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Shortcut Trick

Walking time = 40 ÷ 5 = 8 hours.

Running time = 30 ÷ 10 = 3 hours.

Ratio of walking time to running time = 8 : 3.

∴ The correct answer is 8 : 3.

Alternate Method

Given:

Walking distance = 40 km, Walking speed = 5 km/hr.

Running distance = 30 km, Running speed = 10 km/hr.

Formula Used:

Time = Distance ÷ Speed

⇒ Walking time (T_w) = 40 ÷ 5 = 8 hours

⇒ Running time (T_r) = 30 ÷ 10 = 3 hours

⇒ Ratio = T_w : T_r = 8 : 3

∴ The correct answer is 8 : 3.

Additional Information

Average Speed

The average speed for the entire journey is calculated as Total Distance ÷ Total Time. Here, (40 + 30) ÷ (8 + 3) = 70 ÷ 11 km/hr.

Speed Conversion

To convert speed from km/hr to m/s, multiply the value by 5/18. For example, 10 km/hr = 10 × (5/18) m/s.

Inverse Proportion

When distance is constant, the time taken is inversely proportional to the speed (T ∝ 1/S).

Shortcut Trick

Let the difference between Savings and Expenditure be 1 unit.

Then, Expenditure = 2 × 1 = 2 units.

Savings = Expenditure + Difference = 2 + 1 = 3 units.

Total Salary = Expenditure + Savings = 2 + 3 = 5 units.

Expenditure % = (2 ÷ 5) × 100 = 40%.

∴ The correct answer is 40%.

Alternate Method

Given:

Expenditure (E) = 2 × (Savings (V) − Expenditure (E))

Formula Used:

Salary = Expenditure + Savings

Expenditure Percentage = (Expenditure ÷ Salary) × 100

⇒ E = 2V − 2E

⇒ 3E = 2V

⇒ V = (3 ÷ 2)E = 1.5E

⇒ Salary = E + V = E + 1.5E = 2.5E

⇒ Expenditure Percentage = (E ÷ 2.5E) × 100

⇒ Expenditure Percentage = (100 ÷ 2.5) = 40%

∴ The correct answer is 40%.

Additional Information

Income Expenditure Relation

The basic formula for income problems is: Salary = Expenditure + Savings.

Percentage Calculation

Percentage of any component is calculated as: (Component ÷ Total Salary) × 100.

Ratio and Proportion

When components are given in linear equations, converting them into Ratios (e.g., E:V = 2:3) simplifies calculations.

Given:

0.375 + 3/8 − 0.125

Formula used:

Decimal conversion: 3/8 = 0.375

Calculations:

0.375 + 3/8 − 0.125

⇒ 0.375 + 0.375 − 0.125

⇒ 0.750 − 0.125

⇒ 0.625

∴ The correct answer is option (2).

Shortcut Trick

For the number 12A916B to be divisible by 24, it must be divisible by both 8 and 3.

Divisibility by 8: Last three digits 16B must be divisible by 8. Possible values for B are 0 or 8. To maximize (A + B), let B = 8.

Divisibility by 3: Sum of digits (1 + 2 + A + 9 + 1 + 6 + 8 = 27 + A) must be divisible by 3.

Since 27 is already divisible by 3, A can be 0, 3, 6, or 9. To maximize (A + B), let A = 9.

Maximum value of (A + B) = 9 + 8 = 17.

∴ The correct answer is 17.

Alternate Method

Given: The 7-digit number is 12A916B and it is divisible by 24.

For divisibility by 24, the number must satisfy divisibility rules for 3 and 8 simultaneously.

Step 1: Apply divisibility rule of 8.

⇒ The last three digits 16B must be divisible by 8.

⇒ 160 + B must be divisible by 8.

⇒ Since 160 is divisible by 8, B must be 0 or 8.

⇒ For maximum (A + B), we choose B = 8.

Step 2: Apply divisibility rule of 3.

⇒ Sum of digits = 1 + 2 + A + 9 + 1 + 6 + B

⇒ Sum = 19 + A + B

⇒ Substitute B = 8: Sum = 19 + A + 8 = 27 + A

⇒ For (27 + A) to be divisible by 3, A can be 0, 3, 6, or 9.

⇒ For maximum value, we choose A = 9.

Step 3: Calculate the maximum value.

⇒ Maximum (A + B) = 9 + 8 = 17.

∴ The correct answer is 17.

Additional Information

Divisibility Rule of 8

A number is divisible by 8 if its last three digits are divisible by 8 or if the last three digits are zero.

Divisibility Rule of 3

A number is divisible by 3 if the sum of its digits is a multiple of 3.

Divisibility of Composite Numbers

To check divisibility by a composite number like 24, we check divisibility by its co-prime factors (3 and 8 in this case, because 3 × 8 = 24 and HCF(3, 8) = 1).

Shortcut Trick

Convert the red marble ratios into percentages to compare the proportion directly.

Bag A: 2 ÷ (2 + 3) = 2/5 = 40%

Bag B: 4 ÷ (4 + 1) = 4/5 = 80%

Bag C: 3 ÷ (3 + 2) = 3/5 = 60%

Comparing the values: 80% > 60% > 40%.

∴ The correct answer is Bag B.

Alternate Method

Given:

Ratio of red marbles to blue marbles in:

Bag A = 2 : 3

Bag B = 4 : 1

Bag C = 3 : 2

Formula Used:

Proportion of red marbles = Red Marbles ÷ (Red Marbles + Blue Marbles)

Calculations:

⇒ Red proportion in Bag A = 2 ÷ (2 + 3) = 2/5 = 0.40

⇒ Red proportion in Bag B = 4 ÷ (4 + 1) = 4/5 = 0.80

⇒ Red proportion in Bag C = 3 ÷ (3 + 2) = 3/5 = 0.60

⇒ Comparing the fractions: 0.80 > 0.60 > 0.40

⇒ Bag B has the highest proportion of red marbles at 0.80 (or 80%).

∴ The correct answer is Bag B.

Additional Information

Ratio to Proportion Conversion

To find the proportion of a specific component in a ratio x : y, use the formula x ÷ (x + y).

Comparing Fractions

Fractions can be compared by converting them to a common denominator or by converting them into decimal or percentage forms.

Percentage Calculation

The percentage of a component is (Value ÷ Total) × 100. This is useful for standardized comparison across different total quantities.

Shortcut Trick

Net interest percentage = Rate × Time = 5.5% × 5.5 = 30.25%

Simple Interest = 30.25% of ₹55,000

SI = (30.25 ÷ 100) × 55,000 = 0.3025 × 55,000 = 16,637.5

∴ The correct answer is ₹16,637.5.

Alternate Method

Given:

Principal (P) = ₹55,000

Rate (R) = 5.5% per annum

Time (T) = 5.5 years

Formula Used:

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

Calculations:

⇒ SI = (55,000 × 5.5 × 5.5) ÷ 100

⇒ SI = 550 × 5.5 × 5.5

⇒ SI = 550 × 30.25

⇒ SI = 16,637.5

∴ The correct answer is ₹16,637.5.

Additional Information

Amount Formula

The total Amount (A) to be paid back after the time period is the sum of the Principal and the Simple Interest: A = P + SI.

Simple Interest vs Compound Interest

In Simple Interest, the interest is calculated only on the initial principal, whereas in Compound Interest, it is calculated on the principal plus any accumulated interest.

Rate of Interest p.a.

The Rate of Interest is usually expressed as a percentage per annum (p.a.). If the time is in months, it must be divided by 12 to convert it to years.

Shortcut Trick

Net filling rate of cistern = 1/4 + 1/20 − 1/20 = 1/4 cistern/hr.

Target work is to fill one-fourth of the empty cistern.

Time taken = Work ÷ Net Rate = (1/4) ÷ (1/4) = 1 hour.

∴ The correct answer is 1 hr.

Alternate Method

Given:

Time taken by Tap 1 (Filling) = 4 hr

Time taken by Tap 2 (Filling) = 20 hr

Time taken by Tap 3 (Emptying) = 20 hr

Calculations:

⇒ Total Capacity of cistern = LCM(4, 20, 20) = 20 units

⇒ Target to fill = 1/4 × 20 = 5 units

⇒ Combined Efficiency of all taps = 5 + 1 − 1 = 5 units/hr

⇒ Time required = Target Work ÷ Combined Efficiency

⇒ Time required = 5 ÷ 5 = 1 hour

∴ The correct answer is 1 hr.

Additional Information

Negative Efficiency

In Pipes and Cisterns, the efficiency of an outlet/emptying tap is always taken as negative while the efficiency of an inlet/filling tap is positive.

Work-Time Relation

Total Work = Efficiency × Time. For fractional work, Time = (Total Work × Fraction) ÷ Combined Efficiency.

Combined Rate Formula

If two taps fill in X and Y hours and one empties in Z hours, the net rate is (1/X + 1/Y − 1/Z) parts per hour.

Shortcut Trick

Find the LCM of 6, 10, and 14, which is 210.

Divide the largest 4-digit number (9999) by 210; the remainder is 129.

Largest multiple = 9999 − 129 = 9870.

Add the required remainder: 9870 + 3 = 9873.

∴ The correct answer is 9873.

Alternate Method

Given: Divisors = 6, 10, 14 and Remainder = 3.

Formula Used: Required Number = (LCM(Divisors) × k) + Remainder

⇒ First, find the prime factors of the divisors:

⇒ 6 = 2 × 3

⇒ 10 = 2 × 5

⇒ 14 = 2 × 7

⇒ LCM = 2 × 3 × 5 × 7 = 210

⇒ The largest 4-digit number is 9999.

⇒ 9999 ÷ 210 ⇒ Quotient = 47, Remainder = 129.

⇒ Largest multiple of 210 = 9999 − 129 = 9870.

⇒ Required number = 9870 + 3 = 9873.

∴ The correct answer is 9873.

Additional Information

General Form of Numbers

Any number that leaves a common remainder r when divided by x, y, z is written in the form [LCM(x, y, z) × k] + r.

LCM Calculation

LCM is calculated by taking the highest power of all prime factors involved in the prime factorization of the given numbers.

Divisibility Verification

To verify, subtract the remainder from the result; the value 9870 is exactly divisible by 6, 10, and 14.

Shortcut Trick

For a Building of height H and a Tower of height h, if the angles of depression to the top and bottom are θ₁ and θ₂:

Height of Tower h = H × [1 − (tan θ₁ / tan θ₂)]

Given: H = 20 m, θ₁ = 30°, θ₂ = 60°

h = 20 × [1 − (tan 30° / tan 60°)]

h = 20 × [1 − (1/√3 / √3)] = 20 × [1 − 1/3] = 20 × 2/3 = 13.33 m

∴ The correct answer is 13.33 m.

Alternate Method

Given:

Height of the Building (AB) = 20 m

Angle of depression of the Tower top (C) = 30°

Angle of depression of the Tower bottom (D) = 60°

Let the Height of the Tower be h and the distance between the building and the tower be x.

⇒ In right ▵ABD, tan 60° = AB / BD

⇒ √3 = 20 / x

⇒ x = 20 / √3

⇒ Draw a horizontal line from C to the building meeting it at E. Then, AE = 20 − h and CE = x.

⇒ In right ▵ACE, tan 30° = AE / CE

⇒ 1 / √3 = (20 − h) / (20 / √3)

⇒ 1 / √3 = (20 − h) × √3 / 20

⇒ 20 = 3 × (20 − h)

⇒ 20 = 60 − 3h

⇒ 3h = 40

⇒ h = 40 / 3 = 13.33 m

∴ The correct answer is 13.33 m.

Additional Information

Angle of Depression

The angle between the horizontal line and the line of sight when an observer looks at an object situated at a lower level.

Trigonometric Ratios

For any right-angled triangle, tan θ = Perpendicular / Base. Common values: tan 30° = 1/√3 and tan 60° = √3.

Relationship between H and h

If angles of depression from height H to the top and bottom of another object of height h are α and β respectively, then h = H(1 − tan α / tan β).

Calculation:

Arrange the terms in ascending order based on their coefficients: x/5, x/4, x/3, 2x/5, 2x/3, 3x/4, x.

Equate the median-representing term 2x/5 to the given median value:

2x/5 = 8.4

x = (8.4 × 5) ÷ 2 = 21

∴ The correct answer is 21.