Quantitative Aptitude PYQs

Shortcut Trick

Ratio of Profit = Σ(Capital × Time)

A : B : C = (8 × 8 + 6 × 4) : (6 × 12) : (5 × 8 + 6 × 4)

A : B : C = (64 + 24) : 72 : (40 + 24) = 88 : 72 : 64

Simplified Ratio = 11 : 9 : 8. Total units = 11 + 9 + 8 = 28 units.

28 units = Rs. 14000 ⇒ 1 unit = Rs. 500. Profit of C = 8 × 500 = Rs. 4000. ∴ The correct answer is Rs. 4000.

Alternate Method

Given → Initial Capital Ratio A:B:C = 8:6:5. Time = 12 months.
Formulas → Profit Share = Capital × Time Period

⇒ A's investment: (8 × 8 months) + (8 × 75% × 4 months) = 64 + 24 = 88.
⇒ B's investment: (6 × 12 months) = 72.
⇒ C's investment: (5 × 8 months) + (5 × 120% × 4 months) = 40 + 24 = 64.
⇒ Total Investment Ratio A:B:C = 88 : 72 : 64 = 11 : 9 : 8.
⇒ Sum of ratios = 11 + 9 + 8 = 28.
⇒ C's share = (8 ÷ 28) × 14000 = 8 × 500 = 4000.

∴ The correct answer is Rs. 4000.

Additional Information

Partnership Basics

Profit is distributed based on the product of Capital and Time. If time is constant, profit ratio equals capital ratio.

Percentage Changes

To calculate a 25% decrease, multiply by 0.75. For a 20% increase, multiply by 1.20.

Shortcut Trick

Let the total number of applicants be 100 units.

Eligible candidates = 95% of 100 = 95 units.

Other categories = (100% − 85%) of eligible = 15% of 95 = 14.25 units.

Given: 14.25 units = 4275 ⇒ 1 unit = 300.

Total applicants = 100 × 300 = 30,000.

∴ The correct answer is 30,000.

Alternate Method

Given:

Ineligible = 5% ⇒ Eligible = 95%

General Category = 85% of eligible ⇒ Other Categories = 15% of eligible

Other Categories candidates = 4275

Calculations:

Let the total number of voters be x.

⇒ Eligible candidates = x × (95/100)

⇒ Other categories = x × (95/100) × (15/100) = 4275

⇒ x × 0.95 × 0.15 = 4275

⇒ x × 0.1425 = 4275

⇒ x = 4275 ÷ 0.1425

⇒ x = 30,000

∴ The correct answer is 30,000.

Additional Information

Successive Percentage change

When a value is reduced by a% and then further by b%, the remaining value is P × (1 − a/100) × (1 − b/100).

Base change in Election problems

Always identify the base for each percentage. Here, 85% is of the eligible candidates, not the total applicants.

Ratio Method

Fractions can be used for faster calculation: 95% = 19/20 and 15% = 3/20. Total = 4275 × (20/19) × (20/3).

Shortcut Trick

Initial quantity = 100 litres

Replacement fraction = 10% = 1/10

Remaining fraction after 1 process = 1 − 1/10 = 9/10

Since the process is repeated two times more (total 3 times):

Final Milk = 100 × (9/10)³ = 100 × 729/1000 = 72.9 litres

∴ The correct answer is 72.9 litres.

Alternate Method

Given:

Initial quantity of milk (x) = 100 litres

Quantity replaced (y) = 10% of 100 = 10 litres

Number of times the process is performed (n) = 3

Formula Used:

Final quantity = x × (1 − y/x)ⁿ

Calculations:

⇒ Final Quantity = 100 × (1 − 10/100)³

⇒ Final Quantity = 100 × (1 − 1/10)³

⇒ Final Quantity = 100 × (9/10) × (9/10) × (9/10)

⇒ Final Quantity = 100 × 0.9 × 0.9 × 0.9

⇒ Final Quantity = 100 × 0.729

⇒ Final Quantity = 72.9 litres

∴ The correct answer is 72.9 litres.

Additional Information

Successive Replacement Formula

The formula Final Quantity = Initial × (1 − r/100)ⁿ is used when a fixed percentage r is replaced n times.

Mixture Replacement with Different Quantities

If different amounts a, b, c are replaced each time, the final quantity is x × (1 − a/x) × (1 − b/x) × (1 − c/x).

Ratio of Milk to Water

After the process, the quantity of water = Total Capacity − Final Milk. In this case, 100 − 72.9 = 27.1 litres.

Concept Used:

Follow the BODMAS rule according to the table given below:

Calculation:

(5.96)3 ÷ 17.98 + 47.88 – ? × 1.95 = 19.93

⇒ 6³ ÷ 18 + 48 – ? × 2 = 20

⇒ 216 ÷ 18 + 48 – ? × 2 = 20

⇒ 12 + 48 – ? × 2 = 20

⇒ – ?× 2 = 20 – 60

⇒ -2 × ? = -40

⇒ ? = 20

∴ The value of ? is 20

Solution:

Rules of Approximation:

1. If a number has digits to the right of the decimal less than 5, then just drop the digits to the right of the decimal. The number so obtained will be the approximated value.
2. If a number has digits to the right of the decimal more than 5, then just drop the digits to the right of the decimal and raise the remaining number by '1'.The number so obtained will be the approximated value.

Concept Used:       

Follow BODMAS rule to solve this question, as per the order given below,

Calculation:

351 ÷ 27 + (14)² ÷ 28 = ?

⇒ 351 ÷ 27 + 196 ÷ 28 = ?

⇒ 13 + 7 = ?

∴ ? = 20

Concept used:

Follow BODMAS rule to solve this question, as per the order given below,

Calculations:

{(17.09 + 2.01) × 107.6 ÷ 75.6} + 71.40 × 11.90 = ?

⇒ {(17 + 2) × 108 ÷ 76} + 71 × 12 = ?

⇒ 19 × 108 ÷ 76 + 852 = ?

⇒ 27 + 852 = ?

⇒ ? = 879

∴ The value of ? is 879

Rules of Approximation:

1. If a number has digits to the right of the decimal less than 5, then just drop the digits to the right of the decimal. The number so obtained will be the approximated value.
2. If a number has digits to the right of the decimal more than 5, then just drop the digits to the right of the decimal and raise the remaining number by '1'.The number so obtained will be the approximated value.

Concept Used:       

Follow BODMAS rule to solve this question, as per the order given below,

Calculation:

{(784)^1/2 × 4} + (29 × 13) = (?)² – (8 × 5)

⇒ {28 × 4} + (29 × 13) = (?)² – (8 × 5)

⇒ {28 × 4} + 377 = (?)² – 40

⇒ 112 + 377 = (?)² – 40

⇒ 489 = (?)² – 40

⇒ (?)² = 489 + 40

⇒ (?)² = 529

∴ ? = 23

Concept used:

Follow BODMAS rule to solve this question, as per the order given below:

Calculation:

49.99% of 799.897 + 24.99% of 200.012 = 9 × ?

On taking the approximate values, we get:

⇒ 50% of 800 + 25% of 200 = 9 × ?

⇒ (50/100) × 800 + (25/100) × 200 = 9 × ?

⇒ 400 + 50 = 9 × ?

⇒ ? = 450/9 = 50

∴ The value of ? is 50

Shortcut Trick

For two equal successive discounts of X%, the ratio of Marked Price (MP) to Selling Price (SP) is (100)² : (100 − X)².

Given MP = 2500 and SP = 1764 ⇒ √2500 : √1764 = 50 : 42.

Discount X% = (50 − 42) ÷ 50 × 100 = 16%.

New discount rate = 16% + 2% = 18%.

New SP = 2500 × (82 ÷ 100) × (82 ÷ 100) = 25 × 0.82 × 82 = 1681.

∴ The correct answer is Rs 1681.

Alternate Method

Formulas: SP = MP × (1 − d₁/100) × (1 − d₂/100) | Net Discount = a + b − ab/100

⇒ 1764 = 2500 × (1 − X/100)²

⇒ (1 − X/100)² = 1764 ÷ 2500 = 0.7056

⇒ 1 − X/100 = √0.7056 = 0.84 ⇒ X/100 = 0.16 ⇒ X = 16%

⇒ New Discount rate = X + 2 = 18%

⇒ New SP = 2500 × (1 − 18/100)² = 2500 × (0.82)²

⇒ New SP = 2500 × 0.6724 = 1681

∴ The correct answer is Rs 1681.

Additional Information

Successive Discounts

When two discounts 'a' and 'b' are given one after another, the single equivalent discount is (a + b − ab/100)%.

Relation between MP and SP

SP is always calculated on the Marked Price. SP = MP × (100 − D%)/100.

Shortcut Trick

Net change % = x + y + (x × y)/100 where x = −20% (reduction) and y = +80% (increase).

m = −20 + 80 + (−20 × 80)/100 = 60 − 16 = 44%.

Required value = √(m + 5) = √(44 + 5) = √49 = 7.

∴ The correct answer is 7.

Alternate Method

Given:

Initial Price = 100, Initial Sale = 100 ⇒ Initial Revenue = 100 × 100 = 10000

Calculations:

⇒ New Price = 100 − 20% of 100 = 80

⇒ New Sale = 100 + 80% of 100 = 180

⇒ New Revenue = 80 × 180 = 14400

⇒ Increase in Revenue = 14400 − 10000 = 4400

⇒ Percentage increase (m) = (4400 ÷ 10000) × 100 = 44%

⇒ Value to find = √(m + 5) = √(44 + 5) = √49 = 7

∴ The correct answer is 7.

Additional Information

Successive Percentage Change

When a quantity is changed by x% and then by y%, the overall change is given by [x + y + (xy/100)]%.

Revenue Relationship

Revenue is the product of Price and Consumption (or Sale). If both change, the product change follows the successive percentage rule.

Percentage to Fraction

20% reduction = 4/5 multiplier; 80% increase = 9/5 multiplier. Net multiplier = (4/5) × (9/5) = 36/25. Increase = 11/25 = 44%.

General solution:

Bank A:

Number of persons invested in (Fixed deposit + Mutual Funds) = 600

Number of persons invested in Fixed deposit = 400

Number of persons invested in Mutual Funds = (600 - 400) =200

 

Bank B:

Number of persons invested in (Fixed deposit + Mutual Funds) = 450

Number of persons invested in Fixed deposit = 300

Number of persons invested in Mutual Funds = (450 - 300) =150

 

Bank C:

Number of persons invested in (Fixed deposit + Mutual Funds) = 750

Number of persons invested in Fixed deposit = 500

Number of persons invested in Mutual Funds = (750 - 500) =250

 

Bank D:

Number of persons invested in (Fixed deposit + Mutual Funds) = 700

Number of persons invested in Fixed deposit = 250

Number of persons invested in Mutual Funds = (700 - 250) = 450

 

Bank E:

Number of persons invested in (Fixed deposit + Mutual Funds) = 650

Number of persons invested in Fixed deposit = 350

Number of persons invested in Mutual Funds = (650 - 350) = 300 

 

Solution:

Number of persons invested in Fixed deposition Bank A = 400

Number of females invested in Fixed deposit in Bank A= 400 ×  20/100 = 80

Number of males invested in Fixed deposit in Bank A = (400 - 80) = 320

Number of males invested in Mutual funds in Bank A = 200 ×  (30/100) = 60

Total males invested in both Mutual funds and Fixed deposit in Bank A = (320 + 60) = 380

General solution:

Bank A:

Number of persons invested in (Fixed deposit + Mutual Funds) = 600

Number of persons invested in Fixed deposit = 400

Number of persons invested in Mutual Funds = (600 - 400) =200

 

Bank B:

Number of persons invested in (Fixed deposit + Mutual Funds) = 450

Number of persons invested in Fixed deposit = 300

Number of persons invested in Mutual Funds = (450 - 300) =150

 

Bank C:

Number of persons invested in (Fixed deposit + Mutual Funds) = 750

Number of persons invested in Fixed deposit = 500

Number of persons invested in Mutual Funds = (750 - 500) =250

 

Bank D:

Number of persons invested in (Fixed deposit + Mutual Funds) = 700

Number of persons invested in Fixed deposit = 250

Number of persons invested in Mutual Funds = (700 - 250) = 450

 

Bank E:

Number of persons invested in (Fixed deposit + Mutual Funds) = 650

Number of persons invested in Fixed deposit = 350

Number of persons invested in Mutual Funds = (650 - 350) = 300 

 

Solution:

Number of persons invested in Fixed deposit in Bank B = 300

Number of persons invested in Fixed deposit in Bank C = 500

Number of persons invested in Fixed deposit in Bank B for 2 years = 300 ×  1/3 = 100

Number of persons invested in Fixed deposit in Bank B for 5 years = (300 - 100) = 200

Number of persons invested in Fixed deposit in Bank C for 2 years = 500 ×  1/5 = 100

Number of persons invested in Fixed deposit in Bank C for 5 years = (500 - 100) = 400

Total number of persons invested in Mutual funds in Bank B and C = (150 + 250) = 400

Required difference = (600 - 400) = 200

General solution:

Bank A:

Number of persons invested in (Fixed deposit + Mutual Funds) = 600

Number of persons invested in Fixed deposit = 400

Number of persons invested in Mutual Funds = (600 - 400) =200

 

Bank B:

Number of persons invested in (Fixed deposit + Mutual Funds) = 450

Number of persons invested in Fixed deposit = 300

Number of persons invested in Mutual Funds = (450 - 300) =150

 

Bank C:

Number of persons invested in (Fixed deposit + Mutual Funds) = 750

Number of persons invested in Fixed deposit = 500

Number of persons invested in Mutual Funds = (750 - 500) =250

 

Bank D:

Number of persons invested in (Fixed deposit + Mutual Funds) = 700

Number of persons invested in Fixed deposit = 250

Number of persons invested in Mutual Funds = (700 - 250) = 450

 

Bank E:

Number of persons invested in (Fixed deposit + Mutual Funds) = 650

Number of persons invested in Fixed deposit = 350

Number of persons invested in Mutual Funds = (650 - 350) = 300 

 

Solution:

Number of persons invested in Fixed deposit from Bank D = 250

Number of persons invested  in Mutual deposits from Bank D = 450

Number of persons invested in Bank D is increased by x in Fixed deposit then the ratio becomes 13: 18

(250 + x)/(450) = 13/18

(4500 +18x) = 5850

1350/18 = x

x = 75

General solution:

Bank A:

Number of persons invested in (Fixed deposit + Mutual Funds) = 600

Number of persons invested in Fixed deposit = 400

Number of persons invested in Mutual Funds = (600 - 400) =200

 

Bank B:

Number of persons invested in (Fixed deposit + Mutual Funds) = 450

Number of persons invested in Fixed deposit = 300

Number of persons invested in Mutual Funds = (450 - 300) =150

 

Bank C:

Number of persons invested in (Fixed deposit + Mutual Funds) = 750

Number of persons invested in Fixed deposit = 500

Number of persons invested in Mutual Funds = (750 - 500) =250

 

Bank D:

Number of persons invested in (Fixed deposit + Mutual Funds) = 700

Number of persons invested in Fixed deposit = 250

Number of persons invested in Mutual Funds = (700 - 250) = 450

 

Bank E:

Number of persons invested in (Fixed deposit + Mutual Funds) = 650

Number of persons invested in Fixed deposit = 350

Number of persons invested in Mutual Funds = (650 - 350) = 300 

 

Solution:

 Number of persons invested in Fixed deposits in Bank B = 300

Number of persons invested in Fixed deposits in Bank D = 250

Number of persons invested in Fixed deposits in Bank E = 350

Average number of persons  invested in Fixed deposits in Bank B, D and E = (300 + 250 + 350)/3 = 900/3 = 300

Number of persons invested in Mutual deposits in Bank  A = 200

Number of persons invested in Mutual deposits in Bank C = 250

Number of persons invested in Mutual deposits in Bank E = 300

Average  number of persons invested in Mutual funds in Bank A, C and E = (200 + 250 + 300) = 750/3 = 250

Required difference = (300 - 250) = 50

General solution:

Bank A:

Number of persons invested in (Fixed deposit + Mutual Funds) = 600

Number of persons invested in Fixed deposit = 400

Number of persons invested in Mutual Funds = (600 - 400) =200

 

Bank B:

Number of persons invested in (Fixed deposit + Mutual Funds) = 450

Number of persons invested in Fixed deposit = 300

Number of persons invested in Mutual Funds = (450 - 300) =150

 

Bank C:

Number of persons invested in (Fixed deposit + Mutual Funds) = 750

Number of persons invested in Fixed deposit = 500

Number of persons invested in Mutual Funds = (750 - 500) =250

 

Bank D:

Number of persons invested in (Fixed deposit + Mutual Funds) = 700

Number of persons invested in Fixed deposit = 250

Number of persons invested in Mutual Funds = (700 - 250) = 450

 

Bank E:

Number of persons invested in (Fixed deposit + Mutual Funds) = 650

Number of persons invested in Fixed deposit = 350

Number of persons invested in Mutual Funds = (650 - 350) = 300 

 

Solution:

Number of persons invested in Mutual funds in Bank E = 300

Number of persons invested in Fixed deposits in Bank E = 350

Required ratio = 300: 350 = 6: 7

Calculation

The difference between CI and SI for 2 years at rate r% on principal P:

CI - SI (2 years) = [P × r2 / 1002]

For 3 years:

CI - SI (3 years) = [P × r2 / 1002] × [(300 + r) / 100]

Take ratio:

[ CI - SI for 3 years]/[CI - SI for 2 years] = 25/8

[(300 + r) / 100] ​= 25/8​

So, 2400 + 8r = 2500

So, ⇒ r = 100 / 8 = 12.5%​

Now rate of interest is 12.5 + 7.5 = 20%

Successive CI for 2 years at 20% per annum is 44%

So, required interest amount is 3200 × 44 /100 = 1408

Calculation

Volume of cylinder = 9240 m³

Radius of cylinder = r+4

Height of cylinder = 15 m

Volume of cylinder = πr2h

Or, π(r + 4)2 × 15 = 9240

Or, ⇒ r + 4 = √196​ = 14

⇒ r = 14 – 4 =10​

Side of cube = 2r – 4 = 2 × 10 – 4 = 16

Volume = s³ =163 = 4096 m³

Radius = r + 11 = 10 + 11 = 21 m

Volume = [4/3] π r³ = [4/3] × [22/7] × 213 = 38808 m³

Total Volume (Cube + Sphere) = 4096 + 38808 = 42904 m³

Given:

Cost of cap = ₹140

Cost of sunglass = ₹110

Discount on cap = 20%

Discount on sunglass = 30%

Overall profit = 20%

Formula used:

Marked price (M) = Cost price / (1 - Discount%)

Overall profit = (Total selling price - Total cost price) / Total cost price

Calculations:

Let the marked price of both cap and sunglass be M.

Selling price of cap = M × (1 - 0.20) = 0.80M

Selling price of sunglass = M × (1 - 0.30) = 0.70M

Overall profit = 20% ⇒ (Total selling price - Total cost price) / Total cost price = 0.20

⇒ (0.80M + 0.70M - 140 - 110) / (140 + 110) = 0.20

⇒ (1.50M - 250) / 250 = 0.20

⇒ 1.50M - 250 = 50

⇒ 1.50M = 300

⇒ M = 300 / 1.50 = 200

∴ The marked price of the sunglass is ₹200.

Common Calculation

Shop A produced = 120

Shop A sold = 2× unsold

Let unsold = x ⇒ sold = 2x

So, total = x + 2x = 3x ⇒ 3x = 120 ⇒ x = 40, sold = 80

Shop B:

Produced = A + 20 = 120 + 20 = 140

Unsold in B = 25% more than unsold in A = 40 + 25% of 40 = 40 + 10 = 50

Sold = 140 - 50 = 90

Shop C:

Produced = 33.33% less than A = 120 - (1/3 × 120) = 120 - 40 = 80

Now:

Total unsold in A + B = 40 + 50 = 90

Sold in C = (2/3) × 90 = 60

So, Unsold = 80 - 60 = 20

Shop D:

Unsold = 2 × unsold in C = 2 × 20 = 40

Produced = 12.5% more than produced in C = 80 + 12.5% of 80 = 80 + 10 = 90

Sold = 90 - 40 = 50

Calculation

In shop A,  Gadget : Plastic = 5 : 3 in 120 items

Total parts = 5 + 3 = 8

Gadget items = (5/8) × 120 = 75

Plastic items = (3/8) × 120 = 45

Plastic sold = 12 ⇒ Plastic unsold = 45 - 12 = 33

Total unsold in A = 40

⇒ Gadget unsold = 40 - 33 = 7

Common Calculation

Shop A produced = 120

Shop A sold = 2× unsold

Let unsold = x ⇒ sold = 2x

So, total = x + 2x = 3x ⇒ 3x = 120 ⇒ x = 40, sold = 80

Shop B:

Produced = A + 20 = 120 + 20 = 140

Unsold in B = 25% more than unsold in A = 40 + 25% of 40 = 40 + 10 = 50

Sold = 140 - 50 = 90

Shop C:

Produced = 33.33% less than A = 120 - (1/3 × 120) = 120 - 40 = 80

Now:

Total unsold in A + B = 40 + 50 = 90

Sold in C = (2/3) × 90 = 60

So, Unsold = 80 - 60 = 20

Shop D:

Unsold = 2 × unsold in C = 2 × 20 = 40

Produced = 12.5% more than produced in C = 80 + 12.5% of 80 = 80 + 10 = 90

Sold = 90 - 40 = 50

Calculation

Given:

Number of items sold by Shop B = 90 (from the table)

Number of items sold by Shop C = 60 (from the table)

Number of items sold by Shop F = (2/3) × (Total items sold by Shop B and C together)

Shop F sold 50% of the items it produced

Calculations:

Total items sold by Shop B and C together = 90 + 60 = 150

Number of items sold by Shop F = (2/3) × 150 = 100

Since Shop F sold 50% of the items it produced, let the total number of items produced by Shop F be x:

50% of x = 100

⇒ x = 100 × 2 = 200

The total number of items produced in Shop F is 200.