Search Previous Year Questions

Given:

A and B can do a piece of work in 36 days,

B and C can do the same work in 60 days,

A and C can do the same work in 45 days.

Concept used:

Efficiency = 1/(Total time)

Calculation:

(A + B)’s 1 day work = 1/36      ----(i) 

(B + C)’s 1 day work = 1/60      ----(ii) 

(C + A)’s 1 day work = 1/45      ----(iii)

From equations (i), (ii) and (iii),

2(A + B + C)’s 1 day work = 1/36 + 1/60 + 1/45

⇒ (5 + 3 + 4)/180 = 1/15

⇒ (A + B + C)’s 1 day work = 1/(15 × 2) = 1/30      ----(iv)

From equations (ii) and (iv) we get

A’s 1 day work = 1/30 – 1/60 = 1/60 = 1/60

Time taken by A alone to complete the work = 60

∴ A alone can complete the same work in 60 days

Given:

In a school, the total number of boys is 20% more than the total number of girls.

Formula Used:

Let the number of girls be G.

Number of boys = G + 20% of G

Calculation:

Number of boys = G + 0.2 × G

⇒ Number of boys = 1.2G

The ratio of the number of boys to that of girls = 1.2G : G

⇒ $\dfrac{1.2G}{G}$

⇒ 1.2 : 1

⇒ 6 : 5

∴ The correct answer is option 1.

Given:

The difference between any two consecutive numbers is 5.

The average of the five numbers is 215.

Calculation:

Let the first number = x

Second number = x + 5

Third number = x + 10

Fourth number = x + 15

Fifth number = x + 20

Average = (x + x + 5 + x + 10 + x + 15 + x + 20) / 5 = 215

⇒ (5x + 50) / 5 = 215

⇒ 5x + 50 = 215 × 5

⇒ 5x + 50 = 1075

⇒ 5x = 1075 - 50 = 1025

⇒ x = 1025 / 5

⇒ x = 205

The least number is 205.

Given:

Volume of the hemisphere = 486π cm³

Formula Used:

Volume of a hemisphere = (2/3)πr³

Curved Surface Area (CSA) of a hemisphere = 2πr²

Calculation:

Given Volume,

(2/3)πr³ = 486π

⇒ r³ = (3/2) × 486

⇒ r³ = 729

⇒ r = 9 cm

If the radius is tripled, new radius = 3r = 3 × 9 = 27 cm

Curved Surface Area (CSA) = 2π(27)²

⇒ CSA = 2π × 729

⇒ CSA = 1458π cm²

The correct answer is option 3.

Shortcut Trick

Ratio of numbers = 3 : 5 : 7 (all are co-prime)

LCM = Product of Ratios × HCF

25200 = (3 × 5 × 7) × HCF ⇒ 25200 = 105 × HCF

HCF = 25200 ÷ 105 = 240

Sum of smallest (3) and largest (7) = (3 + 7) × HCF = 10 × 240 = 2400

∴ The correct answer is 2400.

Alternate Method

Given: Ratio = 3 : 5 : 7, LCM = 25200

Formula Used: Numbers are ax, bx, cx where x is HCF. LCM of (ax, bx, cx) = x × LCM(a, b, c)

Let the numbers be 3x, 5x, and 7x, where x is the HCF.

⇒ LCM(3x, 5x, 7x) = x × 3 × 5 × 7 = 105x

⇒ 105x = 25200

⇒ x = 25200 ÷ 105

⇒ x = 240

⇒ Smallest number = 3x = 3 × 240 = 720

⇒ Largest number = 7x = 7 × 240 = 1680

⇒ Sum = 720 + 1680 = 2400

∴ The correct answer is 2400.

Additional Information

HCF and LCM Relation

For two numbers a and b, Product of numbers = HCF × LCM. For more than two numbers, this direct product rule does not apply.

Co-prime Numbers

If numbers are co-prime (HCF = 1), then their LCM is simply the product of the numbers.

Ratio and HCF

If the ratio of numbers is a : b : c and their HCF is x, then the numbers are ax, bx, and cx.

Given:

Length of train = 140 m

Speed of train = 54 km/hr = 54 × (5/18) = 15 m/sec

Time required to cover the bridge = 30seconds

Concept Used:

Distance covered = Speed × Time

1 km/hr = 5/18 m/sec

Calculation:

Let, the length of the bridge is x m

To cross the bridge the train have to cross (140 + x) m

Accordingly,

140 + x = 15 × 30

⇒ x = 450 - 140

⇒ x = 310

∴ The length of the bridge is 310 m.

Shortcut Trick

Let the Savings be 100%. Then Expenditure = 100% + 40% = 140%.

Ratio of Savings : Expenditure = 100 : 140 = 5 : 7

Total Income = Savings + Expenditure = 5 + 7 = 12 units.

Given 12 units = ₹72,000 ⇒ 1 unit = ₹6,000

Savings = 5 units = 5 × 6,000 = ₹30,000

∴ The correct answer is ₹30000.

Alternate Method

Given:

Total Income = ₹72,000

Expenditure is 40% higher than Savings.

Formula Used:

Income = Expenditure + Savings

Calculations:

⇒ Let his Savings be x.

⇒ Expenditure = x + (40% of x) = x + 0.4x = 1.4x

⇒ Total Income = x + 1.4x = 2.4x

⇒ According to the question, 2.4x = 72000

⇒ x = 72000 ÷ 2.4

⇒ x = 720000 ÷ 24

⇒ x = 30000

∴ The correct answer is ₹30000.

Additional Information

Income, Expenditure & Savings

The fundamental equation of personal finance is: Income = Expenditure + Savings. Any total change in income is always equal to the sum of the changes in expenditure and savings.

Ratio Method for Percentages

When a value is P% higher than another, their ratio becomes (100 + P) : 100. This heavily simplifies calculations by avoiding decimals and allowing you to solve the question completely using proportional unit parts.

Base Value Identification

It is crucial to identify the correct base. Here, Expenditure is 40% higher than Savings, meaning Savings is the base (100%). Misinterpreting the base value is the most common error in percentage problems.

Given:

When 80% of a number is added to 80, the result is the same number.

Formula used:

Let the number be x.

0.8x + 80 = x

Calculations:

0.8x + 80 = x

⇒ x - 0.8x = 80

⇒ 0.2x = 80

⇒ x = 400

∴ The correct answer is option (4).

As per pie chart,

Percentage fund allotted for Sevage = 16%

Percentage fund allotted for Road = 22%

Total percentage fund allocated for Sevage and Road area = 16 + 22 = 38%

Fund allocated to a village under a special village development scheme = Rs. 1200 Crore

The total fund allocation for Road and Sevage construction together

= 1200 × (38/100) = 456

∴ the total fund allocation for Road and Sevage construction together is Rs.456 crore.

As per pie chart,

Percentage fund allotted for electrification = 10%

Fund allocated to a village under a special village development scheme = Rs. 1200 Crore

Total budget for electrification =

= 1200 × (10/100)

= 120

∴ total budget for electrification is Rs. 120 crore.