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Shortcut Trick

Find the factors of the product 45 that add up to the sum 14:

Factors of 45: 9 × 5 = 45

Sum of these factors: 9 + 5 = 14

Therefore, the two positive numbers are 9 and 5.

Their positive difference is: 9 − 5 = 4

∴ The correct answer is 4.

Alternate Method

Given:

Sum of two numbers (a + b) = 14

Product of two numbers (ab) = 45

A = 9

Shortcut Trick

The time difference of 42 minutes occurs only during the 98 km segment of the journey.

Let the original speed be s km/hr. Equation: (98 ÷ s) − (98 ÷ (s + 7)) = 42/60 = 0.7 hours.

Testing the provided answer 28 km/hr from the options:

(98 ÷ 28) − (98 ÷ 35) = 3.5 − 2.8 = 0.7 hours.

The condition is satisfied, thus the original speed is 28 km/hr.

∴ The correct answer is 28 km/hr.

Alternate Method

Given:

Total distance from Town A to Town B = 182 km

First segment distance = 84 km

Second segment distance = 98 km

Time saved = 42 minutes = 42 ÷ 60 hours = 0.7 hours

Formula Used:

Time = Distance ÷ Speed

Calculations:

⇒ Let the original speed of the cyclist be s km/hr.

⇒ Hypothetical time to cover the entire distance at original speed = 182 ÷ s

⇒ Actual time taken = (84 ÷ s) + (98 ÷ (s + 7))

⇒ According to the question: (182 ÷ s) − [(84 ÷ s) + (98 ÷ (s + 7))] = 0.7

⇒ (182 − 84) ÷ s − 98 ÷ (s + 7) = 0.7

⇒ 98 ÷ s − 98 ÷ (s + 7) = 0.7

⇒ 98 [ (s + 7 − s) ÷ (s(s + 7)) ] = 0.7

⇒ (98 × 7) ÷ (s² + 7s) = 0.7

⇒ 686 ÷ 0.7 = s² + 7s

⇒ s² + 7s − 980 = 0

⇒ s² + 35s − 28s − 980 = 0

⇒ s(s + 35) − 28(s + 35) = 0

⇒ (s − 28)(s + 35) = 0

⇒ Since speed cannot be negative, s = 28.

∴ The correct answer is 28 km/hr.

Additional Information

Speed-Distance-Time Relation

The fundamental relation is Speed = Distance ÷ Time. For a constant distance, speed is inversely proportional to time.

Average Speed

For any journey, Average Speed is calculated as the Total Distance divided by the Total Time taken for all segments.

Quadratic Equations in Motion

Problems involving speed changes and time savings frequently result in quadratic equations of the form s² + as − k = 0, where s is the speed.

Shortcut Trick

Apply the divisibility rule of 9: the sum of the digits must be a multiple of 9.

Sum of digits of 95061 = 9 + 5 + 0 + 6 + 1 = 21.

The nearest multiples of 9 to 21 are 18 (21 − 3) and 27 (21 + 6).

Subtracting 3 from the number: 95061 − 3 = 95058.

Adding 6 to the number: 95061 + 6 = 95067. Since 3 < 6, 95058 is the nearest.

∴ The correct answer is 95058.

Alternate Method

Given: Target number = 95061, Divisor = 9.

⇒ Divide 95061 by 9 to find the remainder.

⇒ 95061 ÷ 9 = 10562 with a remainder of 3.

⇒ Lower multiple = 95061 − 3 = 95058.

⇒ Higher multiple = 95061 + (9 − 3) = 95061 + 6 = 95067.

⇒ Distance to 95058 = |95061 − 95058| = 3.

⇒ Distance to 95067 = |95067 − 95061| = 6.

⇒ Comparing distances: 3 < 6.

∴ The correct answer is 95058.

Additional Information

Divisibility Rule of 9

A number is divisible by 9 if and only if the sum of its digits is divisible by 9. For example, in 95058, 9+5+0+5+8 = 27, which is divisible by 9.

Nearest Number Logic

To find the nearest number divisible by n, calculate the remainder r when the number N is divided by n. Compare r and (n − r). The smaller value determines whether to subtract or add.

Divisibility Rule of 3

Similar to the rule for 9, a number is divisible by 3 if the sum of its digits is divisible by 3.

Given:

\(\dfrac{(3^2 \times 9)^3}{27^2 \times 3^4}\) = ?

Formula used:

1. (a^m)ⁿ = a^{m × n}
2. a^m × aⁿ = a^{m + n}
3. a^m / aⁿ = a^{m − n}

Calculations:

To Simplify the expression, convert all terms to Base 3:

9 = 3²

27 = 3³

Substituting these values in the Numerator and Denominator:

⇒ \(\dfrac{(3^2 \times 3^2)^3}{(3^3)^2 \times 3^4}\)

Using the Laws of Exponents:

⇒ \(\dfrac{(3^{2+2})^3}{3^{3 \times 2} \times 3^4}\)

⇒ \(\dfrac{(3^4)^3}{3^6 \times 3^4}\)

⇒ \(\dfrac{3^{4 \times 3}}{3^{6+4}}\)

⇒ \(\dfrac{3^{12}}{3^{10}}\)

⇒ 3^{12 − 10}

⇒ 3²

⇒ 9

∴ The correct answer is option (1).

Shortcut Trick

If every number in a data set is decreased by a constant value k, then the average of the set also decreases by the same value k.

Given Old Average = 43 and Decrease (k) = 7.

New Average = Old Average − k

⇒ 43 − 7 = 36

∴ The correct answer is 36.

Alternate Method

Given:

Total numbers = 8

Original average = 43

Decrease in each number = 7

Formula Used:

Average = Sum ÷ Total Numbers

New Average = (Original Sum − Total Decrease) ÷ Total Numbers

Calculations:

⇒ Original Sum = 8 × 43 = 344

⇒ Total Decrease = 8 × 7 = 56

⇒ New Sum = 344 − 56 = 288

⇒ New Average = 288 ÷ 8 = 36

∴ The correct answer is 36.

Additional Information

Multiplication and Division Rule

If each observation is multiplied or divided by a non-zero constant a, the new average becomes the old average multiplied or divided by a.

Addition Rule

If a constant value k is added to every number in a series, the average of that series also increases exactly by k.

Average of Consecutive Numbers

For any arithmetic progression, the average is simply the middle term or (First term + Last term) ÷ 2.

Shortcut Trick

Effective rate for compounded half-yearly is 8% ÷ 2 = 4% per half-year.

Ratio of Principal to Amount for 4% (1/25) for 2 half-years = 25² : 26² = 625 : 676.

Given 676 units = ₹10,816 ⇒ 1 unit = 10816 ÷ 676 = ₹16.

Principal = 625 units = 625 × 16 = ₹10,000.

∴ The correct answer is ₹10,000.

Alternate Method

Given:

Total Amount (A) = ₹10,816

Annual Rate of Interest (R) = 8% per annum

Time (T) = 1 year

Compounding: Half-yearly

Formula Used: A = P(1 + r/100)ⁿ

Where r = Annual Rate ÷ 2 and n = Time in years × 2.

Calculations:

⇒ Half-yearly rate (r) = 8 ÷ 2 = 4%

⇒ Number of half-years (n) = 1 × 2 = 2

⇒ 10816 = P(1 + 4/100)²

⇒ 10816 = P(1 + 1/25)²

⇒ 10816 = P(26/25)²

⇒ 10816 = P × (676 ÷ 625)

⇒ P = (10816 × 625) ÷ 676

⇒ P = 16 × 625

⇒ P = ₹10,000

∴ The correct answer is ₹10,000.

Additional Information

Effective Rate of Interest

For two successive growth periods of r%, the effective rate R_eff = r + r + (r × r)/100. In this case, 4 + 4 + (4 × 4)/100 = 8.16%.

Quarterly Compounding

If interest is compounded quarterly, the rate becomes R/4 and the number of periods becomes T × 4.

Compound Interest vs Simple Interest

For 2 periods, the difference between CI and SI is P(r/100)², where r is the rate per period.

Shortcut Trick

Present sum of ages = Sum after 6 years − (Number of people × Years passed)

Present sum = 60 − (2 × 6) = 48 years

Ratio of A : B is 5 : 7, so total parts = 12

12 parts = 48 ⇒ 1 part = 4

A’s present age = 5 parts = 5 × 4 = 20 years

∴ The correct answer is 20 years.

Alternate Method

Given: Present ratio of ages of A and B = 5 : 7. Sum of ages after 6 years = 60 years.

Let the present ages of A and B be 5x and 7x respectively.

⇒ Age of A after 6 years = 5x + 6

⇒ Age of B after 6 years = 7x + 6

⇒ (5x + 6) + (7x + 6) = 60

⇒ 12x + 12 = 60

⇒ 12x = 60 − 12

⇒ 12x = 48

⇒ x = 4

⇒ A’s present age = 5x = 5 × 4 = 20

∴ The correct answer is 20 years.

Additional Information

Sum of Ages in the Future/Past

If the sum of ages of 'n' persons is S now, then after 't' years, the sum will be S + (n × t). Similarly, 't' years ago, it was S − (n × t).

Ratio of Ages

If the ages are in ratio a : b, their actual values can be taken as ax and bx. The difference between their ages (b−a)x remains constant regardless of time.

Proportional Division

To find A's share from total sum S with ratio A:B = a:b, use the formula: A = [a ÷ (a + b)] × S.

Shortcut Trick

Let the total amount be T. The sum of money remains constant regardless of transfers.

Condition 1: Juhi has 2 parts out of 3 (total 2+1=3) ⇒ Juhi's amount = (2/3)T + 30 (since she gave 30).

Condition 2: Juhi has 4 parts out of 5 (total 4+1=5) ⇒ Juhi's amount = (4/5)T − 10 (since she received 10).

Equating Juhi's original amount: (4/5)T − 10 = (2/3)T + 30 ⇒ (4/5 − 2/3)T = 40.

(2/15)T = 40 ⇒ T = 20 × 15 = 300.

∴ The correct answer is ₹300.

Alternate Method

Given: Juhi (J) and Mamta (M) exchange money in two scenarios. Scenario 1: J gives ₹30 to M. New Ratio J:M = 2:1. Scenario 2: M gives ₹10 to J. New Ratio J:M = 4:1.

Step-by-step Calculation:

⇒ From scenario 1: (J − 30) = 2(M + 30)

⇒ J − 30 = 2M + 60 ⇒ J − 2M = 90 —— (Equation 1)

⇒ From scenario 2: (J + 10) = 4(M − 10)

⇒ J + 10 = 4M − 40 ⇒ J − 4M = −50 —— (Equation 2)

⇒ Subtracting (Equation 2) from (Equation 1): (J − 2M) − (J − 4M) = 90 − (−50)

⇒ 2M = 140 ⇒ M = 70

⇒ Substituting M = 70 in Equation 1: J − 2(70) = 90 ⇒ J − 140 = 90 ⇒ J = 230

⇒ Total initial amount = J + M = 230 + 70 = 300

∴ The correct answer is ₹300.

Additional Information

Simultaneous Linear Equations

Used to find the values of two variables when two distinct conditions are provided. The method of elimination or substitution is typically employed.

Constant Sum Property

When money is transferred internally between two individuals, the sum of their assets (total money) remains unchanged. This allows for quick ratio comparisons.

Ratio Balancing

If the total units in two ratios differ (e.g., 3 units and 5 units), they can be equated by multiplying each ratio by the total of the other to find a common denominator.

Shortcut Trick

The ratio of height to shadow is given as √3 : 1.

In trigonometry, the tangent of the angle of elevation (θ) is the ratio of height (perpendicular) to shadow (base).

⇒ tan θ = √3 / 1 = √3.

Since we know that tan 60° = √3, the value of θ must be 60°.

∴ The correct answer is 60°.

Alternate Method

Given: Ratio of the height of the tower to the length of its shadow = √3 : 1.

Formula Used: tan θ = Perpendicular (Height) ÷ Base (Shadow).

⇒ Let the height of the tower (AB) be √3x.

⇒ Let the length of the shadow (BC) be 1x.

⇒ Let the angle of elevation of the sun be θ.

⇒ In right-angled triangle ABC, tan θ = AB ÷ BC

⇒ tan θ = √3x ÷ 1x

⇒ tan θ = √3

⇒ tan θ = tan 60°

⇒ θ = 60°

∴ The correct answer is 60°.

Additional Information

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when the object being viewed is above the horizontal level.

Standard Trigonometric Ratios

For height-shadow problems, remember: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Sun Altitude Relation

As the angle of elevation of the sun increases (from 0° to 90°), the length of the shadow of a fixed object decreases.

Shortcut Trick

The difference between the direct profit percentage and the first profit percentage equals Q's profit relative to P's Cost Price.

Difference in profit = 25% − 12% = 13%

This 13% of CP is equal to Q's profit of ₹26.

13% of Cost Price = ₹26 ⇒ 1% = ₹2

100% (Cost Price) = 2 × 100 = ₹200

∴ The correct answer is ₹200.

Alternate Method

Given:

Profit by P selling to Q = 12%

Profit by Q selling to R = ₹26

Direct profit if P sells to R at same price = 25%

Calculations:

⇒ Let the Cost Price (CP) of the item for P be ₹x.

⇒ Selling Price for P (which is CP for Q) = x + 12% of x = 1.12x

⇒ Selling Price for Q (which is the price R pays) = 1.12x + 26

⇒ According to the condition, if P sells to R at 1.12x + 26, profit = 25%.

⇒ 1.25x = 1.12x + 26

⇒ 1.25x − 1.12x = 26

⇒ 0.13x = 26

⇒ x = 26 ÷ 0.13 = 2600 ÷ 13

⇒ x = 200

∴ The correct answer is ₹200.

Additional Information

Successive Profit/Loss

When an item is sold through multiple hands, the effective profit is calculated by multiplying the factors: Final Price = CP × (1 + r₁/100) × (1 + r₂/100)...

Profit Percentage Formula

Profit % = [(Selling Price − Cost Price) ÷ Cost Price] × 100.

Difference between Two Profits

If SP₁ and SP₂ are selling prices at profit % p₁ and p₂ respectively, then SP₂ − SP₁ = (p₂ − p₁)% of Cost Price.