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Shortcut Trick

Since the given ratios 17, 5, and 11 are pairwise co-prime numbers, their LCM is simply the product of their ratios multiplied by their HCF.

⇒ LCM = (Product of ratios) × HCF

⇒ 3740 = (17 × 5 × 11) × HCF

⇒ 3740 = 935 × HCF

⇒ HCF = 3740 ÷ 935 = 4

∴ The correct answer is 4.

Alternate Method

Given:

Ratio of the three numbers = 17 : 5 : 11

LCM of the three numbers = 3740

Formula Used:

Let the common factor (which is also the HCF) of the numbers be x.

The three numbers are 17x, 5x, and 11x.

Since 1

Shortcut Trick

According to the ratio 1 : 2, for every 1 part of the first ingredient, there are 2 parts of the second ingredient.

If 1 unit = 3 cups,

Then 2 units = 2 × 3 = 6 cups.

∴ The correct answer is 6 cups.

Alternate Method

Given: Ratio of ingredients = 1 : 2; Quantity of first ingredient = 3 cups.

⇒ Let the quantity of the second ingredient be x cups.

⇒ Ratio = (Ingredient 1) ÷ (Ingredient 2)

⇒ 1 ÷ 2 = 3 ÷ x

⇒ 1 × x = 3 × 2

⇒ x = 6 cups

∴ The correct answer is 6 cups.

Additional Information

Direct Proportion

Two quantities are in direct proportion if the ratio of their values remains constant (x/y = k). If one increases, the other increases accordingly.

Fourth Proportional

If four quantities a, b, c, d are such that a:b = c:d, then d is called the fourth proportional to a, b, and c, calculated as d = (b × c) ÷ a.

Scaling Ratios

To maintain the same ratio when one part is multiplied by a factor, all other parts of the ratio must be multiplied by the same scaling factor.

Shortcut Trick

Net Profit% = Mark-up% − Discount% − (Mark-up × Discount)/100

Net Profit% = 40 − 25 − (40 × 25)/100 = 15 − 10 = 5%

Selling Price = 105% of Cost Price ⇒ 21,000 = 1.05 × CP

CP = 21,000 ÷ 1.05 = 20,000

∴ The correct answer is ₹20,000.

Alternate Method

Given:

Mark-up Percentage = 40%

Discount Percentage = 25%

Selling Price (SP) = ₹21,000

⇒ Let Cost Price (CP) = 100 units

⇒ Marked Price (MP) = 100 + 40% of 100 = 140 units

⇒ Selling Price (SP) = 140 − 25% of 140

⇒ SP = 140 − (25 ÷ 100) × 140 = 140 − 35 = 105 units

⇒ Given, 105 units = ₹21,000

⇒ 1 unit = 21,000 ÷ 105 = 200

⇒ Cost Price (CP) = 100 units = 100 × 200 = ₹20,000

∴ The correct answer is ₹20,000.

Additional Information

Mark-up Percentage

The percentage by which the cost price is increased to set the marked price. Formula: Mark-up% = (MP − CP)/CP × 100.

Relationship between CP and MP

CP × (100 + Profit%) = MP × (100 − Discount%). This ratio can be used to directly find the relation between CP and MP.

Successive Percentage Change

If a value is increased by x% and then decreased by y%, the net change is [x − y − (xy/100)]%.

Given:

$(\frac{28}{8})^{17} \times (\frac{8}{28})^{20} \times (\frac{28}{8})^{3} = (\frac{8}{28})^{9m+14}$

Formula used:

1. (a/b)ⁿ = (b/a)⁻ⁿ
2. a^x × a^y × a^z = a^{x + y + z}
3. If a^x = a^y, then x = y

Calculations:

According to the given equation:

$(\frac{28}{8})^{17} \times (\frac{8}{28})^{20} \times (\frac{28}{8})^{3} = (\frac{8}{28})^{9m+14}$

To solve for m, express all terms with the same base (8/28):

⇒ (8/28)⁻¹⁷ × (8/28)²⁰ × (8/28)⁻³ = (8/28)^{9m + 14}

Applying the law of exponents on the left-hand side:

⇒ (8/28)^{(−17 + 20 − 3)} = (8/28)^{9m + 14}

⇒ (8/28)⁰ = (8/28)^{9m + 14}

Now, equating the exponents on both sides of the equation:

⇒ 0 = 9m + 14

⇒ 9m = −14

⇒ m = −14/9

∴ The correct answer is option (3).

Shortcut Trick

Apply quadrant rules: cos(90° − A) = sin A, cot(90° + A) = −tan A, and cos(180° + A) = −cos A.

Numerator = sin A × sin A = sin² A.

Denominator = (−tan A) × (−cos A) = tan A × cos A = (sin A / cos A) × cos A = sin A.

Expression = sin² A / sin A = sin A.

∴ The correct answer is sin A.

Alternate Method

Given: [sin A × cos(90° − A)] / [cot(90° + A) × cos(180° + A)]

Trigonometric Transformations:

Step-by-step Simplification:

⇒ Numerator: sin A × cos(90° − A)

⇒ In 1^st quadrant, cos(90° − A) = sin A

⇒ Numerator = sin A × sin A = sin² A

⇒ Denominator: cot(90° + A) × cos(180° + A)

⇒ In 2^nd quadrant, cot(90° + A) = −tan A

⇒ In 3^rd quadrant, cos(180° + A) = −cos A

⇒ Denominator = (−tan A) × (−cos A) = tan A × cos A

⇒ Denominator = (sin A / cos A) × cos A = sin A

⇒ Final Expression: sin² A / sin A

⇒ Expression = sin A

∴ The correct answer is sin A.

Additional Information

ASTC Rule

It determines the sign of trigonometric ratios: 1^st Quadrant (All positive), 2^nd (Sine/Cosec positive), 3^rd (Tan/Cot positive), and 4^th (Cos/Sec positive).

Complementary Ratio Changes

Ratios change to their co-ratios at (90° ± θ) or (270° ± θ): sin ↔ cos, tan ↔ cot, and sec ↔ cosec.

Tangent Identity

tan A = sin A / cos A and cot A = cos A / sin A. These are fundamental for simplifying product-form trigonometric expressions.

Shortcut Trick

Ratio of Initial Price to New Price = 400 : 500 = 4 : 5.

Increase = 5 − 4 = 1 unit.

Percentage Increase = (Increase ÷ Initial Ratio) × 100 = (1 ÷ 4) × 100 = 25%.

∴ The correct answer is 25%.

Alternate Method

Given:

Original Price of the shirt = ₹400

Increased Price of the shirt = ₹500

Formula Used:

Percentage Increase = [(New Price − Old Price) ÷ Old Price] × 100

Calculations:

⇒ Total price increase = 500 − 400 = ₹100

⇒ Percentage increase = (100 ÷ 400) × 100

⇒ Percentage increase = (1 ÷ 4) × 100

⇒ Percentage increase = 25%

Therefore, Option (3) is the correct answer.

∴ The correct answer is 25%.

Additional Information

Percentage Change Formula

Percentage Increase or Decrease = (Difference ÷ Original Value) × 100. Always use the starting (initial) value as the base.

Successive Percentage Change

If a value increases by a% and then by b%, the net percentage change is given by [a + b + (a × b ÷ 100)]%.

Fraction to Percentage Conversion

Common fractions help in fast calculations: 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, and 1/8 = 12.5%.

Given:

(3/4) + 0.25 × 2

Formula used:

BODMAS rule: Perform Multiplication before Addition.

Fraction conversion: 3/4 = 0.75

Calculations:

(3/4) + 0.25 × 2

⇒ 0.75 + 0.25 × 2

⇒ 0.75 + 0.50

⇒ 1.25

∴ The correct answer is option (3).

Shortcut Trick

Total Distance = 20 km (Going) + 20 km (Returning) = 40 km.

Total Time = (20 ÷ 10) + 1 (Stay) + (20 ÷ 20) = 2 + 1 + 1 = 4 hours.

Average Speed = Total Distance ÷ Total Time = 40 ÷ 4 = 10 km/h.

∴ The correct answer is 10 km/h.

Alternate Method

Given:

Distance from Home to Market = 20 km

Speed while walking = 10 km/h

Stay time at market = 1 hour

Speed while returning by e-rickshaw = 20 km/h

Formula used:

Time = Distance ÷ Speed

Average Speed = (Total Distance) ÷ (Total Time)

Calculations:

⇒ Time taken to reach the market = 20 ÷ 10 = 2 hours

⇒ Time taken for returning = 20 ÷ 20 = 1 hour

⇒ Total time taken for the entire trip = 2 (walking) + 1 (stay) + 1 (returning) = 4 hours

⇒ Total distance covered = 20 (going) + 20 (returning) = 40 km

⇒ Average Speed = 40 ÷ 4 = 10 km/h

∴ The correct answer is 10 km/h.

Additional Information

Average Speed (Equal Distances)

If a person covers equal distances at speeds x and y, the average speed (excluding stay time) is given by 2xy ÷ (x + y).

Stoppage Time Effect

When stay time or stoppage time is included, the total time of the journey increases, which subsequently reduces the average speed of the trip.

Unit Conversion

To convert speed from km/h to m/s, multiply the value by 5 ÷ 18. To convert m/s to km/h, multiply by 18 ÷ 5.

Shortcut Trick

Identify the numbers as a = 22, b = −13, and c = −9.

Check the sum: a + b + c = 22 + (−13) + (−9) = 0.

Using the Algebraic Identity: If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Calculation: 3 × 22 × (−13) × (−9) = 66 × 117 = 7722.

∴ The correct answer is 7722.

Alternate Method

Given:

The expression to evaluate is 22³ + (−13)³ + (−9)³.

Formula Used:

The conditional identity states that if a + b + c = 0, then a³ + b³ + c³ = 3abc.

⇒ Let a = 22, b = −13, and c = −9.

⇒ a + b + c = 22 + (−13) + (−9)

⇒ a + b + c = 22 − 22 = 0.

⇒ Since the sum of the terms is zero, apply the identity:

⇒ Value = 3 × (22) × (−13) × (−9)

⇒ Value = 66 × 117

⇒ Value = 7722

Therefore, the calculated value is 7722, which matches option (2).

∴ The correct answer is 7722.

Additional Information

General Algebraic Identity

The complete identity is a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

Negative Number Cubes

The cube of a negative number is always negative, for example, (−13)³ = −2197.

Applications of Identity

This identity is primarily used to simplify expressions involving cubes without performing large multiplications manually.

Shortcut Trick

Total units initially sold = 120 + 150 + 110 + 130 + 140 = 650 units

Initial Arithmetic Mean = 650 ÷ 5 = 130 units

A return of 10 units reduces the total sum by 10.

Reduction in mean = Total Reduction ÷ Number of items = 10 ÷ 5 = 2 units

New Arithmetic Mean = 130 − 2 = 128 units

∴ The correct answer is 128 units.

Alternate Method

Given:

Units sold for 5 items = 120, 150, 110, 130, 140

Return from the first item = 10 units

Formula Used:

Arithmetic Mean = Sum of all observations ÷ Total number of observations

Calculations:

⇒ New value of the first item = 120 − 10 = 110 units

⇒ Sum of new observations = 110 + 150 + 110 + 130 + 140

⇒ Total Sum = 640 units

⇒ New Arithmetic Mean = 640 ÷ 5

⇒ New Mean = 128 units

∴ The correct answer is 128 units.

Additional Information

Arithmetic Mean (Average)

The average of a set of numbers is the sum of the numbers divided by the count. Formula: x̄ = (Σx_i) ÷ n.

Property of Change in Mean

If every observation in a data set is decreased by a constant value k, then the mean of the new observations is decreased by k. If only one observation is decreased by k, the mean decreases by k/n.

Sum of Deviations

The sum of the deviations of all observations from their arithmetic mean is always equal to zero, i.e., Σ(x_i − x̄) = 0.