Quantitative Aptitude PYQs

Shortcut Trick

Let the middle odd number between the two consecutive even numbers be n.

The numbers can be written as (n − 1) and (n + 1). Their sum is 2n.

Given: (n − 1)² + (n + 1)² = 1924

⇒ 2(n² + 1) = 1924 ⇒ n² + 1 = 962 ⇒ n² = 961 ⇒ n = 31

The sum of the two numbers is 2n = 2 × 31 = 62.

∴ The correct answer is 62.

Alternate Method

Given:

The sum of the squares of two consecutive even natural numbers is 1924.

Calculations:

Let the two consecutive even natural numbers be x and x + 2.

⇒ x² + (x + 2)² = 1924

Using the identity (a + b)² = a² + 2ab + b²:

⇒ x² + x² + 4x + 4 = 1924

⇒ 2x² + 4x + 4 − 1924 = 0

⇒ 2x² + 4x − 1920 = 0

Divide the entire equation by 2 to simplify:

⇒ x² + 2x − 960 = 0

Factorizing the quadratic equation by splitting the middle term:

⇒ x² + 32x − 30x − 960 = 0

⇒ x(x + 32) − 30(x + 32) = 0

⇒ (x − 30)(x + 32) = 0

⇒ x = 30 or x = −32

Since the question specifies natural numbers, we ignore the negative value and take x = 30.

The consecutive even numbers are 30 and 32.

⇒ Sum of the numbers = 30 + 32 = 62

∴ The correct answer is 62.

Additional Information

Consecutive Numbers

Consecutive even or odd numbers always have a common difference of 2. They can be generally represented as x and x + 2 to set up algebraic equations.

Simplifying Sum of Squares

When calculating the sum of squares of two consecutive numbers differing by 2, assuming them as (n − 1) and (n + 1) simplifies the equation to 2(n² + 1), canceling out the linear terms completely.

Nature of Roots

A quadratic equation ax² + bx + c = 0 yields two roots. In practical word problems involving natural numbers, dimensions, or time, negative or fractional roots are discarded.

Shortcut Trick

For any regular polygon, the exterior angle is directly calculated as 360° ÷ number of sides.

⇒ Exterior Angle = 360° ÷ 15 = 24°

The interior and exterior angles always form a linear pair (sum = 180°).

⇒ Interior Angle = 180° − Exterior Angle

⇒ Interior Angle = 180° − 24° = 156°

∴ The correct answer is 156°.

Alternate Method

Given:

Number of sides of the regular polygon (n) = 15

Formula Used:

Sum of interior angles of an n-sided polygon = (n − 2) × 180°

Each interior angle = [(n − 2) × 180°] ÷ n

Interior = 156°
Exterior = 24°
Sum of Interior & Exterior Angle = 180°

Calculations:

⇒ Each interior angle = [(15 − 2) × 180°] ÷ 15

⇒ Each interior angle = [13 × 180°] ÷ 15

⇒ Each interior angle = 13 × 12°

⇒ Each interior angle = 156°

∴ The correct answer is 156°.

Additional Information

Sum of Interior Angles

For a polygon with n sides, the sum of all interior angles is given by (n − 2) × 180°.

Sum of Exterior Angles

The sum of exterior angles of any convex polygon is always 360°, regardless of the number of sides.

Number of Diagonals

The number of diagonals that can be drawn in an n-sided polygon is given by n(n − 3) ÷ 2.

Shortcut Trick

4% increase = +1/25 ⇒ Multiplying factor = 26/25

24% increase = +6/25 ⇒ Multiplying factor = 31/25

Total multiplying factor = (26/25) × (31/25) = 806/625

Initial Sales = Current Sales × (625/806)

Initial Sales = 1,61,200 × (625/806) = 200 × 625 = 1,25,000

∴ The correct answer is ₹1,25,000.

Alternate Method

Given:

First year increase = 4%

Second year increase = 24%

Current sales (after 2 years) = ₹1,61,200

Formula Used:

Final Value = Initial Value × (1 + R₁/100) × (1 + R₂/100)

Successive Percentage IncreaseInitial? (x)+4%× 26/25After 1 yr1.04x+24%× 31/25Current1,61,200x × (26/25) × (31/25) = 1,61,200

Calculations:

⇒ Let the initial sales two years ago be x.

⇒ Final Sales = x × (1 + 4/100) × (1 + 24/100)

⇒ 1,61,200 = x × (104/100) × (124/100)

⇒ 1,61,200 = x × (26/25) × (31/25)

⇒ 1,61,200 = x × (806/625)

⇒ x = 1,61,200 × (625/806)

⇒ x = 200 × 625

⇒ x = 1,25,000

∴ The correct answer is ₹1,25,000.

Additional Information

Successive Percentage Change

When a value undergoes successive percentage changes of x% and y%, the effective net percentage change is given by the formula: (x + y + xy/100)%.

Multiplying Factor Method

For an increase of R%, the multiplier is (1 + R/100). For successive changes, simply multiply all individual factors: Final Value = Initial Value × Factor₁ × Factor₂.

Compound Interest Analogy

Successive percentage increase follows the exact same logical framework as compound interest, representing cumulative growth upon previous growth over multiple periods.

Shortcut Trick

According to the BODMAS rule, perform division and multiplication first:

(−60) ÷ (−12) = 5

(−2) × 8 = −16

Substitute these results back into the original expression:

⇒ −9 − 5 − 16 = −30

∴ The correct answer is -30.

Alternate Method

Given:

(−9) − (−60) ÷ (−12) + (−2) × 8

Concept Used:

BODMAS Rule: Priority of mathematical operations follows the order Bracket, Order, Division, Multiplication, Addition, and Subtraction.

Calculations:

⇒ (−9) − [(−60) ÷ (−12)] + [(−2) × 8]

⇒ −9 − (5) + (−16)

⇒ −9 − 5 − 16

⇒ −14 − 16

⇒ −30

∴ The correct answer is -30.

Additional Information

BODMAS Rule

It specifies the correct order of operations to solve a mathematical expression: Brackets, Order (powers and roots), Division, Multiplication, Addition, and Subtraction.

Sign Convention for Multiplication and Division

The product or quotient of two numbers with the same sign (both negative or both positive) is always positive. The product or quotient of one positive and one negative number is always negative.

Sign Convention for Addition

When adding numbers with the same sign, you simply add their absolute values and keep the common sign. For example, −A − B = −(A + B).

Shortcut Trick

Total work = LCM of 30, 16, and 15 = 240 units.

Efficiency of Anjani (A) = 240 ÷ 30 = 8 units/day.

Efficiency of (A + K) = 240 ÷ 16 = 15 units/day, and (A + K + S) = 240 ÷ 15 = 16 units/day.

Efficiency of Sushmita (S) = 16 − 15 = 1 unit/day.

Time taken by A + S = 240 ÷ (8 + 1) = 240 ÷ 9 = 80/3 days.

∴ The correct answer is 80/3 days.

Alternate Method

Given:

Time taken by Anjani (A) = 30 days.

Time taken by Anjani and Khushbu (A + K) = 16 days.

Time taken by Anjani, Khushbu, and Sushmita (A + K + S) = 15 days.

Formula Used:

Total Work = Efficiency × Time taken

Efficiency is inversely proportional to time.

Calculations:

⇒ Efficiency of Sushmita (S) = Efficiency of (A + K + S) − Efficiency of (A + K)

⇒ Efficiency of S = 16 − 15 = 1 unit/day

⇒ Efficiency of Anjani (A) = 8 units/day

⇒ Combined efficiency of Anjani and Sushmita (A + S) = 8 + 1 = 9 units/day

⇒ Time taken by A and S together = Total Work ÷ Combined Efficiency

⇒ Time taken = 240 ÷ 9 = 80/3 days

∴ The correct answer is 80/3 days.

Additional Information

Pipes & Cisterns

Similar to the time and work concept, an inlet pipe has a positive filling efficiency while an outlet pipe (or leak) has a negative emptying efficiency. Time to fill = Total Capacity ÷ Net Efficiency.

Work & Wages

Total wages are distributed strictly in proportion to the actual amount of work done by each individual. If they work for the exact same number of days, the wages are divided in the ratio of their efficiencies.

Alternate Day Work

When individuals work on alternate days, find the total work done in one complete cycle (e.g., 2 days for 2 people) and divide the total work by the cycle work to find the number of full cycles needed.

Shortcut Trick

Remaining Distance = 886 − (53 × 8) − 150 = 312 km

Remaining Time = 17 − 8 − (150 ÷ 50) = 6 hours

Required Speed = 312 ÷ 6 = 52 km/hr

∴ The correct answer is 52 km/hr.

Alternate Method

Given:

Total distance = 886 km

Total time = 17 hours

Total Distance: 886 km | Total Time: 17 hours

424 km
150 km
312 km
8 hrs
3 hrs
6 hrs
53 km/hr
50 km/hr
? km/hr

Formula Used:

Distance = Speed × Time

Speed = Distance ÷ Time

Calculations:

⇒ Distance covered in first 8 hours = 53 × 8 = 424 km

⇒ Time taken to cover next 150 km = 150 ÷ 50 = 3 hours

⇒ Total distance covered so far = 424 + 150 = 574 km

⇒ Total time used so far = 8 + 3 = 11 hours

⇒ Remaining distance = 886 − 574 = 312 km

⇒ Remaining time = 17 − 11 = 6 hours

⇒ Required speed = 312 ÷ 6 = 52 km/hr

∴ The correct answer is 52 km/hr.

Additional Information

Average Speed

Average Speed is the ratio of total distance covered to the total time taken. Formula: Average Speed = Total Distance ÷ Total Time.

Relative Speed

When two bodies are moving in the same direction, relative speed is the difference of their speeds (S₁ − S₂). In opposite directions, it is the sum of their speeds (S₁ + S₂).

Conversion of Units

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

Shortcut Trick

In compound interest, if a sum becomes 'x' times in 'T' years, it will become xⁿ times in n × T years.

Given, the sum becomes 3 times in 5 years.

We need to find the time it takes to become 9 times, which is 3² times.

Here, x = 3, T = 5 years, and n = 2.

Required time = n × T = 2 × 5 = 10 years.

∴ The correct answer is 10 years.

Alternate Method

P

3P

9P

5 years
× 3

5 years
× 3

Total Time = 10 years

Given:

Amount becomes 3 times the principal in 5 years at compound interest.

Formula Used:

A = P(1 + R/100)^t

Calculations:

⇒ 3P = P(1 + R/100)⁵

⇒ 3 = (1 + R/100)⁵     ... (Equation 1)

We need to find the time 't' when the amount becomes 9P.

⇒ 9P = P(1 + R/100)^t

⇒ 9 = (1 + R/100)^t

Since 9 = 3², we can substitute the value of 3 from Equation 1.

⇒ ((1 + R/100)⁵)² = (1 + R/100)^t

⇒ (1 + R/100)¹⁰ = (1 + R/100)^t

By comparing the powers on both sides:

⇒ t = 10

∴ The correct answer is 10 years.

Additional Information

Simple Interest vs Compound Interest

In Simple Interest, the principal remains constant. In Compound Interest, the interest generated is added back to the principal to form a new base for the next compounding period.

Compound Interest Formula

A = P(1 + R/100)^t, where A is the Final Amount, P is the Principal sum, R is the Rate of interest per annum, and t is the time period in years.

Multiplying Factor in CI

For equal time intervals, the amounts in compound interest are in a geometric progression. The ratio between amounts over any equal continuous time intervals remains strictly constant.

Shortcut Trick

Let Cost Price (CP) = 100%, then Marked Price (MP) = 136%.

Selling Price (SP) after 25% discount = 136% × 0.75 = 102%.

Profit = SP − CP = 102% − 100% = 2%.

Given 2% = ₹171 ⇒ MP (136%) = 171 × 68 = ₹11,628.

∴ The correct answer is ₹11,628.

Alternate Method

CP = 100x

SP = 102x

MP = 136x

+36% Markup

Profit = 2x

−25% Discount

Given:

⇒ Markup percentage = 36%

⇒ Discount percentage = 25%

⇒ Profit earned = ₹171

Let the Cost Price (CP) of the fan be 100x.

⇒ Marked Price (MP) = 100x + 36% of 100x = 136x

⇒ Selling Price (SP) = MP − 25% of MP = 136x × 0.75 = 102x

⇒ Profit = SP − CP = 102x − 100x = 2x

According to the given data, profit is ₹171:

⇒ 2x = 171

⇒ x = 85.5

⇒ List Price (MP) = 136x = 136 × 85.5 = 11,628

∴ The correct answer is ₹11,628.

Additional Information

1. Discount and Marked Price

Discount is always calculated on the Marked Price (List Price). Formula: Selling Price (SP) = Marked Price (MP) × (1 − Discount%/100). 2. Relation between CP and MP

When Profit percentage and Discount percentage are given, the direct ratio is: CP / MP = (100 − Discount%) / (100 + Profit%). 3. Successive Discounts

When two successive discounts of a% and b% are applied, the net equivalent discount is given by: [a + b − (a × b)/100]%.

Shortcut Trick

To check divisibility by 7, 11, and 13 simultaneously, group the digits into blocks of three from right to left.

The blocks of 788973289 are: 788, 973, 289.

Difference between the sum of odd-placed blocks and even-placed blocks = (788 + 289) − 973 = 1077 − 973 = 104.

Since 104 is exactly divisible by 13 (13 × 8 = 104), the entire number is perfectly divisible by 13.

∴ The correct answer is 13.

Alternate Method

Given:

Number = 788973289

Method Used:

Test each option systematically using standard divisibility rules.

Calculations:

⇒ Option 1 (18): A number divisible by 18 must be even. Since 788973289 ends with the digit 9, it is an odd number and not divisible by 18.

⇒ Option 2 (12): A number divisible by 12 must be divisible by 4 (hence even). Thus, it is not divisible by 12.

⇒ Option 3 (7) and Option 4 (13): Apply the block divisibility rule for 7, 11, and 13.

⇒ Split the number into groups of three digits from right to left: 289, 973, 788.

⇒ Sum of alternating groups = 788 + 289 = 1077.

⇒ Difference = 1077 − 973 = 104.

⇒ Check divisibility of 104 by 7: 104 ÷ 7 = 14 with a remainder of 6. So, not divisible by 7.

⇒ Check divisibility of 104 by 13: 104 ÷ 13 = 8 with a remainder of 0. So, it is divisible by 13.

⇒ Therefore, 788973289 is exactly divisible by 13.

∴ The correct answer is 13.

Additional Information

Divisibility Rule of 7, 11, and 13

Make groups of 3 digits starting from the rightmost digit. Find the difference between the sum of the odd-placed groups and the even-placed groups. If this final difference is divisible by 7, 11, or 13, then the original number is also perfectly divisible by 7, 11, or 13 respectively.

Divisibility Rule of 12

A number is divisible by 12 if it is perfectly divisible by both 3 and 4. Consequently, any number divisible by 12 must be an even number.

Divisibility Rule of 18

A number is exactly divisible by 18 if it is an even number (ends in 0, 2, 4, 6, 8) and the sum of its digits is a multiple of 9.

Shortcut Trick

Given 7A = 4B = 2C, we can find the ratio using the cover-up method.

A : B : C = (4 × 2) : (7 × 2) : (7 × 4)

⇒ A : B : C = 8 : 14 : 28 = 4 : 7 : 14

Total parts = 4 + 7 + 14 = 25 parts

A's share = (4 ÷ 25) × 15500 = 4 × 620 = 2480

∴ The correct answer is ₹2480.

Alternate Method

Given:

Total Amount = ₹15,500

Condition: 7A = 4B = 2C

Total Amount = ₹15,500 (25 units)

A
4 units

B
7 units

C
14 units

A's share = 4 / 25 × 15500 = ₹2480

Calculations:

⇒ Let 7A = 4B = 2C = k

⇒ A = k/7, B = k/4, C = k/2

⇒ Ratio A : B : C = 1/7 : 1/4 : 1/2

⇒ To simplify, multiply by the LCM of 7, 4, and 2 (which is 28).

⇒ A : B : C = (1/7 × 28) : (1/4 × 28) : (1/2 × 28)

⇒ A : B : C = 4 : 7 : 14

⇒ Sum of the ratio parts = 4 + 7 + 14 = 25

⇒ Value of 1 part = 15500 ÷ 25 = 620

⇒ A's share = 4 parts = 4 × 620

⇒ A's share = 2480

∴ The correct answer is ₹2480.

Additional Information

Cover-Up Method for Ratios

When given an equation like aA = bB = cC, the ratio A : B : C can be directly found by hiding the coefficient of the variable and multiplying the other two coefficients. A = (b × c), B = (a × c), C = (a × b).

Inverse Proportion to Direct Ratio

If variables are multiplied by constants equating each other, they are inversely proportional to those constants. Taking the reciprocal of each constant directly gives the actual ratio components.

LCM Method for Fractions

To simplify a fractional ratio like 1/x : 1/y : 1/z into integers, simply multiply each term by the LCM of the denominators x, y, and z.

Shortcut Trick

Apply the BODMAS rule directly to the expression to solve quickly:

⇒ 9 × (19 × 16) ÷ 14 + 13 − 86

⇒ 9 × 304 ÷ 14 − 73

⇒ 2736 ÷ 14 − 73 ⇒ 1368/7 − 73

⇒ (1368 − 511) / 7 ⇒ 857/7

∴ The correct answer is 857/7.

Alternate Method

Given:

Expression: 9 × (19 × (4²)) ÷ 14 + 13 − 86

Formula Used:

BODMAS Rule: Follow the strict sequence of Brackets, Orders (powers), Division, Multiplication, Addition, and Subtraction.

Calculation:

Solve the innermost bracket (power) first:

⇒ 9 × (19 × 16) ÷ 14 + 13 − 86

Solve the remaining bracket (multiplication):

⇒ 9 × 304 ÷ 14 + 13 − 86

Perform multiplication next:

⇒ 2736 ÷ 14 + 13 − 86

Perform division and simplify the fraction:

⇒ 1368/7 + 13 − 86

Perform subtraction of integers first for easier calculation:

⇒ 1368/7 − 73

Take the lowest common multiple (LCM) to subtract the integer from the fraction:

⇒ (1368 − 73 × 7) / 7

⇒ (1368 − 511) / 7

⇒ 857/7

∴ The correct answer is 857/7.

Additional Information

BODMAS Rule

BODMAS dictates the correct order of operations in math: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction. Expressions must be evaluated strictly in this sequence to avoid calculation errors.

Fraction Simplification

When adding or subtracting whole numbers and fractions, convert the whole number to a fraction by taking the LCM of the denominators to combine the terms effectively.

Rules of Exponents

An exponent signifies repeated multiplication of the base. For example, a² = a × a. Always calculate the exponent before attempting multiplication or division connected to it.

Shortcut Trick

A discount of 20% is equivalent to a fraction of 1/5.

This means if the Marked Price is 5 units, the Selling Price is 4 units (5 − 1).

Given, 4 units = ₹1,800.

1 unit = 1800 ÷ 4 = ₹450.

Marked Price (5 units) = 5 × 450 = ₹2,250.

∴ The correct answer is ₹2,250.

Alternate Method

Given:

Selling Price (SP) of the gift article = ₹1,800

Discount percentage given on the article = 20%

Formula Used:

SP = MP × [(100 − Discount%) ÷ 100]

MP = ?
100% (Base)

20% Discount

SP = ₹1,800
80% of MP
80% = ₹1,800 → 100% = ₹2,250

Calculations:

⇒ Let the Marked Price of the article be MP.

⇒ 1800 = MP × [(100 − 20) ÷ 100]

⇒ 1800 = MP × (80 ÷ 100)

⇒ 1800 = MP × (4 ÷ 5)

⇒ MP = (1800 × 5) ÷ 4

⇒ MP = 450 × 5

⇒ MP = 2250

∴ The correct answer is ₹2,250.

Additional Information

Marked Price (MP) vs Cost Price (CP)

Discount is always calculated on the Marked Price set by the shopkeeper, whereas Profit or Loss is calculated on the actual Cost Price.

Successive Discounts

If two successive discounts of a% and b% are given on an article, the equivalent single discount percentage is (a + b − ab ÷ 100)%.

Direct Relation Formula

The direct ratio of Cost Price to Marked Price can be represented using the formula: CP / MP = (100 − Discount%) / (100 + Profit%).

Shortcut Trick

To find the divisor quickly, check the divisibility rule for 3.

Sum of digits of 342165 = 3 + 4 + 2 + 1 + 6 + 5 = 21.

Since 21 is exactly divisible by 3, the number 342165 is also divisible by 3.

∴ The correct answer is 3.

Alternate Method

Given:

Number = 342165

Concept Used:

Divisibility rules of numbers.

Calculations:

⇒ Let us check the divisibility rules for the given values:

⇒ Divisibility by 2: The given number ends in 5, which is an odd digit. Hence, it is not divisible by 2.

⇒ Divisibility by 4: A number is divisible by 4 if its last two digits form a number divisible by 4. Here, the last two digits are 65. Since 65 is not completely divisible by 4, the number is not divisible by 4.

⇒ Divisibility by 7: Dividing the number manually: 342165 ÷ 7 = 48880 with a remainder of 5. Since the remainder is not 0, it is not divisible by 7.

⇒ Divisibility by 3: A number is exactly divisible by 3 if the sum of all its digits is a multiple of 3.

⇒ Sum of digits = 3 + 4 + 2 + 1 + 6 + 5 = 21.

⇒ 21 ÷ 3 = 7 (exactly divisible).

⇒ Therefore, the number 342165 is completely divisible by 3.

∴ The correct answer is 3.

Additional Information

Divisibility by 6

A number is divisible by 6 if it successfully satisfies the divisibility rules of both 2 and 3.

Divisibility by 9

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

Divisibility by 11

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

Shortcut Trick

Convert the given decimals into simplified fractions:

3.8 = 38/10 = 19/5

0.052 = 52/1000 = 13/250

Formula: LCM of fractions = LCM of Numerators ÷ HCF of Denominators

⇒ LCM(19, 13) ÷ HCF(5, 250) = 247 ÷ 5 = 49.4

∴ The correct answer is 49.4.

Alternate Method

Given:

The numbers are 3.8 and 0.052.

Concept Used:

To find the LCM of decimals, equate the number of decimal places by appending zeros, compute the LCM of the integer values without decimals, and then place the decimal point back at the same original position.

Calculations:

⇒ Make the decimal places equal: 3.8 becomes 3.800 and 0.052 remains 0.052.

⇒ Treat the numbers as integers by ignoring the decimals: 3800 and 52.

⇒ Prime factorization of 3800 = 2³ × 5² × 19

⇒ Prime factorization of 52 = 2² × 13

⇒ LCM(3800, 52) = Product of the maximum power of each prime factor

⇒ LCM(3800, 52) = 2³ × 5² × 13 × 19 = 8 × 25 × 13 × 19

⇒ LCM(3800, 52) = 200 × 247 = 49400

⇒ Place the decimal point 3 places from the right (as there are 3 decimal places in 3.800 and 0.052).

⇒ Final LCM = 49400 ÷ 1000 = 49.4

∴ The correct answer is 49.4.

Additional Information

LCM of Fractions

The LCM of two or more fractions is calculated as: LCM = LCM of Numerators ÷ HCF of Denominators. Always ensure the fractions are reduced to their simplest form before applying the formula.

HCF of Fractions

The HCF of two or more fractions is calculated as: HCF = HCF of Numerators ÷ LCM of Denominators.

LCM of Decimals

Make the number of decimal places identical by appending zeros to the right end. Compute the LCM of the integer values ignoring the decimal point completely, then put the decimal point back to match the equalized decimal places in the final result.

Shortcut Trick

When Simple Interest is equal for two investments made for the same duration, the ratio of their principals is inversely proportional to their respective interest rates.

P₁ : P₂ = R₂ : R₁ = 4 : 10 = 2 : 5

Total ratio parts = 2 + 5 = 7

Sum invested at 10% (P₁) = (2 ÷ 7) × 20300 = ₹5800

∴ The correct answer is ₹5800.

Alternate Method

Given:

Total sum invested = ₹20,300

Rate for Part 1 (R₁) = 10% per annum, Time = 4 years

Rate for Part 2 (R₂) = 4% per annum, Time = 4 years

Interest from Part 1 = Interest from Part 2

Formula Used:

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

Total Investment = ₹20,300

Part 1 (P1)
R = 10%, T = 4 yrs

Part 2 (P2)
R = 4%, T = 4 yrs

Interest 1

Interest 2

=

Calculations:

⇒ Let the sum invested at 10% be P₁ and at 4% be P₂.

⇒ SI₁ = (P₁ × 10 × 4) ÷ 100

⇒ SI₂ = (P₂ × 4 × 4) ÷ 100

⇒ According to the question, SI₁ = SI₂

⇒ (P₁ × 40) ÷ 100 = (P₂ × 16) ÷ 100

⇒ 40P₁ = 16P₂

⇒ P₁ ÷ P₂ = 16 ÷ 40 = 2 ÷ 5

⇒ Ratio of investments P₁ : P₂ = 2 : 5

⇒ Sum of ratio parts = 2 + 5 = 7

⇒ Sum invested at 10% (P₁) = (2 ÷ 7) × 20,300

⇒ P₁ = 2 × 2900 = ₹5800

∴ The correct answer is ₹5800.

Additional Information

Equal Simple Interest Rule

If a certain sum is divided into multiple parts such that their simple interests are equal, the ratio of the principal amounts is given by: P₁ : P₂ : P₃ = 1 ÷ (R₁T₁) : 1 ÷ (R₂T₂) : 1 ÷ (R₃T₃).

Equal Amount Rule

If a sum is divided into multiple parts such that the final amounts (Principal + Interest) are equal, then the principal ratio is: P₁ : P₂ = 1 ÷ (100 + R₁T₁) : 1 ÷ (100 + R₂T₂).

Simple vs Compound Interest Difference

For a sum invested over 2 years at rate R%, the difference between CI and SI is: Difference = P(R ÷ 100)².

Shortcut Trick

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

Mean proportional = √(98 × 18)

Mean proportional = √(49 × 2 × 9 × 2)

Mean proportional = 7 × 3 × 2 = 42

∴ The correct answer is 42.

Alternate Method

98 : x
x : 18
::

Product of Means = Product of Extremes
x × x = 98 × 18

Given:

Two numbers: 98 and 18

Formula Used:

If x is the mean proportional between a and b, then a : x = x : b, which implies x² = a × b.

Calculations:

⇒ Let the mean proportional be x.

⇒ 98 ÷ x = x ÷ 18

⇒ x² = 98 × 18

⇒ x² = (49 × 2) × (9 × 2)

⇒ x² = 49 × 9 × 4

⇒ x = √(49 × 9 × 4)

⇒ x = 7 × 3 × 2

⇒ x = 42

∴ The correct answer is 42.

Additional Information

Third Proportional

If c is the third proportional to a and b, then a : b :: b : c, which gives c = b² ÷ a.

Fourth Proportional

If d is the fourth proportional to a, b and c, then a : b :: c : d, which gives d = (b × c) ÷ a.

Duplicate & Sub-duplicate Ratio

The duplicate ratio of a : b is a² : b², and the sub-duplicate ratio is √a : √b.

Shortcut Trick

In a 'Buy X Get Y Free' scheme, the actual discount amount given to the customer is strictly the total marked price of the free items.

Number of free items = 2

Marked Price per item = ₹250

Total Discount Amount = 2 × 250 = ₹500

∴ The correct answer is ₹500.

Alternate Method

Given:

Paid items = 5

Free items = 2

Marked Price (MP) per item = ₹250

Formula Used:

Total MP = Total items × MP per item

Total Selling Price (SP) = Paid items × MP per item

Discount Amount = Total MP − Total SP

5 Paid Items

2 Free Items

₹250

₹250

₹250

₹250

₹250

₹250

₹250

Total MP = 7 × 250 = ₹1750
Customer pays = 5 × 250 = ₹1250
Discount Amount = 1750 − 1250 = ₹500

Calculations:

⇒ Total items taken by customer = 5 + 2 = 7

⇒ Total MP of 7 items = 7 × 250 = ₹1750

⇒ Total SP (since customer only pays for 5 items) = 5 × 250 = ₹1250

⇒ Discount Amount = Total MP − Total SP

⇒ Discount Amount = 1750 − 1250 = ₹500

∴ The correct answer is ₹500.

Additional Information

Buy X Get Y Free %

To calculate the actual percentage discount equivalent to the scheme: Discount % = [Free Items ÷ (Paid + Free Items)] × 100.

Marked Price (MP)

The labeled or written price of an article. Any discount or scheme offered by the seller is always calculated on the MP, not the Cost Price (CP).

Successive Discounts

When two successive discounts of d₁% and d₂% are given, the effective single discount is calculated as (d₁ + d₂ − d₁×d₂/100)%.

Shortcut Trick

Total Work = LCM of (21, 15, 2) = 210 units.

Efficiency of A = 210 ÷ 21 = 10 units/day.

Efficiency of B = 210 ÷ 15 = 14 units/day.

Combined Efficiency of (A + B + C) = 210 ÷ 2 = 105 units/day.

Time for C = 210 ÷ (105 − 10 − 14) = 210 ÷ 81 = 70 ÷ 27 = 2¹⁶/₂₇ days.

∴ The correct answer is 2¹⁶/₂₇ days.

Alternate Method

Given:

Time taken by A = 21 days

Time taken by B = 15 days

Time taken by A, B, and C together = 2 days

Calculations:

⇒ Efficiency of A = 10 units/day

⇒ Efficiency of B = 14 units/day

⇒ Efficiency of A + B + C = 105 units/day

⇒ Efficiency of C = Efficiency of (A + B + C) − Efficiency of A − Efficiency of B

⇒ Efficiency of C = 105 − 10 − 14 = 81 units/day

⇒ Time taken by C alone = Total Work ÷ Efficiency of C

⇒ Time taken by C alone = 210 ÷ 81

⇒ Time taken by C alone = 70 ÷ 27 = 2¹⁶/₂₇ days

∴ The correct answer is 2¹⁶/₂₇ days.

Additional Information

Pipes & Cisterns

Works exactly like Time & Work. Filling pipes have a positive efficiency, and emptying pipes (leaks) have a negative efficiency.

Work & Wages

Wages are distributed in the ratio of the total work done by each person. If they work for the same number of days, wages are distributed directly in the ratio of their individual efficiencies.

Alternate Day Work

When persons work on alternate days, find the work done in one complete cycle (e.g., 2 days for 2 persons). Then find the closest multiple to the total work before calculating the final remaining days.

Shortcut Trick

For a circle, when Circumference = 44 cm, Radius = 7 cm and Base Area = 154 cm² (standard fixed values).

Volume of Cylinder = Base Area × Height

Volume = 154 × 20 = 3080 cm³.

∴ The correct answer is 3080 cu. cm.

Alternate Method

Given:

Height of the cylinder (h) = 20 cm

Circumference of the base (C) = 44 cm

Formula Used:

Circumference of a circle = 2πr

Volume of a cylinder = πr²h

r

h = 20 cm

Circumference = 44 cm

Right Circular Cylinder

Calculations:

⇒ 2 × (22/7) × r = 44

⇒ (44/7) × r = 44

⇒ r = 44 × (7/44)

⇒ r = 7 cm

Now, substitute the values of r and h into the volume formula:

⇒ Volume = (22/7) × 7² × 20

⇒ Volume = 22 × 7 × 20

⇒ Volume = 154 × 20

⇒ Volume = 3080 cm³

∴ The correct answer is 3080 cu. cm.

Additional Information

Curved Surface Area of a Cylinder

The area of the curved surface is given by the formula CSA = 2πrh.

Total Surface Area of a Cylinder

It includes the curved surface and the two circular bases: TSA = 2πr(r + h).

Volume of a Cone vs Cylinder

A cone with the same base radius and height as a cylinder has exactly one-third of its volume: V = (1/3)πr²h.

Shortcut Trick

If x + 1/x = 2, then x = 1.

Since cosec θ = 1 / sin θ, the given equation is sin θ + 1 / sin θ = 2.

This implies sin θ = 1 and cosec θ = 1.

For any power n, sinⁿθ + cosecⁿθ = 1ⁿ + 1ⁿ = 2.

∴ The correct answer is 2.

Alternate Method

Given:

sin θ + cosec θ = 2

Formula Used:

cosec θ = 1 / sin θ

Calculation:

⇒ sin θ + (1 / sin θ) = 2

Let sin θ = x. Then the equation becomes:

⇒ x + 1/x = 2

Multiplying both sides by x:

⇒ x² + 1 = 2x

⇒ x² − 2x + 1 = 0

⇒ (x − 1)² = 0

⇒ x = 1

Since x = sin θ, we have sin θ = 1.

⇒ cosec θ = 1 / sin θ = 1 / 1 = 1

Now, substitute these values into the required expression:

⇒ sin¹²⁰θ + cosec¹²⁰θ

⇒ (1)¹²⁰ + (1)¹²⁰

⇒ 1 + 1 = 2

∴ The correct answer is 2.

Additional Information

Algebraic identity x + 1/x = −2

If x + 1/x = −2, then x = −1. For this case, xⁿ + 1/xⁿ will be 2 if n is even, and −2 if n is odd.

Similar Trigonometric Pairs

This property applies to other reciprocal trigonometric ratios. If cos θ + sec θ = 2, then cos θ = 1 and sec θ = 1. Similarly, if tan θ + cot θ = 2, then tan θ = 1 and cot θ = 1.

Algebraic identity x + 1/x = k

If x + 1/x = k, then x² + 1/x² = k² − 2, and x³ + 1/x³ = k³ − 3k.