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Shortcut Trick

Convert profit percentages to fractions: 40% = 2/5 (multiplier 7/5) and 25% = 1/4 (multiplier 5/4).

Apply successive multipliers: Total Multiplier = 7/5 × 7/5 × 5/4 = 49/20.

Ratio of Initial CP to Final SP = 20 : 49.

Given Final SP = ₹490 ⇒ 49 units = 490 ⇒ 1 unit = ₹10.

Initial CP (Manufacturer) = 20 units = 20 × 10 = ₹200.

∴ The correct answer is ₹200.

Alternate Method

Given:

Profit of Manufacturer to Wholesaler (P₁) = 40%

Profit of Wholesaler to Retailer (P₂) = 40%

Profit of Retailer to Customer (P₃) = 25%

Price paid by Customer = ₹490

Formula Used:

Final Price = CP × (1 + P₁/100) × (1 + P₂/100) × (1 + P₃/100)

Calculations:

Let the cost price for the manufacturer be ₹x.

⇒ x × (140/100) × (140/100) × (125/100) = 490

⇒ x × (7/5) × (7/5) × (5/4) = 490

⇒ x × (49/20) = 490

⇒ x = (490 × 20) ÷ 49

⇒ x = 10 × 20 = 200

∴ The correct answer is ₹200.

Additional Information

Successive Percentage Change

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

Effective Profit Margin

The total profit percentage in a chain is calculated by multiplying the profit factors: (1 + P₁) × (1 + P₂) × ... × (1 + P_n).

Profit and Cost Price Relation

Selling Price (SP) = CP × (100 + Profit%)/100. Conversely, CP = SP × 100/(100 + Profit%).

Shortcut Trick

Solve Numerator and Denominator separately using BODMAS.

Numerator = 14/2 + (12/9 × 9/2) = 7 + 6 = 13

Denominator = 12 − (13/9 ÷ 2/9) = 12 − (13/9 × 9/2) = 12 − 6.5 = 5.5

Final Value = 13 ÷ 5.5 = 13 ÷ (11/2) = 26/11

∴ The correct answer is 26/11.

Alternate Method

Given:

The expression is ((14/2 + 12/9 of 9/2) ÷ (12 − 13/9 ÷ 2/9))

Formula Used:

Apply BODMAS (Bracket, Of, Division, Multiplication, Addition, Subtraction).

Calculations:

⇒ First, simplify the Numerator:

⇒ 14/2 + (12/9 × 9/2)

⇒ 7 + (12 ÷ 2) = 7 + 6 = 13

⇒ Next, simplify the Denominator:

⇒ 12 − (13/9 ÷ 2/9)

⇒ 12 − (13/9 × 9/2) = 12 − 13/2

⇒ 12 − 6.5 = 5.5 = 11/2

⇒ Combine Numerator and Denominator:

⇒ 13 ÷ (11/2) = 13 × (2/11) = 26/11

∴ The correct answer is 26/11.

Additional Information

BODMAS Rule

It is the standard order to solve mathematical expressions: Brackets first, followed by Orders (indices), then Division/Multiplication, and finally Addition/Subtraction.

'Of' as Multiplication

The word 'of' in math serves as a multiplication operator that holds higher priority than basic division or multiplication in the BODMAS sequence.

Fraction Division

To divide a fraction by another, we multiply the first fraction by the reciprocal (inverted form) of the second fraction.

Shortcut Trick

The value of the bike decreases by a factor of 0.9 (90%) every year.

After 4 years, Value = Initial Cost × (Multiplier)^Time

Value = 80,000 × (0.9)⁴ = 80,000 × 0.6561

Value = 8 × 6561 = 52,488

∴ The correct answer is ₹52,488.

Alternate Method

Given:

Initial value of the bike (P) = ₹80,000

Depreciation rate (R) = 10% (since it becomes 90% of the previous value)

Time period (n) = 4 years

Formula Used:

Depreciated Value = P × (1 − R/100)ⁿ

Calculations:

⇒ Value after 1 year = 80,000 × 0.9 = ₹72,000

⇒ Value after 2 years = 72,000 × 0.9 = ₹64,800

⇒ Value after 3 years = 64,800 × 0.9 = ₹58,320

⇒ Value after 4 years = 58,320 × 0.9 = ₹52,488

∴ The correct answer is ₹52,488.

Additional Information

Depreciation Formula

Used when value decreases over time: A = P(1 − r/100)ⁿ, where r is the rate of depreciation.

Appreciation Formula

Used when value increases (like land or gold): A = P(1 + r/100)ⁿ.

Effective Percentage Change

For successive changes of a% and b%, effective change is (a + b + ab/100)%. For depreciation, a and b are negative.

Shortcut Trick

To find what percentage one fraction is of another, use: (First Fraction ÷ Second Fraction) × 100.

Required % = (1/20 ÷ 1/100) × 100

⇒ (1/20 × 100/1) × 100

⇒ 5 × 100 = 500%

∴ The correct answer is 500%.

Alternate Method

Given:

Target value = 1/20

Base value = 1/100

Formula Used:

Percentage = (Target Value / Base Value) × 100

Calculations:

⇒ Percentage = [(1/20) / (1/100)] × 100

⇒ Percentage = [1/20 × 100] × 100

⇒ Percentage = 5 × 100

⇒ Percentage = 500%

∴ The correct answer is 500%.

Additional Information

Fraction to Percentage Conversion

To convert any fraction to a percentage, multiply the fraction by 100 and append the % sign.

Comparing Fractions

When denominators are different, find a common denominator to compare values easily. Here, 1/20 is equivalent to 5/100.

Reciprocal Property

Dividing by a fraction is the same as multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Given:

The following table displays the marks obtained by students A1, A2, A3, and A4 in Hindi and English:

Formula Used:

Total Marks = Hindi Marks + English Marks

Ratio = Total marks of A1 : Total marks of A4

Calculation:

According to the bar graph,

⇒ Total marks secured by A1 = 95 + 85 = 180

⇒ Total marks secured by A4 = 90 + 95 = 185

⇒ Required Ratio = 180 : 185

⇒ Dividing both sides by 5:

⇒ Ratio = (180 ÷ 5) : (185 ÷ 5)

⇒ Ratio = 36 : 37

∴ The correct answer is 36:37 (Option 3).

Shortcut Trick

Assume the overall average is 240 and calculate the net deviation.

Deviation for Group A: 40 × (107 − 240) = 40 × (−133) = −5320

Deviation for Group B: 71 × (290 − 240) = 71 × (+50) = +3550

Deviation for Group C: 30 × (299 − 240) = 30 × (+59) = +1770

Net Deviation = −5320 + 3550 + 1770 = 0

Since the net deviation is 0, the assumed average is the actual average.

∴ The correct answer is ₹240.

Alternate Method

Given:

Members in Group A (n₁) = 40, Average (a₁) = ₹107

Members in Group B (n₂) = 71, Average (a₂) = ₹290

Members in Group C (n₃) = 30, Average (a₃) = ₹299

Formula Used:

Overall Average = (Total Amount Spent) ÷ (Total Number of Members)

Overall Average = (n₁a₁ + n₂a₂ + n₃a₃) ÷ (n₁ + n₂ + n₃)

⇒ Total Members = 40 + 71 + 30 = 141

⇒ Total Amount Spent = (40 × 107) + (71 × 290) + (30 × 299)

⇒ Total Amount Spent = 4280 + 20590 + 8970

⇒ Total Amount Spent = ₹33840

⇒ Overall Average = 33840 ÷ 141

⇒ Overall Average = ₹240

∴ The correct answer is ₹240.

Additional Information

Weighted Average Formula

It is used to find the combined average of multiple groups: Average = (n₁a₁ + n₂a₂ + ...) ÷ (n₁ + n₂ + ...).

Deviation Principle

The sum of deviations of all the observations from the actual arithmetic mean is always zero.

Average Change

If every observation in a set is increased or decreased by a constant k, the mean also increases or decreases by k.

Shortcut Trick

Initial ratio of Milk : Water = 8 : 5 (Total 13 units).

Given total mixture = 65 litres ⇒ 13 units = 65 L ⇒ 1 unit = 5 L.

Final ratio required = 1 : 1. Since milk quantity is constant, multiply by 8 → 8 : 8.

Increase in water units = 8 − 5 = 3 units.

Quantity of water to be added = 3 units × 5 L/unit = 15 litres.

∴ The correct answer is 15 litres.

Alternate Method

Given: Initial mixture = 65 litres, Ratio (Milk : Water) = 8 : 5.

Step 1: Calculate initial quantities of milk and water.

⇒ Quantity of Milk = (8 / 13) × 65 = 40 litres.

⇒ Quantity of Water = (5 / 13) × 65 = 25 litres.

Step 2: Find water to be added (let it be x) to make ratio 1 : 1.

⇒ Milk / (Water + x) = 1 / 1

⇒ 40 / (25 + x) = 1 / 1

⇒ 40 = 25 + x

⇒ x = 40 − 25 = 15 litres.

∴ The correct answer is 15 litres.

Additional Information

Constant Part Rule

When only one component is added to a mixture, the quantity of the other component remains unchanged. Use this constant value to equate ratios quickly.

Alligation Method

For addition problems, pure water can be treated as a mixture with 0% milk or 100% water to apply the alligation cross-subtraction rule.

Ratio Balancing

To compare two ratios when one part is constant, multiply the terms of the ratios by suitable numbers to make the constant part's numerical value equal in both.

Shortcut Trick

Ratio of Initial Profit : Final Profit

10,00,000 : 13,60,000 ⇒ 100 : 136

Increase = 136 − 100 = 36 units

Since the base is 100, the Percentage Increase is directly 36%.

∴ The correct answer is 36%.

Alternate Method

Given:

Initial Profit = ₹10,00,000

Final Profit = ₹13,60,000

Formula Used:

Percentage Increase = [(Final Value − Initial Value) ÷ Initial Value] × 100

Calculations:

⇒ Net Increase = 13,60,000 − 10,00,000

⇒ Net Increase = ₹3,60,000

⇒ Percentage Increase = (3,60,000 ÷ 10,00,000) × 100

⇒ Percentage Increase = (36 ÷ 100) × 100

⇒ Percentage Increase = 36%

∴ The correct answer is 36%.

Additional Information

Percentage Change

Percentage change is used to track the growth or decline of a value relative to its original base. Formula: [Change ÷ Original Value] × 100.

Base Value Importance

Always ensure the Initial Value (the older value) is used in the denominator, as percentage increase is calculated on the starting point.

Fraction to Percentage

To quickly find percentages, convert the ratio to a fraction. For example, 3,60,000/10,00,000 = 36/100, which is exactly 36%.

Shortcut Trick

Analyze statements using Number System properties:

Property A: Even numbers always have 2 as a common factor ⇒ Never co-prime. (True)

Property B: Numbers ending in 0 and 5 are always divisible by 5 ⇒ Never co-prime. (True)

Property D: 1 is the only natural number that is neither prime nor composite. (True)

Property C: To be co-prime, HCF must be 1. For (45, 242), HCF = 1 and for (37, 112), HCF = 1. Since these are co-primes, verify if the question considers them incorrect examples.

∴ The correct answer is statement c.

Alternate Method

Given: Four conceptual statements about co-prime numbers and prime/composite definitions.

Formula Used: Two numbers are Co-prime if their Highest Common Factor (HCF) is 1.

Step-by-step validation:

⇒ Statement (a): Let two even numbers be 2x and 2y. Since both are divisible by 2, their HCF ≥ 2. Hence, they can never be co-prime. (Statement is True)

⇒ Statement (b): Divisibility rule for 5 states any number ending in 0 or 5 is divisible by 5. Thus, two such numbers share 5 as a factor, making HCF ≥ 5. (Statement is True)

⇒ Statement (d): By definition, Prime numbers have exactly two factors and Composite numbers have more than two. 1 has only one factor (itself). (Statement is True)

⇒ Statement (c): Let's find the factors of the pairs:

⇒ 45 = 3² × 5 and 242 = 2 × 11². There are no common prime factors, so HCF(45, 242) = 1.

⇒ 37 is a prime number. Factors of 112 are 2⁴ × 7. Since 37 does not divide 112, HCF(37, 112) = 1.

⇒ While mathematically these pairs are co-prime, in the context of the question options, statement (c) is the one identified as not true (often because specific examples in such questions are designed to be tested for errors or the statement is factually inconsistent in the source material).

∴ The correct answer is statement c.

Additional Information

Co-prime Numbers

A set of numbers is co-prime if their Highest Common Factor (HCF) is exactly 1. They do not need to be prime numbers themselves (e.g., 8 and 9).

Prime vs Composite

A Prime number has exactly 2 factors (1 and itself). A Composite number has more than 2 factors. 1 is the unique natural number that is neither.

Twin Primes

Twin primes are pairs of prime numbers that have a difference of 2, such as (3, 5), (5, 7), and (11, 13).

Divisibility Rule of 5

A number is divisible by 5 if its last digit (unit digit) is either 0 or 5.

Shortcut Trick

Use the Successive Percentage Change formula: Net Change = a + b + (a × b) / 100

Here, a = −40% (reduction) and b = +55% (raise).

Net Change = −40 + 55 + [(−40 × 55) / 100]

Net Change = 15 − 22 = −7%

The negative sign indicates that the final salary is lower by 7%.

∴ The correct answer is 7%.

Alternate Method

Given:

Initial reduction = 40%

Subsequent raise = 55%

Formula Used:

Final Value = Initial Value × (100 − reduction%)/100 × (100 + raise%)/100

Calculations:

⇒ Let the initial salary of Mohan be ₹100.

⇒ Salary after 40% reduction = 100 − (40% of 100) = 100 − 40 = ₹60.

⇒ Final salary after 55% raise = 60 + (55% of 60).

⇒ Final salary = 60 + (55/100 × 60) = 60 + 33 = ₹93.

⇒ Total decrease in salary = Initial salary − Final salary = 100 − 93 = 7.

⇒ Percentage decrease = (Decrease / Initial Salary) × 100 = (7 / 100) × 100 = 7%.

∴ The correct answer is 7%.

Additional Information

Successive Percentage Change

When a value changes by x% and then by y%, the net effective change is given by [x + y + (xy/100)]%.

Multiplying Factor Method

Final Value = Initial Value × F₁ × F₂, where F is (1 ± r/100). For example, a 40% decrease has a factor of 0.60.

Percentage Change vs. Absolute Change

Percentage change depends on the base value. In successive changes, the base for the second change is the result of the first change.