Step 1: Understanding the Concept:

Profit occurs when the Selling Price (SP) is higher than the Cost Price (CP).

The percentage is calculated by comparing the profit made to the original investment (CP).


Step 2: Key Formula or Approach:

The profit percentage formula is:
\[ Profit % = \left( \frac{Selling Price - Cost Price}{Cost Price} \right) \times 100 \]


Step 3: Detailed Explanation:

Given:

Cost Price (\( CP \)) =
(450

Selling Price (\( SP \)) =
)540



First, determine the absolute profit:
\[ Profit = SP - CP = 540 - 450 = 90 \]

Now, apply the profit percentage formula:
\[ Profit % = \left( \frac{90}{450} \right) \times 100 \]
\[ Profit % = \frac{1}{5} \times 100 = 20% \]


Step 4: Final Answer:

The profit percentage is 20%.
Quick Tip: Always reduce the fraction \( \frac{Profit}{CP} \) to its simplest form before multiplying by 100.
For example, \( \frac{90}{450} \) easily simplifies to \( \frac{1}{5} \), and \( \frac{1}{5} \) is a standard fraction for 20%.

Step 1: Understanding the Concept:

A ratio represents the proportional distribution of a total sum among different entities.


Step 2: Key Formula or Approach:

Let the ages be \( 3x \) and \( 4x \).

The formula for the sum is: \( (a + b)x = Total Sum \).


Step 3: Detailed Explanation:

Let A's age be \( 3x \) and B's age be \( 4x \).

According to the problem:
\[ 3x + 4x = 28 \]
\[ 7x = 28 \]
\[ x = \frac{28}{7} = 4 \]

Now, find the age of B:
\[ B's age = 4x = 4 \times 4 = 16 years. \]


Step 4: Final Answer:

The age of B is 16 years.
Quick Tip: To find a single person's share directly:
\[ Value = \frac{Their Ratio}{Sum of Ratios} \times Total \]
B's age = \( \frac{4}{7} \times 28 = 16 \).

Step 1: Understanding the Concept:

Speed is the rate of travel per unit time. To convert km/h to m/s, specific conversion factors must be applied.


Step 2: Key Formula or Approach:

1. \( Speed = \frac{Distance}{Time} \)

2. To convert km/h \(\rightarrow\) m/s, multiply by \( \frac{5}{18} \).


Step 3: Detailed Explanation:

Step 1: Calculate initial speed in km/h.
\[ Speed = \frac{120 km}{3 hours} = 40 km/h \]

Step 2: Convert to meters per second.
\[ Speed in m/s = 40 \times \frac{5}{18} \]
\[ Speed in m/s = \frac{200}{18} = \frac{100}{9} \approx 11.11 m/s. \]


Step 4: Final Answer:

The speed is 11.11 m/s.
Quick Tip: Always remember the 5/18 factor. It comes from:
\[ \frac{1000 meters}{3600 seconds} = \frac{5}{18} \]
Knowing this derivation helps if you ever forget the specific numbers.

Step 1: Understanding the Concept:

Simple interest is the interest calculated only on the principal amount for the total duration.


Step 2: Key Formula or Approach:
\[ S.I. = \frac{P \times R \times T}{100} \]


Step 3: Detailed Explanation:

Given:

Principal (\( P \)) =
(2000

Rate (\( R \)) = 5%

Time (\( T \)) = 3 years



Substituting the values:
\[ S.I. = \frac{2000 \times 5 \times 3}{100} \]
\[ S.I. = 20 \times 15 = 300 \].


Step 4: Final Answer:

The simple interest is
)300.
Quick Tip: Think of simple interest as a "Percentage of Principal":
Total % = \( R \times T = 5 \times 3 = 15% \).
15% of 2000 = 300.

Step 1: Understanding the Concept:

In a number series, the relationship between consecutive terms can be found by examining their differences.


Step 2: Key Formula or Approach:

Identify the "Difference of Differences" or the mathematical progression pattern.


Step 3: Detailed Explanation:

Looking at the differences between consecutive terms:
\( 6 - 2 = 4 \)
\( 12 - 6 = 6 \)
\( 20 - 12 = 8 \)
\( 30 - 20 = 10 \)

The sequence of differences is 4, 6, 8, 10.

These differences are increasing by 2 at each step.

The next difference should be \( 10 + 2 = 12 \).

The next term = \( 30 + 12 = 42 \).


Step 4: Final Answer:

The next number in the series is 42.
Quick Tip: This series follows the pattern \( n^2 + n \) where \( n \) starts from 1:
\( 1^2 + 1 = 2 \), \( 2^2 + 2 = 6 \), \( 3^2 + 3 = 12 \), \( 4^2 + 4 = 20 \), \( 5^2 + 5 = 30 \).
Next is \( 6^2 + 6 = 36 + 6 = 42 \).

Step 1: Understanding the Concept:

Verbs must agree with the timeline of the sentence. Words like "yesterday" indicate that the action is completed in the past.


Step 2: Key Formula or Approach:

For completed actions in the past with a specific time mention, use the Simple Past Tense (\( V_2 \)).


Step 3: Detailed Explanation:

1. go: Present tense (base form).

2. goes: Present tense (singular subject).

3. gone: Past participle (requires "has/have/had").

4. went: Simple past form of 'go'.

5. going: Present participle (continuous).

Since "yesterday" is used, "went" is the only grammatically correct option.


Step 4: Final Answer:

The correct word is "went".
Quick Tip: Whenever you see "yesterday", "last week", or "ago", look for the second form of the verb (\( V_2 \)) in the options.

Step 1: Understanding the Concept:

Area of a circle is the total flat surface contained within its perimeter.


Step 2: Key Formula or Approach:
\[ Area = \pi r^2 \]


Step 3: Detailed Explanation:

Given: \( r = 7 cm \) and \( \pi = \frac{22}{7} \).

Substitute these values into the area formula:
\[ Area = \frac{22}{7} \times 7 \times 7 \]

Cancel the \( 7 \) in the denominator with one \( 7 \) in the numerator:
\[ Area = 22 \times 7 = 154 sq cm. \]


Step 4: Final Answer:

The area is 154 sq cm.
Quick Tip: For radius 7 cm:
Circumference = 44 cm.
Area = 154 sq cm.
Memorizing this pair helps with many geometry problems.

Step 1: Understanding the Concept:

Successive percentage changes (markup and discount) do not sum up linearly.


Step 2: Key Formula or Approach:

Net Change = \( a + b + \frac{ab}{100} \)

(Markup \( a \) is positive, Discount \( b \) is negative).


Step 3: Detailed Explanation:

Let markup \( a = +40 \) and discount \( b = -20 \).
\[ Net change = 40 - 20 + \frac{40 \times (-20)}{100} \]
\[ Net change = 20 - \frac{800}{100} \]
\[ Net change = 20 - 8 = 12% \].


Step 4: Final Answer:

The overall profit is 12%.
Quick Tip: Assume CP = 100.
MP = 140 (40% increase).
Discount = 20% of 140 = 28.
SP = \( 140 - 28 = 112 \).
Profit = \( 112 - 100 = 12% \).

Step 1: Understanding the Concept:

Work capacity is inversely proportional to the time taken. More workers mean fewer days.


Step 2: Key Formula or Approach:

Use the "Man-Days" constant formula:
\[ M_1 \times D_1 = M_2 \times D_2 \]


Step 3: Detailed Explanation:

Given: \( M_1 = 15, D_1 = 20, M_2 = 25 \).

Find \( D_2 \):
\[ 15 \times 20 = 25 \times D_2 \]
\[ 300 = 25 \times D_2 \]
\[ D_2 = \frac{300}{25} = 12 days. \]


Step 4: Final Answer:

The task will be completed in 12 days.
Quick Tip: You can also solve by ratios:
Ratio of men = 15:25 = 3:5.
Ratio of days (inverse) = 5:3.
If 5 units = 20 days, then 1 unit = 4.
So, 3 units = 12 days.

Step 1: Understanding the Concept:

A change in average weight during replacement is caused by the difference between the weight of the incoming person and the outgoing person.


Step 2: Key Formula or Approach:
\[ New person's weight = Old weight + (Number of people \times Increase in Average) \]


Step 3: Detailed Explanation:

Old weight = 65 kg.

Total people = 8.

Average increase = 2.5 kg.

Total weight increase = \( 8 \times 2.5 = 20 kg. \)

New person's weight = \( 65 + 20 = 85 kg. \)


Step 4: Final Answer:

The new person weighs 85 kg.
Quick Tip: If the average had decreased, you would subtract the total weight difference from the old person's weight.

Step 1: Understanding the Concept:

Blood relation logic requires substituting relationships relative to the speaker.


Step 2: Key Formula or Approach:

Break down the complex phrase "my father's son" while accounting for the speaker's siblings.


Step 3: Detailed Explanation:

1. "I have no brother or sister" \(\rightarrow\) This means the speaker is the "only child."

2. "My father's son" \(\rightarrow\) Since he is the only child, "my father's son" can only be the speaker himself.

3. Now the statement becomes: "That man's father is ME."

4. If the man in the photo has the speaker as his father, then the photo is of the speaker's son.


Step 4: Final Answer:

The photograph belongs to his son.
Quick Tip: Always simplify "My father's only son" to "Me."
It turns complex riddles into simple statements.

Step 1: Understanding the Concept:

Permutations of words with repeating letters require dividing by the factorial of the frequencies of repeated letters.


Step 2: Key Formula or Approach:
\[ Total Arrangements = \frac{n!}{p! \times q! ...} \]

where \( n \) is total letters and \( p, q \) are counts of repeating letters.


Step 3: Detailed Explanation:

Word: APPLE.

Total letters (\( n \)) = 5 (A, P, P, L, E).

Repeating letter: 'P' appears 2 times.

Arrangements = \( \frac{5!}{2!} \)
\[ Arrangements = \frac{120}{2} = 60 \].


Step 4: Final Answer:

The letters can be arranged in 60 ways.
Quick Tip: Identify repeating letters first. If no letters repeat, it's simply \( 5! = 120 \).
If two letters repeat (like in 'PEPPER'), the calculation would be \( \frac{6!}{3!2!} \).

Step 1: Understanding the Concept:

Probability is the ratio of favorable outcomes to the total sample space.


Step 2: Key Formula or Approach:
\[ P(E) = \frac{Number of favorable outcomes}{Total outcomes} \]


Step 3: Detailed Explanation:

Sample space when two coins are tossed: \( \{HH, HT, TH, TT\} \).

Total outcomes = 4.

"At least one head" means 1 or 2 heads.

Favorable outcomes: \( HH, HT, TH \).

Count = 3.
\[ P = \frac{3}{4} \].


Step 4: Final Answer:

The probability is 3/4.
Quick Tip: For "at least 1" questions, use: \( 1 - P(None) \).
\( P(No heads) = P(TT) = 1/4 \).
\( 1 - 1/4 = 3/4 \).

Step 1: Understanding the Concept:

Linear equation solving involves grouping variable terms on one side and constants on the other.


Step 2: Key Formula or Approach:

Transposition method: Change signs when moving terms across the '=' sign.


Step 3: Detailed Explanation:
\[ 3x + 5 = 2x + 12 \]

Subtract \( 2x \) from both sides:
\[ 3x - 2x + 5 = 12 \]
\[ x + 5 = 12 \]

Subtract 5 from both sides:
\[ x = 12 - 5 \]
\[ x = 7 \].


Step 4: Final Answer:

The value of \( x \) is 7.
Quick Tip: To verify, plug the answer back in: \( 3(7) + 5 = 26 \) and \( 2(7) + 12 = 26 \). Both match!

Step 1: Understanding the Concept:

Synonyms are words that share very similar meanings. 'Abandon' implies leaving something permanently.


Step 2: Key Formula or Approach:

Contextual matching: Compare the definitions of the options with the core meaning of the given word.


Step 3: Detailed Explanation:

1. Abandon: To leave, desert, or relinquish.

2. Keep: To retain (Antonym).

3. Forsake: To renounce or leave entirely (Synonym).

4. Adopt: To take up/start using (Opposite).

5. Cherish: To value highly (Opposite).

6. Maintain: To continue/keep up (Opposite).


Step 4: Final Answer:

The synonym is Forsake.
Quick Tip: Look for the "odd one out." Here, options A, C, D, and E all involve keeping or holding something, while B involves leaving it.

Step 1: Understanding the Concept:

Syllogisms use formal logic to determine if conclusions necessarily follow from premises.


Step 2: Key Formula or Approach:

Venn Diagram Representation: Draw concentric circles representing each category.


Step 3: Detailed Explanation:

1. "All cats are dogs": Place 'Cats' circle inside 'Dogs' circle.

2. "All dogs are animals": Place 'Dogs' circle inside 'Animals' circle.

Result: The 'Cats' circle is inside the 'Animals' circle.

Conclusion I: "All cats are animals" - True, as the Cats circle is entirely within Animals.

Conclusion II: "All animals are cats" - False, as the Animals circle is larger and contains area outside the Cats circle.


Step 4: Final Answer:

Only conclusion I follows.
Quick Tip: Moving from "Inner to Outer" in Venn diagrams follows the "All" rule. Moving "Outer to Inner" follows the "Some" rule.

Step 1: Understanding the Concept:

Divisibility rules simplify checking for factors without full division.


Step 2: Key Formula or Approach:

A number is divisible by 9 if the sum of its digits is divisible by 9.


Step 3: Detailed Explanation:

Check each option:

(A) \( 1+2+3+4+5 = 15 \) (Not divisible by 9)

(B) \( 6+7+8+1+2 = 24 \) (Not divisible by 9)

(C) \( 4+5+6+3+9 = 27 \). Since 27 is divisible by 9, the whole number is divisible.

(D) \( 8+9+1+2+4 = 24 \) (Not divisible by 9)

(E) \( 5+5+5+5+5 = 25 \) (Not divisible by 9)


Step 4: Final Answer:

The number is 45639.
Quick Tip: To sum digits faster, ignore any digits that are already 9 or add up to 9.
For 45639: (4+5=9 ignore), (6+3=9 ignore), (9 ignore). Sum effectively becomes 0 or 9.

Step 1: Understanding the Concept:

Filling pipes have positive efficiency, and emptying pipes (leaks) have negative efficiency.


Step 2: Key Formula or Approach:

Combined Rate = \( \frac{1}{A} - \frac{1}{B} \)


Step 3: Detailed Explanation:

Rate of Pipe A = \( 1/10 \) tank per hour.

Rate of Pipe B = \( 1/15 \) tank per hour (negative).

Combined Rate = \( \frac{1}{10} - \frac{1}{15} \)

Find LCM of 10 and 15, which is 30:
\[ Rate = \frac{3 - 2}{30} = \frac{1}{30} \]

Time = \( 1 / Rate = 30 hours. \)


Step 4: Final Answer:

The tank will fill in 30 hours.
Quick Tip: Short formula: \( Time = \frac{xy}{y - x} = \frac{10 \times 15}{15 - 10} = \frac{150}{5} = 30 \).

Step 1: Understanding the Concept:

The product of two numbers is always equal to the product of their HCF and LCM.


Step 2: Key Formula or Approach:
\[ a \times b = HCF \times LCM \]


Step 3: Detailed Explanation:

Given: \( a = 24, HCF = 12, LCM = 72 \).
\[ 24 \times b = 12 \times 72 \]
\[ b = \frac{12 \times 72}{24} \]

Simplify: \( \frac{12}{24} = 1/2 \).
\[ b = \frac{72}{2} = 36 \].


Step 4: Final Answer:

The second number is 36.
Quick Tip: Never multiply the numbers out fully first. Always set up the fraction and simplify to avoid long division or multiplication.

Step 1: Understanding the Concept:

Percentage calculations are fundamental for simplification problems. Convert percentages to fractions or decimals.


Step 2: Key Formula or Approach:

Apply: \( \frac{x}{100} \times y \).


Step 3: Detailed Explanation:

1. \( 25% of 400 = \frac{25}{100} \times 400 = \frac{1}{4} \times 400 = 100 \).

2. \( 10% of 500 = \frac{10}{100} \times 500 = 50 \).

3. Total = \( 100 + 50 = 150 \).


Step 4: Final Answer:

The result is 150.
Quick Tip: 25% is a quarter (divide by 4).
10% is easy (shift decimal one place left).

Step 1: Understanding the Concept:

Expenditure is constant, so Price and Consumption are inversely related.


Step 2: Key Formula or Approach:

Reduction in Consumption = \( \left( \frac{r}{100 + r} \right) \times 100% \).


Step 3: Detailed Explanation:

Given increase \( r = 25 \).
\[ Reduction = \left( \frac{25}{100 + 25} \right) \times 100 \]
\[ Reduction = \frac{25}{125} \times 100 \]
\[ Reduction = \frac{1}{5} \times 100 = 20% \].


Step 4: Final Answer:

The reduction is 20%.
Quick Tip: If price increases by \( 1/x \), consumption must decrease by \( 1/(x+1) \).
Here: \( 25% = 1/4 \). Decrease = \( 1/(4+1) = 1/5 = 20% \).

Step 1: Understanding the Concept:

Profit distribution in a partnership is based on the investment ratio if the time period is the same.


Step 2: Key Formula or Approach:

Profit Share \(\propto\) Capital.


Step 3: Detailed Explanation:

Investment ratio A:B = 20,000 : 30,000 = 2:3.

Total parts = 5.

B's share = \( \frac{3}{5} \times 10,000 \).

B's share = \( 3 \times 2,000 = 6,000 \).


Step 4: Final Answer:

B's share is
(6,000.
Quick Tip: Cancel all the zeros in the investment amount first to get the simplest ratio immediately.

Step 1: Understanding the Concept:

This idiom highlights the futility of regretting past events that cannot be changed.


Step 2: Key Formula or Approach:

Figurative Language Analysis: Connect the literal image of the phrase to its practical social application.


Step 3: Detailed Explanation:

Once milk is spilt on the floor, it cannot be gathered back into the container. Therefore, crying about it is a waste of energy.

Similarly, in life, if an error has occurred and cannot be undone, we use this phrase to suggest moving forward instead of dwelling on the past.


Step 4: Final Answer:

The idiom means complaining about a past loss that is unfixable.
Quick Tip: Think of other "regret" idioms like "water under the bridge" to group similar meanings together.

Step 1: Understanding the Concept:

The probability of "NOT" an event is found by excluding the specified outcome from the favorable set.


Step 2: Key Formula or Approach:
\[ P(Not Blue) = \frac{Red balls + Green balls}{Total balls} \]


Step 3: Detailed Explanation:

Total balls = \( 5 + 3 + 2 = 10 \).

Favorable outcomes (Not blue) = Red (5) + Green (2) = 7.

Probability = \( \frac{7}{10} \).


Step 4: Final Answer:

The probability is 7/10.
Quick Tip: Always double-check if the question says "is blue" or "is NOT blue." Misreading "NOT" is the most common error in probability exams.