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Given:

(1/4)th of the tank fills in 3 hours

Calculation:

Remaining portion of a tank = 1 – (1/4)

= 3/4

(1/4)th of a tank = 3 hours

Full container fill = 3 × 4

⇒ 12 hours

(3/4)th tank fills = 12 × (3/4)

⇒ 9 hours

∴ 9 hours more required by pipe to fill the tank.

Given:

The ratio of the number of red and blue coins in a bag is 5 : 4.

There are 16 blue coins.

Formula Used:

Number of red coins = (Ratio of red coins / Ratio of blue coins) × Number of blue coins

Calculation:

Ratio of red coins = 5

Ratio of blue coins = 4

Number of blue coins = 16

Number of red coins = (5 / 4) × 16

⇒ Number of red coins = 5 × 4

⇒ Number of red coins = 20

The number of red coins in the bag is 20.

Given:

Five years ago, the average age of a couple was 24 years.

At present, the average age of the couple and a child is 20 years.

Formula Used:

Average age = Sum of ages / Number of individuals

Calculation:

Let the ages of the couple five years ago be x and y.

Given that 5 years ago, their average age was 24:

$\frac{x + y}{2} = 24$

⇒ x + y = 48 (Equation 1)

At present, their ages would be x + 5 and y + 5.

Let the child's present age be z.

Given that the current average age of the couple and the child is 20:

$\frac{(x + 5) + (y + 5) + z}{3} = 20$

⇒ x + y + 10 + z = 60

Substitute Equation 1 into the above equation:

⇒ 48 + 10 + z = 60

⇒ 58 + z = 60

⇒ z = 60 - 58

⇒ z = 2

The child's age is 2 years.

Shortcut Trick

Given: (2/5) × (3/8) × (1/6) × Number = 52

⇒ (1/5) × (1/8) × (1/1) × Number = 52 × 3 (Simplifying 2/6 to 1/3 and cancelling 3)

⇒ (1/40) × Number = 52

⇒ Number = 52 × 40 = 2080

⇒ 40% of 2080 = 0.4 × 2080 = 832

∴ The correct answer is 832.

Alternate Method

Given:

Fractional value = 52

Fractions involved = 2/5, 3/8, and 1/6

Calculations:

⇒ Let the certain number be x.

⇒ (2/5) × (3/8) × (1/6) × x = 52

⇒ (6/240) × x = 52

⇒ (1/40) × x = 52

⇒ x = 52 × 40 = 2080

⇒ Required value = 40% of 2080

⇒ Required value = (40/100) × 2080

⇒ Required value = 4 × 208 = 832

∴ The correct answer is 832.

Additional Information

Percentage to Fraction Conversion

To convert a percentage to a fraction, divide by 100. Example: 40% = 40/100 = 2/5.

Product of Fractions

When dealing with 'of' in mathematics, it signifies multiplication. (a/b) of (c/d) = (a×c)/(b×d).

Base Change Rule

If a% of b = c, then b = (c × 100)/a. This helps in finding the original number quickly.

Shortcut Trick

Since Expenditure = 2/3 of Income, the Ratio of Expenditure to Savings = 2 : 1.

Total Income ratio = 2 + 1 = 3.

Using the weighted average method for percentage changes (Income = Exp + Sav):

3 × 18% = 2 × 24% + 1 × x%

54% = 48% + x% ⇒ x = 6%

∴ The correct answer is 6%.

Alternate Method

Given:

Expenditure = 2/3 of Income

Income increases by 18% and Expenditure increases by 24%.

Calculations:

⇒ Let the initial income be 300 units.

⇒ Initial Expenditure = (2 ÷ 3) × 300 = 200 units

⇒ Initial Savings = 300 − 200 = 100 units

⇒ New Income = 300 + (18% of 300) = 300 + 54 = 354 units

⇒ New Expenditure = 200 + (24% of 200) = 200 + 48 = 248 units

⇒ New Savings = New Income − New Expenditure

⇒ New Savings = 354 − 248 = 106 units

⇒ Increase in Savings = 106 − 100 = 6 units

⇒ Percentage Increase = (Increase ÷ Initial Savings) × 100%

⇒ Percentage Increase = (6 ÷ 100) × 100% = 6%

∴ The correct answer is 6%.

Additional Information

Income, Expenditure & Savings Relation

Income = Expenditure + Savings. The percentage change in total income is always equal to the weighted average of the percentage changes in expenditure and savings.

Percentage Change Formula

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

Base Value Concept

When calculating percentage increase or decrease, the original or initial value is always taken as the base value (100%).

Given:- 

A sum of ₹7,000 deposited at compound interest becomes triple of itself after 4 years

Formula used:-

(1) Amount = Principal + Compound Interest   

(2)   $C.I = P(1 + \frac{R}{100} )^T$ 

Where C. I = Compound Interest,

P = Principal

R = Rate

T = Time 

Calculation:-

 A sum of ₹7,000 deposited at compound interest

becomes triple of itself after 4 years.

So money after 4 years = 7000 × 3  = 21000 

We know that  Amount = Principal + $P(1 + \frac{R}{100} )^T$   

⇒ 21000 = 7000( 1 + (R/100)⁴ 

⇒ ( 1 + (R/100)4 = 3 

Now apply the cube in the above equation.

⇒ [ ( 1 + (R/100)4]³ = 3³  

.

⇒ [ ( 1 + (R/100)]¹² = 27  

So from the above equation,

We can say that after 12 years money will be 27 times itself   

Hence money after 12 years = 27 × 7000 = 189000 

∴ The required answer is 189000

Shortcut Trick LCM of 20 and 12 = 60 eggs. Cost Price (CP) of 60 eggs = 3 × ₹7 = ₹21. Selling Price (SP) of 60 eggs = 5 × ₹7.20 = ₹36. Profit earned on 60 eggs = ₹36 − ₹21 = ₹15. Since ₹15 profit is on 60 eggs, ₹30 profit will be on (60 ÷ 15) × 30 = 120 eggs. ∴ The correct answer is 120 eggs.

Alternate Method Given: Cost Price (CP) of 20 eggs = ₹7 Selling Price (SP) of 12 eggs = ₹7.20 Total Profit earned = ₹30

Formula: Profit = Selling Price (SP) − Cost Price (CP) ⇒ CP of 1 egg = 7 ÷ 20 = ₹0.35 ⇒ SP of 1 egg = 7.20 ÷ 12 = ₹0.60 ⇒ Profit earned on 1 egg = 0.60 − 0.35 = ₹0.25 ⇒ Let the number of eggs bought be 'n'. ⇒ Total Profit = n × Profit per egg ⇒ 30 = n × 0.25 ⇒ n = 30 ÷ 0.25 ⇒ n = 3000 ÷ 25 ⇒ n = 120 Thus, the total number of eggs bought by Meena is 120, which corresponds to Option (2). ∴ The correct answer is 120 eggs.

Additional Information Unitary Method in Profit & Loss Always calculate the CP and SP for a single unit to simplify calculations when quantities for buying and selling are different. Profit Percentage Formula Profit % = (Profit ÷ Cost Price) × 100. In this case, Profit % = (0.25 ÷ 0.35) × 100 ≈ 71.42%. Relationship between CP, SP and Profit If SP > CP, it is a Profit. If CP > SP, it is a Loss. Total Profit = Quantity × (SP per unit − CP per unit).

Shortcut Trick

Successive discount formula = a + b − (a × b) ÷ 100

Net discount = 12 + 12 − (12 × 12) ÷ 100 = 24 − 1.44 = 22.56%

Selling Price (SP) = 100% − 22.56% = 77.44% of Marked Price (MP)

SP = 12500 × 0.7744 = 9680

Alternatively, using multiplier: SP = 12500 × (88 ÷ 100) × (88 ÷ 100) = 125 × 0.88 × 88 = 9680

∴ The correct answer is ₹9680.

Alternate Method

Given: Marked Price (MP) = ₹12,500 | Discount₁ = 12% | Discount₂ = 12%

Formulas: SP = MP × (100 − D₁)% × (100 − D₂)%

⇒ Price after 1^st discount = 12500 − (12% of 12500)

⇒ Price after 1^st discount = 12500 − 1500 = ₹11,000

⇒ Final Selling Price = 11000 − (12% of 11000)

⇒ Final Selling Price = 11000 − 1320 = ₹9,680

∴ The correct answer is ₹9680.

Additional Information

Successive Discounts

When two or more discounts are applied one after another, the second discount is calculated on the reduced price. Formula: Net D% = (d₁ + d₂ − d₁d₂/100)%.

Marked Price vs Selling Price

The Marked Price (MP) is the tag price, while the Selling Price (SP) is the actual amount paid after subtracting the Discount.

Shortcut Trick

When an item is replaced, use the net deviation method: New Value = Old Value + (Total Number × Change in Average)

Given, Total Number of boys = 5

Average weight increases by 9 kg, so Total Increase = 5 × 9 = 45 kg

Weight of the new boy = Weight of replaced boy + Total Increase

Weight of the new boy = 39 + 45 = 84 kg

∴ The correct answer is 84 kg.

Alternate Method

Given:

Total number of boys = 5

Average increase in weight = 9 kg

Weight of the boy who is replaced = 39 kg

Formula Used:

Average = Sum of all values ÷ Total number of values

Calculations:

⇒ Let the initial average weight of 5 boys be 'A' kg.

⇒ Total weight of 5 boys initially = 5 × A = 5A kg

⇒ Total weight after replacement = 5A − 39 + Weight of new boy

⇒ The new average weight becomes = A + 9

⇒ New total weight = 5 × (A + 9) = 5A + 45

⇒ Equating the new total weights:

⇒ 5A − 39 + Weight of new boy = 5A + 45

⇒ Weight of new boy = 45 + 39

⇒ Weight of new boy = 84 kg

∴ The correct answer is 84 kg.

Additional Information

Average Formula

Average = Sum of observations ÷ Total number of observations.

Net Deviation Method (Replacement Method)

When an entity is replaced without changing the total count, the change in total sum is strictly equal to (Number of observations × Change in Average).

Combined Average

If two groups with sizes n₁ and n₂ have averages A₁ and A₂, the Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Given:

Distance = 20 km

Time = 15 minutes

Calculation:

Speed of car = Distance/Time = 20/(15/60) = 80 km/hr

When speed is decreased by 10 km/hr,

Speed of car = 80 – 10 = 70 km/hr

∴ Time taken to cover same distance = 20/70 = 2/7 hrs = (2/7) × 60 = 120/7 minutes