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Shortcut Trick

The third proportional to any two numbers a and b is directly given by b² ÷ a.

Here, a = 16 and b = 20.

Third proportional = 20² ÷ 16

⇒ 400 ÷ 16 = 25

∴ The correct answer is 25.

Alternate Method

Given:

First number (a) = 16

Second number (b) = 20

Formula Used:

If three numbers a, b, and x are in continued proportion, then a : b :: b : x, which means x = b² ÷ a.

Continued Proportion: Constant Multiplier

× 1.25 × 1.25

16 First

20 Second

x Third

Ratio: 16 : 20 = 20 : x

Calculations:

⇒ Let the third proportional be x.

⇒ 16 : 20 :: 20 : x

⇒ 16 ÷ 20 = 20 ÷ x

⇒ 16 × x = 20 × 20

⇒ 16x = 400

⇒ x = 400 ÷ 16

⇒ x = 25

∴ The correct answer is 25.

Additional Information

Mean Proportional

The mean proportional between two quantities a and b is calculated as √(a × b).

Fourth Proportional

The fourth proportional to three numbers a, b, and c is x such that a : b :: c : x, giving x = (b × c) ÷ a.

Continued Proportion

Three non-zero numbers a, b, and c are in continued proportion if a : b = b : c. Here, b is called the mean proportional and c is the third proportional.

Shortcut Trick

For three numbers a, b, and c, the Fourth Proportional is calculated as (b × c) ÷ a.

Given values are 15, 10, and 12.

Fourth Proportional = (10 × 12) ÷ 15 = 120 ÷ 15 = 8.

∴ The correct answer is 8.

Alternate Method

Given: First term (a) = 15, Second term (b) = 10, Third term (c) = 12.

Formula Used: If a, b, c, d are in proportion, then a : b = c : d or a × d = b × c.

Let the fourth proportional be x.

⇒ 15 : 10 = 12 : x

⇒ 15 / 10 = 12 / x

⇒ 15 × x = 10 × 12

⇒ 15x = 120

⇒ x = 120 ÷ 15

⇒ x = 8

∴ The correct answer is 8.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Third Proportional

The third proportional to two numbers a and b is defined as c such that a : b = b : c, given by c = b² ÷ a.

Compounded Ratio

The compounded ratio of (a : b) and (c : d) is (a × c) : (b × d).

Given:

Simplify the expression: 25 ÷ 5 + 3 of 2 – 4 + 6 of 3 ÷ 2 – 8 + 4

Formula Used:

Order of operations (BODMAS/BIDMAS): Brackets, Orders (i.e., powers and roots, etc.), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right)

Calculation:

Expression: 25 ÷ 5 + 3 of 2 – 4 + 6 of 3 ÷ 2 – 8 + 4

⇒ 5 + 6 – 4 + 18 ÷ 2 – 8 + 4

⇒ 11 – 4 + 9 – 8 + 4

⇒ 12

The correct answer is option 4.

Total work = 96 units (LCM of 24, 16 and 32)

Hence, efficiency of A = 4 units, B = 6 units and C = 3 units

Work done by A and C in 6 days = (4 + 3) × 6 = 42 units

Remaining work = 96 – 42 = 54 units

Remaining work, B complete in = 54/6 = 9 days

Shortcut Trick

The sum of reciprocals of two numbers equals their sum divided by their product.

We know, Product of two numbers = HCF × LCM = 5 × 175 = 875.

Given the sum of the numbers = 60.

Sum of reciprocals = 60 ÷ 875 = 12 / 175.

∴ The correct answer is 12/175.

Alternate Method

Given:

Sum of the two numbers (a + b) = 60

Highest Common Factor (HCF) = 5

Least Common Multiple (LCM) = 175

Formula Used:

Product of two numbers (a × b) = HCF × LCM

Sum of reciprocals = 1/a + 1/b = (a + b) / (a × b)

Sum (a + b) 60 Product (a × b) HCF × LCM 875 1/a + 1/b = (a+b) / ab = 60/875

Calculations:

⇒ Product of numbers (a × b) = 5 × 175 = 875

⇒ Sum of reciprocals = (a + b) / (a × b)

⇒ Sum of reciprocals = 60 / 875

⇒ Dividing numerator and denominator by 5, we get:

⇒ (60 ÷ 5) / (875 ÷ 5) = 12 / 175

∴ The correct answer is 12/175.

Additional Information

Product of Numbers Rule

For any two numbers, the product of the numbers is always equal to the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM). This means a × b = HCF × LCM.

Sum of Reciprocals

The sum of reciprocals of two numbers directly relates to their sum and their product. The generic formula representation is: 1/a + 1/b = (a + b) / (a × b).

Co-prime Numbers

Two numbers are co-prime if their HCF is 1. For such pairs, the LCM is exactly equal to the product of the two numbers.

Given:

Cost of first variety of pulses = Rs. 60 per kg

Cost of second variety of pulses = Rs. 75 per kg

Cost of the mixture = Rs. 65 per kg

Concept:

To find the ratio in which two varieties should be mixed to get a mixture of a given price, we use the rule of alligation.

Formula Used:

If two ingredients are mixed, their ratio is given by:

Ratio = (Price of second ingredient - Price of mixture) : (Price of mixture - Price of first ingredient)

Calculation:

We have,

⇒ Price of first variety = Rs. 60 per kg

⇒ Price of second variety = Rs. 75 per kg

⇒ Price of mixture = Rs. 65 per kg

⇒ Ratio = (75 - 65) : (65 - 60)

⇒ Ratio = 10 : 5

⇒ Ratio = 2 : 1

∴ The grocer must mix the pulses in the ratio 2:1.

Let the initial price of sugar be 100x.

Price of sugar after increased = 100x × [124/100] = 124x

Person wants his expenditure increase by 18% only,

So, his expenditure should be = 100x × [118/100] = 118x

Required percentage = [(124 – 118x)/124] × 100 = 4.8% (approx)

To convert a number into percentage it will be multiplied by 100.

2.4 × 100 = 240%

Additional Information ∶ To convert percentage into a number it will be divided by 100.

240% = 240/100 = 2.4

Shortcut Trick

Net Simple Interest (SI) rate for 2.4 years at 10% = 10% × 2.4 = 24%.

Successive rates for Compound Interest (CI): 10%, 10%, and (10 × 2/5)% = 4%.

Net CI rate = 10 + 10 + (10 × 10)/100 = 21%; then 21 + 4 + (21 × 4)/100 = 25.84%.

Difference in percentage = 25.84% − 24% = 1.84%.

Difference in value = 1.84% of ₹8000 = 80 × 1.84 = ₹147.20.

∴ The correct answer is ₹147.20.

Alternate Method

Given: Principal (P) = ₹8000, Rate (R) = 10% p.a., Time (T) = 2²/₅ years = 2.4 years.

Formula Used: SI = (P × R × T) / 100 | CI Amount (A) = P(1 + R/100)ⁿ

⇒ Simple Interest (SI) = (8000 × 10 × 2.4) / 100

⇒ SI = 80 × 24 = ₹1920

⇒ Compound Interest Amount (A) = 8000 × (1 + 10/100) × (1 + 10/100) × (1 + 4/100)

⇒ A = 8000 × 1.1 × 1.1 × 1.04

⇒ A = 8000 × 1.21 × 1.04 = 9680 × 1.04 = ₹10067.20

⇒ Compound Interest (CI) = 10067.20 − 8000 = ₹2067.20

⇒ Difference = 2067.20 − 1920 = ₹147.20

∴ The correct answer is ₹147.20.

Additional Information

CI and SI Difference (2 Years)

For a period of 2 years, the difference is calculated as: Diff = P(R/100)².

CI and SI Difference (3 Years)

For a period of 3 years, the difference is: Diff = P(R/100)² × (3 + R/100).

Fractional Time Period CI

If time is T = n + a/b years, Amount = P(1 + R/100)ⁿ × (1 + (a/b × R)/100).

Let the length of the train X be l m and speed of train Y be x km/hr

As we know,

Time = Distance/speed

$72 = \frac{{225 + l}}{{\left( {60 - x} \right)\frac{5}{{18}}}}$     …1)

$18 = \frac{{225 + l}}{{\left( {60 + x} \right)\frac{5}{{18}}}}$     …2)

Multiply by 4 in equation (2)

$72 = \frac{{4\left( {225 + l} \right)}}{{\left( {60 + x} \right)\frac{5}{{18}}}}$     …3)

From equation (1) and equation (3)

$\frac{{4\left( {225 + l} \right)}}{{\left( {60 + x} \right)\frac{5}{{18}}}} = \frac{{225 + l}}{{\left( {60 - x} \right)\frac{5}{{18}}}}$

⇒ 4 (60 – x) = (60 + x)

⇒ 240 – 4x = 60 + x

⇒ 4x + x = 240 – 60

⇒ 5x = 180

⇒ x = 36

∴ Speed of train Y is 36 km/hr

From equation (1)

$72 = \frac{{225 + l}}{{\left( {60 - 36} \right)\frac{5}{{18}}}}$

⇒ 72 × 24 × 5/18 = 225 + l

⇒ 480 = 225 + l

⇒ l = 480 – 225

⇒ l = 255 m

∴ Length of train X is 255 m.