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Shortcut Trick

Total Expenditure = (4 × 3000) + (8 × 2000) = 12000 + 16000 = ₹28000.

Total Income = Total Expenditure + Total Savings = 28000 + 8000 = ₹36000.

Average Monthly Income = Total Income ÷ 12 = 36000 ÷ 12 = ₹3000.

∴ The correct answer is ₹3000.

Alternate Method

Given:

Average Expenditure (Months 1-4) = ₹3000

Average Expenditure (Months 5-12) = ₹2000

Total Savings for 12 months = ₹8000

Formula Used:

Total Income = Total Expenditure + Savings

Average Monthly Income = Total Income ÷ 12

Calculations:

⇒ Total Expenditure = (4 × 3000) + (8 × 2000)

⇒ Total Expenditure = 12000 + 16000 = ₹28000

⇒ Total Income = 28000 + 8000 = ₹36000

⇒ Average Monthly Income = 36000 ÷ 12

⇒ Average Monthly Income = ₹3000

∴ The correct answer is ₹3000.

Additional Information

Income-Expenditure Relation

Income is always the sum of Expenditure and Savings. Income = Expenditure + Savings.

Average Income Formula

Average Monthly Income = (Total Annual Expenditure + Total Annual Savings) ÷ 12.

Weighted Average

If expenditure varies, total expenditure = Σ(Number of months × Average expenditure of those months).

Shortcut Trick

Successive multiplier method: Let CP of A be x.

x × 1.25 × 1.20 × 0.70 = 2100.

x × (5/4) × (6/5) × (7/10) = 2100 ⇒ x × 21/20 = 2100 ⇒ x = 2000.

CP of B = SP of A = 2000 × 1.25 = 2500.

CP of C = SP of B = 2500 × 1.20 = 3000.

Difference = 3000 − 2500 = 500.

∴ The correct answer is Rs.500.

Alternate Method

Given → Profit A = 25%, Profit B = 20%, Loss C = 30%, Final Price (D) = Rs.2,100. Formulas: SP = CP × (100 + P%)/100 | SP = CP × (100 − L%)/100.

⇒ Let CP of A = 100 units. ⇒ CP of B (SP of A) = 100 + 25% of 100 = 125 units. ⇒ CP of C (SP of B) = 125 + 20% of 125 = 125 + 25 = 150 units. ⇒ CP of D (SP of C) = 150 − 30% of 150 = 150 − 45 = 105 units. ⇒ Given 105 units = Rs.2,100 ⇒ 1 unit = Rs.20. ⇒ CP of B = 125 × 20 = Rs.2,500. ⇒ CP of C = 150 × 20 = Rs.3,000. ⇒ Difference = 3000 − 2500 = Rs.500.

∴ The correct answer is Rs.500.

Additional Information

Successive Percentage Changes

When a value is changed multiple times, the final value is CP × (1 ± r₁/100) × (1 ± r₂/100)...

Chain Rule in Transactions

The selling price for one person becomes the cost price for the next person in the transaction chain.

Shortcut Trick

Combine the ratios: A : B = 2 : 5 and B : C = 8 : 11.

Equalize B by multiplying first ratio by 8 and second by 5 → A : B : C = 16 : 40 : 55.

Sum of A and C parts = 16 + 55 = 71 units.

Given 71 units = 71 years → 1 unit = 1 year.

Age of B = 40 units = 40 years.

∴ The correct answer is 40 years.

Alternate Method

Given:

Ratio of A : B = 2 : 5

Ratio of B : C = 8 : 11

Sum of ages of A and C = 71 years

Formula Used:

If A : B = a : b and B : C = c : d, then A : B : C = (a × c) : (b × c) : (b × d)

Calculations:

⇒ To combine ratios, make B common: A : B = (2 × 8) : (5 × 8) = 16 : 40

⇒ B : C = (8 × 5) : (11 × 5) = 40 : 55

⇒ A : B : C = 16 : 40 : 55

⇒ Let ages be 16x, 40x, and 55x

⇒ Sum of A and C = 16x + 55x = 71x

⇒ 71x = 71

⇒ x = 1

⇒ Age of B = 40x = 40 × 1 = 40 years

∴ The correct answer is 40 years.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the result is the compound ratio. For A:B and C:D, the compound ratio is (A×C):(B×D).

Ratio Proportion Property

If a/b = c/d, then a×d = b×c (Product of extremes = Product of means).

Ages after 'n' years

If current age is x, then age after n years is x + n and age n years ago was x − n.

Common Calculation

Ratio of milk to water = 9:4

(4x + 5)/(x + 10) = 9/4

4(4x + 5) = 9(x + 10)

16x + 20 = 9x + 90

16x - 9x = 90 - 20

7x = 70

=> x = 10

Common Calculation

Ratio of milk to water = 9:4

(4x + 5)/(x + 10) = 9/4

4(4x + 5) = 9(x + 10)

16x + 20 = 9x + 90

16x - 9x = 90 - 20

7x = 70

=> x = 10

y = Milk - Water = (4x + 5) - (x + 10) = 3x - 5 = 3(10) - 5 = 25

Total mixture = x + y = 10 + 25 = 35 litres

Milk:Water ratio = 4:3

Milk = (4/7) × 35 = 20 litres

Water = (3/7) × 35 = 15 litres

Difference z = 20 - 15 = 5 litres

Shortcut Trick

Observe the proportionality between the first group and the target group.

First Group: 5 men + 6 women → 12 days

Target Group: 15 men + 18 women = 3 × (5 men + 6 women)

Since the number of workers is multiplied by 3, the time taken will be divided by 3 (inverse proportion).

New Time = 12 ÷ 3 = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Step 1: Find the relation between Men (M) and Women (W)

⇒ (5M + 6W) × 12 = (8M + 4W) × 8

⇒ (5M + 6W) × 3 = (8M + 4W) × 2

⇒ 15M + 18W = 16M + 8W

⇒ 1M = 10W

Step 2: Calculate Total Work in terms of Women

⇒ Total Work = (5 × 10W + 6W) × 12

⇒ Total Work = 56W × 12 = 672W

Step 3: Efficiency of 15 men and 18 women

⇒ Efficiency = 15M + 18W = 15 × (10W) + 18W = 150W + 18W = 168W

Step 4: Calculate Time Required

⇒ Time = Total Work ÷ Efficiency

⇒ Time = 672W ÷ 168W = 4 days

∴ The correct answer is 4 days.

Additional Information

Chain Rule Formula

The general formula is M₁D₁H₁/W₁ = M₂D₂H₂/W₂, where M is persons, D is days, H is hours, and W is work.

Inverse Proportionality

Time taken to complete a work is inversely proportional to the efficiency or the number of workers. If workers become 'n' times, time becomes 1/n.

Efficiency Comparison

To solve group work problems, always equate the total work (M₁D₁ = M₂D₂) to find the ratio of individual efficiencies.

Shortcut Trick

Upstream speed = 12 / 2 = 6 kmph.

Stream speed = Still water speed − Upstream speed = 16 − 6 = 10 kmph.

Downstream speed = Still water speed + Stream speed = 16 + 10 = 26 kmph.

Downstream time = 52 / 26 = 2 hours.

∴ The correct answer is 2 hr.

Alternate Method

Given:

Speed of boat in still water (u) = 16 kmph

Upstream distance = 12 km, Upstream time = 2 hours

Downstream distance = 52 km

Formula Used:

Speed = Distance ÷ Time

Upstream Speed (v_u) = u − v (where v is stream speed)

Downstream Speed (v_d) = u + v

⇒ Upstream speed = 12 / 2 = 6 kmph

⇒ 16 − v = 6

⇒ v = 16 − 6 = 10 kmph (Speed of stream)

⇒ Downstream speed = u + v = 16 + 10 = 26 kmph

⇒ Time to travel 52 km downstream = 52 / 26 = 2 hours

∴ The correct answer is 2 hr.

Additional Information

Still Water Speed Calculation

Still water speed (u) = 1/2 × (Downstream Speed + Upstream Speed).

Stream Speed Calculation

Stream speed (v) = 1/2 × (Downstream Speed − Upstream Speed).

Relative Speed Concept

In downstream, the stream helps the boat (add speeds); in upstream, the stream opposes the boat (subtract speeds).

Shortcut Trick

Ratio of saving (Kunal : Karan) = 1 : 3

Difference in ratio units = 3 − 1 = 2 units

Given 2 units = ₹28,000 ⇒ 1 unit = ₹14,000

Total saving (1 + 3) = 4 units = 4 × ₹14,000 = ₹56,000

∴ The correct answer is ₹56,000.

Alternate Method

Given: Saving ratio of Kunal and Karan = 1 : 3

Formula: Value of one unit = Actual Difference ÷ Difference in Ratio

⇒ Let saving of Kunal = 1x and Karan = 3x

⇒ 3x − 1x = 28,000

⇒ 2x = 28,000

⇒ x = 14,000

⇒ Total saving = 1x + 3x = 4x

⇒ Total saving = 4 × 14,000 = 56,000

∴ The correct answer is ₹56,000.

Additional Information

Ratio and Proportion Basics

If a : b is the ratio of two quantities, the total sum is represented by (a + b) units and the difference by |a − b| units.

Income, Expenditure and Saving Relation

Income = Expenditure + Savings. This fundamental formula is widely used in financial ratio problems.

Unitary Method in Ratios

To find the actual value of any part, divide the known total or difference by its corresponding ratio units to find the value of 1 unit.

Common Calculation

Calculation

Ratio of boys in 2021 and 2022 = 2:3;

Girls in 2022 are 10% less than boys.

Boys 2022 = 160 × (3/2) = 240

Girls 2022 = 240 × 0.9 = 216

Answer: 216 girls

Common Calculation

Calculation

Given: Difference between boys in 2020 and 2019 = 5x

Difference = 150 − 140 = 10

5x = 10 → x = 2

7x + 22 = 7 × 2 + 22 = 36

Answer: 36