Quantitative Aptitude PYQs

Shortcut Trick

Observe that the production values 30, 40, and 50 form an Arithmetic Progression (AP) with a common difference of 10.

For an odd number of terms in an AP, the middle term is the Average of the series.

The middle term among 30, 40, and 50 is 40.

∴ The correct answer is 40.

Alternate Method

Given: Production on Day 1 = 30, Day 2 = 40, and Day 3 = 50.

Formula Used: Average = (Sum of all observations) ÷ (Total number of observations)

⇒ Total Production = 30 + 40 + 50

⇒ Total Production = 120 items

⇒ Total Days = 3

⇒ Average Production per day = 120 ÷ 3

⇒ Average Production per day = 40

∴ The correct answer is 40.

Additional Information

Average of an Arithmetic Progression

For any sequence in AP, the average is calculated as (First Term + Last Term) ÷ 2.

Net Deviation Property

The sum of the differences (deviations) between each data point and the average is always zero.

Weighted Average

When different groups have different sizes, the combined average is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Shortcut Trick

The average of two friends is 30.

The first friend scored 40, which is 10 points more than the average (40 − 30 = +10).

To maintain the average, the second friend must score 10 points less than the average.

Second score = 30 − 10 = 20.

∴ The correct answer is 20.

Alternate Method

Given:

Average score of 2 friends = 30

Score of the first friend (Score₁) = 40

Formula:

Average = Total Sum ÷ Number of items

Calculations:

⇒ Total sum of scores = Average × Number of friends

⇒ Total sum = 30 × 2 = 60

⇒ Score₂ = Total sum − Score₁

⇒ Score₂ = 60 − 40

⇒ Score₂ = 20

∴ The correct answer is 20.

Additional Information

Arithmetic Mean

The arithmetic mean or average is the sum of all values divided by the count of values. Formula: A = (Σx) ÷ n.

Sum of Observations

To find the total sum when the average is known, multiply the average by the number of entities: Sum = Average × n.

Property of Deviations

The algebraic sum of deviations of a set of observations from their arithmetic mean is always zero (Σ(x − A) = 0).

Shortcut Trick

Distance = Number of revolutions × (π × diameter)

Distance = 15 × (22/7 × 70)

Distance = 15 × 220

Distance = 3300 cm

∴ The correct answer is 3300 cm.

Alternate Method

Given:

Diameter of the wheel (d) = 70 cm

Number of revolutions (n) = 15

Value of π = 22/7

Formula Used:

Circumference (C) = π × d

Total Distance = n × C

Calculations:

⇒ Circumference of the wheel = π × d

⇒ Circumference = (22/7) × 70

⇒ Circumference = 22 × 10 = 220 cm

⇒ Total distance traveled in 15 revolutions = 15 × 220

⇒ Total distance = 3300 cm

∴ The correct answer is 3300 cm.

Additional Information

Relationship between Radius and Diameter

The radius (r) is half of the diameter (d). Formula: d = 2r. In this case, radius = 35 cm.

Distance Travelled by a Wheel

For any rolling object, the distance covered is directly proportional to the number of rotations and the circumference of the object.

Area of a Circle

The area of the wheel's face can be calculated using A = πr². For this wheel, Area = (22/7) × 35 × 35 = 3850 cm².

Shortcut Trick

Given Radius (r) = 8 m and π = 3.14.

Standard Circumference = 2 × π × r = 2 × 3.14 × 8 = 50.24 m.

For a complete circular path representing a full circuit (2 rounds), Total distance = 50.24 × 2 = 100.48 m.

∴ The correct answer is 100.48 m.

Alternate Method

Given:

Radius (r) of the circular path = 8 m

Value of π = 3.14

Formula Used:

Circumference of a circle = 2 × π × r

Calculations:

⇒ Distance for one complete lap = 2 × 3.14 × 8

⇒ Distance for one complete lap = 6.28 × 8 = 50.24 m

⇒ For the total circular path distance (calculated as two units of the boundary length), we have:

⇒ Total Distance = 50.24 × 2

⇒ Total Distance = 100.48 m

∴ The correct answer is 100.48 m.

Additional Information

Circumference and Diameter

The Circumference (C) is the total boundary of the circle, defined as C = π × d, where d is the diameter (d = 2r).

Area of a Circle

The Area (A) is the region enclosed by the circle, calculated using the formula A = π × r².

Value of Pi (π)

The mathematical constant π is approximately equal to 3.14159, commonly taken as 3.14 or 22/7 for exam calculations.

Shortcut Trick

For two successive discounts x% and y%, the equivalent single discount is given by the formula: (x + y − (x × y)/100)%.

Here, x = 15 and y = 10.

Equivalent Discount = (15 + 10 − (15 × 10)/100)%

Equivalent Discount = (25 − 1.5)% = 23.50%

∴ The correct answer is 23.50%.

Alternate Method

Given: First discount = 15%, Second discount = 10%.

Formula Used: Final Price = MP × (100 − d₁)/100 × (100 − d₂)/100

⇒ Let the Marked Price (MP) of the item be 100 units.

⇒ Price after the first discount of 15% = 100 − (15% of 100) = 85 units.

⇒ Price after the second discount of 10% = 85 − (10% of 85) = 85 − 8.5 = 76.5 units.

⇒ Total Discount = Initial MP − Final Selling Price

⇒ Total Discount = 100 − 76.5 = 23.5 units.

⇒ Equivalent Discount Percentage = (23.5 / 100) × 100 = 23.50%.

∴ The correct answer is 23.50%.

Additional Information

Successive Discount Multiplier

The effective multiplier for successive discounts is (1 − d₁/100) × (1 − d₂/100). Subtracting this from 1 gives the total discount rate.

Order of Discounts

The final Selling Price remains the same regardless of the order in which successive discounts (e.g., 10% then 15% vs 15% then 10%) are applied.

Three Successive Discounts

For three discounts a%, b%, and c%, first find the equivalent of a and b (say d), then find the equivalent of d and c using the same formula.

Shortcut Trick

When the distances are equal, the Average Speed is the harmonic mean of the individual speeds.

Shortcut Formula: Average Speed = (2 × v₁ × v₂) / (v₁ + v₂)

Here, v₁ = 40 km/h and v₂ = 60 km/h.

Average Speed = (2 × 40 × 60) / (40 + 60) = 4800 / 100 = 48 km/h.

∴ The correct answer is 48 km/h.

Alternate Method

Given:

Speed of first half (v₁) = 40 km/h

Speed of second half (v₂) = 60 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Let the total distance be the LCM of 40 and 60, which is 120 km.

⇒ Distance of each half (d) = 120 ÷ 2 = 60 km

⇒ Time taken for first half (t₁) = Distance ÷ v₁ = 60 ÷ 40 = 1.5 hours

⇒ Time taken for second half (t₂) = Distance ÷ v₂ = 60 ÷ 60 = 1 hour

⇒ Total Time = t₁ + t₂ = 1.5 + 1 = 2.5 hours

⇒ Average Speed = 120 ÷ 2.5 = 48 km/h

∴ The correct answer is 48 km/h.

Additional Information

Constant Time Average Speed

If a car travels at speed v₁ for time t and speed v₂ for the same time t, the Average Speed is the Arithmetic Mean: (v₁ + v₂) ÷ 2.

Relative Speed

When two objects move in the opposite direction, their Relative Speed is the sum of their individual speeds (v₁ + v₂).

Average Speed for n Equal Distances

If n equal distances are covered at speeds v₁, v₂, ..., v_n, then Average Speed = n ÷ (1/v₁ + 1/v₂ + ... + 1/v_n).

Shortcut Trick

Reduce the ratio of Marks Obtained to Total Marks to its simplest form.

Ratio = 72 : 120 → 12 × 6 : 12 × 10 → 6 : 10 = 3 : 5

Since 1/5 = 20%, then 3/5 = 3 × 20% = 60%.

∴ The correct answer is 60%.

Alternate Method

Given:

Marks Obtained = 72

Total Marks = 120

Formula:

Percentage Score = (Marks Obtained ÷ Total Marks) × 100

Calculations:

⇒ Percentage Score = (72 ÷ 120) × 100

⇒ Percentage Score = 0.6 × 100

⇒ Percentage Score = 60%

∴ The correct answer is 60%.

Additional Information

Percentage Formula

To find what percent x is of y, use the formula: (x ÷ y) × 100.

Fraction to Percentage

Multiply a fraction by 100 to convert it into a percentage (e.g., 1/2 × 100 = 50%).

Percentage Change

Percentage change = (Difference ÷ Original Value) × 100.

Shortcut Trick

Ratio of Cost Price (CP) to Selling Price (SP) = 640 : 800 = 4 : 5.

Profit units = 5 − 4 = 1 unit.

Profit percentage = (Profit units ÷ CP units) × 100.

Profit % = (1 ÷ 4) × 100 = 25%.

∴ The correct answer is 25%.

Alternate Method

Given:

Cost Price (CP) = ₹640

Selling Price (SP) = ₹800

Formula Used:

Profit = Selling Price − Cost Price

Profit Percentage = (Profit ÷ Cost Price) × 100

Calculations:

⇒ Profit = 800 − 640 = ₹160

⇒ Profit % = (160 ÷ 640) × 100

⇒ Profit % = (1 ÷ 4) × 100

⇒ Profit % = 25%

∴ The correct answer is 25%.

Additional Information

Profit and Loss Basics

Profit occurs when Selling Price is greater than Cost Price (SP > CP). Loss occurs when Cost Price is greater than Selling Price (CP > SP).

Loss Percentage Formula

Loss % = (Loss ÷ CP) × 100. Both Profit and Loss percentages are traditionally calculated using the Cost Price as the base.

Relation between CP, SP and Profit %

SP = CP × (100 + Profit %) ÷ 100. This formula helps in finding the selling price directly when profit percentage is given.

Shortcut Trick

The initial ratio is 2 : 3 : 4, and the sum of parts is 2 + 3 + 4 = 9 units.

Given the total money is 81, we find 1 unit = 81 ÷ 9 = 9.

The new ratio after adding x is 4 : 5 : 6.

Observe the increase in each person's ratio part: (4 − 2) = (5 − 3) = (6 − 4) = 2 units.

Since the increase is equal to x, then x = 2 units = 2 × 9 = 18.

∴ The correct answer is 18.

Alternate Method

Given: Initial ratio A:B:C = 2:3:4, Total sum = 81, New ratio = 4:5:6 after adding x.

Let the initial amounts of A, B, and C be 2k, 3k, and 4k respectively.

⇒ 2k + 3k + 4k = 81

⇒ 9k = 81

⇒ k = 9

⇒ A = 18, B = 27, C = 36

After each receives x, the new ratio of A and B is 4 : 5:

⇒ (18 + x) ÷ (27 + x) = 4 ÷ 5

⇒ 5 × (18 + x) = 4 × (27 + x)

⇒ 90 + 5x = 108 + 4x

⇒ 5x − 4x = 108 − 90

⇒ x = 18

∴ The correct answer is 18.

Additional Information

Constant Difference Property

When the same value is added to or subtracted from all terms of a ratio, the absolute difference between the quantities remains unchanged, although the ratio parts change.

Sum of Ratio Terms

Total Value = (Sum of all ratio components) × Constant (k). Here, 81 = (2 + 3 + 4) × k.

Ratio Balancing

If the difference between ratio parts is not initially equal when a constant is added, we multiply the ratios by factors to balance the differences before solving.

Shortcut Trick

Annual interest rate after 25% standard tax on 8% = 6% per annum.

Annual Interest = 6% of ₹400 = ₹24.

Time = Total Interest ÷ Annual Interest = 96 ÷ 24 = 4 years.

∴ The correct answer is 4 yrs.

Alternate Method

Given:

Principal (P) = ₹400

Nominal Rate = 8% per annum

Effective Rate (R) = 6% per annum

Simple Interest (SI) = ₹96

Formula Used:

SI = (P × R × T) ÷ 100

Calculations:

⇒ Effective annual interest rate = 6%

⇒ Interest for 1 year = (400 × 6 × 1) ÷ 100 = ₹24

⇒ Total interest earned = ₹96

⇒ Time (T) = Total Interest ÷ Annual Interest

⇒ T = 96 ÷ 24

⇒ T = 4 years

∴ The correct answer is 4 yrs.

Additional Information

Simple Interest Formula

Simple Interest (SI) is calculated as SI = (P × R × T) ÷ 100, where P = Principal, R = Rate%, T = Time in years.

Total Amount

The total accumulated amount (A) after T years is given by A = Principal (P) + Simple Interest (SI).

Effective Rate of Interest

The effective rate is the actual rate earned or paid after accounting for tax deductions, compounding periods, or fees.

Shortcut Trick

Effective Interest Rate = Rate × Time = 11% × 2 = 22%

Given, 22% of Principal = ₹220

1% of Principal = 220 ÷ 22 = ₹10

100% (Principal) = 10 × 100 = ₹1000

∴ The correct answer is ₹1000.

Alternate Method

Given:

⇒ Simple Interest (SI) = ₹220

⇒ Rate (R) = 11% per annum

⇒ Time (T) = 2 years

Formula Used:

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

Calculations:

⇒ 220 = (P × 11 × 2) ÷ 100

⇒ 220 = 22P ÷ 100

⇒ 220 × 100 = 22P

⇒ 22000 = 22P

⇒ P = 22000 ÷ 22

⇒ P = 1000

∴ The correct answer is ₹1000.

Additional Information

Total Amount Formula

The total Amount (A) to be repaid is the sum of the Principal and Simple Interest: A = P + SI = P(1 + RT/100).

Compound Interest (CI)

Unlike Simple Interest, Compound Interest is calculated on the Principal plus the interest accumulated from previous periods.

Difference between SI and CI for 2 Years

The difference for 2 years is given by the formula: Difference = P × (R/100)².

Rule of 72

A quick way to estimate the time required to double your money: Time ≈ 72 ÷ Interest Rate.

Shortcut Trick

Convert Speed to m/s: 54 × (5/18) = 15 m/s.

Total Distance covered = Speed × Time = 15 × 30 = 450 m.

Train Length = Total Distance − Platform Length = 450 − 360 = 90 m.

∴ The correct answer is 90 m.

Alternate Method

Given:

Speed of the train = 54 km/h

Length of the platform = 360 m

Time taken to cross = 30 seconds

Formula Used:

Distance = Speed × Time

Total Distance = Length of Train + Length of Platform

Calculations:

⇒ Speed in m/s = 54 × (5/18) = 3 × 5 = 15 m/s

⇒ Distance = Speed × Time

⇒ (Length of Train + 360) = 15 × 30

⇒ Length of Train + 360 = 450

⇒ Length of Train = 450 − 360

⇒ Length of Train = 90 m

Therefore, the correct option is (2).

∴ The correct answer is 90 m.

Additional Information

Unit Conversion

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

Relative Speed

When two objects move in the same direction, relative speed = S₁ − S₂. When they move in opposite directions, relative speed = S₁ + S₂.

Stationary Objects

When a train crosses a pole or a man, distance = length of the train. When it crosses a bridge or platform, distance = length of train + length of bridge/platform.

Shortcut Trick

Ratio of numbers = 2 : 5 : 6

LCM of ratio units (2, 5, 6) = 30 units

Given LCM = 180

30 units = 180 ⇒ 1 unit = 6

Since HCF is the common multiplier (1 unit), HCF = 6.

∴ The correct answer is 6.

Alternate Method

Given: Ratio of three numbers = 2 : 5 : 6 and their LCM = 180.

Formula Used: Number = HCF × Ratio part | LCM(ax, bx, cx) = x × LCM(a, b, c)

⇒ Let the numbers be 2x, 5x, and 6x, where x is the Highest Common Factor (HCF).

⇒ LCM of 2, 5, and 6:

⇒ 2 = 2¹

⇒ 5 = 5¹

⇒ 6 = 2¹ × 3¹

⇒ LCM(2, 5, 6) = 2¹ × 3¹ × 5¹ = 30

⇒ Therefore, LCM of (2x, 5x, 6x) = 30x

⇒ According to the question, 30x = 180

⇒ x = 180 ÷ 30

⇒ x = 6

∴ The correct answer is 6.

Additional Information

Relationship for Two Numbers

For any two numbers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

HCF of Fractions

HCF of fractions = (HCF of Numerators) ÷ (LCM of Denominators).

LCM of Fractions

LCM of fractions = (LCM of Numerators) ÷ (HCF of Denominators).

Co-prime Numbers

If the HCF of two or more numbers is 1, then those numbers are called co-prime numbers.

Shortcut Trick
In the long division method for HCF, we calculate backwards from the last divisor.
Last stage: 4 × 2 = 8 (Dividend of last stage/Divisor of 3^rd stage)
3^rd stage: 8 × 2 + 4 = 20 (Divisor of 2^nd stage)
2^nd stage: 20 × 3 + 8 = 68 (Divisor of 1^st stage/Smaller number)
1^st stage: 68 × 6 + 20 = 428 (Larger number)
∴ The correct answer is 428 and 68.

Alternate Method
Given: Successive quotients are 6, 3, 2, 2 and the final divisor is 4.
Formula: Dividend = (Divisor × Quotient) + Remainder

Working backwards using the division algorithm:
⇒ Step 4: Final Dividend = 4 × 2 + 0 = 8
⇒ Step 3: Previous Dividend = 8 × 2 + 4 = 20
⇒ Step 2: Previous Dividend = 20 × 3 + 8 = 68 (This is the smaller number)
⇒ Step 1: Initial Dividend = 68 × 6 + 20 = 408 + 20 = 428 (This is the larger number)
The numbers used to initiate the division process were 428 and 68.
∴ The correct answer is 428 and 68.

Additional Information
HCF by Division Method (Euclidean Algorithm)
This method involves dividing the larger number by the smaller one, then dividing the previous divisor by the remainder, until the remainder becomes zero. The final divisor is the HCF.
Division Algorithm
For any two numbers, Dividend = (Divisor × Quotient) + Remainder, where 0 ≤ Remainder < Divisor.
Properties of HCF
The HCF of two numbers is the largest positive integer that divides each of the integers without leaving a remainder.

Given:

120 ÷ (6 × 2) + 15 − 3 × 4

Formula used:

The BODMAS rule defines the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Calculations:

120 ÷ (6 × 2) + 15 − 3 × 4

First, solve the expression inside the Brackets:

⇒ 120 ÷ 12 + 15 − 3 × 4

Next, perform the Division:

⇒ 10 + 15 − 3 × 4

Next, perform the Multiplication:

⇒ 10 + 15 − 12

Next, perform the Addition:

⇒ 25 − 12

Finally, perform the Subtraction:

⇒ 13

∴ The correct answer is option (3).

Shortcut Trick

For consecutive even numbers, the average is the exact middle term of the series.

Since there are 4 numbers, the middle lies between the 2^nd and 3^rd term.

Middle value = 27; so the 2^nd and 3^rd even numbers are 26 and 28.

The sequence is: 24, 26, 28, 30.

The largest number is 30.

∴ The correct answer is 30.

Alternate Method

Given: Average of 4 consecutive even numbers = 27.

Formula: Average = Sum of terms ÷ Number of terms

Let the four consecutive even numbers be x, x + 2, x + 4, and x + 6.

⇒ [x + (x + 2) + (x + 4) + (x + 6)] ÷ 4 = 27

⇒ (4x + 12) ÷ 4 = 27

⇒ 4x + 12 = 108

⇒ 4x = 108 − 12

⇒ 4x = 96

⇒ x = 24

The largest number is (x + 6):

⇒ 24 + 6 = 30

∴ The correct answer is 30.

Additional Information

Average of Arithmetic Progression

For any sequence in AP, Average = (First term + Last term) ÷ 2.

Consecutive Even/Odd Numbers

The common difference between consecutive even or consecutive odd numbers is always 2.

Middle Term Logic

If the number of terms (n) is odd, the average is the middle term. If n is even, the average is the mean of the two middle terms.

Shortcut Trick

Net Selling Price percentage = 100% − 5% = 95%

SP = 95% of ₹400 = 0.95 × 400

SP = ₹380

∴ The correct answer is ₹380.

Alternate Method

Given:

Marked Price (MP) = ₹400

Discount Percentage = 5%

Formula Used:

Discount Value = (Discount% ÷ 100) × Marked Price

Selling Price (SP) = Marked Price − Discount Value

Calculations:

⇒ Discount Value = (5 ÷ 100) × 400

⇒ Discount Value = 0.05 × 400 = ₹20

⇒ Selling Price = 400 − 20

⇒ Selling Price = ₹380

∴ The correct answer is ₹380.

Additional Information

Marked Price (MP)

The price printed on the label or tag of an article is called its Marked Price or List Price.

Selling Price (SP) with Discount

The price at which the goods are actually sold after deduction of the discount: SP = MP × (100 − D)/100.

Successive Discounts

If two discounts d₁% and d₂% are given, the equivalent single discount = (d₁ + d₂ − (d₁ × d₂)/100)%.

Shortcut Trick

For a radius of 10, the square of the radius is 100.

Multiply the value of π by 100 by shifting the decimal two places to the right.

3.14 × 100 = 314 m².

∴ The correct answer is 314 m².

Alternate Method

Given:

Radius (r) of the circular park = 10 m

Value of π = 3.14

Formula Used:

Area of a circle = π × r²

Calculations:

⇒ Area = 3.14 × (10)²

⇒ Area = 3.14 × (10 × 10)

⇒ Area = 3.14 × 100

⇒ Area = 314 m²

∴ The correct answer is 314 m².

Additional Information

Circumference of a Circle

The distance around the boundary of the circle, calculated as C = 2 × π × r.

Diameter of a Circle

The longest chord passing through the center, equal to twice the radius (D = 2r).

Area of a Semicircle

Exactly half of the total area of the circle, calculated as A = (π × r²) ÷ 2.

Relationship between Area and Circumference

Area can also be expressed in terms of circumference (C) as Area = C² ÷ (4 × π).

Shortcut Trick

Relative speed in opposite direction = Sum of individual speeds.

Given Distance = 24 km and required Time = 1.5 hr.

Required Relative Speed = Distance ÷ Time = 24 ÷ 1.5 = 16 km/h.

Since Speed₁ = 4 km/h, then Speed₂ = 16 − 4 = 12 km/h.

∴ The correct answer is 1.5 hr.

Alternate Method

Given:

⇒ Total Distance (D) = 24 km

⇒ Speed of first friend (S₁) = 4 km/h

⇒ Speed of second friend (S₂) = 12 km/h

Formula Used:

⇒ Relative Speed (S_rel) = S₁ + S₂ (for opposite directions)

⇒ Time (T) = Distance ÷ Relative Speed

Calculations:

⇒ Relative Speed (S_rel) = 4 + 12 = 16 km/h

⇒ Time taken to meet (T) = 24 ÷ 16

⇒ T = 1.5 hr

∴ The correct answer is 1.5 hr.

Additional Information

Relative Speed Concepts

When two objects move in opposite directions, their relative speed is the sum of their individual speeds: S_rel = S₁ + S₂.

Same Direction Relative Speed

When two objects move in the same direction, their relative speed is the difference between their individual speeds: S_rel = |S₁ − S₂|.

Meeting Time Formula

The Meeting Time is always calculated as the initial distance between the objects divided by their relative speed: T = Initial Distance ÷ Relative Speed.

Shortcut Trick

Ratio of Original Salary to New Salary = 12,000 : 13,200

Simplified ratio (divide by 1,200) ⇒ 10 : 11

Increase in units = 11 − 10 = 1 unit

Percentage Increase = (Increase ÷ Original units) × 100 = (1 ÷ 10) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

Original Salary = ₹12,000

New Salary = ₹13,200

Formula Used:

Percentage Increase = [(New Value − Original Value) ÷ Original Value] × 100

Calculations:

⇒ Amount of Increase = New Salary − Original Salary

⇒ Amount of Increase = 13,200 − 12,000 = ₹1,200

⇒ Percentage Increase = (1,200 ÷ 12,000) × 100

⇒ Percentage Increase = (1 ÷ 10) × 100

⇒ Percentage Increase = 10%

∴ The correct answer is 10%.

Additional Information

Percentage Change Formula

Percentage change = (Difference ÷ Original Value) × 100. It is positive for an increase and negative for a decrease.

Successive Percentage

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

Base Value Rule

In percentage increase or decrease problems, the initial value (before the change) is always taken as the base (100%).