Search Previous Year Questions

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).