Quantitative Aptitude PYQs

Shortcut Trick

Convert all units to cm: 735 cm, 945 cm, and 1050 cm.

To find the greatest length, find the HCF using the difference method.

Difference 1 = 945 − 735 = 210; Difference 2 = 1050 − 945 = 105.

Since 105 divides 735 (105 × 7), 945 (105 × 9), and 1050 (105 × 10), the HCF is 105 cm.

105 cm = 100 cm + 5 cm = 1 m 5 cm.

∴ The correct answer is 1 m 5 cm.

Alternate Method

Given: Fabric lengths are 7 m 35 cm, 9 m 45 cm, and 10 m 50 cm.

Formula Used: The greatest length that can exactly measure several lengths is their Highest Common Factor (HCF).

⇒ Conversion to cm: (1 m = 100 cm)

⇒ L₁ = 7 × 100 + 35 = 735 cm

⇒ L₂ = 9 × 100 + 45 = 945 cm

⇒ L₃ = 10 × 100 + 50 = 1050 cm

⇒ Prime Factorization:

⇒ 735 = 3 × 5 × 7 × 7

⇒ 945 = 3 × 3 × 3 × 5 × 7

⇒ 1050 = 2 × 3 × 5 × 5 × 7

⇒ Common factors = 3, 5, 7

⇒ HCF = 3 × 5 × 7 = 105 cm

⇒ Converting back to meters: 105 cm = 100 cm + 5 cm = 1 m 5 cm

∴ The correct answer is 1 m 5 cm.

Additional Information

Highest Common Factor (HCF)

The HCF of two or more numbers is the largest number that divides each of them without leaving a remainder. In measurement problems, the greatest possible size is always the HCF.

Least Common Multiple (LCM)

The LCM is the smallest number that is a multiple of all the given numbers. It is used for finding simultaneous occurrences or the smallest common container.

Unit Conversion

Always ensure all measurements are in the same standard unit before performing calculations. 1 meter = 100 centimeters; 1 kilometer = 1000 meters.

Shortcut Trick

Assume the base average marks is 72.

Deviation for 3 additional subjects = 3 × (84 − 72) = +36.

Distribute this extra value across all 8 subjects = 36 ÷ 8 = 4.5.

New overall average = 72 + 4.5 = 76.5.

∴ The correct answer is 76.5.

Alternate Method

Given:

Number of subjects in first group (n₁) = 5

Average marks of first group (A₁) = 72

Number of additional subjects (n₂) = 3

Average marks of additional group (A₂) = 84

Formula Used:

Overall Average = (Total Marks in all subjects) ÷ (Total number of subjects)

Calculations:

⇒ Total marks in 5 subjects = 5 × 72 = 360

⇒ Total marks in 3 additional subjects = 3 × 84 = 252

⇒ Grand total marks = 360 + 252 = 612

⇒ Total number of subjects = 5 + 3 = 8

⇒ Overall Average = 612 ÷ 8

⇒ Overall Average = 76.5

Comparing this value with the given options, we find it matches Option (4).

∴ The correct answer is 76.5.

Additional Information

Weighted Average Formula

When two groups are combined, the new average is given by: Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Deviation Method

Overall Average = Assumed Average + (Total Deviation ÷ Total Count). This minimizes calculation steps for large datasets.

Concept of Net Change

In averages, the sum of deviations of all observations from their arithmetic mean is always zero.

Shortcut Trick

Direct identity: (Difference)² = (Sum)² − 4 × (Product)

Difference² = 67² − 4 × 1012

Difference² = 4489 − 4048 = 441

Difference = √441 = 21

∴ The correct answer is 21.

Alternate Method

Given:

Sum of two numbers (x + y) = 67

Product of two numbers (xy) = 1012

Formula Used:

The relationship between sum, product, and difference of two numbers is given by the algebraic identity: (x − y)² = (x + y)² − 4xy

Calculations:

⇒ Let the two positive numbers be x and y.

⇒ (x − y)² = (67)² − 4 × 1012

⇒ (x − y)² = 4489 − 4048

⇒ (x − y)² = 441

⇒ (x − y) = √441

⇒ x − y = 21

∴ The correct answer is 21.

Additional Information

Algebraic Identity for Squares

The square of a sum and square of a difference are related as: (a + b)² − (a − b)² = 4ab.

Quadratic Equation Roots

If the sum (S) and product (P) of two numbers are known, the numbers are the roots of the equation: t² − St + P = 0.

Difference of Squares

The product of the sum and the difference of two numbers equals the difference of their squares: (a + b)(a − b) = a² − b².

Shortcut Trick

The ratio of the first two numbers is derived from the fourth proportional: a : b = 20 : 36 = 5 : 9.

Sum of the ratio parts = 5 + 9 = 14.

Since the given sum is exactly 14, the actual numbers are 5 and 9.

The Geometric Mean = √(5 × 9) = √45 = 3√5.

∴ The correct answer is 3√5.

Alternate Method

Given:

The fourth proportional to a, b, and 20 is 36.

The sum of the first two numbers (a + b) is 14.

Formula Used:

If a, b, c, d are in proportion, then a/b = c/d.

Geometric Mean of two numbers a and b = √(a × b).

⇒ a/b = 20/36

⇒ a/b = 5/9

⇒ Let a = 5k and b = 9k

⇒ a + b = 14 ⇒ 5k + 9k = 14

⇒ 14k = 14 ⇒ k = 1

⇒ Therefore, a = 5 and b = 9

⇒ Geometric Mean = √(5 × 9) = √45

⇒ Geometric Mean = 3√5

∴ The correct answer is 3√5.

Additional Information

Fourth Proportional

In a proportion sequence a, b, c, d where a:b = c:d, the value d is the fourth proportional, calculated as d = (b × c) ÷ a.

Geometric Mean (GM)

For any two positive numbers x and y, the geometric mean is the square root of their product: GM = √(x × y).

Mean Proportional

The mean proportional between two numbers a and c is a number b such that a:b = b:c, which simplifies to b² = ac or b = √(ac).

Shortcut Trick

Identify prime numbers a such that a² is less than 20: only a = 2 and a = 3 are possible.

If a = 2, a² = 4. Possible primes b such that 4 + b is a prime < 20 are 3, 7, and 13.

If a = 3, a² = 9. The only prime b such that 9 + b is a prime < 20 is 2 (Sum = 11).

The set of all possible b values is {2, 3, 7, 13}.

Total count of b values = 4.

∴ The correct answer is 4.

Alternate Method

Given: a and b are prime numbers, and a² + b is a prime number < 20.

Step-by-step evaluation:

⇒ Prime numbers are {2, 3, 5, 7, 11, 13, 17, 19, ...}.

⇒ Since a² + b < 20, and b ≥ 2, we must have a² < 18. Thus a can only be 2 or 3.

⇒ For a = 2: 4 + b must be in {5, 7, 11, 13, 17, 19}.

⇒ 4 + b = 7 ⇒ b = 3; 4 + b = 11 ⇒ b = 7; 4 + b = 17 ⇒ b = 13.

⇒ For a = 3: 9 + b must be in {11, 13, 17, 19}.

⇒ 9 + b = 11 ⇒ b = 2; 9 + b = 13 ⇒ b = 4 (Not prime); 9 + b = 17 ⇒ b = 8 (Not prime); 9 + b = 19 ⇒ b = 10 (Not prime).

⇒ Distinct values of b = {2, 3, 7, 13}.

∴ The correct answer is 4.

Additional Information

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The sequence begins 2, 3, 5, 7, 11, ...

Even Prime Property

The number 2 is the only even prime number. All other prime numbers are odd, which means their squares are also odd.

Parity in Sums

For the sum of two numbers to be an odd prime (like 7, 11, 17), one number must be even and the other must be odd.

Shortcut Trick

Total Work = LCM(10, 12, 15) = 60 units.

Efficiencies: A = 6, B = 5, C = 4.

Work done by A and B in 3 days = (6 + 5) × 3 = 33 units.

Remaining Work = 60 − 33 = 27 units.

Days required by B and C = 27 ÷ (5 + 4) = 3 days.

∴ The correct answer is 3 days.

Alternate Method

Given:

Time taken by A = 10 days

Time taken by B = 12 days

Time taken by C = 15 days

Calculations:

⇒ Work done by A and B together in 1 day = 6 + 5 = 11 units.

⇒ Work completed by A and B in 3 days = 11 × 3 = 33 units.

⇒ Remaining Work = Total Work − Completed Work = 60 − 33 = 27 units.

⇒ Combined Efficiency of B and C = 5 + 4 = 9 units/day.

⇒ Time required for B and C to complete the Remaining Work = 27 ÷ 9

⇒ Time = 3 days.

∴ The correct answer is 3 days.

Additional Information

Pipes and Cisterns

Similar to time and work, where filling pipes perform positive work and outlet/leakage pipes perform negative work.

Work and Wages

Wages are generally distributed in the ratio of Efficiency × Time (total work done by each individual).

Alternate Day Work

When people work on alternate days, work is calculated in cycles (e.g., 2-day or 3-day blocks) based on the rotation.

Man-Days Formula

The relationship between men, days, and hours is given by (M₁ × D₁ × H₁)/W₁ = (M₂ × D₂ × H₂)/W₂.

Given:

Expression = { (20/33) ÷ (20/7) } ÷ ( (3/5) × (10/9) + (2/11) ) + (7/4) ÷ (28/23) of (23/4)

Formula used:

The VBODMAS rule is applied to solve this expression, following the order: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Calculations:

First, solve the operations inside the brackets:

⇒ { (20/33) ÷ (20/7) } = (20/33) × (7/20) = 7/33

⇒ ( (3/5) × (10/9) + (2/11) ) = ( (1/1) × (2/3) + (2/11) ) = (2/3 + 2/11)

⇒ (22 + 6) / 33 = 28/33

Now, substitute these values back into the expression:

⇒ (7/33) ÷ (28/33) + (7/4) ÷ (28/23) of (23/4)

Next, evaluate the of operator:

⇒ (28/23) of (23/4) = (28/23) × (23/4) = 7

The expression now becomes:

⇒ (7/33) ÷ (28/33) + (7/4) ÷ 7

Perform the division operations from left to right:

⇒ (7/33 × 33/28) + (7/4 × 1/7)

⇒ 1/4 + 1/4

Finally, perform the addition:

⇒ 2/4 = 1/2

∴ The correct answer is option (3).

Shortcut Trick

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

For the number 2145:

Sum of digits at odd places (1^st and 3^rd) = 2 + 4 = 6.

Sum of digits at even places (2^nd and 4^th) = 1 + 5 = 6.

Difference = 6 − 6 = 0.

∴ The correct answer is 2145.

Alternate Method

Given: Numbers to check are 3522, 4735, 2145, and 5281.

Method: Verify each number using the alternating sum property.

⇒ Sum of odd-placed digits (from left) = 2 + 4 = 6

⇒ Sum of even-placed digits (from left) = 1 + 5 = 6

⇒ Difference = |6 − 6| = 0

⇒ Since the result is 0, the number 2145 is divisible by 11.

Checking other options:

⇒ 3522: (3 + 2) − (5 + 2) = 5 − 7 = −2 (No)

⇒ 4735: (4 + 3) − (7 + 5) = 7 − 12 = −5 (No)

⇒ 5281: (5 + 8) − (2 + 1) = 13 − 3 = 10 (No)

∴ The correct answer is 2145.

Additional Information

Divisibility by 3 and 9

A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9 respectively.

Divisibility by 7

Subtract twice the last digit from the remaining number; if the result is divisible by 7, the original number is too.

Divisibility by 11 (General Case)

For any integer, the difference between the sum of alternating digits must be a multiple of 11 (e.g., ..., −22, −11, 0, 11, 22, ...).

Shortcut Trick

In such cases, the absolute increase in tax equals the absolute decrease in net income.

Tax Rate = [Reduction in Net Income / (Increase in Tax + Reduction in Net Income)] × 100

Rate = [2 / (18 + 2)] × 100

Rate = (2 / 20) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Increase in Income Tax = 18%

Reduction in Net Income = 2%

Calculations:

⇒ Let Income Tax be T and Net Income be NI.

⇒ 18% of T = 2% of NI

⇒ 18/100 × T = 2/100 × NI

⇒ 18 × T = 2 × NI

⇒ T / NI = 2 / 18 = 1 / 9

⇒ Gross Income = Tax + Net Income = 1 + 9 = 10

⇒ Original Rate of Tax = (Income Tax / Gross Income) × 100

⇒ Rate = (1 / 10) × 100 = 10%

∴ The correct answer is 10%.

Additional Information

Net Income Formula

Net Income is the remaining income after deducting Income Tax from Gross Income. Formula: Net Income = Gross Income − Income Tax.

Tax and Net Income Relation

For a fixed Gross Income, any change in the amount of Income Tax results in an equal and opposite change in the amount of Net Income.

Tax Rate Calculation

The Tax Rate is always calculated as a percentage of the Gross Income. Formula: (Income Tax ÷ Gross Income) × 100.

Shortcut Trick

If an object travels different distances at the same uniform speed, the average speed for the entire journey is equal to that uniform speed itself.

Since the scooter travels both segments at 20 km/hr, the average speed is 20 km/hr.

∴ The correct answer is 20 km/hr.

Alternate Method

Given:

Distance 1 (d₁) = 40 km, Speed 1 (s₁) = 20 km/hr

Distance 2 (d₂) = 60 km, Speed 2 (s₂) = 20 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

Calculations:

⇒ Total Distance = 40 + 60 = 100 km

⇒ Time for first part (t₁) = 40 ÷ 20 = 2 hours

⇒ Time for second part (t₂) = 60 ÷ 20 = 3 hours

⇒ Total Time = 2 + 3 = 5 hours

⇒ Average Speed = 100 ÷ 5 = 20 km/hr

∴ The correct answer is 20 km/hr.

Additional Information

Average Speed for Equal Distances

If a journey is divided into two equal distances traveled at speeds x and y, the average speed = 2xy ÷ (x + y).

Relative Speed

When two objects move in the opposite direction, relative speed = s₁ + s₂. In the same direction, relative speed = |s₁ − s₂|.

Unit Conversion

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

Shortcut Trick

Calculate the effective interest rate for the given period: 8% × 3 years = 24%.

The total simple interest earned is 24% of the principal sum.

Given 24% = ₹1,440, then 1% = 1,440 ÷ 24 = ₹60.

The principal (100%) = 60 × 100 = ₹6,000.

∴ The correct answer is ₹6,000.

Alternate Method

Given:

Simple Interest (SI) = ₹1,440

Rate of Interest (R) = 8% per annum

Time (T) = 3 years

Formula Used:

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

Calculations:

⇒ 1,440 = (P × 8 × 3) ÷ 100

⇒ 1,440 = 24P ÷ 100

⇒ 24P = 1,440 × 100

⇒ 24P = 144,000

⇒ P = 144,000 ÷ 24

⇒ P = 6,000

∴ The correct answer is ₹6,000.

Additional Information

Amount in Simple Interest

The total sum payable at the end of the period is called Amount (A). Formula: A = Principal (P) + Interest (SI) = P × (1 + RT/100).

Difference between SI and CI

For a period of 2 years, the difference between Compound Interest (CI) and Simple Interest (SI) is given by: D = P × (R/100)².

Calculation of Rate or Time

Using the basic formula, we can derive: R = (SI × 100) ÷ (P × T) and T = (SI × 100) ÷ (P × R).

Shortcut Trick

Let initial Cost Price (CP) = 100% and Selling Price (SP) = 158%.

New CP = 100% + ₹20 and New SP = 158% − ₹74.

According to 52.5% profit: (158% − 74) = 1.525 × (100% + 20)

158% − 74 = 152.5% + 30.5 ⇒ 5.5% = 104.5

Original CP (100%) = (104.5 ÷ 5.5) × 100 = 1900

∴ The correct answer is ₹1,900.

Alternate Method

Given:

Initial Profit = 58%, CP increase = ₹20, SP decrease = ₹74, New Profit = 52.5%.

Formula Used:

Selling Price (SP) = Cost Price (CP) × (1 + Profit%/100)

Calculations:

Let the original cost price of the article be ₹x.

⇒ Original Selling Price = x + 58% of x = 1.58x

⇒ New Cost Price = x + 20

⇒ New Selling Price = 1.58x − 74

As per the question, new profit is 52.5%:

⇒ New SP = New CP × (100 + 52.5)/100

⇒ 1.58x − 74 = (x + 20) × 1.525

⇒ 1.58x − 74 = 1.525x + (20 × 1.525)

⇒ 1.58x − 74 = 1.525x + 30.5

⇒ 1.58x − 1.525x = 30.5 + 74

⇒ 0.055x = 104.5

⇒ x = 104.5 ÷ 0.055

⇒ x = 104500 ÷ 55

⇒ x = 1900

∴ The correct answer is ₹1,900.

Additional Information

Profit Percentage on Cost Price

Profit % = [(SP − CP) ÷ CP] × 100. By default, profit is always calculated on the Cost Price unless stated otherwise.

Selling Price Multiplier

If an article is sold at a profit of P%, then SP = CP × (100 + P)/100. For example, a 58% profit means SP = 1.58 × CP.

Successive Changes in CP and SP

When both CP and SP change by absolute values, the new profit percentage is applied to the updated CP to find the new SP.

Shortcut Trick

In a trapezoid where two sides are parallel (AB || DC), the length of the segment joining the midpoints of the diagonals (EF) is given by the formula:

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

⇒ EF = (AB − DC) ÷ 2

⇒ EF = (60 − 18) ÷ 2

⇒ EF = 42 ÷ 2 = 21 cm

∴ The correct answer is 21 cm.

Alternate Method

Given:

Parallel sides: AB = 60 cm and DC = 18 cm.

Non-parallel sides: BC = 65 cm and AD = 10 cm.

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

Theorem:

In any quadrilateral ABCD where AB || DC, let E be the midpoint of diagonal AC and F be the midpoint of diagonal BD. The segment EF is parallel to AB and DC, and its length is half the absolute difference of the lengths of the parallel sides.

Calculation:

⇒ Length of EF = 1/2 × |AB − DC|

⇒ EF = 1/2 × (60 − 18)

⇒ EF = 1/2 × 42

⇒ EF = 21 cm

∴ The correct answer is 21 cm.

Additional Information

Median of a Trapezoid

The segment joining the midpoints of the non-parallel sides is called the median. Its length is the arithmetic mean of the parallel sides: Median = (AB + DC) ÷ 2.

Area of a Trapezoid

The area is calculated as: Area = 1/2 × (Sum of parallel sides) × Height. The segment joining the midpoints of diagonals is always collinear with the median.

Property of Diagonals

In a trapezoid, the diagonals intersect each other in the same ratio as the lengths of the parallel sides. Specifically, AO/OC = BO/OD = AB/DC, where O is the intersection point.

Shortcut Trick

The mode of a dataset is the value that appears most frequently.

Counting the occurrences: 52 appears 3 times, while all other numbers appear 2 times or once.

Since 52 has the highest count, it is the mode.

∴ The correct answer is 52.

Alternate Method

Given: Data observations = {40, 47, 54, 45, 50, 43, 53, 52, 53, 49, 40, 50, 42, 52, 44, 52, 47, 55}

Formula: Mode = Observation with the maximum frequency.

⇒ Organize data into a frequency table: ⇒ Value 40: 2 times | Value 42: 1 time | Value 43: 1 time | Value 44: 1 time ⇒ Value 45: 1 time | Value 47: 2 times | Value 49: 1 time | Value 50: 2 times ⇒ Value 52: 3 times | Value 53: 2 times | Value 54: 1 time | Value 55: 1 time ⇒ Maximum frequency observed = 3. ⇒ The corresponding value for maximum frequency is 52.

∴ The correct answer is 52.

Additional Information

Measures of Central Tendency

Central tendency includes the Mean (sum ÷ total count), Median (middle value of sorted data), and Mode (most frequent value).

Empirical Relationship

For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Types of Mode

A dataset can be Unimodal (one mode), Bimodal (two modes), or Multimodal (more than two modes).

Shortcut Trick

To find the compounded ratio of the first and last terms (a : e), simply multiply all the given intermediate ratios.

Required ratio (a/e) = (a/b) × (b/c) × (c/d) × (d/e)

a/e = (2/3) × (4/5) × (3/2) × (5/4)

By observing the fractions, we see that 2/3 cancels with 3/2 and 4/5 cancels with 5/4, resulting in 1/1.

∴ The correct answer is 1 : 1.

Alternate Method

Given:

a : b = 2 : 3 | b : c = 4 : 5 | c : d = 3 : 2 | d : e = 5 : 4

Formula Used:

Compound Ratio = Product of individual ratios = (a₁/b₁) × (a₂/b₂) × ... × (a_n/b_n)

Calculations:

⇒ a/b = 2/3

⇒ b/c = 4/5

⇒ c/d = 3/2

⇒ d/e = 5/4

⇒ a/e = (a/b) × (b/c) × (c/d) × (d/e)

⇒ a/e = (2/3) × (4/5) × (3/2) × (5/4)

⇒ a/e = (2 × 4 × 3 × 5) / (3 × 5 × 2 × 4)

⇒ a/e = 120 / 120

⇒ a/e = 1 / 1

∴ The correct answer is 1 : 1.

Additional Information

Compound Ratio

For two or more ratios, the ratio of the product of the antecedents to the product of the consequents is called the compounded ratio.

Duplicate and Triplicate Ratios

The duplicate ratio of a : b is a² : b², and the triplicate ratio is a³ : b³.

Reciprocal Ratio

The reciprocal or inverse ratio of a : b is 1/a : 1/b, which simplifies to b : a.

Continued Proportion

Three quantities a, b, and c are in continued proportion if a : b = b : c, which implies b² = ac.

Shortcut Trick

Ratio of time taken to cross the two persons = 9 : 10.

Since Length is constant, Relative Speed ratio = 10 : 9.

Difference in Relative Speeds = 4 km/hr − 2 km/hr = 2 km/hr.

Difference in ratio parts = 10 − 9 = 1 unit ⇒ 1 unit = 2 km/hr.

Relative Speed with 1^st person = 10 units = 20 km/hr.

Length of train = 20 × (5/18) × 9 = 50 m.

∴ The correct answer is 50 m.

Alternate Method

Given:

Speed of 1^st person = 2 km/hr | Time taken (t₁) = 9 sec

Speed of 2^nd person = 4 km/hr | Time taken (t₂) = 10 sec

Formula Used:

Distance (Length) = Relative Speed × Time

Relative Speed (Same Direction) = Speed of Train − Speed of Person

Calculations:

⇒ Let Speed of Train be V km/hr and Length be L meters.

⇒ L = (V − 2) × (5/18) × 9 = (V − 2) × 5/2

⇒ L = (V − 4) × (5/18) × 10 = (V − 4) × 25/9

⇒ (V − 2) × 5/2 = (V − 4) × 25/9

⇒ (V − 2) ÷ 2 = (V − 4) × 5 ÷ 9

⇒ 9 × (V − 2) = 10 × (V − 4)

⇒ 9V − 18 = 10V − 40

⇒ V = 22 km/hr

⇒ Length (L) = (22 − 2) × (5/18) × 9 = 20 × 5/2 = 50 m.

∴ The correct answer is 50 m.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed is (u − v). When they move in opposite directions, relative speed is (u + v).

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.

Train Crossing a Person

The distance covered by a train while crossing a person (standing or moving) is equal to the length of the train itself.

Shortcut Trick

Remove pencils of 10 extra boys: 170 − (10 × 2) = 150.

Combined pencil units for one pair (1 Girl + 1 Boy) = 3 + 2 = 5.

Number of Girls = 150 ÷ 5 = 30.

Number of Boys = 30 + 10 = 40.

Required Ratio (Girls : Boys) = 30 : 40 = 3 : 4.

∴ The correct answer is 3:4.

Alternate Method

Given:

Extra Boys = 10 | Pencils per Girl = 3 | Pencils per Boy = 2 | Total Pencils = 170

Let the number of Girls be G.

Then the number of Boys = G + 10.

⇒ Total Pencils = (3 × G) + [2 × (G + 10)]

⇒ 170 = 3G + 2G + 20

⇒ 170 − 20 = 5G

⇒ 150 = 5G

⇒ G = 30

⇒ Boys = 30 + 10 = 40

⇒ Ratio of Girls to Boys = 30 : 40 = 3 : 4

∴ The correct answer is 3:4.

Additional Information

Ratio and Proportion

A ratio is a comparison of two quantities of the same kind. If two ratios a:b and c:d are equal, they are said to be in proportion (a:b :: c:d).

Linear Equation Formulation

To solve word problems, translate phrases like more than or times into mathematical operators (+, −, ×) to form an equation.

Weighted Average/Sum

The total value is the sum of (Quantity × Value per Unit) for all individual categories involved.

Shortcut Trick

For two objects on the same side of a tower with angles θ₁ and θ₂, the distance d between them is d = h (cot θ₂ − cot θ₁).

Here d = 50 m, θ₁ = 45°, and θ₂ = 30°.

⇒ 50 = h (cot 30° − cot 45°)

⇒ 50 = h (√3 − 1) ⇒ h = 50 / (√3 − 1)

⇒ h = 25 (√3 + 1) m (after rationalizing the denominator).

∴ The correct answer is 25 (√3 + 1) m.

Alternate Method

Let the height of the lighthouse be h meters and the distance of the first ship from the lighthouse be x.

In triangle ABC (nearer ship):

⇒ tan 45° = h / x

⇒ 1 = h / x ⇒ x = h

In triangle ABD (farther ship):

⇒ tan 30° = h / (x + 50)

⇒ 1 / √3 = h / (h + 50)

⇒ h + 50 = h × √3

⇒ h√3 − h = 50

⇒ h (√3 − 1) = 50

⇒ h = 50 / (√3 − 1)

To rationalize, multiply numerator and denominator by (√3 + 1):

⇒ h = [50 × (√3 + 1)] / [(√3 − 1) × (√3 + 1)]

⇒ h = 50(√3 + 1) / (3 − 1)

⇒ h = 50(√3 + 1) / 2

⇒ h = 25 (√3 + 1) m

∴ The correct answer is 25 (√3 + 1) m.

Additional Information

Distance Formula (Opposite Sides)

If two ships are on the opposite sides of the lighthouse, the distance d between them is d = h(cot θ₁ + cot θ₂).

Trigonometric Ratios

For standard angles: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Cotangent is the reciprocal of tangent.

Angle of Depression

The angle of depression from the top is numerically equal to the angle of elevation from the ground point due to alternate interior angles.

Shortcut Trick

Assume Total Work = 60 units (LCM of 12 and 15).

Efficiency: A = 5, B = 4; Combined efficiency = 9 units/day.

Work done in 4 days = 9 × 4 = 36 units.

Remaining work = 60 − 36 = 24 units.

Time taken by B = 24 ÷ 4 = 6 days.

∴ The correct answer is 6 days.

Alternate Method

Given:

Time taken by A = 12 days

Time taken by B = 15 days

Work duration for both = 4 days

Formula Used:

Efficiency = Total Work ÷ Time Taken

Calculations:

⇒ Total work to be done = 60 units

⇒ Efficiency of A and B together = 5 + 4 = 9 units/day

⇒ Work done by A and B in 4 days = 9 × 4 = 36 units

⇒ Remaining work = 60 − 36 = 24 units

⇒ Time taken by B to finish the remaining work = Remaining work ÷ Efficiency of B

⇒ Time = 24 ÷ 4 = 6 days

∴ The correct answer is 6 days.

Additional Information

Work and Efficiency Relationship

The Efficiency of a person is inversely proportional to the Time taken to complete the work (Efficiency ∝ 1 ÷ Time).

Combined Rate Rule

When multiple people work together, their net efficiency is the sum of their individual efficiencies (E_total = E₁ + E₂ + ... + E_n).

Remaining Work Calculation

Remaining Work = Total Work − (Rate × Time Spent). This is crucial when one person leaves mid-way through the task.

Shortcut Trick

Third proportional of two numbers a and b is calculated as b² ÷ a.

First, find p using a = 6 and b = 18: p = (18 × 18) ÷ 6 = 3 × 18 = 54.

Now, find q using p = 54 and second value 54: q = (54 × 54) ÷ 54 = 54.

∴ The correct answer is 54.

Alternate Method

Given: First pair for third proportional p is (6, 18). Second pair for third proportional q is (p, 54).

Formula Used: If a, b, and c are in continued proportion, then a : b = b : c, which means c = b² ÷ a.

Calculations:

⇒ 6 : 18 = 18 : p

⇒ 6 ÷ 18 = 18 ÷ p

⇒ 1 ÷ 3 = 18 ÷ p

⇒ p = 18 × 3 = 54

⇒ Now, for q: p : 54 = 54 : q

⇒ 54 : 54 = 54 : q

⇒ 1 = 54 ÷ q

⇒ q = 54

∴ The correct answer is 54.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is x such that x² = ab or x = √(ab).

Fourth Proportional

If four quantities a, b, c, d are in proportion (a:b = c:d), then d is the fourth proportional: d = (b × c) ÷ a.

Compound Ratio

For two ratios a:b and c:d, the compound ratio is (a × c) : (b × d).