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Shortcut Trick

Ratio of CP : MP = (100 − Discount%) : (100 + Profit%)

⇒ CP : MP = (100 − 25) : (100 + 25) = 75 : 125 = 3 : 5

Let CP = 300 and MP = 500.

New Selling Price at 10% discount = 500 − 10% of 500 = 450.

Profit% = [(450 − 300) ÷ 300] × 100 = 150/300 × 100 = 50%.

∴ The correct answer is 50%.

Alternate Method

Given: First discount = 25%, First profit = 25%, Second discount = 10%.

Formula: MP/CP = (100 + Profit%) ÷ (100 − Discount%)

⇒ Let Cost Price (CP) = x

⇒ Marked Price (MP) = CP × (100 + 25) ÷ (100 − 25)

⇒ MP = x × (125 ÷ 75) = (5/3)x

⇒ New Selling Price (SP_new) = MP × (100 − 10) ÷ 100

⇒ SP_new = (5/3)x × (90 ÷ 100) = (5/3)x × 0.9

⇒ SP_new = 1.5x

⇒ Profit = SP_new − CP = 1.5x − x = 0.5x

⇒ Profit% = (Profit ÷ CP) × 100 = (0.5x ÷ x) × 100 = 50%

∴ The correct answer is 50%.

Additional Information

CP and MP Relationship

The direct formula connecting Cost Price and Marked Price is MP/CP = (100 + Profit%) / (100 − Discount%).

Successive Discounts

If two discounts d₁ and d₂ are given, the equivalent discount is d₁ + d₂ − (d₁×d₂)/100.

Effective Profit

Profit % is always calculated on the Cost Price unless stated otherwise, while Discount % is calculated on the Marked Price.

Shortcut Trick

Cost Price (CP) of 1 orange = 1000 ÷ 200 = Rs. 5.

To gain 25% profit, Selling Price (SP) of 1 orange = 5 × 1.25 = Rs. 6.25.

Number of oranges for Rs. 100 = Total Amount ÷ SP per orange.

Number of oranges = 100 ÷ 6.25 = 16 oranges.

∴ The correct answer is 16.

Alternate Method

Given: Total CP = Rs. 1000, Total Quantity = 200, Profit% = 25%.

Formulas: SP = CP × (100 + P%) ÷ 100 | Quantity = Total SP ÷ SP per unit.

⇒ CP of 200 oranges = Rs. 1000.

⇒ CP of 1 orange = 1000 ÷ 200 = Rs. 5.

⇒ SP of 1 orange = 5 × (125 ÷ 100) = Rs. 6.25.

⇒ Number of oranges for Rs. 100 = 100 ÷ 6.25 = 16.

∴ The correct answer is 16.

Additional Information

Unitary Method in Profit and Loss

Always calculate the price per unit to simplify calculations when the quantity changes between buying and selling scenarios.

Profit Percentage

Profit is always calculated on the Cost Price (CP) unless stated otherwise. Formula: SP = CP × (1 + Profit%/100).

Given:

The speed of car = 45 km/h

Formula used:

Distance = Speed × Time

Calculation:

According to the question

Time is taken by car = 2 hours 20 min   [10:40 AM to 1 PM]

⇒ 2 + \(\frac{20}{{60}}\) = \(\frac{7}{{3}}\) hours

Now, Distance = Speed × Time

⇒ (45 × \(\frac{7}{{3}}\)) = 105 km

∴ The speed of the car is 105 km.

Given:

Downstream = 32 km

Upstream = 14 km

Time taken to cover each distance = 6 hours

Formula used:

Speed of current = \(\frac{1}{{2}}\)(Downstream - Upstream)

Calculation:

According to the question

Rate of downstream = (\(\frac{32}{{6}}\)) km/hr

Rate of Upstream = (\(\frac{14}{{6}}\)) km/hr

Speed of current = \(\frac{1}{{2}}\)(Downstream - Upstream)

⇒ \(\frac{1}{{2}}\)(\(\frac{32}{{6}}\) - \(\frac{14}{{6}}\))

⇒ (\(\frac{1}{{2}}\) × \(\frac{18}{{6}}\)) 

⇒ \(\frac{3}{{2}}\) = 1.5 km/h

∴ The speed of the current is 1.5 km/h.

Given:

Speed of boy when he is going to school = 12 km/h

Speed of the boy when he returned to his house = 8 km/h

Time = 50 min

Formula used:

Speed = Distance/Time

Calculation:

Let the distance between his school and house be x.

According to the question

Total time taken = Time taken from school to house + Time taken from house to school

⇒ \(\frac{50}{{60}}\) h = \(\frac{x}{{12}}\) + \(\frac{x}{{8}}\)

⇒ \(\frac{50}{{60}}\) h = (\(\frac{2x+3x}{{24}}\))

⇒ \(\frac{50}{{60}}\) h = \(\frac{5x}{{24}}\)

⇒ x = 4 km/h

Now, the total distance walked = 2x = (2 × 4) = 8 km

∴  The total distance walked is 8 km.

Mistake PointsDon't mark option 1 i.e. 4 km/h since it is the distance covered in one way only.

Shortcut Trick

Total work = LCM(15, 20, 25) = 300 units.

Efficiency ratio (A : B : C) = 300/15 : 300/20 : 300/25 = 20 : 15 : 12.

Since they worked together for the entire duration, their wages are distributed directly in the ratio of their efficiencies.

C's share = [12 ÷ (20 + 15 + 12)] × 4700 = (12 ÷ 47) × 4700 = ₹1200

∴ The correct answer is ₹1200.

Alternate Method

Given:

A can finish the work in 15 days.

B can finish the work in 20 days.

C can finish the work in 25 days.

Total wages earned = ₹4700.

Concept Used:

Wages are distributed in proportion to the total work done by each person. If they work together for the same number of days, the wages are divided strictly in the ratio of their daily efficiencies.

Calculation:

⇒ Ratio of their wages = Efficiency of A : Efficiency of B : Efficiency of C

⇒ Wage ratio = 20 : 15 : 12

⇒ Sum of ratio parts = 20 + 15 + 12 = 47 parts

⇒ Total wages = 47 parts = ₹4700

⇒ Value of 1 part = 4700 ÷ 47 = ₹100

⇒ Share of C = 12 parts

⇒ Share of C = 12 × 100 = ₹1200

∴ The correct answer is ₹1200.

Additional Information

Work and Wages

Wages are always distributed in the ratio of the total work done by each individual. If persons work for an equal number of days from start to finish, the wages are divided proportionally to their per-day efficiency.

Leaving Mid-way or Joining Later

If someone leaves the work midway or joins late, their total share is strictly proportional to the actual units of work they have personally completed (Efficiency × Number of days worked).

Pipes and Cisterns

This concept operates identically to Time & Work. Filling pipes have a positive efficiency (similar to completing a task), whereas leakages or emptying pipes have a negative efficiency (similar to destroying work).

Given:

5 tractors can plough 5 hectares of land in 5 days

Concept used:

M₁D₁/W1​ = M₂D₂/W2

Where, M = Number of tractors,

D = Number of days, W = Hectares

Calculation:

Let the number of required tractors be y

According to the above concept,

⇒ M1D₁/W₁​ = M2D₂/W₂

⇒ (\(\frac{5\times 5}{{5}}\)) = (\(\frac{y\times 50}{{100}}\))

⇒ 5 = \(\frac{y}{{2}}\)

⇒ y = 10

∴ The required number of tractors is 10.

Given:

9√2 - √8 - 4√2

Calculation:

9√2 - √8 - 4√2

⇒ 9√2 - 2√2 - 4√2

⇒ 3√2

∴ The required value is 3√2.

Given:

The shadow of a pole 6 m high and 15 m long.

At the same time, the shadow of a tree is 25 m long.

Calculation:

Let the height of the tree be H.

According to the question

⇒ \(\frac{6}{{H}}\) = \(\frac{15}{{25}}\)

⇒ 15H = 150

⇒ H = 10 m

∴ The required height of the tree is 10 m.

Shortcut Trick

Let the two-digit number be x.

Difference between 1/3 and 1/4 of the number is 8.

⇒ (1/3 − 1/4) × x = 8

⇒ (1/12) × x = 8

⇒ x = 96

Sum of digits = 9 + 6 = 15.

∴ The correct answer is 15.

Alternate Method

Given:

One-third of the number = x/3

One-fourth of the number = x/4

Calculations:

⇒ x/3 − x/4 = 8

⇒ (4x − 3x) / 12 = 8

⇒ x / 12 = 8

⇒ x = 8 × 12

⇒ x = 96

The two-digit number is 96.

⇒ Sum of digits = 9 + 6 = 15

∴ The correct answer is 15.

Additional Information

Two-Digit Number Representation

A two-digit number with tens digit 'a' and units digit 'b' is represented as 10a + b.

Sum of Digits

The sum of digits for a number 10a + b is simply a + b.

Fractional Difference

To find the whole when the difference between two fractions is given: Value = Difference ÷ (1/n₁ − 1/n₂).