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Shortcut Trick

Total work (LCM of 20, 25) = 100 units.

Efficiency: A = 5 units/day, B = 4 units/day.

Together in 5 days = (5 + 4) × 5 = 45 units.

Remaining work = 100 − 45 = 55 units.

Time for A = 55 ÷ 5 = 11 days.

∴ The correct answer is 11 days.

Alternate Method

Given:

Time taken by A = 20 days

Time taken by B = 25 days

Work duration together = 5 days

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Total Work = LCM(20, 25) = 100 units

⇒ Efficiency of A = 100 ÷ 20 = 5 units/day

⇒ Efficiency of B = 100 ÷ 25 = 4 units/day

⇒ Combined Efficiency of A + B = 5 + 4 = 9 units/day

⇒ Work completed in 5 days = 9 × 5 = 45 units

⇒ Remaining Work = 100 − 45 = 55 units

⇒ Extra time taken by A = Remaining Work ÷ Efficiency of A

⇒ Time = 55 ÷ 5 = 11 days

∴ The correct answer is 11 days.

Additional Information

Pipes and Cisterns

Similar to Time & Work, but includes negative efficiency for outlet pipes that empty the tank.

Work and Wages

Wages are distributed in the ratio of the actual work done by each person, which is proportional to their efficiencies if they work for the same duration.

Alternate Day Work

Calculate work done in one full cycle (e.g., 2 days) and then divide the total work by the cycle efficiency.

Shortcut Trick

The third proportional (c) for two numbers a and b is calculated using the formula: c = b² ÷ a.

Given: a = 6 and b = 12.

Substituting the values: c = 12² ÷ 6

c = 144 ÷ 6 = 24.

∴ The correct answer is 24.

Alternate Method

Given: First number (a) = 6, Second number (b) = 12.

Formula Used: If three numbers a, b, and c are in continued proportion, then a : b :: b : c or a/b = b/c.

⇒ Let the third proportional be c.

⇒ According to the definition: 6 : 12 :: 12 : c

⇒ 6 ÷ 12 = 12 ÷ c

⇒ 1 ÷ 2 = 12 ÷ c

⇒ c = 12 × 2

⇒ c = 24.

∴ The correct answer is 24.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is given by the square root of their product: Mean = √(a × b).

Fourth Proportional

For four numbers a, b, c, and d to be in proportion (a : b :: c : d), the fourth proportional d is calculated as (b × c) ÷ a.

Continued Proportion

A series of numbers are in continued proportion if the ratio of any term to the next term is constant (a/b = b/c = c/d ...).

Shortcut Trick

Initial sum of ratios: 5 + 7 = 12 units; Final sum: 7 + 5 = 12 units.

Since the total marbles remain constant, compare the ratio parts directly.

Change in Bag 2: 7 units → 5 units = 2 units decrease.

Given transfer = 12 marbles ⇒ 2 units = 12 marbles.

1 unit = 6 marbles; Original Bag 2 = 7 units × 6 = 42 marbles.

∴ The correct answer is 42 marbles.

Alternate Method

Given:

Initial ratio of marbles in Bag 1 and Bag 2 = 5 : 7

Number of marbles transferred from Bag 2 to Bag 1 = 12

Final ratio of marbles = 7 : 5

Formula Used:

If original values are ax and bx, and k is transferred: (ax + k) / (bx − k) = m / n

Calculations:

⇒ Let the original number of marbles be 5x and 7x.

⇒ According to the question: (5x + 12) / (7x − 12) = 7 / 5

⇒ 5 × (5x + 12) = 7 × (7x − 12)

⇒ 25x + 60 = 49x − 84

⇒ 49x − 25x = 60 + 84

⇒ 24x = 144

⇒ x = 144 ÷ 24 = 6

⇒ Original marbles in Bag 2 = 7x = 7 × 6 = 42

∴ The correct answer is 42 marbles.

Additional Information

Ratio and Constant Sum

When items are transferred between two bags, the total quantity remains unchanged. Scaling ratios to have the same sum allows for direct comparison of units.

Inverse Proportion

If the product of two variables is constant, they are in inverse proportion. In this case, the total count is fixed, so an increase in one bag must result in a decrease in the other.

Proportionality Constant (x)

The constant x converts a ratio into actual values. Multiplying each term of the ratio by x gives the real quantity of the objects involved.

Shortcut Trick

The diagonal of a square always acts as the angle bisector of the 90° vertex angle.

Therefore, the angle between the diagonal and any adjacent side is exactly 45°.

Since an extension of a straight line does not change its orientation, the acute angle formed between the extension of the diagonal and the adjacent side remains 45°.

In standard geometry, the angle between two intersecting lines is typically measured as the acute angle unless specified otherwise.

∴ The correct answer is 45 degrees.

Alternate Method

Given:

A square ABCD where the diagonal AC is extended by 6 units to point E.

Formula Used:

1. Interior angle of a square = 90°
2. Angle between the diagonal and side of a square = 45°
3. Angles on a straight line (linear pair) sum to 180°

Calculations:

⇒ In square ABCD, the interior angle ∠BCD = 90°.

⇒ The diagonal AC bisects ∠BCD, so ∠ACB = 45° and ∠ACD = 45°.

⇒ Line ACE is a straight line because it is an extension of the diagonal.

⇒ The adjacent side to the extension at vertex C is either BC or CD.

⇒ The angle between the extension CE and side BC is ∠BCE.

⇒ Since ∠ACB + ∠BCE = 180° (Linear Pair), then ∠BCE = 180° − 45° = 135°.

⇒ In geometry, the angle between two lines is often considered the acute angle.

⇒ Acute angle = 180° − 135° = 45°.

∴ The correct answer is 45 degrees.

Additional Information

Diagonal of a Square

The diagonal of a square with side s is given by d = s√2 and always bisects the vertex angles into 45° each.

Linear Pair of Angles

Angles on a straight line that share a common vertex and side sum up to 180°, forming a supplementary pair.

Angle between Lines

When two lines intersect, they form two pairs of angles; the smaller angle (0° to 90°) is typically defined as the angle between the lines.

Shortcut Trick

Total sum of ages after 4 years = 76 + (2 × 4) = 84 years.

Ratio of Mother to Daughter after 4 years = 3 : 1.

Total units = 3 + 1 = 4 units → 84 years.

1 unit = 84 ÷ 4 = 21 years.

Mother's age after 4 years = 3 × 21 = 63 years.

Present age of Mother = 63 − 4 = 59 years.

∴ The correct answer is 59 years.

Alternate Method

Given:

Sum of present ages = 76 years.

Ratio after 4 years = 3 : 1 (Mother : Daughter).

Let the present age of the Mother be M and the Daughter be D.

⇒ M + D = 76 --- (Eq. 1)

After 4 years, ages will be (M + 4) and (D + 4).

⇒ (M + 4) = 3 × (D + 4)

⇒ M + 4 = 3D + 12

⇒ M − 3D = 8 --- (Eq. 2)

Subtracting (Eq. 2) from (Eq. 1):

⇒ (M + D) − (M − 3D) = 76 − 8

⇒ 4D = 68

⇒ D = 17 years

Substituting D in (Eq. 1):

⇒ M + 17 = 76

⇒ M = 76 − 17 = 59 years

∴ The correct answer is 59 years.

Additional Information

Age Sum Concept

If the sum of ages of 'n' people is S, then after 't' years, the new sum will be S + (n × t).

Constant Age Difference

The difference between the ages of two individuals always remains constant, regardless of the number of years passed.

Ratio Method for Ages

When a ratio is given for a future or past time, calculate the sum or difference for that specific time to use the unitary method effectively.

Shortcut Trick

In an isosceles triangle with equal sides a, base b, and altitude h, the relationship is (2a − b) = 4h² ÷ Perimeter.

Given h = 12√2 and Perimeter (2a + b) = 48.

⇒ (2a − b) = (4 × 288) ÷ 48 = 1152 ÷ 48 = 24.

⇒ Base b = (48 − 24) ÷ 2 = 12 cm.

⇒ Area = 1/2 × base × height = 1/2 × 12 × 12√2 = 72√2 cm².

∴ The correct answer is 72√2 cm².

Alternate Method

Given:

Altitude (h) = 12√2 cm

Perimeter (P) = 48 cm

Formula Used:

For an isosceles triangle, Perimeter = 2a + b and a² = h² + (b/2)², where a is the equal side and b is the base.

Calculations:

⇒ a² − (b/2)² = h²

⇒ a² − b²/4 = (12√2)²

⇒ (4a² − b²) / 4 = 144 × 2

⇒ (2a − b)(2a + b) = 1152

Substitute Perimeter (2a + b) = 48:

⇒ (2a − b) × 48 = 1152

⇒ 2a − b = 1152 ÷ 48 = 24

Now, we have two equations: 2a + b = 48 and 2a − b = 24.

⇒ (2a + b) − (2a − b) = 48 − 24

⇒ 2b = 24 ⇒ b = 12 cm

⇒ Area = 1/2 × base × height = 1/2 × 12 × 12√2

⇒ Area = 6 × 12√2 = 72√2 cm²

∴ The correct answer is 72√2 cm².

Additional Information

Properties of Isosceles Triangle

The altitude drawn from the vertex of the equal sides to the base bisects the base and the vertex angle.

Area Formula

The area of an isosceles triangle can also be calculated as (b/4) × √(4a² − b²), where a is the length of the equal sides and b is the base.

Pythagorean Relation

In an isosceles triangle, the hypotenuse of the right-angled triangle formed by the altitude is the equal side (a), and the legs are the altitude (h) and half the base (b/2).

Shortcut Trick

Net Profit % = x + y + (xy ÷ 100)

Net Profit % = 10 + 15 + (10 × 15 ÷ 100) = 25 + 1.5 = 26.5%

Final Selling Price = 200 × (100 + 26.5)%

Final Selling Price = 200 × 1.265 = ₹253

∴ The correct answer is ₹253.

Alternate Method

Given: Cost Price (CP) of the chair = ₹200, First Profit = 10%, Second Profit = 15%

Formula: Selling Price (SP) = CP × (100 + Profit%) ÷ 100

⇒ First Selling Price (SP₁) = 200 + (10% of 200)

⇒ SP₁ = 200 + (10 ÷ 100 × 200) = 200 + 20 = ₹220

⇒ This SP₁ becomes the Cost Price for the buyer.

⇒ Final Selling Price (SP₂) = 220 + (15% of 220)

⇒ SP₂ = 220 + (15 ÷ 100 × 220) = 220 + 33 = ₹253

∴ The correct answer is ₹253.

Additional Information

Successive Percentage Change

When a value is changed by x% and then by y%, the net effective percentage change is given by [x + y + (xy ÷ 100)]%.

Profit and Loss Formulas

Profit = Selling Price − Cost Price. Profit percentage is always calculated on the Cost Price (CP) unless stated otherwise.

Successive Discounts

If two successive discounts d₁% and d₂% are offered, the equivalent single discount is [d₁ + d₂ − (d₁ × d₂ ÷ 100)]%.

Shortcut Trick

Total Distance = 40 + (40 × 3) = 40 + 120 = 160 km

Total Time = (40 ÷ 20) + 3 = 2 + 3 = 5 hours

Average Speed = Total Distance ÷ Total Time = 160 ÷ 5 = 32 km/h

∴ The correct answer is 32 km/h.

Alternate Method

Given:

Case 1: Distance₁ = 40 km, Speed₁ = 20 km/h

Case 2: Speed₂ = 40 km/h, Time₂ = 3 hours

Formula:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed | Distance = Speed × Time

Calculations:

⇒ Time₁ = Distance₁ ÷ Speed₁ = 40 ÷ 20 = 2 hours

⇒ Distance₂ = Speed₂ × Time₂ = 40 × 3 = 120 km

⇒ Total Distance = Distance₁ + Distance₂ = 40 + 120 = 160 km

⇒ Total Time = Time₁ + Time₂ = 2 + 3 = 5 hours

⇒ Average Speed = 160 ÷ 5 = 32 km/h

∴ The correct answer is 32 km/h.

Additional Information

Average Speed for Equal Distances

If a person covers two equal distances at speeds x and y, then the average speed = 2xy ÷ (x + y).

Average Speed for Equal Time Intervals

If a person travels for equal time intervals at speeds x and y, then the average speed = (x + y) ÷ 2.

Unit Conversion

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

Shortcut Trick

If every number in a set is decreased by a constant value k, the new average is simply the old average minus k.

Here, Old Average = 45 and Constant Decrease (k) = 4.

New Average = 45 − 4 = 41.

∴ The correct answer is 41.

Alternate Method

Given: Total numbers = 8, Average = 45, Decrease per number = 4.

Formula Used: Average = Sum of observations ÷ Total number of observations

⇒ Original Sum = Average × Count = 45 × 8 = 360

⇒ Total decrease across 8 numbers = 4 × 8 = 32

⇒ New Sum = 360 − 32 = 328

⇒ New Average = 328 ÷ 8

⇒ New Average = 41

∴ The correct answer is 41.

Additional Information

Effect of Operations on Average

If each term is multiplied by x, the average is multiplied by x. Similarly, if each term is divided by x (x ≠ 0), the average is divided by x.

Arithmetic Mean Formula

Mean (&xmacr;) = (Σx_i) ÷ n, where Σx_i is the sum of observations and n is the total number of observations.

Sum of First n Natural Numbers

The average of the first n natural numbers is (n + 1) ÷ 2.

Shortcut Trick

In an Average problem, the sum of deviations from the mean must be zero.

Deviations from 15: (7−15) = −8, (11−15) = −4, (15−15) = 0, (23−15) = +8.

Net deviation of known numbers = −8 + (−4) + 0 + 8 = −4.

To balance the Average, the deviation of x must be +4.

x = 15 + 4 = 19.

∴ The correct answer is 19.

Alternate Method

Given: Numbers = 7, 11, 15, x, 23 | Total count (n) = 5 | Average = 15

Formula Used: Average = (Sum of all observations) ÷ (Total number of observations)

⇒ Total Sum = Average × Number of terms

⇒ 7 + 11 + 15 + x + 23 = 15 × 5

⇒ 56 + x = 75

⇒ x = 75 − 56

⇒ x = 19

∴ The correct answer is 19.

Additional Information

Arithmetic Mean Definition

The mean is the sum of all data points divided by the count of data points: Mean = (Σx_i) ÷ n.

Properties of Deviations

The algebraic sum of deviations of a set of observations from their Arithmetic Mean is always zero.

Average of AP

If numbers are in an Arithmetic Progression (AP), the Average is equal to the middle term (for odd n) or the mean of the two middle terms (for even n).