Quantitative Aptitude PYQs

Shortcut Trick

Use the Deviation Method: The 25^th observation accounts for the total change in the mean.

Change in Mean = 32.6 − 30 = 2.6

Total Deviation = 24 × 2.6 = 62.4

25^th observation = New Mean + Total Deviation = 32.6 + 62.4 = 95

∴ The correct answer is 95.

Alternate Method

Given:

Arithmetic Mean of 25 observations = 32.6

Arithmetic Mean of the first 24 observations = 30

Formula Used:

Sum of observations = Mean × Total Number of observations

Calculations:

⇒ Total sum of 25 observations = 25 × 32.6 = 815

⇒ Total sum of first 24 observations = 24 × 30 = 720

⇒ 25^th observation = Total Sum (25) − Total Sum (24)

⇒ 25^th observation = 815 − 720 = 95

∴ The correct answer is 95.

Additional Information

Arithmetic Mean Formula

The average or Mean (x¯) is calculated as the sum of all values divided by the total number of values: x¯ = (∑x_i) ÷ n.

Effect of Inclusion/Exclusion

When a new value is added, the New Sum = (Old Average × Old Count) + New Value. Conversely, New Value = (New Average × New Count) − (Old Average × Old Count).

Deviation Concept

The sum of deviations of all observations from their arithmetic mean is always zero (∑(x_i − x¯) = 0).

Shortcut Trick

Use the Effective Multiplier logic: Total Gain Factor = Price Factor × Weight Factor.

Total Gain 85% → Factor = 1.85 | Loss 35% → Factor = 0.65.

Weight Factor = 1000 / Actual Weight (x).

1.85 = 0.65 × (1000 / x) ⇒ x = 650 / 1.85.

x = 351.35... ≈ 351.4 gm.

∴ The correct answer is 351.4 gm.

Alternate Method

Given: Professed Loss = 35%, Overall Gain = 85%, Standard weight = 1000 gm.

Let the Actual Weight used by the shopkeeper be x gm.

Let the Cost Price (CP) of 1 gm be ₹1.

⇒ CP of 1000 gm = ₹1000.

⇒ Selling Price (SP) at 35% loss = 1000 − (35% of 1000) = ₹650.

⇒ The shopkeeper actually gives x gm for ₹650.

⇒ Actual CP of x gm = ₹x.

⇒ Overall Gain% = [(SP − CP) / CP] × 100

⇒ 85 = [(650 − x) / x] × 100

⇒ 0.85 = (650 / x) − 1

⇒ 1.85 = 650 / x

⇒ x = 650 / 1.85 = 351.351...

⇒ x ≈ 351.4 gm (rounded to one decimal place).

∴ The correct answer is 351.4 gm.

Additional Information

Dishonest Dealer Formula

Profit % = [(True Weight − False Weight) / False Weight] × 100, assuming goods are sold at cost price.

Overall Profit with Price Change

Final Gain/Loss % = [((100 ± P/L%) / 100) × (True Weight / False Weight) − 1] × 100.

Multiplier Method

Effective Multiplier (M_total) = M_price × M_weight. If M > 1, it is a profit; if M < 1, it is a loss.

Shortcut Trick

Convert the fraction 11/16 into its decimal equivalent: 11 ÷ 16 = 0.6875.

Perform the addition with the given decimal: 0.6875 + 0.4375 = 1.125.

∴ The correct answer is 1.125.

Alternate Method

Given:

Fraction = 11/16

Decimal = 0.4375

Formula Used:

Decimal to Fraction conversion: Value / 10ⁿ (where n is the number of decimal places).

Calculations:

⇒ Convert 0.4375 to a fraction: 0.4375 = 4375 ÷ 10000

⇒ Simplify the fraction: 4375/10000 = 875/2000 = 175/400 = 35/80 = 7/16

⇒ Calculate the Sum: 11/16 + 7/16 = (11 + 7) ÷ 16

⇒ Sum = 18/16 = 9/8

⇒ Convert 9/8 to decimal form: 9 ÷ 8 = 1.125

∴ The correct answer is 1.125.

Additional Information

Fraction to Decimal Conversion

A fraction is converted to a decimal by dividing the numerator by the denominator. If the denominator is in the form 2^m × 5ⁿ, the decimal will be terminating.

Adding Decimals

To add decimals, align the decimal points and add as whole numbers, then place the decimal point directly below the others in the sum.

Like Fractions

Fractions with the same denominator are called like fractions. Their addition or subtraction is done by performing the operation on the numerators while keeping the denominator constant.

Shortcut Trick

Effective Rate of Interest for 2 years = x + y + (xy/100)%

⇒ 20 + 20 + (20 × 20)/100 = 44%

Compound Interest (CI) = 44% of ₹2000

⇒ 0.44 × 2000 = ₹880

∴ The correct answer is ₹880.

Alternate Method

Given: Principal (P) = ₹2000, Rate (R) = 20%, Time (n) = 2 years

Formula Used: Amount (A) = P(1 + R/100)ⁿ

⇒ A = 2000(1 + 20/100)²

⇒ A = 2000(1.2)²

⇒ A = 2000 × 1.44 = ₹2880

⇒ Compound Interest = Amount − Principal

⇒ CI = 2880 − 2000 = ₹880

∴ The correct answer is ₹880.

Additional Information

Difference between CI and SI

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

Effective Rate Formula

To find the equivalent single percentage for two successive increases of x% and y%, use the formula: x + y + (xy/100)%.

Compound Interest Ratio Method

For 2 years, the ratio 2:1 can be used. CI = 2 × (Interest on P) + 1 × (Interest on Interest).

Shortcut Trick

Use the Successive Multiplier Method to find the final price ratio.

Final Price Factor = 0.45 × 0.65 × 0.55 × 1.4 = 0.225225

Net Percentage Decrease = (1 − 0.225225) × 100 = 77.4775%

Rounding to 2 decimal places, we get 77.48%.

∴ The correct answer is 77.48%.

Alternate Method

Given: Successive decreases of 55%, 35%, 45% and an increase of 40%.

Formula Used: Final Price = Initial Price × (1 ± r₁/100) × (1 ± r₂/100) ...

Let the original price be 100.

⇒ Price after 1^st month = 100 − 55% of 100 = 45

⇒ Price after 2^nd month = 45 − 35% of 45 = 45 × 0.65 = 29.25

⇒ Price after 3^rd month = 29.25 − 45% of 29.25 = 29.25 × 0.55 = 16.0875

⇒ Price after 4^th month = 16.0875 + 40% of 16.0875 = 16.0875 × 1.4 = 22.5225

⇒ Total decrease from original = 100 − 22.5225 = 77.4775

⇒ Percentage decrease = (77.4775 / 100) × 100 = 77.4775%

∴ The correct answer is 77.48%.

Additional Information

Successive Percentage Change

When a value changes by a% and then by b%, the net change is [a + b + (ab/100)]%. For decreases, use negative values.

Multiplier Method

For a decrease of x%, the multiplier is (100 − x)/100. For an increase of y%, it is (100 + y)/100. Multiply all factors to get the final ratio.

Rounding Rule

When rounding to 2 decimal places, check the 3^rd decimal. If it is 5 or greater, round up the 2^nd decimal.

Shortcut Trick

The mean proportional between two numbers a and b is calculated as √(a × b).

Given a = 6 m and b = 24 m.

Height (h) = √(6 × 24) = √144 = 12 m.

∴ The correct answer is 12 m.

Alternate Method

Given:

First value (a) = 6 m

Second value (b) = 24 m

The height (h) of the ladder on the wall is the mean proportional.

Formula Used:

If h is the mean proportional between a and b, then:

a : h = h : b ⇒ h² = a × b ⇒ h = √(a × b)

Calculations:

⇒ h = √(6 × 24)

⇒ h = √144

⇒ h = 12 m

∴ The correct answer is 12 m.

Additional Information

Mean Proportional (Geometric Mean)

The mean proportional between two numbers a and b is the value x such that a/x = x/b, which gives x = √(ab).

Third Proportional

If a and b are two numbers, then their third proportional c is such that a/b = b/c, which gives c = b²/a.

Fourth Proportional

If a, b, and c are three numbers, then their fourth proportional d is such that a/b = c/d, which gives d = (b × c)/a.

Shortcut Trick

Effective Simple Interest rate for 2 years = 12% × 2 = 24%.

According to the question, 24% of the Principal = ₹1,440.

1% of the Principal = 1440 ÷ 24 = 60.

Total Principal (100%) = 60 × 100 = ₹6,000.

∴ The correct answer is ₹6,000.

Alternate Method

Given:

Simple Interest (SI) = ₹1,440

Rate of Interest (R) = 12% per annum

Time Period (T) = 2 years

Formula Used:

Simple Interest (SI) = (Principal × Rate × Time) ÷ 100

Calculations:

⇒ 1440 = (P × 12 × 2) ÷ 100

⇒ 1440 = (P × 24) ÷ 100

⇒ 1440 × 100 = 24P

⇒ P = 144000 ÷ 24

⇒ P = 6000

∴ The correct answer is ₹6,000.

Additional Information

Simple Interest (S.I.)

The interest remains constant every year as it is calculated only on the original principal. S.I. = (P × R × T) ÷ 100.

Amount (A)

Total sum of money (Principal + Interest) to be paid back at the end of the time period. A = P + S.I. = P[1 + (RT ÷ 100)].

Difference between C.I. and S.I.

For a period of 2 years, the difference is given by D = P × (R ÷ 100)².

Rate and Time Calculation

Rate R = (SI × 100) ÷ (P × T) and Time T = (SI × 100) ÷ (P × R).

Shortcut Trick

For a regular octagon with circumradius R, the side length a is given by:

a = 2 × R × sin(180° / 8) = 2 × R × sin(22.5°)

Given: R = 10 m and sin(22.5°) ≈ 0.3827

a = 2 × 10 × 0.3827 ⇒ 20 × 0.3827

a = 7.654 m ≈ 7.65 m

∴ The correct answer is 7.65 m.

Alternate Method

Given: Circumradius (R) = 10 m, Number of sides (n) = 8.

Formula used: Law of Cosines in the triangle formed by the center and two adjacent vertices.

The central angle subtended by each side of the regular octagon at the center O is:

θ = 360° ÷ n ⇒ 360° ÷ 8 = 45°

In ΔOAB, let side AB = a, and OA = OB = R = 10 m.

Using Law of Cosines:

⇒ a² = R² + R² − 2 × R × R × cos(45°)

⇒ a² = 2R² − 2R² × (1/√2)

⇒ a² = 2R² (1 − 1/√2)

⇒ a² = 2 × 10² × (1 − 0.7071)

⇒ a² = 200 × 0.2929

⇒ a² = 58.58

⇒ a = √58.58 ≈ 7.6537 m

∴ The correct answer is 7.65 m.

Additional Information

Area of a Regular Octagon

The area can be calculated as Area = 2(1 + √2)a², where a is the side length.

Interior and Exterior Angles

For a regular octagon, each exterior angle is 45° and each interior angle is 135°.

Inradius of a Regular Polygon

The distance from the center to the midpoint of a side is r = R × cos(180°/n).

Shortcut Trick

Convert speed from km/hr to m/min directly.

Speed = 54 × (1000 ÷ 60) m/min

Speed = 54 × (50 ÷ 3) = 18 × 50 = 900 m/min

In 1 minute, the bus covers 900 meters.

∴ The correct answer is 900 m.

Alternate Method

Given:

Speed of the bus = 54 km/hr

Time taken = 1 min

Formula Used:

Distance = Speed × Time

Calculations:

⇒ 1 km = 1000 m and 1 hr = 60 min

⇒ Speed in m/min = (54 × 1000) ÷ 60

⇒ Speed = 54000 ÷ 60 = 900 m/min

⇒ Distance = 900 m/min × 1 min

⇒ Distance = 900 m

∴ The correct answer is 900 m.

Additional Information

Unit Conversion Trick

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

Distance Formula

Distance = Speed × Time. Always ensure that the units of speed and time are consistent (e.g., both in seconds or both in hours) before calculation.

Uniform Speed

When an object covers equal distances in equal intervals of time, it is said to be moving with uniform speed.

Shortcut Trick

The average speed for the onward journey is the weighted average of speeds with respect to time.

Onward Average Speed = (1/3 × 30) + (2/3 × 60) = 10 + 40 = 50 km/hr.

Since the distance for the onward journey and return journey is equal, use the harmonic mean formula:

Average Speed = (2 × V_on × V_ret) ÷ (V_on + V_ret) = (2 × 50 × 75) ÷ (50 + 75) = 7500 ÷ 125 = 60 km/hr.

∴ The correct answer is 60 km/hr.

Alternate Method

Given:

Stage 1: Speed = 30 km/hr, Time = 1/3 of total onward time.

Stage 2: Speed = 60 km/hr, Time = 2/3 of total onward time.

Return Speed: 75 km/hr.

Calculations:

⇒ Let the total time for the onward journey be 3 hours.

⇒ Time for Stage 1 = 1/3 × 3 = 1 hour.

⇒ Time for Stage 2 = 2/3 × 3 = 2 hours.

⇒ Distance in Stage 1 = Speed × Time = 30 × 1 = 30 km.

⇒ Distance in Stage 2 = Speed × Time = 60 × 2 = 120 km.

⇒ Total onward distance = 30 + 120 = 150 km.

⇒ Return journey distance = 150 km.

⇒ Return time = Distance ÷ Speed = 150 ÷ 75 = 2 hours.

⇒ Total Trip Distance = Onward + Return = 150 + 150 = 300 km.

⇒ Total Trip Time = Onward Time + Return Time = 3 + 2 = 5 hours.

⇒ Average speed = Total Distance ÷ Total Time = 300 ÷ 5 = 60 km/hr.

∴ The correct answer is 60 km/hr.

Additional Information

Average Speed (Equal Distance)

When two equal distances are covered at speeds V₁ and V₂, the average speed is 2V₁V₂ ÷ (V₁ + V₂).

Average Speed (Equal Time Intervals)

If a journey is divided into equal time intervals at different speeds, the average speed is the simple arithmetic mean: (V₁ + V₂ + ... + V_n) ÷ n.

General Average Speed Formula

Average Speed = Total Distance ÷ Total Time. This applies to all scenarios regardless of uniformity in time or distance.

Shortcut Trick

Apply the Successive Percentage formula: Net % change = a + b + (ab/100)

Here, Mark-up (a) = +40 and Discount (b) = −25

Net change = 40 − 25 + [40 × (−25) ÷ 100]

Net change = 15 − 10 = 5%

Positive result indicates Profit.

∴ The correct answer is 5% profit.

Alternate Method

Given:

Mark-up Percentage = 40%

Discount Percentage = 25%

Calculations:

⇒ Let the Cost Price (CP) of the article be 100.

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

⇒ Discount = 25% of MP = (25/100) × 140 = 35.

⇒ Selling Price (SP) = MP − Discount = 140 − 35 = 105.

⇒ Profit = SP − CP = 105 − 100 = 5.

⇒ Profit Percentage = (Profit ÷ CP) × 100 = (5 ÷ 100) × 100 = 5%.

∴ The correct answer is 5% profit.

Additional Information

CP-MP Relation Formula

CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%). This is extremely useful for finding any one unknown variable when others are given.

Markup Percentage

Markup% = [(Marked Price − Cost Price) ÷ Cost Price] × 100. It represents how much the price is increased over the cost before selling.

Successive Discounts

If two successive discounts d₁% and d₂% are given, the equivalent single discount is [d₁ + d₂ − (d₁ × d₂) ÷ 100]%.

Shortcut Trick

Ratio of areas: (Base₁ × Height₁) ÷ (Base₂ × Height₂) = Area₁ ÷ Area₂

Given: Area₂ = 2 × Area₁ ⇒ Area₁ : Area₂ = 1 : 2

(18 × 20) ÷ (24 × H₂) = 1 ÷ 2

360 ÷ (24 × H₂) = 1 ÷ 2

H₂ = (360 × 2) ÷ 24 = 720 ÷ 24 = 30 cm

∴ The correct answer is 30 cm.

Alternate Method

Given:

Base of first triangle (b₁) = 18 cm

Height of first triangle (h₁) = 20 cm

Base of second triangle (b₂) = 24 cm

Area of second triangle (A₂) = 2 × Area of first triangle (A₁)

Formula Used:

Area of a triangle = (1 ÷ 2) × base × height

Calculations:

⇒ Area₁ = (1 ÷ 2) × 18 × 20

⇒ Area₁ = 9 × 20 = 180 cm²

⇒ Area₂ = 2 × Area₁ = 2 × 180 = 360 cm²

⇒ 360 = (1 ÷ 2) × 24 × h₂

⇒ 360 = 12 × h₂

⇒ h₂ = 360 ÷ 12

⇒ h₂ = 30 cm

∴ The correct answer is 30 cm.

Additional Information

Heron's Formula

When all three sides a, b, and c are known, Area = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a+b+c)/2.

Equilateral Triangle Area

For a triangle with all sides equal to 'a', the area is given by (√3 ÷ 4) × a².

Right-Angled Triangle

The area is (1 ÷ 2) × base × perpendicular, where the two sides forming the 90° angle act as base and height.

Shortcut Trick

Net efficiency = Efficiency of Tap 1 + Efficiency of Tap 2 − Efficiency of Tap 3.

Since Tap 1 fills in 4 hr and Tap 3 empties in 4 hr, their rates cancel each other out completely.

The net filling rate is solely determined by Tap 2, which takes 16 hr to fill the full cistern.

Time to fill 1/4 of the cistern = 1/4 × 16 hr = 4 hr.

∴ The correct answer is 4 hr.

Alternate Method

Given: Filling time for Tap A = 4 hr, Tap B = 16 hr, and Emptying time for Tap C = 4 hr.

Formula Used: Total Work = L.C.M. of time taken | Efficiency = Total Work ÷ Time.

⇒ Total efficiency when all taps are open = 4 + 1 − 4 = 1 unit/hr

⇒ Required capacity to fill = 1/4 × Total Capacity = 1/4 × 16 = 4 units

⇒ Time taken = Required Capacity ÷ Total Efficiency

⇒ Time taken = 4 units ÷ 1 unit/hr = 4 hr

∴ The correct answer is 4 hr.

Additional Information

Efficiency of Emptying Pipe

The efficiency of an outlet or emptying pipe is always taken as negative because it reduces the total volume of the cistern.

Net Work Formula

If pipe A fills in x hr and pipe B empties in y hr, the net part filled in 1 hr is (1/x − 1/y). If y > x, the tank will eventually fill.

Fractional Filling

To find the time for a fractional part of the tank, multiply the total time required for a full tank by that specific fraction.

Shortcut Trick

By the Remainder Theorem, if a polynomial is divisible by (x − 5), substituting x = 5 must yield zero.

Put x = 5 in the expression: 5³ − 3(5²) − 7(5) − k = 0

125 − 75 − 35 − k = 0

15 − k = 0 ⇒ k = 15

∴ The correct answer is 15.

Alternate Method

Given:

Polynomial P(x) = x³ − 3x² − 7x − k

Divisor = (x − 5)

Formula Used:

According to the Factor Theorem, if (x − a) is a factor of P(x), then P(a) = 0.

Calculations:

⇒ For the expression to be divisible by (x − 5), the remainder must be 0.

⇒ P(5) = 0

⇒ (5)³ − 3 × (5)² − 7 × (5) − k = 0

⇒ 125 − 3 × 25 − 35 − k = 0

⇒ 125 − 75 − 35 − k = 0

⇒ 50 − 35 − k = 0

⇒ 15 − k = 0

⇒ k = 15

∴ The correct answer is 15.

Additional Information

Remainder Theorem

When a polynomial f(x) is divided by (x − a), the remainder is always f(a).

Factor Theorem

A linear polynomial (x − a) is a factor of a polynomial f(x) if and only if f(a) = 0.

Divisibility Rule for Polynomials

If P(x) is divisible by Q(x), then every root of Q(x) must also be a root of P(x).

Shortcut Trick

Given cos A = 4/5, we identify the Pythagorean triplet (3, 4, 5) where Base = 4, Hypotenuse = 5, and Perpendicular = 3.

Then sin A = 3/5, tan A = 3/4, and sec A = 5/4.

Substitute values into the expression: ((3/5)² − 5×(4/5)² + (5/4)²) ÷ (3/4)².

Numerator = (9/25 − 80/25 + 25/16) = −71/25 + 25/16 = (−1136 + 625)/400 = −511/400.

Result = (−511/400) × (16/9) = −511/(25×9) = −511/225.

∴ The correct answer is −511/225.

Alternate Method

Given: cos A = 4/5

Formula Used: In a right-angled triangle, cos A = Base/Hypotenuse. By Pythagoras theorem, Perpendicular² = Hypotenuse² − Base². Also, sin A = P/H, tan A = P/B, and sec A = H/B.

Calculations:

⇒ Let Base (B) = 4 and Hypotenuse (H) = 5.

⇒ Perpendicular (P) = √(5² − 4²) = √(25 − 16) = √9 = 3.

⇒ sin A = P/H = 3/5

⇒ tan A = P/B = 3/4

⇒ sec A = H/B = 5/4

⇒ Value = (sin² A − 5 cos² A + sec² A) / tan² A

⇒ Value = ((3/5)² − 5×(4/5)² + (5/4)²) / (3/4)²

⇒ Value = (9/25 − 5×(16/25) + 25/16) / (9/16)

⇒ Value = (9/25 − 80/25 + 25/16) / (9/16) = (−71/25 + 25/16) / (9/16)

⇒ Value = [(−71×16 + 25×25) / (25×16)] ÷ (9/16)

⇒ Value = [(−1136 + 625) / 400] ÷ (9/16) = (−511/400) × (16/9)

⇒ Value = −511 / (25 × 9) = −511/225.

∴ The correct answer is −511/225.

Additional Information

Trigonometric Identities

Standard identities include sin² A + cos² A = 1, 1 + tan² A = sec² A, and 1 + cot² A = cosec² A.

Pythagorean Triplets

Common integer sets satisfying a² + b² = c² like (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) are extremely useful in exam time-saving.

Reciprocal Ratios

The reciprocal ratios are defined as sec A = 1/cos A, cosec A = 1/sin A, and cot A = 1/tan A.

Shortcut Trick

Simplify the first bracket: {15/42 × 3/15} = 3/42 = 1/14.

Simplify the second bracket: (2/3 + 5/14) = (28 + 15)/42 = 43/42.

Divide the results: (1/14) ÷ (43/42) = 1/14 × 42/43 = 3/43.

Simplify the final term using BODMAS: 1/2 ÷ (43/6) = 1/2 × 6/43 = 3/43.

Combine: 3/43 + 3/43 = 6/43.

∴ The correct answer is 6/43.

Alternate Method

Given: { (15/42) ÷ (15/3) } ÷ ( (3/10) × (20/9) + (5/14) ) + (3/6) ÷ (43/20) of (20/6)

Formula Used: VBODMAS (Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction)

⇒ Step 1 (Solve Brackets): { 15/42 × 3/15 } ÷ ( 2/3 + 5/14 ) + 3/6 ÷ ( 43/20 of 20/6 )

⇒ Step 2 (Simplify Fractions): { 1/14 } ÷ ( 43/42 ) + 1/2 ÷ ( 43/20 × 20/6 )

⇒ Step 3 (Process 'Of' and Divisions): (1/14 × 42/43) + 1/2 ÷ 43/6

⇒ Step 4 (Multiplication and Reciprocal): 3/43 + (1/2 × 6/43)

⇒ Step 5 (Final Addition): 3/43 + 3/43 = 6/43

∴ The correct answer is 6/43.

Additional Information

VBODMAS Rule

This rule defines the precedence of operations: Vinculum (Bar), Brackets (), Of, Division (÷), Multiplication (×), Addition (+), and Subtraction (−).

Fraction Reciprocal Rule

To divide by a fraction, multiply by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

'Of' vs Multiplication

The operator 'of' behaves as multiplication but holds a higher priority than division and regular multiplication in the order of operations.

Shortcut Trick

The ratio of the perimeter of an outer square to the inner square (formed by joining midpoints) is always √2 : 1.

Given, Perimeter of inner square = 40 cm.

Perimeter of outer square = Perimeter of inner square × √2

Perimeter of outer square = 40 × √2 = 40√2 cm.

∴ The correct answer is 40√2 cm.

Alternate Method

Given:

Perimeter of inner square = 40 cm

Formula Used:

Perimeter of a square = 4 × Side

In a right-angled triangle, Pythagoras Theorem: Hypotenuse² = Base² + Perpendicular²

Calculations:

⇒ Side of the inner square (s) = 40 ÷ 4 = 10 cm

⇒ Let the side of the outer square be a. Since the inner vertices are midpoints, the side of the inner square forms a triangle with legs a/2.

⇒ By Pythagoras Theorem: (a/2)² + (a/2)² = 10²

⇒ a²/4 + a²/4 = 100

⇒ 2a²/4 = 100

⇒ a²/2 = 100 ⇒ a² = 200

⇒ a = √200 = 10√2 cm

⇒ Perimeter of outer square = 4 × a = 4 × 10√2 = 40√2 cm

∴ The correct answer is 40√2 cm.

Additional Information

Area Ratio of Midpoint Squares

The area of a square formed by joining the midpoints of another square is exactly half (1/2) the area of the original square.

Successive Midpoint Squares

If the process is repeated, the perimeters form a Geometric Progression (GP) with a common ratio of 1/√2.

Diagonal Relationship

The side of the inner square is equal to half the diagonal of the outer square: s = d/2 = (a√2)/2 = a/√2.

Shortcut Trick

Price Ratio = 200 : 180 → 10 : 9

Reduction in units = 10 − 9 = 1 unit

Percentage Decrease = (1 ÷ 10) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Initial shirt's price = ₹200

Final shirt's price = ₹180

Formula Used:

Percentage Decrease = (Decrease ÷ Original Value) × 100

Calculations:

⇒ Price Decrease = ₹200 − ₹180 = ₹20

⇒ Percentage Decrease = (20 ÷ 200) × 100

⇒ Percentage Decrease = (1 ÷ 10) × 100

⇒ Percentage Decrease = 10%

∴ The correct answer is 10%.

Additional Information

Percentage Change Formula

Percentage change = ((New Value − Old Value) ÷ Old Value) × 100. A negative result indicates a decrease, while a positive result indicates an increase.

Fraction to Percentage Conversion

Common fractions: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/10 = 10%, and 1/20 = 5%.

Successive Percentage Change

If a value changes by a% and then by b%, the effective change is (a + b + ab/100)%.

Shortcut Trick

Fourth proportional m = (Product of middle terms) ÷ (First term)

⇒ m = (6 × 8) ÷ 4 = 12

Third proportional n = (Square of second term) ÷ (First term)

⇒ n = (12 × 12) ÷ 8 = 144 ÷ 8 = 18

∴ The correct answer is 18.

Alternate Method

Given:

Numbers for fourth proportional: 4, 6, 8, and m

Numbers for third proportional: 8, m, and n

Formula Used:

1. If a, b, c, d are in proportion, then a/b = c/d
2. If a, b, c are in continued proportion, then a/b = b/c (where c is the third proportional)

Calculations:

Step 1: Finding m

⇒ 4 ÷ 6 = 8 ÷ m

⇒ 4 × m = 6 × 8

⇒ 4 × m = 48

⇒ m = 12

Step 2: Finding n

⇒ 8 ÷ 12 = 12 ÷ n

⇒ 8 × n = 12 × 12

⇒ 8 × n = 144

⇒ n = 144 ÷ 8 = 18

∴ The correct answer is 18.

Additional Information

Fourth Proportional

For four numbers a, b, c, and d to be in proportion, the relation is a/b = c/d, which gives d = (b × c) / a.

Third Proportional

For three numbers a, b, and c to be in continued proportion, the relation is a/b = b/c, which gives c = b² / a.

Mean Proportional

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

Shortcut Trick

First Number = 2 × 50 = 100.

Product of numbers = HCF × LCM.

100 × Second Number = 25 × 500.

Second Number = 12500 ÷ 100 = 125.

∴ The correct answer is 125.

Alternate Method

Given:

H.C.F. = 25

L.C.M. = 500

First number when divided by 2 gives quotient = 50 and remainder = 0.

Formula Used:

Dividend = (Divisor × Quotient) + Remainder

Product of two numbers = H.C.F. × L.C.M.

Calculations:

⇒ First number (N₁) = (2 × 50) + 0

⇒ N₁ = 100

⇒ N₁ × N₂ = 25 × 500

⇒ 100 × N₂ = 12500

⇒ N₂ = 12500 ÷ 100

⇒ N₂ = 125

∴ The correct answer is 125.

Additional Information

Product Rule

For any two numbers a and b, the relationship is: a × b = HCF(a, b) × LCM(a, b).

Highest Common Factor (HCF)

The HCF is the greatest number that divides each of the two or more numbers without leaving a remainder.

Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest number (not zero) that is a multiple of all the numbers.

Division Algorithm

The rule states that Dividend = (Divisor × Quotient) + Remainder, where 0 ≤ Remainder < Divisor.