Quantitative Aptitude PYQs

To solve questions of this type, follow the laws of “Surds and indices’’ given below -

Laws of Indices:

1. a^m × aⁿ = a^{{m + n}}
2. a^m ÷ aⁿ = a^{{m - n}}
3. (a^m)ⁿ = a^mn
4. (a)^-m = 1/a^m
5. (a)^m/n = ⁿ√a^m
6. (a)⁰ = 1

By using this laws:

⇒ (216)^7/3 × 64² × 81^5/4 = 3^-7 × 6^?

⇒ (6³)^7/3 × (2⁶)² × (3⁴)^5/4 = 3^-7 × 6^?

⇒ 6⁷ × 2¹² × 3⁵ = 3^-7 × 6^?

⇒ 6⁷ × 2¹² × 3⁵ = 3^-7 × 6^?

⇒ 6⁷ × 2¹² × 3⁵ ÷ 3^-7 = 6^?

⇒ 6⁷ × 2¹² × 3¹² = 6⁷

⇒ 6⁷ × 6¹² = 6^?

⇒ ? = 19

[3132 ÷ 29 × 48] ÷ 18 = 12 × ?

⇒ (3132/29) × 48 × ( 1/18) = 12 × ?

⇒ (108 × 48)/18 = 12 × ?

⇒ 12 × ? = 6 × 48

⇒ ? = 24

∴ ? = 24.

⇒ 76% of 395 + 48% of 135 =?

⇒ ? = 76/100 × 395 + 48/100 × 135

⇒ ? = 300.2 + 64.8

⇒ ? = 365

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,

⇒ 3 + {3 + (4 - 1 × 2)} × 4 = 759 ÷ ?

⇒ 3 + {3 + (4 - 2)} × 4 = 759 ÷ ?

⇒ 3 + {3 + 2} × 4 = 759 ÷ ?

⇒ 3 + 5 × 4 = 759 ÷ ?

⇒ 3 + 20 = 759 ÷ ?

⇒ 23 = 759 ÷ ?

⇒ ? = 759/23 = 33

⇒ ? = 33

Given expression is,

⇒ 12% of (75% of (25% of (8% of 4800))) = ?

\(\begin{array}{l} \Rightarrow \frac{{12}}{{100}} \times \left( {\frac{{75}}{{100}} \times \left( {\frac{{25}}{{100}} \times \left( {\frac{8}{{100}} \times 4800} \right)} \right)} \right) = ;?\ \Rightarrow \frac{3}{{25}} \times \left( {\frac{3}{4} \times \left( {\frac{1}{4} \times \left( {48 \times 8} \right)} \right)} \right) = ;?\ \Rightarrow \frac{3}{{25}} \times \left( {\frac{3}{4} \times 96} \right) = ;?\ \Rightarrow \frac{3}{{25}} \times 3 \times 24 = ;? \end{array}\)

⇒ ? = 216/25 = 8.64

⇒ ? = 8.64

Shortcut Trick

When Simple Interest is a fraction (1/n) of Principal and Time (T) equals Rate (R).

Formula: Rate (R) = √(Fraction × 100)

R = √(1/16 × 100) = √(100/16)

R = 10/4 = 2.5%

∴ The correct answer is 2.5%.

Alternate Method

Given:

Simple Interest (SI) = 1/16 × Principal (P)

Number of years (T) = Rate per annum (R)

Formula Used:

SI = (P × R × T) ÷ 100

Calculations:

⇒ P/16 = (P × R × R) ÷ 100

⇒ 1/16 = R² ÷ 100

⇒ R² = 100 ÷ 16

⇒ R = √(100/16)

⇒ R = 10/4

⇒ R = 2.5%

∴ The correct answer is 2.5%.

Additional Information

Simple Interest Formula

The basic formula is SI = (P × R × T) / 100, where P is Principal, R is Rate, and T is Time.

Case when R = T

If numerical values of Rate and Time are equal, then R = T = 10 × √(SI/P).

Amount Calculation

Total Amount (A) is the sum of Principal and Simple Interest: A = P + SI.

New mines in A = 150 + 20% of 150 = 150 + 30 = 180

New mines in D = 300 − 10% of 300 = 300 − 30 = 270

Ratio (A:D) = 180 : 270 = 2 : 3

Hence, the correct answer is 2 : 3.

Original mines in C = 200

If doubled ⇒ New mines in C = 200 × 2 = 400

New total mines = 150 + 250 + 400 + 300 = 1100

Required percentage = (400 ÷ 1100) × 100 = 36.36%

Hence, the correct answer is 36.36%.

Average number of mines in Place A and B = (150 + 250) ÷ 2 = 400 ÷ 2 = 200

Average number of mines in Place C and D = (200 + 300) ÷ 2 = 500 ÷ 2 = 250

Difference = 250 − 200 = 50

Hence, the correct answer is 50.

Mines in Place A = 150

Mines in Place B = 250

Difference = 250 − 150 = 100

Percentage increase = (Difference ÷ Place A) × 100

= (100 ÷ 150) × 100 = 66.67%

Hence, the correct answer is 66.67%.

Mines in Place A = 15 hundreds = 150

Mines in Place C = 20 hundreds = 200

Total (A + C) = 150 + 200 = 350

Mines in Place B = 250

Mines in Place D = 300

Total (B + D) = 250 + 300 = 550

Ratio (A + C) : (B + D) = 350 : 550 → 7 : 11

Hence, the correct answer is 7 : 11.

Shortcut Trick

Side of Square = √1764 = 42 cm.

Radius of Semicircle (r) = 42 ÷ 2 = 21 cm.

Perimeter of Semicircle = r × (36 ÷ 7).

Perimeter = 21 × (36 ÷ 7) = 3 × 36 = 108 cm.

∴ The correct answer is 108 cm.

Alternate Method

Given: Area of Square = 1764 cm²; Radius = Side / 2.

Formula Used: Area of Square = a²; Perimeter of Semicircle = πr + 2r.

⇒ Side of Square (a) = √1764

⇒ a = 42 cm

⇒ Radius of Semicircle (r) = 42 ÷ 2 = 21 cm

⇒ Perimeter = (π × r) + (2 × r)

⇒ Perimeter = (22/7 × 21) + (2 × 21)

⇒ Perimeter = 66 + 42 = 108 cm

∴ The correct answer is 108 cm.

Additional Information

Area of Semicircle

The Area is half of the circle's area, given by (πr²) ÷ 2.

Diagonal of a Square

The Diagonal of a square with side 'a' is calculated as a√2.

Simplified Perimeter Constant

The Perimeter of a semicircle can be expressed as r × (36/7) when using π = 22/7.

Shortcut Trick

Profit ratio = (Capital_A × Time_A) : (Capital_B × Time_B)

Ratio of Capitals A : B = 76,000 : 57,000 = 4 : 3

Let B joined for x months. A was there for 12 months.

(4 × 12) : (3 × x) = 2 : 1 ⇒ 48 : 3x = 2 : 1

3x × 2 = 48 × 1 ⇒ 6x = 48 ⇒ x = 8 months.

B joined after = 12 − 8 = 4 months.

∴ The correct answer is 4 months.

Alternate Method

Given → Investment of A = Rs. 76,000, Investment of B = Rs. 57,000, Profit Ratio = 2 : 1, Time for A = 12 months. Formulas: Profit = Investment × Time | Investment₁ × T₁ ÷ Investment₂ × T₂ = P₁ ÷ P₂

⇒ Let B invested for t months. ⇒ (76000 × 12) ÷ (57000 × t) = 2 ÷ 1 ⇒ (4 × 12) ÷ (3 × t) = 2 ÷ 1 ⇒ 48 ÷ 3t = 2 ÷ 1 ⇒ 16 ÷ t = 2 ⇒ t = 8 months. ⇒ Since the business ran for 12 months and B was in for 8 months, B joined after 12 − 8 = 4 months.

∴ The correct answer is 4 months.

Additional Information

Partnership Basics

In a partnership, the profit is distributed based on the weighted investment, which is the product of Capital and Time Period.

Time Calculation

Always distinguish between the duration of investment and the time after which a partner joins. Time joined after = Total time − Duration of investment.

General Solution:

Given:

Total visitors on Monday = 390

Females on Monday = 62.5% of Males on Monday

Males on Tuesday = Females on Monday + 35

Let MMon = males on Monday, FMon = females on Monday.

Calculations:

FMon = 62.5% of MMon = (5/8)·MMon

MMon + FMon = 390 ⇒ MMon + (5/8)MMon = 390

⇒ (13/8)MMon = 390 ⇒ MMon = 390 × 8 / 13 = 240

⇒ FMon = 390 − 240 = 150

Males on Tuesday = FMon + 35 = 150 + 35 = 185

Thus,

Males on Monday = 240

Females on Monday = 150

Males on Tuesday = 185

Calculations:

Original males on Monday = 240

Reduced by 20% ⇒ New males = 240 − 0.2 × 240 = 240 − 48 = 192

Females remain the same = 150

New total visitors = 192 + 150 = 342

Hence, the correct answer is 342.

General Solution:

Given:

Total visitors on Monday = 390

Females on Monday = 62.5% of Males on Monday

Males on Tuesday = Females on Monday + 35

Let MMon = males on Monday, FMon = females on Monday.

Calculations:

FMon = 62.5% of MMon = (5/8)·MMon

MMon + FMon = 390 ⇒ MMon + (5/8)MMon = 390

⇒ (13/8)MMon = 390 ⇒ MMon = 390 × 8 / 13 = 240

⇒ FMon = 390 − 240 = 150

Males on Tuesday = FMon + 35 = 150 + 35 = 185

Thus,

Males on Monday = 240

Females on Monday = 150

Males on Tuesday = 185

Calculations:

Males on Monday = 240

Males on Tuesday = 185

Ratio = Males(Monday) : Males(Tuesday) = 240 : 185 → 48 : 37

Hence, the correct answer is 48 : 37.

General Solution:

Given:

Total visitors on Monday = 390

Females on Monday = 62.5% of Males on Monday

Males on Tuesday = Females on Monday + 35

Let MMon = males on Monday, FMon = females on Monday.

Calculations:

FMon = 62.5% of MMon = (5/8)·MMon

MMon + FMon = 390 ⇒ MMon + (5/8)MMon = 390

⇒ (13/8)MMon = 390 ⇒ MMon = 390 × 8 / 13 = 240

⇒ FMon = 390 − 240 = 150

Males on Tuesday = FMon + 35 = 150 + 35 = 185

Thus,

Males on Monday = 240

Females on Monday = 150

Males on Tuesday = 185

Calculations:

Males on Tuesday = 185

Total visitors on Monday = 390

Percentage = (185 ÷ 390) × 100

= 47.44% (approx)

Hence, the correct answer is 47.44%.

General Solution:

Given:

Total visitors on Monday = 390

Females on Monday = 62.5% of Males on Monday

Males on Tuesday = Females on Monday + 35

Let M_Mon = males on Monday, F_Mon = females on Monday.

Calculations:

F_Mon = 62.5% of M_Mon = (5/8)·M_Mon

M_Mon + F_Mon = 390 ⇒ M_Mon + (5/8)M_Mon = 390

⇒ (13/8)M_Mon = 390 ⇒ M_Mon = 390 × 8 / 13 = 240

⇒ F_Mon = 390 − 240 = 150

Males on Tuesday = F_Mon + 35 = 150 + 35 = 185

Thus,

Males on Monday = 240

Females on Monday = 150

Males on Tuesday = 185

Calculations:

Total visitors on Monday = 390

Total visitors on Monday + Tuesday = 755

Males on Tuesday = 185 

Total visitors on Tuesday = 755 − 390 = 365

Females on Tuesday = Total visitors − Males on Tuesday = 365 − 185 = 180

Hence, the correct answer is 180.

Shortcut Trick

Efficiency ratio E : F = (1 + 1/4) : 1 = 5/4 : 1 = 5 : 4

Total Work = Efficiency of E × Days taken by E = 5 × 15 = 75 units

Combined efficiency of E + F = 5 + 4 = 9 units/day

Days taken together = 75 ÷ 9 = 25/3 days

∴ The correct answer is 25/3 days.

Alternate Method

Given:

Efficiency of E is 1/4 more than F.

Time taken by E = 15 days.

Calculations:

⇒ Let efficiency of F = 4 units/day

⇒ Efficiency of E = 4 + (1/4 × 4) = 5 units/day

⇒ Total Work = Efficiency × Time = 5 × 15 = 75 units

⇒ Time taken by E and F together = Total Work ÷ (Eff_E + Eff_F)

⇒ Time = 75 ÷ (5 + 4) = 75/9 days

⇒ Time = 25/3 days

∴ The correct answer is 25/3 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a work. If the ratio of efficiency is x : y, the ratio of time taken is y :

x. Work Formula

Total Work = Efficiency × Time. Combined time for two people working together is (Total Work) ÷ (Sum of individual efficiencies).

Shortcut Trick

Combine the ratio of A : B : C : D using the common terms.

A : B = 3 : 4 | B : C = 1 : 2 (or 4 : 8) | C : D = 4 : 3 (or 8 : 6).

Combined ratio A : B : C : D = 3 : 4 : 8 : 6.

Given D = 6 units = 72 years → 1 unit = 12 years.

Oldest is C = 8 units = 8 × 12 = 96 years.

∴ The correct answer is 96 yrs.

Alternate Method

Given:

Ratio A : B = 3 : 4

Ratio B : C = 1 : 2

Ratio C : D = 4 : 3

Age of D = 72 years

Calculations:

⇒ To combine, make B same: A : B = 3 : 4, B : C = 4 : 8

⇒ Now make C same: C : D is 4 : 3, but in combined it is 8. So, C : D = 8 : 6

⇒ Combined Ratio A : B : C : D = 3 : 4 : 8 : 6

⇒ Value of D = 6x = 72

⇒ x = 72 ÷ 6 = 12

⇒ Oldest is C (highest ratio part)

⇒ Age of C = 8 × 12 = 96 years

∴ The correct answer is 96 yrs.

Additional Information

Combined Ratio Rule

To combine A:B and B:C, multiply the ratios so that the common term (B) has the same value in both sets.

Ratio & Proportion Basics

If a : b = c : d, then the product of extremes equals the product of means (a × d = b × c).

Age Related Problems

The difference between the ages of two persons remains constant throughout their lives regardless of the time elapsed.

Shortcut Trick

By Alligation Method, the ratio of Solution 1 to Solution 2 is calculated using the water concentrations.

Ratio = (35 − 23) : (23 − 15) = 12 : 8 = 3 : 2

Given: 2 units (Solution 2) = 40 L

⇒ 1 unit = 20 L

⇒ 3 units (Solution 1) = 3 × 20 L = 60 L

∴ The correct answer is 60 L.

Alternate Method

Given:

Water in Solution 1 = 15%

Water in Solution 2 = 35%

Target water concentration = 23%

Quantity of Solution 2 = 40 L

Calculations:

⇒ Let the quantity of Solution 1 be x L.

⇒ By Alligation Rule: (Quantity of 1) / (Quantity of 2) = (35 − 23) / (23 − 15)

⇒ x / 40 = 12 / 8

⇒ x / 40 = 3 / 2

⇒ x = (3 × 40) ÷ 2

⇒ x = 120 ÷ 2 = 60 L

∴ The correct answer is 60 L.

Additional Information

Rule of Alligation

Used to find the ratio in which two ingredients at given prices or concentrations must be mixed to produce a mixture at a desired mean value.

Weighted Average Formula

Mean Concentration = (n₁c₁ + n₂c₂) ÷ (n₁ + n₂), where n is quantity and c is concentration.

Successive Dilution

Final Quantity = Initial Quantity × (1 − x/V)ⁿ, where x is quantity replaced and V is total volume.