All practice questions

What is the value of the positive integer $n$ ?

Statement I: $n + 8 = 15$
Statement II: $3n = 21$

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I nor Statement II alone is sufficient, but both statements together are sufficient to answer the question.

Neither Statement I nor Statement II, even when taken together, is sufficient to answer the question.

Show answer & explanation

Answer: Either Statement I alone or Statement II alone is sufficient to answer the question.

Answer

Either Statement I alone or Statement II alone is sufficient to answer the question.

Evaluating Statement I alone gives

n = 15 - 8 = 7

, which uniquely determines

n

. Evaluating Statement II alone gives

n = \frac{21}{3} = 7

, which also uniquely determines

n

. Since each statement independently provides a unique answer, either statement alone is sufficient.

Step-by-Step Solution

1

Evaluate Statement I independently

From

n + 8 = 15

, subtracting

8

from both sides gives

n = 7

.

This yields a unique value for

n

, so Statement I alone is sufficient.

2

Evaluate Statement II independently

From

3n = 21

, dividing both sides by

3

gives

n = 7

.

This also yields a unique value for

n

, so Statement II alone is sufficient.

3

Determine overall sufficiency choice

Since both Statement I alone and Statement II alone yield the exact unique answer, either statement alone is sufficient.

This matches the standard option where either statement independently suffices.

Key Concept

Single-Variable Linear Data Sufficiency

Estimated Time:45s

Question 302Question →

Evaluate the following complex mathematical expression strictly adhering to the hierarchical order of operations (BODMAS rule):

\left\{ 64\% \text{ of } 125 + \left[ \frac{13}{3} \div \left\{ 2\frac{1}{6} - \left( 1.4 \times 2.5 - \overline{\frac{3}{5} + \frac{7}{10}} \right) \right\} \right] \right\} \times 0.375

What is the exact numerical value of the final simplified expression?

Show answer & explanation

Answer: -18.75

Answer

The simplified final value of the expression is -18.75.

Following the BODMAS rule strictly: first simplify under the vinculum (3/5 + 7/10 = 1.3), then inside the round brackets (1.4 * 2.5 - 1.3 = 2.2). Next, inside the curly brackets (13/6 - 2.2 = -1/30), then the division in the square brackets ((13/3) / (-1/30) = -130). Evaluating 64% of 125 gives 80. Combining inside the main bracket gives 80 + (-130) = -50. Finally, multiplying by 0.375 yields -18.75.

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar

\frac{3}{5} + \frac{7}{10} = 1.3

The vinculum (bar bracket) has the highest priority under BODMAS and must be evaluated first.

2

Simplify the innermost round brackets

1.4 \times 2.5 - 1.3 = 3.5 - 1.3 = 2.2

Perform multiplication inside the round brackets before subtraction, then subtract the vinculum result.

3

Evaluate the expression inside the inner curly brackets

2\frac{1}{6} - 2.2 = \frac{13}{6} - \frac{11}{5} = -\frac{1}{30}

Convert the mixed fraction and decimal to improper fractions and subtract.

4

Evaluate the square bracket division

\frac{13}{3} \div \left( -\frac{1}{30} \right) = \frac{13}{3} \times (-30) = -130

Dividing by a fraction is equivalent to multiplying by its reciprocal.

5

Calculate the percentage term '64% of 125'

\frac{64}{100} \times 125 = 80

'Of' operation takes precedence over addition.

6

Combine the terms within the main outer brackets

80 + (-130) = -50

Add the result of the percentage term to the square bracket result.

7

Multiply by the outer factor 0.375

-50 \times 0.375 = -18.75

Perform final multiplication to complete the simplification.

Key Concept

BODMAS order of operations with nested brackets, vinculum, fractions, decimals, and percentage operations.

Estimated Time:3m 0s

Question 303Question →

Find the unit digit of the numerical expression $N = (567^{63} \times 234^{45}) - 343^{37}$ .

Show answer & explanation

Answer: 9

Answer

9

The unit digit of

567^{63}

is 3 and for

234^{45}

it is 4, making their product's unit digit 2. The unit digit of

343^{37}

is 3. Subtracting 3 from 2 with regrouping (

12 - 3

) yields 9.

Step-by-Step Solution

1

Determine the unit digit of

567^{63}

3

Base unit digit is 7 with cyclicity 4. Since

63 \pmod 4 = 3

,

7^3

gives a unit digit of 3.

2

Determine the unit digit of

234^{45}

4

Base unit digit is 4 with cyclicity 2. An odd exponent yields a unit digit of 4.

3

Multiply the unit digits of the first two terms

2

The unit digit of the product is

(3 \times 4) \pmod{10} = 2

.

4

Determine the unit digit of

343^{37}

3

Base unit digit is 3 with cyclicity 4. Since

37 \pmod 4 = 1

,

3^1

gives a unit digit of 3.

5

Subtract the unit digit of the second part from the first part

9

Subtracting 3 from 2 requires borrowing 10 (

12 - 3 = 9

) to yield a valid positive unit digit.

Key Concept

Unit Digit and Cyclicity of Powers

Question 304Question →

What is the average age of a class of 20 students?

Statement I: The sum of the ages of all 20 students in the class is 300 years.
Statement II: The age of the youngest student in the class is 12 years.

Statement I alone is sufficient to answer the question, while Statement II alone is not sufficient.

Statement II alone is sufficient to answer the question, while Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient to answer the question.

Neither Statement I nor Statement II is sufficient to answer the question.

Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Show answer & explanation

Answer: Statement I alone is sufficient to answer the question, while Statement II alone is not sufficient.

Answer

Statement I alone is sufficient to answer the question, while Statement II alone is not sufficient.

The correct choice highlights that Statement I alone provides the total sum of ages for the 20 students. Since average equals total sum divided by the number of items (

\text{Average} = \frac{300}{20} = 15

), Statement I alone is sufficient. Statement II only tells us the age of one specific student, which does not allow us to compute the average for all 20 students.

Step-by-Step Solution

1

Analyze the core formula needed to solve the question

The average age of a group is calculated using the formula:

\text{Average Age} = \frac{\text{Sum of ages of all students}}{\text{Total number of students}}

. The total number of students is given as

20

.

Understanding the required parameters helps evaluate statement sufficiency.

2

Evaluate Statement I individually

Statement I gives the sum of ages

= 300

years.

\text{Average Age} = \frac{300}{20} = 15

years. Thus, Statement I alone yields a unique answer and is sufficient.

Determines if Statement I provides enough data on its own.

3

Evaluate Statement II individually

Statement II gives the age of the youngest student

= 12

years. This tells us nothing about the remaining

19

students, so we cannot determine the sum or the average age. Thus, Statement II alone is not sufficient.

Determines if Statement II provides enough data on its own.

Key Concept

Data Sufficiency - Evaluation of Individual Statements in Average Calculations

Question 305Question →

What is the simplified value of the following mathematical expression when evaluated using the standard BODMAS rule?

60 - [15 + \{20 \div 5 + 3 \times (8 - \overline{4 + 2})\}]

35

40

23

31

Show answer & explanation

Answer: $35$

Answer

The simplified value of the expression is

35

.

Following the BODMAS rule strictly: first resolve the vinculum

\overline{4 + 2} = 6

, then the round brackets

(8 - 6) = 2

. Inside the curly brackets, division and multiplication are evaluated before addition (

20 \div 5 + 3 \times 2 = 4 + 6 = 10

). Adding this to

15

inside the square brackets gives

25

. Subtracting

25

from

60

gives the correct simplified result of

35

.

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar.

\overline{4 + 2} = 6

Operations under a vinculum (line bracket) take highest priority.

2

Simplify the innermost round brackets

(8 - 6)

.

8 - 6 = 2

Parentheses must be evaluated next.

3

Evaluate the terms inside the curly brackets

\{20 \div 5 + 3 \times 2\}

. Perform division and multiplication before addition.

20 \div 5 = 4

and

3 \times 2 = 6

, so

4 + 6 = 10

According to BODMAS, division and multiplication precede addition.

4

Simplify the square brackets

[15 + 10]

.

15 + 10 = 25

Resolve the remaining bracket structure.

5

Perform the final subtraction

60 - 25

.

35

Complete the outermost operation.

Key Concept

Simplification using hierarchical order of operations (BODMAS / Vinculum rule)

Question 306Question →

A rectangular garden is surrounded on all four sides by a concrete walking path of uniform width $w$ meters. What is the area of the concrete walking path in square meters?

Statement I: The perimeter of the outer boundary of the walking path is $80\text{ meters}$ .
Statement II: The area of the rectangular garden is $300\text{ square meters}$ , and its length is $5\text{ meters}$ greater than its width.

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient, but Statement II alone is not sufficient.

Statement II alone is sufficient, but Statement I alone is not sufficient.

Either Statement I alone or Statement II alone is sufficient.

Neither Statement I nor Statement II is sufficient, even when combined.

Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

Show answer & explanation

Answer: Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

Answer

Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

The correct answer states that both statements together are sufficient, but neither alone is sufficient. Statement I leaves the path width and garden dimensions undetermined, while Statement II gives the garden dimensions without the path width. Combining them allows the path width and consequently the exact path area to be calculated uniquely.

Step-by-Step Solution

1

Analyze Statement I independently

Let the length and width of the garden be

L

and

W

meters, and the uniform path width be

w

meters. The outer dimensions are

(L + 2w)

and

(W + 2w)

. The outer perimeter is

2[(L + 2w) + (W + 2w)] = 2(L + W + 4w) = 80

, which simplifies to

L + W + 4w = 40

. The area of the path is

(L + 2w)(W + 2w) - LW = 2w(L + W + 2w)

. Since

L + W

and

w

can take multiple values satisfying

L + W + 4w = 40

, the area of the path cannot be uniquely determined. Hence, Statement I alone is NOT sufficient.

Evaluating whether Statement I alone fixes a single numerical value for the path area.

2

Analyze Statement II independently

Statement II states that

L \cdot W = 300

and

L = W + 5

. Substituting gives

(W + 5)W = 300

, so

W^2 + 5W - 300 = 0

. Factoring gives

(W + 20)(W - 15) = 0

, yielding

W = 15\text{ m}

and

L = 20\text{ m}

. However, no information is given about the path width

w

. Thus, the path area

2w(35 + 2w)

cannot be computed without

w

. Hence, Statement II alone is NOT sufficient.

Evaluating whether Statement II alone fixes a single numerical value for the path area.

3

Analyze Statements I and II combined

From Statement II,

L = 20\text{ m}

and

W = 15\text{ m}

, so

L + W = 35\text{ m}

. Substitute this into the relation from Statement I:

35 + 4w = 40 \Rightarrow 4w = 5 \Rightarrow w = 1.25\text{ meters}

. The area of the path is

2(1.25)(35 + 2.5) = 2.5 \times 37.5 = 93.75\text{ square meters}

, which is a unique value. Therefore, both statements together are sufficient.

Determining if combining both statements resolves all unknown variables uniquely.

Key Concept

Evaluating individual statement sufficiency for multi-variable geometric systems prior to combination.

Question 307Question →

Three automatic signaling beacons in a traffic control system flash at regular time intervals of $\frac{4}{15}$ hours, $\frac{9}{20}$ hours, and $\frac{14}{35}$ hours, respectively. If all three beacons flash simultaneously at 12:00 noon, after how many hours will they all flash together again for the first time?

Show answer & explanation

Answer: 7.2

Answer

7.2 hours

To find when events occurring at fractional time intervals coincide, compute the LCM of the fractions. Crucially, each fraction must first be simplified to its lowest terms.

\frac{14}{35}

reduces to

\frac{2}{5}

. Taking

\frac{\text{LCM}(4, 9, 2)}{\text{HCF}(15, 20, 5)}

gives

\frac{36}{5} = 7.2

hours.

Step-by-Step Solution

1

Simplify all given fractional time intervals to their lowest terms.

The reduced fractions are

\frac{4}{15}

,

\frac{9}{20}

, and

\frac{2}{5}

.

The standard LCM formula for fractions

\frac{\text{LCM of numerators}}{\text{HCF of denominators}}

is mathematically valid only when all fractions are reduced to co-prime numerator-denominator pairs.

2

Calculate the LCM of the numerators.

\text{LCM}(4, 9, 2) = 36

.

The least common multiple of

4 = 2^2

,

9 = 3^2

, and

2 = 2^1

is

2^2 \times 3^2 = 36

.

3

Calculate the HCF of the denominators.

\text{HCF}(15, 20, 5) = 5

.

The highest common factor dividing

15

,

20

, and

5

is

5

.

4

Divide the numerator LCM by the denominator HCF to find the simultaneous flashing interval.

\frac{36}{5} = 7.2

hours.

The LCM of the fractional time intervals determines the minimum duration before all events synchronize.

Key Concept

LCM of Fractions with Mandatory Simplification

Estimated Time:2m 0s

Question 308Question →

What is the unit digit of the composite exponential expression $E = (238^{440} + 547^{321}) \times 929^{103}$ ?

7

2

9

1

Show answer & explanation

Answer: 7

Answer

The unit digit of the given expression is 7.

The unit digit of

238^{440}

is 6 (since

440 \pmod 4 = 0 \implies 8^4 \rightarrow 6

). The unit digit of

547^{321}

is 7 (since

321 \pmod 4 = 1 \implies 7^1 \rightarrow 7

). Their sum has a unit digit of

(6 + 7) \rightarrow 3

. The unit digit of

929^{103}

is 9 (since 103 is odd

\implies 9^1 \rightarrow 9

). Multiplying the components gives

3 \times 9 = 27

, yielding a final unit digit of 7.

Step-by-Step Solution

1

Find the unit digit of

238^{440}

.

The unit digit of base 238 is 8. The cyclicity of 8 is 4 (

8, 4, 2, 6

). Divide the exponent 440 by 4:

440 \pmod 4 = 0

. Since the remainder is 0, we take the 4th power in the cycle (

8^4

), which ends in 6.

When an exponent is completely divisible by the cycle length (remainder 0), the position corresponds to the maximum cycle power (power 4).

2

Find the unit digit of

547^{321}

.

The unit digit of base 547 is 7. The cyclicity of 7 is 4 (

7, 9, 3, 1

). Divide exponent 321 by 4:

321 \pmod 4 = 1

. The 1st power in the cycle (

7^1

) ends in 7.

The unit digit is determined by the remainder of the exponent modulo 4.

3

Calculate the unit digit of the expression inside the parentheses:

(238^{440} + 547^{321})

.

Unit digit =

(6 + 7) \pmod{10} = 13 \pmod{10} = 3

.

Addition of unit digits follows standard modular arithmetic.

4

Find the unit digit of

929^{103}

.

The unit digit of base 929 is 9. The cyclicity of 9 is 2 (

9, 1

). Since the exponent 103 is odd,

9^{\text{odd}}

ends in 9.

Bases ending in 9 have an alternating cyclicity of 2.

5

Compute the final unit digit of

E = (3 \times 9)

.

Unit digit =

(3 \times 9) \pmod{10} = 27 \pmod{10} = 7

.

Multiplying the resultant unit digits gives the final unit digit of the composite product.

Key Concept

Unit digit cyclicity rule and exponent modulo operations

Estimated Time:2m 0s

Question 309Question →

In a performance audit of $200$ public health centers across a administrative division, adoption of three digital portals was evaluated: Tele-consultation ( $T$ ), Electronic Health Records ( $E$ ), and Medicine Inventory Management ( $M$ ). The audit revealed the following findings:

- $105$ centers use Tele-consultation ( $T$ )
- $90$ centers use Electronic Health Records ( $E$ )
- $85$ centers use Medicine Inventory Management ( $M$ )
- $40$ centers use both $T$ and $E$
- $35$ centers use both $E$ and $M$
- $30$ centers use both $T$ and $M$
- $15$ centers use all three platforms

How many public health centers utilize exactly two of the three digital health platforms?

60

105

90

45

Show answer & explanation

Answer: 60

Answer

60 public health centers utilize exactly two of the three digital health platforms.

The number of centers using 'exactly two' platforms consists of three distinct regions: (Centers in

T

and

E

only) + (Centers in

E

and

M

only) + (Centers in

T

and

M

only). Since each given pairwise intersection includes the 15 centers that use all three portals, we subtract 15 from each pairwise value:

(40 - 15) + (35 - 15) + (30 - 15) = 25 + 20 + 15 = 60

.

Step-by-Step Solution

1

Identify given set cardinalities and intersections

Total universe

N = 200

;

n(T \cap E) = 40

,

n(E \cap M) = 35

,

n(T \cap M) = 30

, and

n(T \cap E \cap M) = 15

.

The given pairwise intersections include centers that use all three platforms.

2

Calculate centers using strictly two platforms (excluding the triple overlap)

Centers using ONLY

T

and

E = 40 - 15 = 25

.
Centers using ONLY

E

and

M = 35 - 15 = 20

.
Centers using ONLY

T

and

M = 30 - 15 = 15

.

To find 'exactly two', centers using all three must be removed from each pairwise intersection.

3

Sum the exclusive two-set regions

Total =

25 + 20 + 15 = 60

.

Adding these mutually exclusive regions yields the exact total of centers using exactly two portals.

Key Concept

3-Set Venn Diagram Region Isolation (Inclusion-Exclusion Principle)

Estimated Time:1m 30s

Question 310Question →

What is the simplified value of the following mathematical expression when evaluated using the standard BODMAS rule?

20 + 4 \times \left[ 36 \div \left\{ 12 - \left( 3 + \overline{7 - 4} \right) \right\} \right]

44

144

32

72

Show answer & explanation

Answer: 44

Answer

44

Following the BODMAS order of operations, we first evaluate the vinculum line to get 3, then solve the round bracket (3 + 3 = 6), followed by the curly bracket (12 - 6 = 6), and then the square bracket (36 ÷ 6 = 6). Finally, performing multiplication before addition gives 20 + (4 × 6) = 20 + 24 = 44.

Step-by-Step Solution

1

Simplify the expression under the vinculum (bar)

7 - 4 = 3

The vinculum has the highest priority and must be evaluated first.

2

Simplify the innermost round brackets

3 + 3 = 6

Perform addition inside the parentheses.

3

Simplify the curly brackets

12 - 6 = 6

Subtract the result of the round brackets from 12.

4

Simplify the square brackets

36 ÷ 6 = 6

Perform division inside the square brackets.

5

Apply multiplication before addition in the outer expression

4 × 6 = 24

Multiplication has higher priority than addition according to BODMAS.

6

Perform final addition

20 + 24 = 44

Add 20 to 24 to get the final simplified answer.

Key Concept

Simplification and BODMAS Rule

Question 311Question →

What is the simplified numerical value of the following mathematical expression when evaluated using the correct BODMAS order of operations?

20\% \text{ of } 150 - \left[ 3.5 + 2 \times \left\{ 7.2 - \left( 1.4 + \overline{3.8 - 1.6} \right) \div 0.6 \right\} \right]

24.1

14.5

23.4

28.82

Show answer & explanation

Answer: 24.1

Answer

24.1

Evaluating the expression by strictly following the VBODMAS hierarchy (Vinculum -> Brackets -> Of/Orders -> Division -> Multiplication -> Addition -> Subtraction) yields the step-by-step values: vinculum gives 2.2, round brackets give 3.6, division inside curly braces gives 6, curly braces give 1.2, multiplication inside square brackets gives 2.4, square brackets give 5.9, percentage gives 30, and final subtraction gives 24.1.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{3.8 - 1.6} = 2.2

The vinculum has the highest priority in the hierarchy of brackets.

2

Simplify the terms within the parentheses (round brackets)

1.4 + 2.2 = 3.6

Complete the inner addition inside the round brackets.

3

Perform division inside the braces (curly brackets)

3.6 \div 0.6 = 6

Division takes precedence over subtraction inside the curly brackets.

4

Perform subtraction inside the braces (curly brackets)

7.2 - 6 = 1.2

Complete the remaining operation inside the curly brackets.

5

Perform multiplication inside the square brackets

2 \times 1.2 = 2.4

Multiplication takes precedence over addition inside the square brackets.

6

Perform addition inside the square brackets

3.5 + 2.4 = 5.9

Complete the final operation inside the square brackets.

7

Calculate the percentage value

20\% \text{ of } 150 = \frac{20}{100} \times 150 = 30

Evaluate the 'Of' operation / percentage computation.

8

Subtract the bracket result from the percentage value

30 - 5.9 = 24.1

Execute the final subtraction outside all brackets.

Key Concept

Hierarchy of Order of Operations (BODMAS / VBODMAS)

Question 312Question →

If $x = \sqrt{7 + 4\sqrt{3}}$ , $y = \sqrt{7 - 4\sqrt{3}}$ , and $z = \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$ , what is the value of the expression $\frac{x^3 + y^3}{x^2 + y^2 - z}$ ?

4

\frac{56}{13}

\frac{13}{3}

13

Show answer & explanation

Answer: 4

Answer

4

Expressing

7 \pm 4\sqrt{3}

as perfect squares

(2 \pm \sqrt{3})^2

simplifies

x

to

2 + \sqrt{3}

and

y

to

2 - \sqrt{3}

. This yields

x+y=4

,

xy=1

,

x^2+y^2=14

, and

x^3+y^3=52

. For

z

, using the cubic identity

z^3 = a^3 + b^3 + 3ab z

transforms the expression into

z^3 + 3z - 4 = 0

, yielding the real solution

z = 1

. Substituting these values gives

\frac{52}{14 - 1} = 4

.

Step-by-Step Solution

1

Simplify the square root surds

x

and

y

x = 2 + \sqrt{3}

and

y = 2 - \sqrt{3}

Since

7 + 4\sqrt{3} = 4 + 3 + 2(2)(\sqrt{3}) = (2 + \sqrt{3})^2

, taking the square root gives

2 + \sqrt{3}

. Similarly,

7 - 4\sqrt{3} = (2 - \sqrt{3})^2

.

2

Calculate fundamental algebraic combinations of

x

and

y

x + y = 4

,

xy = 1

,

x^2 + y^2 = 14

, and

x^3 + y^3 = 52

x + y = (2+\sqrt{3}) + (2-\sqrt{3}) = 4

.

xy = (2+\sqrt{3})(2-\sqrt{3}) = 4 - 3 = 1

.

x^2 + y^2 = (x+y)^2 - 2xy = 16 - 2 = 14

.

x^3 + y^3 = (x+y)^3 - 3xy(x+y) = 64 - 3(1)(4) = 52

.

3

Evaluate the nested cube root expression for

z

z = 1

Let

z = a + b

where

a = \sqrt[3]{2+\sqrt{5}}

and

b = \sqrt[3]{2-\sqrt{5}}

. Cubing both sides:

z^3 = a^3 + b^3 + 3ab(a+b) = (2+\sqrt{5}) + (2-\sqrt{5}) + 3\sqrt[3]{(2+\sqrt{5})(2-\sqrt{5})} z = 4 + 3\sqrt[3]{4-5} z = 4 - 3z

. Solving

z^3 + 3z - 4 = 0

gives

(z-1)(z^2 + z + 4) = 0

, whose unique real root is

z = 1

.

4

Substitute all values into the given expression

\frac{x^3 + y^3}{x^2 + y^2 - z}

\frac{52}{14 - 1} = \frac{52}{13} = 4

Direct numerical evaluation of the simplified algebraic components.

Key Concept

Simplification of Nested Square and Cube Surds using Algebraic Identities

Estimated Time:3m 0s

Question 313Question →

The table below presents the quarterly renewable energy generation (in Megawatts, MW) of four geographical zones of a state in 2025:

Zone	Q1 (MW)	Q2 (MW)	Q3 (MW)	Q4 (MW)
North Zone	120	150	180	150
South Zone	200	220	250	210
East Zone	90	110	130	110
West Zone	160	180	200	160

Based on the table, what is the average quarterly renewable energy generation (in MW) for the South Zone in 2025?

Show answer & explanation

Answer: 220

Answer

The average quarterly renewable energy generation for the South Zone in 2025 is

220\text{ MW}

.

The correct average is obtained by summing the four quarterly generation figures of the South Zone (

200 + 220 + 250 + 210 = 880\text{ MW}

) and dividing by

4

, giving

220\text{ MW}

.

Step-by-Step Solution

1

Locate the row for the South Zone and extract data for all four quarters.

Q1 =

200\text{ MW}

, Q2 =

220\text{ MW}

, Q3 =

250\text{ MW}

, Q4 =

210\text{ MW}

.

Data needs to be aggregated across the full year.

2

Calculate total annual generation for the South Zone.

Total =

200 + 220 + 250 + 210 = 880\text{ MW}

.

Summing values is required before finding the average.

3

Divide total generation by the number of quarters.

Average =

\frac{880}{4} = 220\text{ MW}

.

An average across four periods requires dividing the total sum by 4.

Key Concept

Average calculation from a tabular dataset

Question 314Question →

Three storage vessels in a chemical laboratory contain $\frac{12}{35}$ litres, $\frac{18}{25}$ litres, and $\frac{24}{49}$ litres of different reagents. What is the maximum capacity (in litres) of a single measuring container that can be used to measure the exact quantity of reagent from any of the three vessels without leaving any remainder?

\frac{6}{1225}

litres

\frac{72}{1225}

litres

\frac{1225}{6}

litres

72

litres

Show answer & explanation

Answer: $\frac{6}{1225}$ litres

Answer

The maximum capacity of the measuring container is

\frac{6}{1225}

litres.

To find the largest measuring unit that can measure fractional quantities exactly, we need the HCF of those fractions. The HCF of fractions is defined as

\frac{\text{HCF of numerators}}{\text{LCM of denominators}}

. Calculating

\text{HCF}(12, 18, 24) = 6

and

\text{LCM}(35, 25, 49) = 1225

yields

\frac{6}{1225}

litres.

Step-by-Step Solution

1

Identify the mathematical rule required for measuring fractional quantities exactly.

To find the maximum capacity of a container that measures given fractional quantities an exact number of times, compute the HCF of the fractions:

\text{HCF}\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{\text{HCF}(a, c, e)}{\text{LCM}(b, d, f)}

.

The measuring container's capacity must be a common divisor of all three fractional volumes.

2

Calculate the HCF of the numerators

12

,

18

, and

24

.

\text{HCF}(12, 18, 24) = 6

.

The prime factorizations are

12 = 2^2 \times 3

,

18 = 2 \times 3^2

, and

24 = 2^3 \times 3

. The highest common power of prime factors shared by all three numbers is

2^1 \times 3^1 = 6

.

3

Calculate the LCM of the denominators

35

,

25

, and

49

.

\text{LCM}(35, 25, 49) = 1225

.

The prime factorizations are

35 = 5 \times 7

,

25 = 5^2

, and

49 = 7^2

. The LCM takes the highest power of each prime:

5^2 \times 7^2 = 25 \times 49 = 1225

.

4

Substitute the calculated HCF of numerators and LCM of denominators into the fraction HCF formula.

\text{HCF} = \frac{6}{1225}

litres.

Dividing the HCF of the numerators by the LCM of the denominators gives the exact maximum volume.

Key Concept

HCF of Fractions

Estimated Time:2m 0s

Question 315Question →

In an agricultural extension survey conducted among $250$ farmers in an administrative block, awareness of three welfare schemes was evaluated: Crop Insurance ( $C$ ), Soil Health Card ( $S$ ), and Kisan Credit Card ( $K$ ). The survey revealed that $120$ farmers are aware of $C$ , $110$ are aware of $S$ , and $130$ are aware of $K$ . Additionally, $45$ farmers are aware of both $C$ and $S$ , $50$ are aware of both $S$ and $K$ , and $40$ are aware of both $C$ and $K$ . If $15$ farmers are aware of all three schemes, how many farmers are aware of exactly two of these schemes?

Show answer & explanation

Answer: 90

Answer

The total number of farmers aware of exactly two schemes is

90

.

To find the number of farmers aware of exactly two schemes, we isolate the three two-set intersection regions that exclude the three-set intersection. Subtracting the

15

farmers aware of all three schemes from each pairwise intersection yields

30

(for

C

and

S

only),

35

(for

S

and

K

only), and

25

(for

C

and

K

only). Summing these mutually exclusive regions gives

30 + 35 + 25 = 90

.

Step-by-Step Solution

1

Calculate the count of farmers aware of only Crop Insurance and Soil Health Card

45 - 15 = 30

The pairwise intersection includes farmers aware of all three schemes, so subtracting the triple intersection isolates those aware of only these two schemes.

2

Calculate the count of farmers aware of only Soil Health Card and Kisan Credit Card

50 - 15 = 35

Subtract the triple intersection count from the pairwise intersection count of

S

and

K

.

3

Calculate the count of farmers aware of only Crop Insurance and Kisan Credit Card

40 - 15 = 25

Subtract the triple intersection count from the pairwise intersection count of

C

and

K

.

4

Sum the three region counts for exactly two schemes

30 + 35 + 25 = 90

The regions representing 'only C and S', 'only S and K', and 'only C and K' are mutually exclusive.

Key Concept

3-Set Venn Diagram Region Isolation

Question 316Question →

What is the final numerical value obtained by evaluating the following mathematical expression strictly according to the BODMAS rule?

50 - \left[ 52 \div \left\{ 4.5 + \frac{4}{5} \times \left( 3.7 - \overline{1.8 - 0.6} \right) \right\} \right]

Show answer & explanation

Answer: 42

Answer

42

Following VBODMAS rule strictly:
1. Vinculum: 1.8 - 0.6 = 1.2
2. Round brackets: 3.7 - 1.2 = 2.5
3. Multiplication in curly brackets: (4/5) * 2.5 = 2
4. Addition in curly brackets: 4.5 + 2 = 6.5
5. Square bracket division: 52 / 6.5 = 8
6. Outer subtraction: 50 - 8 = 42.

Step-by-Step Solution

1

Evaluate the vinculum (bar line above numbers)

\overline{1.8 - 0.6} = 1.2

According to BODMAS (or VBODMAS), the vinculum takes precedence over round brackets.

2

Simplify the terms inside the parentheses (round brackets)

3.7 - 1.2 = 2.5

Parentheses are evaluated next after the vinculum.

3

Evaluate the multiplication inside the curly brackets

\frac{4}{5} \times 2.5 = 0.8 \times 2.5 = 2.0

Within curly brackets, multiplication precedes addition.

4

Complete the addition inside the curly brackets

4.5 + 2.0 = 6.5

Completing all operations within the curly brackets.

5

Evaluate the division inside the square brackets

52 \div 6.5 = 8

Simplifying the entire square bracket term.

6

Perform the final subtraction outside all brackets

50 - 8 = 42

Final arithmetic step to find the value of the expression.

Key Concept

Order of Operations (BODMAS / VBODMAS)

Question 317Question →

Calculate the exact numerical value of the following expression by applying the standard BODMAS/VBODMAS order of operations:

45\% \text{ of } 160 - \left[ 12.5 + \left\{ \frac{3}{4} \times \left( 36 \div 3 - \overline{4.2 + 1.8} \right) + 5.5 \right\} \div 0.5 \right]

Show answer & explanation

Answer: 39.5

Answer

The simplified numerical value of the given expression is 39.5.

Evaluating step by step according to VBODMAS rules yields 39.5. First, the vinculum gives 6. The round bracket gives 36 ÷ 3 - 6 = 6. The curly bracket gives (3/4) × 6 + 5.5 = 10. The square bracket gives 12.5 + 10 ÷ 0.5 = 32.5. Finally, 45% of 160 = 72, and 72 - 32.5 = 39.5.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{4.2 + 1.8} = 6

The vinculum acts as a top-priority bracket, so the addition under the bar must be executed before other operations.

2

Simplify the innermost round brackets ( )

36 \div 3 - 6 = 12 - 6 = 6

Inside the round bracket, division precedes subtraction according to BODMAS.

3

Simplify the curly brackets { }

\frac{3}{4} \times 6 + 5.5 = 4.5 + 5.5 = 10

Inside the curly bracket, multiplication of fractions precedes addition.

4

Simplify the square brackets [ ]

12.5 + 10 \div 0.5 = 12.5 + 20 = 32.5

Inside the square bracket, division by decimal (10 / 0.5 = 20) takes precedence over addition.

5

Calculate the percentage term ('of' operation)

45\% \text{ of } 160 = 0.45 \times 160 = 72

The 'of' operation represents multiplication for percentage evaluation.

6

Perform the final subtraction

72 - 32.5 = 39.5

Subtract the fully simplified bracket result from the percentage term.

Key Concept

BODMAS / VBODMAS Rule with Nested Brackets, Percentages, and Vinculum

Estimated Time:2m 0s

Question 318Question →

A State Public Service Commission conducted a detailed employment survey across three major districts: District A, District B, and District C. The total workforce across all three districts combined is $100,000$ .

- District A accounts for $40\%$ of the total workforce, and the ratio of male to female workers in District A is $3 : 2$ .
- District B has a total workforce that is $25\%$ greater than the total workforce of District A. Females constitute $44\%$ of District B's workforce.
- District C comprises the remaining workforce across the three districts, with a male-to-female ratio of $3 : 2$ .
- Across all three districts combined, $25\%$ of the total female workforce is employed in the government sector.
- Specifically, $30\%$ of females in District A and $20\%$ of females in District B are employed in the government sector, while the remaining government-employed females work in District C.

Based on the given information, what is the ratio of the number of female government sector workers in District C to the total number of male workers across all three districts combined?

13 : 580

13 : 236

13 : 60

1 : 40

Show answer & explanation

Answer: $13 : 580$

Answer

The ratio of the number of female government sector workers in District C to the total number of male workers across all three districts combined is

13 : 580

.

The total female government workers across all districts is 10,500 (25% of 42,000 total females). Subtracting District A's 4,800 and District B's 4,400 yields 1,300 government females in District C. The total male population across all three districts is 58,000 (24,000 + 28,000 + 6,000). The ratio 1,300 : 58,000 simplifies directly to 13 : 580.

Step-by-Step Solution

1

Calculate total workforce and gender breakdown for District A

Total workforce of District A =

40\%

of

100,000 = 40,000

. With a Male:Female ratio of

3:2

, Male workforce =

\frac{3}{5} \times 40,000 = 24,000

and Female workforce =

\frac{2}{5} \times 40,000 = 16,000

.

Establishing District A numbers is required as the baseline for District B and total female count.

2

Calculate total workforce and gender breakdown for District B

Workforce of District B =

1.25 \times 40,000 = 50,000

. Female workforce =

44\%

of

50,000 = 22,000

. Male workforce =

50,000 - 22,000 = 28,000

.

District B's population is given relative to District A.

3

Determine workforce and gender breakdown for District C

Workforce of District C =

100,000 - (40,000 + 50,000) = 10,000

. With Male:Female ratio of

3:2

, Male workforce =

\frac{3}{5} \times 10,000 = 6,000

and Female workforce =

\frac{2}{5} \times 10,000 = 4,000

.

District C contains the residual workforce from the total

100,000

.

4

Calculate overall male and female workforce totals

Total Male workforce =

24,000 + 28,000 + 6,000 = 58,000

. Total Female workforce =

16,000 + 22,000 + 4,000 = 42,000

.

The total male workforce forms the denominator of the required ratio.

5

Calculate government-employed females in District C

Total Female Government Workers =

25\%

of

42,000 = 10,500

. District A Female Government Workers =

30\%

of

16,000 = 4,800

. District B Female Government Workers =

20\%

of

22,000 = 4,400

. District C Female Government Workers =

10,500 - (4,800 + 4,400) = 1,300

.

Subtracting District A and B government females from the overall total isolates District C's count.

6

Compute the final ratio

Ratio =

\frac{1,300}{58,000} = \frac{13}{580}

or

13 : 580

.

Simplifying the fraction gives the required final ratio.

Key Concept

Multi-step quantitative interpretation of paragraph data involving percentage base shifts and system of equations.

Estimated Time:3m 0s

Question 319Question →

What is the simplified numerical value of the mathematical expression

36 - \left[ 18 \div \left\{ 6 - \left( 4 - \overline{3 - 1} \right) \right\} \times 2 \right]

when evaluated strictly adhering to the VBODMAS rule?

27

30

33.75

21

Show answer & explanation

Answer: $27$

Answer

The simplified numerical value of the given expression is 27.

First resolve the vinculum

\overline{3-1} = 2

. Next, simplify the round bracket

(4-2) = 2

, then the curly bracket

\{6-2\} = 4

. Inside the square bracket, evaluate division before multiplication from left to right:

18 \div 4 = 4.5

, and

4.5 \times 2 = 9

. Finally, subtract from 36 to get

36 - 9 = 27

.

Step-by-Step Solution

1

Evaluate the expression under the bar/vinculum

\overline{3 - 1} = 2

According to VBODMAS rules, vinculum (bar) has highest priority over standard operations.

2

Simplify the innermost round bracket

(4 - 2) = 2

Resolve terms within parenthesis after vinculum.

3

Simplify the curly bracket

\{6 - 2\} = 4

Perform subtraction inside curly braces.

4

Evaluate operations inside the square bracket from left to right

[18 \div 4 \times 2] = 4.5 \times 2 = 9

Division and multiplication have equal priority and must be evaluated left to right.

5

Perform the final outer subtraction

36 - 9 = 27

Complete the remaining operation.

Key Concept

VBODMAS / BODMAS Order of Operations

Question 320Question →

In a state public infrastructure development scheme for the fiscal year 2025–26, a total budget of ₹ $9,000$ crore was allocated across four major sectors: Agriculture Irrigation, Highway Expansion, Solar Infrastructure, and Rural Electrification.

Agriculture Irrigation received $28\%$ of the total allocated budget. The budget allocated to Highway Expansion was $37.5\%$ more than the budget allocated to Agriculture Irrigation. The remaining allocated budget was divided between Solar Infrastructure and Rural Electrification in the ratio $5 : 4$ .

Regarding the actual expenditures recorded at the end of the fiscal year:
- Agriculture Irrigation utilized $85\%$ of its allocated funds.
- Highway Expansion utilized $92\%$ of its allocated funds.
- Solar Infrastructure had an unspent budget of ₹ $315$ crore.
- Rural Electrification spent an amount equal to $1.25$ times the unspent budget of Agriculture Irrigation.

Based on the information provided, what is the total unspent budget across all four sectors combined, expressed in crore rupees?

Show answer & explanation

Answer: 1837.7

Answer

The total unspent budget across all four sectors combined is 1837.7 crore rupees.

By accurately deriving each sector's allocation from the given base percentage, percentage increment, and ratio, and then computing the respective unspent components according to the expenditure conditions, the total unspent amount is determined to be 1837.7 crore rupees.

Step-by-Step Solution

1

Calculate allocations for Agriculture Irrigation and Highway Expansion

Agriculture Irrigation = ₹2,520 crore; Highway Expansion = ₹3,465 crore

Agriculture Irrigation receives 28% of 9,000 crore = 2,520 crore. Highway Expansion gets 37.5% more than Agriculture Irrigation, which is 2,520 * (1 + 0.375) = 3,465 crore.

2

Calculate allocations for Solar Infrastructure and Rural Electrification

Solar Infrastructure = ₹1,675 crore; Rural Electrification = ₹1,340 crore

Remaining budget = 9,000 - (2,520 + 3,465) = 3,015 crore. Dividing 3,015 in ratio 5:4 gives (5/9)*3,015 = 1,675 crore for Solar and (4/9)*3,015 = 1,340 crore for Rural Electrification.

3

Determine unspent amounts for each sector individually

Agriculture Irrigation Unspent = ₹378 crore; Highway Expansion Unspent = ₹277.2 crore; Solar Infrastructure Unspent = ₹315 crore; Rural Electrification Unspent = ₹867.5 crore

15% of Agriculture Irrigation budget is unspent = 0.15 * 2,520 = 378 crore. 8% of Highway Expansion budget is unspent = 0.08 * 3,465 = 277.2 crore. Solar unspent is given as 315 crore. Rural Electrification expenditure is 1.25 * 378 = 472.5 crore, leaving 1,340 - 472.5 = 867.5 crore unspent.

4

Sum unspent amounts across all four sectors

Total Unspent Budget = ₹1,837.7 crore

Summing unspent amounts: 378 + 277.2 + 315 + 867.5 = 1,837.7 crore rupees.

Key Concept

Multi-step caselet data extraction, ratio division, percentage calculation, and aggregation

Answer

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