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Given:

Numan does half the work as Gagan in 4/5 of the time.

Together they take 16 days to complete a piece of work

Calculation:

According to the question,

Numan × 4/5 = Gagan × 1/2

⇒ Numan = Gagan × 5/8

⇒ Numan/Gagan = 5/8 

So, Numan’s efficiency = 5 unit/day. & Gagan’s efficiency = 8 unit/day.

Now,

Total work = 16 × (5 + 8)

⇒ 16 × 13

Time taken by Gagan = (16 × 13)/8

⇒ 26 Days

∴ Gagan will take 26 days to complete the work.

Shortcut Trick

Let B = 100 units, then A = 128 units.

Sum of A and B = 100 + 128 = 228 units.

C = 75% of 228 = 171 units (since C is 25% less than the sum).

Difference (C − A) = 171 − 128 = 43 units.

Required % = (43 ÷ 128) × 100 ≈ 33.6%.

∴ The correct answer is 33.6%.

Alternate Method

Given: A = B + 28% of B; C = (A + B) − 25% of (A + B).

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

⇒ Let B = 100

⇒ A = 100 × (128 ÷ 100) = 128

⇒ Sum (A + B) = 128 + 100 = 228

⇒ C = 228 × (75 ÷ 100) = 171

⇒ Difference between C and A = 171 − 128 = 43

⇒ Required Percentage = (43 ÷ 128) × 100

⇒ Required Percentage = 33.59375% ≈ 33.6%

∴ The correct answer is 33.6%.

Additional Information

Percentage Comparison Base

Always identify the 'than' part in the question to determine the denominator (base). In 'C is more than A', A is the base.

Fractional Equivalents

Using fractions can speed up calculations: 25% less means 3/4 of the value, and 20% more means 6/5 of the value.

Successive Percentage Change

For two successive changes a% and b%, the net change is given by [a + b + (ab ÷ 100)]%.

According to the question

⇒ 960 × [75/100] × [(100 – x)/100] = 612

⇒ 720 × [(100 – x)/100] = 612

⇒ (100 – x)/100 = 612/720

⇒ 100 – x = (17/20) × 100

⇒ 100 – x = 85%

⇒ x = 100 – 85 = 15%

Let its cost price be Rs. x

From the problem’s statement

⇒ 540 - x = x – 370

⇒ 2x = 910

⇒ x = 455

If it was sold for Rs. 910

⇒ Profit = 910 - 455 = 455

⇒ Profit % = Profit × 100/CP = 455 × 100/455

⇒ Profit % = 100

∴ The profit percentage will be 100%

Formula used:

If the mean proportional to a and b is x, then,

a / x = x / b or x2 = ab

Calculation:

Let x be the required mean proportional. Then,

x2 = ab

⇒ x2 = 27 × 300

⇒ x2 = 8100

⇒ x = 90

∴ Mean proportional = 90

Given:

Actual speed = 45 km/h

Reduced speed = 30 km/h

Formula used:

Total distance = $\frac{(S_1 × S_2)}{S_1-S_2} \times (T_1+T_2)$

Here,

S1 = Actual speed

S2 = Reduced speed

T1 = Time1

T2 = Time 2

Calculation:

According to the concept,

Total distance = $\frac{(45 × 30)}{45-30} \times (0+\frac{1}{4})$

⇒ $\frac{(1350)}{15} \times \frac{1}{4}$

⇒ 22.5 km

∴ The distance between A and B is 22.5 km.

Shortcut Trick

Income = Expenditure + Savings (Ratio 80 : 20 → 4 : 1)

Apply Alligation or Weighted Average: % Change in Income = (E% × E_ratio + S% × S_ratio) ÷ (Total Ratio)

12 = (17.5 × 4 + x × 1) ÷ 5

60 = 70 + x ⇒ x = −10%

Negative sign indicates a decrease.

∴ The correct answer is 10%.

Alternate Method

Given: Initial Income = 100, Expenditure = 80, Savings = 20.

Formula Used: Income = Expenditure + Savings

⇒ New Income = 100 + (12% of 100) = 112

⇒ New Expenditure = 80 + (17.5% of 80) = 80 + 14 = 94

⇒ New Savings = New Income − New Expenditure = 112 − 94 = 18

⇒ Decrease in Savings = Old Savings − New Savings = 20 − 18 = 2

⇒ Percentage Decrease = (Decrease ÷ Old Savings) × 100

⇒ Percentage Decrease = (2 ÷ 20) × 100 = 10%

∴ The correct answer is 10%.

Additional Information

Income-Expenditure Relationship

Total Income is always the sum of Expenditure and Savings. Any change in one must be balanced by the others.

Percentage Change Formula

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

Weighted Average in Income

% Change in Income = (% Change in Exp × Exp Ratio + % Change in Sav × Sav Ratio) ÷ (Total Ratio).

The distance between two stations, A and B = 575 km,

Speed of first train = 50 km/hr

Speed of second train = 60 km/hr

Distance covered by first train in 30 min (1/2 hr) = 50 × 1/2 = 25 km

Remaining distance = 575 - 25 = 550 km

Relative speed, if opposite direction = 50 + 60 = 110 km/hr

As we know,

Time = Distance/speed

Both train meet each other in = 550/110 = 5 hrs

Distance covered in 5 hrs by first train = 5 × 50 = 250 km

Total distance covered by first train = 25 + 250 = 275 km

Both trains meet each other 275 km far from station A.

Shortcut Trick

Old ratio Ravi : Sumit = 2 : 3 (Difference = 1)

New ratio Ravi : Sumit = 40 : 57 (Difference = 17)

Balance the differences: Old ratio × 17 → 34 : 51

Change in parts = 40 − 34 = 6 units = Rs. 4000

Sumit’s new salary = 57 units = (4000 ÷ 6) × 57 = Rs. 38000

∴ The correct answer is Rs. 38000.

Alternate Method

Given:

Initial Ratio = 2 : 3

Increase in each salary = Rs. 4000

New Ratio = 40 : 57

Formula Used:

If a : b is the ratio, then values are ax and bx. New ratio = (ax + k) / (bx + k)

Sumit Old: 51 units (34000)

Sumit New: 57 units (38000)

+6 units = 4000

Calculations:

Let the initial salaries of Ravi and Sumit be 2x and 3x respectively.

⇒ (2x + 4000) / (3x + 4000) = 40 / 57

⇒ 57 × (2x + 4000) = 40 × (3x + 4000)

⇒ 114x + 228000 = 120x + 160000

⇒ 120x − 114x = 228000 − 160000

⇒ 6x = 68000

⇒ 3x = 34000

Sumit’s new salary = 3x + 4000

⇒ New salary = 34000 + 4000 = 38000

∴ The correct answer is Rs. 38000.

Additional Information

Balancing Ratios

When a constant value is added to both terms of a ratio, the difference between the terms of the ratio should be made equal by multiplying with the cross-difference.

Duplicate and Sub-duplicate Ratio

The duplicate ratio of a : b is a² : b², and the sub-duplicate ratio is √a : √b.

Mean Proportional

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

Shortcut Trick

For consecutive integers, the Average lies exactly at the mid-point of the series.

Largest Integer = Average + (n − 1) ÷ 2

Largest Integer = 22.5 + (18 − 1) ÷ 2 = 22.5 + 8.5 = 31

∴ The correct answer is 31.

Alternate Method

Given: Number of terms (n) = 18, Average = 22.5

Formula: Average of AP = (First term + Last term) ÷ 2

⇒ Let the 18 consecutive integers be x, x+1, ..., x+17.

⇒ 22.5 = (x + x + 17) ÷ 2

⇒ 22.5 × 2 = 2x + 17

⇒ 45 = 2x + 17

⇒ 2x = 45 − 17 = 28

⇒ x = 14 (Smallest integer)

⇒ Largest integer = x + 17 = 14 + 17 = 31

∴ The correct answer is 31.

Additional Information

Consecutive Integers Average

For any Arithmetic Progression (AP), the average is always the middle term (if n is odd) or the mean of the two middle terms (if n is even).

Smallest and Largest Relation

The difference between the largest and smallest of n consecutive integers is always (n − 1).

Sum of Consecutive Integers

Sum = Average × Number of terms = [n ÷ 2] × [2a + (n − 1)d].