x2 = 3 + 5 + 2√15 ⇒ x2 = 8 + 2√15 ⇒ Simplify and square the value of y: ⇒ y = 2√2 + √2 = 3√2 ⇒ y2 = (3√2)2 = 9 × 2 = 18 ⇒ Compare the squared values: ⇒ We know that √15 is strictly less than √16 (which evaluates to 4). ⇒ So, x2 = 8 + 2√15 < 8 + 2 × 4 ⇒ x2 < 16 ⇒ Since x2 < 16 and y2 = 18, it is completely clear that y2 > x2. ⇒ y > x ∴ The correct answer is y > x. Additional Information Squaring Method to Compare Surds When comparing irrational expressions involving square roots, squaring both sides removes the root operations and provides whole numbers or simplified surds for easier comparison. Simplifying Like Surds Surds that can be reduced to have the exact same irrational part can be added or subtracted directly like standard algebraic terms, for example, √8 + √2 = 2√2 + √2 = 3√2. Comparing Roots of Different Orders For comparing surds of different roots (such as cube roots and square roots), find the LCM of the orders of the roots and raise all surds to that LCM power to compare exact integer values.

Shortcut Trick

By applying the cross-multiplication method on Red and Blue marbles:

Initial Ratio → 3 : 4 | Final Ratio → 4 : 5 | Quantity Added → +20, +15

1 initial ratio unit = [(4 × 15) − (5 × 20)] ÷ [(3 × 5) − (4 × 4)] = (60 − 100) ÷ (15 − 16) = 40.

Initial Green marbles = 6 × 40 = 240.

New Red marbles = (3 × 40) + 20 = 140, which represents 4 parts in the final ratio ⇒ 1 final unit = 140 ÷ 4 = 35.

New Green marbles = 7 × 35 = 245 ⇒ Green marbles added = 245 − 240 = 5.

∴ The correct answer is 5.

Alternate Method

Given:

Initial ratio of red, blue, and green marbles = 3 : 4 : 6

Marbles added = 20 red, 15 blue, and an unknown number of green marbles (y).

Final ratio of red, blue, and green marbles = 4 : 5 : 7

Initial RatioRed = 3xBlue = 4xGreen = 6xFinal RatioRed = 4 partsBlue = 5 partsGreen = 7 parts+20+15+y

Calculations:

⇒ Let the initial number of red, blue, and green marbles be 3x, 4x, and 6x respectively.

⇒ After adding 20 red and 15 blue marbles, the new counts become (3x + 20) and (4x + 15).

⇒ Given the new ratio of red to blue is 4 : 5.

⇒ (3x + 20) ÷ (4x + 15) = 4 ÷ 5

⇒ 5 × (3x + 20) = 4 × (4x + 15)

⇒ 15x + 100 = 16x + 60

⇒ 16x − 15x = 100 − 60

⇒ x = 40

⇒ Initial red marbles = 3 × 40 = 120

⇒ Initial green marbles = 6 × 40 = 240

⇒ New red marbles = 120 + 20 = 140

⇒ Let the number of green marbles added be y. The new count of green marbles is (240 + y).

⇒ Given the new ratio of red to green is 4 : 7.

⇒ 140 ÷ (240 + y) = 4 ÷ 7

⇒ 240 + y = (140 × 7) ÷ 4

⇒ 240 + y = 35 × 7 = 245

⇒ y = 245 − 240 = 5

∴ The correct answer is 5.

Additional Information

Ratio and Proportion

A ratio compares two quantities, indicating how many times one number contains another. For quantities a and b, it is represented as a : b or the fraction a/b.

Cross-Multiplication Method in Ratios

When a given ratio a : b changes to c : d by adding quantities x and y respectively, the value of 1 initial ratio unit is directly obtained using the formula (cy − dx) ÷ (ad − bc).

Shortcut Trick

For continued fractions, solve directly from the bottommost denominator to the top.

Bottom fraction: 4 + 1/2 = 9/2

Middle fraction: 3 + 2/9 = 29/9

Top fraction: 2 + 9/29 = 67/29

∴ The correct answer is 67/29.

Alternate Method

Given:

x = 2 + 1 / [3 + 1 / (4 + 1/2)]

Calculations:

Solve the innermost fraction first (bottom-up approach):

⇒ 4 + 1/2 = (8 + 1) / 2 = 9/2

Substitute this back into the expression:

⇒ x = 2 + 1 / [3 + 1 / (9/2)]

⇒ x = 2 + 1 / (3 + 2/9)

Now, simplify the next denominator:

⇒ 3 + 2/9 = (27 + 2) / 9 = 29/9

Substitute this back into the expression:

⇒ x = 2 + 1 / (29/9)

⇒ x = 2 + 9/29

⇒ x = (58 + 9) / 29

⇒ x = 67/29

∴ The correct answer is 67/29.

Additional Information

BODMAS Rule in Fractions

Always follow the order of Brackets, Orders, Division, Multiplication, Addition, and Subtraction. In continued fractions, treat the bottommost denominator as an implicit bracket.

Reciprocal of a Fraction

Dividing by a fraction is the same as multiplying by its reciprocal. For example, 1 ÷ (a/b) = b/a.

Continued Fractions

A continued fraction is evaluated through a step-by-step upward process of simplifying the lowest mixed fraction and reciprocating it.

Shortcut Trick

Profit Ratio = Capital Ratio (since time is constant) = 40000 : 60000 = 2 : 3

Total profit ratio parts = 2 + 3 = 5 parts

5 parts = ₹50,000 ⇒ 1 part = ₹10,000

A's share (2 parts) = 2 × 10,000 = ₹20,000

∴ The correct answer is ₹20,000.

Alternate Method

Given:

A's investment = ₹40,000

B's investment = ₹60,000

Total Profit = ₹50,000

Formula Used:

Profit Ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

Capital & Profit Distribution

Total Profit = ₹50,000 (5 units)

A ₹40,000

B ₹60,000

Ratio: 2 Units Ratio: 3 Units

A's Share = 2/5 × ₹50,000 = ₹20,000

Calculations:

⇒ Since the investment time is exactly the same for both A and B, their profit ratio will be directly equal to their investment ratio.

⇒ Profit Ratio = 40000 : 60000

⇒ Profit Ratio = 4 : 6 = 2 : 3

⇒ Total profit parts = 2 + 3 = 5

⇒ A's share = (2 ÷ 5) × Total Profit

⇒ A's share = (2 ÷ 5) × 50000

⇒ A's share = 2 × 10000 = 20000

∴ The correct answer is ₹20,000.

Additional Information

Simple Partnership

When partners invest their capital for the exact same time period, all final profits or losses are distributed in the direct ratio of their original investments.

Compound Partnership

When partners invest for varying time periods, their profit distribution ratio is calculated as (Capital₁ × Time₁) : (Capital₂ × Time₂).

Working vs Sleeping Partner

A working partner manages the business operations and generally receives a fixed salary or percentage of the profit before the remaining balance is divided among all partners based on their respective investments.

Shortcut Trick

Total share of A = (Salary as working partner) + (Investment share of remaining profit)

⇒ A's total share = 10% of 64000 + (3 ÷ 8) × 90% of 64000

⇒ A's total share = 6400 + (3 ÷ 8) × 57600

⇒ A's total share = 6400 + 21600 = 28000

∴ The correct answer is ₹28,000.

Alternate Method

Given:

Total profit = ₹64,000

Investment ratio of A and B = 3 : 5

A's extra share (working partner) = 10% of total profit

Total Profit: ₹64,000A's 10% Extra₹6,400Remaining 90%₹57,600A (3 parts)₹21,600B (5 parts)₹36,000A's Total = ₹28,000

Calculations:

⇒ A's extra profit = 10% of ₹64,000

⇒ A's extra profit = (10 ÷ 100) × 64,000 = ₹6,400

⇒ Remaining profit = 64,000 − 6,400 = ₹57,600

⇒ Total ratio parts = 3 + 5 = 8 parts

⇒ A's share from remaining profit = (3 ÷ 8) × 57,600

⇒ A's share from remaining profit = 3 × 7,200 = ₹21,600

⇒ A's total share = Extra profit + Investment share

⇒ A's total share = 6,400 + 21,600 = ₹28,000

∴ The correct answer is ₹28,000.

Additional Information

Active vs Sleeping Partner

An active partner contributes both capital and time to manage the business, often receiving an extra salary or fixed percentage of profit before the remainder is distributed. A sleeping partner only contributes capital.

Profit Distribution Formula

Profit ratio is directly proportional to the product of invested capital and time period. Profit Ratio = (C₁ × T₁) : (C₂ × T₂).

Equivalent Capital

When investments are made for unequal times, we calculate the equivalent capital for 1 unit of time. Equivalent capital = Investment × Time invested.