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Given:

He drives for 4 hours at the speed of 75 kmph

He drives for 6 hours at the speed of 90 kmph

Concept:

Average speed = Total distance / Total time

Calculation:

Total distance = 4 hours ×  75 kmph + 6 hours ×  90 kmph = 300 km + 540 km = 840 km

Total time = 4 hours + 6 hours = 10 hours

 Average speed = 840 km / 10 hours = 84 kmph

Therefore, the average speed of the car during the entire journey is 84 kmph.

Given:

An article is sold at ₹800 by giving the article at 20% discount.

If the article is sold without giving any discount the seller earned 25% profit.

Formula used:

(1) Selling Price = (100% - D%) of Marked Price

Where, 

D, is the value of the discount per cent

(2) Cost price = \(\frac{\text {Selling Price}}{100\text {%} + \text {Profit %}}\)

Calculation:

In the first case:

The selling price of the article = ₹800

The discount (in per cent) given on the article = 20%

The marked price of the article = \(\frac{₹800}{100\text {%} - 20\text {%}}\) = \(\frac{₹800}{0.8}\) = ₹1000

In the second case:

The article is sold at 0% discount.

The marked price of the article = ₹1000

The selling price of the article = ₹1000

The profit per cent earned = 25% 

The cost price of the article =  \(\frac{₹1000}{\text {100%+25%}}\) = ₹800

∴ The cost price of the article is ₹800.

Shortcut Trick

Present average of Husband, Wife, and Child = 26 + 3 = 29 years

Present total age of Husband, Wife, and Child = 29 × 3 = 87 years

Present average of Wife and Child = 20 + 5 = 25 years

Present total age of Wife and Child = 25 × 2 = 50 years

Present age of Husband = 87 − 50 = 37 years

∴ The correct answer is 37 years.

Alternate Method

Given:

Average age of Husband (H), Wife (W), and Child (C) 3 years ago = 26 years

Average age of Wife (W) and Child (C) 5 years ago = 20 years

Formula: Total Age = Average Age × Number of Persons

⇒ Total age of (H + W + C) 3 years ago = 26 × 3 = 78 years ⇒ Present total age of (H + W + C) = 78 + (3 persons × 3 years) = 78 + 9 = 87 years ⇒ Total age of (W + C) 5 years ago = 20 × 2 = 40 years ⇒ Present total age of (W + C) = 40 + (2 persons × 5 years) = 40 + 10 = 50 years ⇒ Present age of Husband = (Present total of H, W, C) − (Present total of W, C) ⇒ Present age of Husband = 87 − 50 = 37 years

∴ The correct answer is 37 years.

Additional Information

Effect of Time on Average Age

If the present average age of a group of n people is A, then after k years, their average age will be A + k, and k years ago, it was A − k.

Sum of Ages

The total sum of ages of n people increases by n × k after k years, as each individual's age increases by k.

Finding Age of a New Member

Age of new member = (New Average × New Strength) − (Old Average × Old Strength).

Shortcut Trick

For n consecutive even or odd numbers, the difference between the largest and smallest number is 2 × (n − 1).

Number of terms (n) = 7

Difference = 2 × (7 − 1)

Difference = 2 × 6 = 12

∴ The correct answer is 12.

Alternate Method

Given: Average of 7 consecutive even numbers = 36

Formula Used: Average of consecutive terms in Arithmetic Progression (AP) = Middle Term (if n is odd)

30

32

34

36 Average

38

40

42

Difference = 42 − 30 = 12

⇒ Let the seven consecutive even numbers be n₁, n₂, n₃, n₄, n₅, n₆, n₇.

⇒ Since there are 7 terms (odd), the middle term n₄ is the average.

⇒ n₄ = 36

⇒ Consecutive even numbers have a common difference of 2.

⇒ Lowest number (n₁) = n₄ − (3 × 2) = 36 − 6 = 30

⇒ Highest number (n₇) = n₄ + (3 × 2) = 36 + 6 = 42

⇒ Difference = 42 − 30 = 12

∴ The correct answer is 12.

Additional Information

Average of AP

For any arithmetic progression, Average = (First Term + Last Term) ÷ 2.

Sum of Consecutive Even Numbers

The sum of first n even numbers is given by n × (n + 1).

General Difference Property

In any AP with n terms and common difference d, the difference between the n^th term and the 1^st term is always (n − 1) × d.

Calculation:

Average consumption of electricity over 5 months

⇒ (165 + 150 + 190 + 230 + 275)/5

⇒ 202 units

Average variable cost

⇒ 202 × 3.5 = Rs. 707

Now, the average monthly electricity charge for all the five months

⇒ 707 + 180 = Rs. 887

∴ The average monthly electricity charge for all five months is Rs. 887.

Given:

P = Rs.8800

R = 20/3%

T = 9 months

Formula Used:

I = PRT/100

I = Simple Interest, P = Principal amount,

R = Rate of Interest, T = Time Period

Calculation:

I = PRT/100 = (8800 × 20 × 9)/(100 × 3 × 12)

⇒ I = 88 × 5 = 440

∴ Simple Interest is Rs.440

Given:

Investment of A = Rs. 10000, after 4 months of A's investment B joined.

Profit is same for both.

Formula used:

Investment × time = profit

Calculations:

Let the profit be 1

Let the investment of B = Rs. x

Time taken by A = 12 months = 1year

Time taken by B = (12 - 4) = 8 months

So, A : B = 10000 × 12 : x × 8

In terms of profit, A : B = \(\dfrac{10000 × 12}{x × 8}\) : \(\dfrac{1}{1}\)

x = \(\dfrac{10000 × 12}{8}\)

x = 5000 × 3 = Rs. 15000

∴ The answer is Rs. 15000

Shortcut Trick

Winning margin % = (1000 ÷ 10000) × 100 = 10%.

In a two-candidate election, Loser's % = (100% − Margin %) ÷ 2.

Candidate B's % = (100% − 10%) ÷ 2

Candidate B's % = 90% ÷ 2 = 45%.

∴ The correct answer is 45%.

Alternate Method

Given:

Total number of votes cast = 10000

Winning margin of Candidate A = 1000 votes

Calculations:

⇒ Let the number of votes acquired by candidate B be x.

⇒ Since candidate A won by 1000 votes, votes acquired by A = x + 1000

⇒ Total votes = Votes of A + Votes of B

⇒ x + x + 1000 = 10000

⇒ 2x + 1000 = 10000

⇒ 2x = 9000

⇒ x = 4500

⇒ Votes acquired by candidate B = 4500

⇒ Percentage of votes acquired by B = (4500 ÷ 10000) × 100

⇒ Percentage = 45%

∴ The correct answer is 45%.

Additional Information

Winning Margin Percentage

Winning margin % is the exact difference between the percentage of the winner and the loser: Margin % = Winner % − Loser %.

Valid vs Invalid Votes

In many election problems, a certain percentage of votes are declared invalid. The winning margin and percentages are always calculated based on valid votes unless stated otherwise.

Successive Percentage Rules

If an electorate has X voters, out of which y% didn't vote and z% votes were invalid, Valid Votes = X × (1 − y/100) × (1 − z/100).

Shortcut Trick

Speed in m/s = 76 × 5/18 = 190/9 m/s

Total Distance = Speed × Time = (190/9) × 27 = 570 m

Train Length = Total Distance − Platform Length

Train Length = 570 − 450 = 120 m

∴ The correct answer is 120 m.

Alternate Method

Given:

Speed of the train = 76 km/h

Length of the platform = 450 m

Time taken to cross = 27 sec

Formula Used:

Total Distance = Speed × Time

When a train crosses a platform, Total Distance = Length of Train + Length of Platform.

Calculations:

⇒ Speed in m/s = 76 × (5/18) m/s

⇒ Speed = 190/9 m/s

⇒ Total Distance = (Length of Train + 450) m

⇒ Length of Train + 450 = Speed × Time

⇒ Length of Train + 450 = (190/9) × 27

⇒ Length of Train + 450 = 190 × 3

⇒ Length of Train + 450 = 570

⇒ Length of Train = 570 − 450

⇒ Length of Train = 120 m

∴ The correct answer is 120 m.

Additional Information

Unit Conversion

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

Train Crossing a Point Object

When a train crosses a pole, tree, or a stationary person, the total distance traveled is equal to the length of the train itself.

Relative Speed

When two objects move in the opposite direction, relative speed = S₁ + S₂. In the same direction, relative speed = S₁ − S₂.

Let's identify the second, fourth, fifth, and sixth letters of the word "HOCKEY":

Second letter: O, Fourth letter: K, Fifth letter: E, Sixth letter: Y

So, the letters we have are: O, K, E, and Y.

Now, let's see if we can form meaningful English words with these letters:

OKEY - Not a standard English word.

YOKE - A meaningful English word (a wooden crosspiece fastened over the necks of two animals and attached to the plow or cart that they are to pull).

KEYO - Not a standard English word.

Therefore, only one meaningful English word can be formed.

Hence, the correct answer is "Option 4".