Quantitative Aptitude PYQs

Shortcut Trick

Sum of present ages = 66 + (7 × 2) = 80 years.

Total ratio parts = 5 + 3 = 8 units.

8 units = 80 years → 1 unit = 10 years.

Gitika's present age = 3 × 10 = 30 years.

∴ The correct answer is 30 years.

Alternate Method

Given: Ratio of present ages of Gauravi and Gitika = 5 : 3.

Calculation:

⇒ Let the present ages be 5x and 3x.

⇒ Sum of ages 7 years ago = (5x − 7) + (3x − 7) = 66

⇒ 8x − 14 = 66

⇒ 8x = 80

⇒ x = 10

⇒ Gitika's present age = 3x = 3 × 10 = 30 years.

∴ The correct answer is 30 years.

Additional Information

Age Sum Translation

If the sum of ages of 'n' people is S years now, the sum 't' years ago was S − (n × t).

Ratio and Proportion in Ages

The difference between the ages of two people remains constant regardless of the number of years passed.

Future Age Calculation

If current age is X, age after 'n' years will be X + n. Sum of ages after 'n' years for 'm' people increases by m × n.

Shortcut Trick

Ratio of Profit = (Capital₁ × Time₁) : (Capital₂ × Time₂)

Given: Raman (C₁=8000, T₁=12) and Lakhan (C₂=x, T₂=9).

Profit Ratio = (8000 × 12) : (x × 9) = 32 : 21.

Simplifying: 96000 : 9x = 32 : 21 ⇒ 96000 ÷ 9x = 32 ÷ 21.

x = (96000 × 21) ÷ (32 × 9) = 3000 × 7 ÷ 3 = 7000.

∴ The correct answer is Rs. 7000.

Alternate Method

Given → Raman Capital = Rs. 8000, Raman Time = 12 months, Lakhan Time = (12 − 3) = 9 months, Profit Ratio = 32 : 21. Formulas: Profit = Capital × Time | Capital = Profit ÷ Time

⇒ Let Lakhan contribution be x. ⇒ Profit of Raman = 8000 × 12 = 96000 units. ⇒ Profit of Lakhan = x × 9 = 9x units. ⇒ 96000 ÷ 9x = 32 ÷ 21. ⇒ 32 × 9x = 96000 × 21. ⇒ 288x = 2016000. ⇒ x = 2016000 ÷ 288 = 7000.

∴ The correct answer is Rs. 7000.

Additional Information

Partnership Basics

In a partnership, the profit is distributed in the ratio of the product of investment and time duration.

Investment Time

If a person joins 'n' months after the start for a year, their money is invested for (12 − n) months.

Shortcut Trick

Relative Speed in same direction = Difference of speeds = 60 − 35 = 25 kmph.

Time taken to catch = Distance ÷ Relative Speed.

Time = 50 ÷ 25 = 2 hr.

∴ The correct answer is 2 hr.

Alternate Method

Given:

Speed of Fast Car (S₁) = 60 km/hr

Speed of Slow Car (S₂) = 35 km/hr

Initial Distance (D) = 50 km

Formula Used:

Relative Speed (Same Direction) = S₁ − S₂

Time = Distance ÷ Relative Speed

Calculations:

⇒ Relative Speed = 60 km/hr − 35 km/hr = 25 km/hr

⇒ Time taken = 50 km ÷ 25 km/hr

⇒ Time taken = 2 hr

∴ The correct answer is 2 hr.

Additional Information

Relative Speed in Opposite Direction

When two objects move towards each other, their relative speed is the sum of their individual speeds: S_R = S₁ + S₂.

Unit Conversion

To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Average Speed

If a certain distance is covered at S₁ and the same distance at S₂, the average speed is (2 × S₁ × S₂) ÷ (S₁ + S₂).

​Calculation

Team X (2020 + 2021) = 425 + 400 = 825

Team Y (2020 + 2021) = 325 + 350 = 675

Ratio = 825 : 675

Both numbers are divisible by 75. 825 / 75 = 11 and 675 / 75 = 9.

→ 11 : 9

​Calculation

Team Y (2022) = 425, and Team Y (2023) = 275

Team X (2022) = 300 Team X ( 2023) = 475

Difference = [475 + 300] – [ 425 + 275] = 775 – 700 = 75

​Calculation

Team X (2021) = 400, Team Y (2021) = 350

Total = 400 + 350 = 750

Percentage = (400 / 750) × 100 = 53.33%

​Calculation

Difference = 375 – 350 = 25

Percentage = (25 / 350) × 100 = 7.14%

Calculation

Team X (2023) = 475 → 20% increase = 475 + 95 = 570

Team Y (2021) = 350 → 10% decrease = 350 – 35 = 315

Difference = 570 – 315 = 255

Shortcut Trick

Difference in Total Marks = (45 − 40) × 3 = 15 marks.

Since A, B, C and B, C, D are compared, (A − D) = 15.

Given (A + D) = 75.

Solving for A: A = (Sum + Difference) ÷ 2 = (75 + 15) ÷ 2 = 45.

∴ The correct answer is 45 marks.

Alternate Method

Given:

Average of A, B, C = 45 marks

Average of B, C, D = 40 marks

Sum of A and D = 75 marks

Calculations:

⇒ Total marks (A + B + C) = 45 × 3 = 135 --- (Eq 1)

⇒ Total marks (B + C + D) = 40 × 3 = 120 --- (Eq 2)

⇒ Subtracting Eq 2 from Eq 1: (A + B + C) − (B + C + D) = 135 − 120

⇒ A − D = 15 --- (Eq 3)

⇒ Given: A + D = 75 --- (Eq 4)

⇒ Adding Eq 3 and Eq 4: 2A = 90

⇒ A = 90 ÷ 2 = 45

∴ The correct answer is 45 marks.

Additional Information

Average Concept

Average = (Sum of all observations) ÷ (Total number of observations). Sum = Average × Number of terms.

Linear Equations Trick

If x + y = S and x − y = D, then x = (S + D) ÷ 2 and y = (S − D) ÷ 2.

Weighted Average

Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), used when two groups are merged.

Shortcut Trick

Combined efficiency of 2(A + B + C) = Efficiency(A+B) + Efficiency(B+C) + Efficiency(A+C)

Total Work = LCM(18, 24, 36) = 72 units.

Efficiencies: (A+B) = 4, (B+C) = 3, (A+C) = 2. Sum = 9 units.

Efficiency of (A+B+C) = 9 ÷ 2 = 4.5 units.

Time taken = 72 ÷ 4.5 = 16 days.

∴ The correct answer is 16 days.

Alternate Method

Given: Time taken by (A + B) = 18 days, (B + C) = 24 days, and (A + C) = 36 days.

Formula Used: Efficiency = Total Work ÷ Time Taken

⇒ Adding the efficiencies: (A + B) + (B + C) + (A + C) = 4 + 3 + 2

⇒ 2(A + B + C) = 9

⇒ Efficiency of (A + B + C) = 9/2 = 4.5

⇒ Time taken by (A + B + C) = Total Work ÷ Efficiency

⇒ Time = 72 ÷ 4.5 = 16 days

∴ The correct answer is 16 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete the work. If time increases, efficiency decreases.

Individual Efficiency Calculation

To find the efficiency of A alone: Efficiency(A) = Efficiency(A+B+C) − Efficiency(B+C).

Total Work Assumption

Total work is usually taken as the LCM of the number of days to keep efficiency values as integers or simple decimals.

Shortcut Trick

Initial Ratio (W:M) = 3 : 2 (Total = 5 units) | Final Ratio (W:M) = 1 : 2 (Total = 3 units)

Since only Milk is added, Water remains constant. Make Water units same: (3:2) and (1:2)×3 → (3:2) and (3:6).

Milk increase = 6 − 2 = 4 units. This 4 units = 5 liters replaced. So, 1 unit = 1.25 liters.

Total capacity = 3 + 6 = 9 units = 9 × 1.25 = 11.25 liters. Initial Milk = (2/5) × 11.25 = 4.5 liters.

∴ The correct answer is 4.5 liters.

Alternate Method

Given: Initial ratio Water : Milk = 3 : 2. Replaced volume = 5 liters. Final ratio = 1 : 2.

Formula Used: Quantity of liquid = Total volume × (Ratio part / Total parts)

⇒ Let total initial volume be 5x. Water = 3x, Milk = 2x.

⇒ After removing 5L: Water = 3x − 5×(3/5) = 3x − 3; Milk = 2x − 5×(2/5) = 2x − 2.

⇒ Adding 5L pure Milk: New Milk = 2x − 2 + 5 = 2x + 3.

⇒ New Ratio: (3x − 3) / (2x + 3) = 1 / 2

⇒ 2(3x − 3) = 1(2x + 3) ⇒ 6x − 6 = 2x + 3 ⇒ 4x = 9 ⇒ x = 2.25.

⇒ Initial Milk = 2x = 2 × 2.25 = 4.5 liters.

∴ The correct answer is 4.5 liters.

Additional Information

Mixture Replacement Rule

When a portion of mixture is removed, the ratio of components in the remaining mixture stays the same as the initial ratio.

Alligation Method for Replacement

Concentration of milk in original × (Volume left) + Concentration of added milk × (Volume added) = Final concentration × Total Volume.

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

⇒ 987 + 23% of 1500 = ? + 21% of 700

\(\Rightarrow 987 + \frac{{23}}{{100}} \times 1500 = ;? + \frac{{21}}{{100}} \times 700\)

⇒ 987 + 23 × 15 = ? + 21 × 7

⇒ 987 + 345 = ? + 147

⇒ 1332 = ? + 147

⇒ ? = 1332 – 147 = 1185

⇒ ? = 1185

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

Step - 1 - Parts of an equation enclosed in 'Brackets' must be solved first, and in the bracket,

Step - 2 - Any mathematical 'Of' or 'Exponent' must be solved next,

Step - 3 - Next, the parts of the equation that contain 'Division' and 'Multiplication' are calculated,

Step - 4 - Last but not least, the parts of the equation that contain 'Addition' and 'Subtraction' should be calculated.

Given expression is,

⇒ 271 + 85 × 3 – 122 = ? + 36 × 9

⇒ 271 + 255 – 122 = ? + 36 × 9

⇒ 526 – 122 = ? + 324

⇒ ? = 404 – 324

∴ ? = 80

Concept used:

Calculation:

Given expression is,

3 – [19 + {14 – (16 – 3 – 21)}]=?

⇒ 3 – [19 + {14 – (13 – 21)}] =?

⇒ 3 – [19 + {14 – (- 8)}] =?

⇒ 3 – [19 + {14 + 8}] =?

⇒ 3 – [19 + 22] =?

⇒ 3 – [41] =?

⇒ ? = -38

∴ The value of ? is -38

Concept used:

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

The given expression is,

(42 – 23) × 6 – 5 × (4² + 5)

= 19 × 6 – 5 × (16 + 5)

= 19 × 6 – 5 × 21

= 114 – 105

= 9

Concept used:

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

Calculation:

599 – 339 + 459 – 319 = ?2 

599 + 459 – (339 + 319) = ?2

1058 – 658 = 400 = ?2

? = √400 = 20

∴ The value of ?' is 20

Given

The value is 3-2 × 272/3 ÷ (729)-1/3

Calculation

3-2 × 272/3 ÷ (729)^-1/3

⇒ (1/3²) × (3³)^2/3 ÷ (9³)^-1/3

⇒ (1/9) × 32 ÷ 1/9

⇒ (1/9) × 9 × 9

⇒  9

∴ the value of 3-2 × 272/3 ÷ (729)^-1/3 is 9

Given:

[90 ÷ 6 × (24 - 8 ÷ 4) ÷ 3] – 90 ÷ 5 + 34

Concept used:

Follow the BODMAS rule according to the table given below:

Calculation:

Using the BODMAS rule

⇒ [90 ÷ 6 × (24 – 8 ÷ 4) ÷ 3] – 90 ÷ 5 + 34 = ?

⇒ [90 ÷ 6 × (24 – 2) ÷ 3] – 90 ÷ 5 + 34 = ?

⇒ [90 ÷ 6 × (22) ÷ 3] – 90 ÷ 5 + 34 = ?

⇒ [90 ÷ 6 × 22/3] – 90 ÷ 5 + 34 = ?

⇒ [15 × 22/3] – 90 ÷ 5 + 34 = ?

⇒ [110] – 90 ÷ 5 + 34 = ?

⇒ 110 – 18 + 34 = ?

⇒ 110 + 16 = ?

⇒ ? = 126

∴ In place of ‘?’ 126 will come in place of ‘?’

Concept used:

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

Calculation:

⇒ 2487 + 4876 – 5645 + 222 = ?

⇒ 7363 – 5645 + 222 = ?

⇒ 1718 + 222 = ?

⇒ ? = 1940

∴ The value of ? is 1940

Calculations:

Considering the given equation,

(27)2 – (13)2 + (12)3 + (23)2 = ?

⇒ 729 – 169 + 1728 + 529 = ?

⇒ 2817 = ?

∴ The value of ? is 2817.