All practice questions

Question 21Question →

The following table provides the annual expenditure (in ₹ Crores) of four administrative departments in a state over three fiscal years, with some values missing (indicated by dashes '—').

Department	2021-22	2022-23	2023-24	Total Expenditure
Agriculture	450	—	520	1450
Education	—	680	740	2000
Healthcare	350	420	—	1250
Infrastructure	820	970	1110	—
Total	2200	—	2850	—

Based on the data provided, what is the difference (in ₹ Crores) between the average department expenditure in 2022-23 and the average department expenditure in 2021-22?

Show answer & explanation

Answer: 87.5

Answer

The difference between the average department expenditure in 2022-23 and 2021-22 is 87.5 ₹ Crores.

To find the difference between the average expenditures, we calculate the missing values first. For 2021-22, Education expenditure is

2000 - (680 + 740) = 580

₹ Crores. Total 2021-22 expenditure is

450 + 580 + 350 + 820 = 2200

₹ Crores, making the average

2200 / 4 = 550

₹ Crores. For 2022-23, Agriculture expenditure is

1450 - (450 + 520) = 480

₹ Crores. Total 2022-23 expenditure across all four departments is

480 + 680 + 420 + 970 = 2550

₹ Crores, making the average

2550 / 4 = 637.5

₹ Crores. The difference between the averages is

637.5 - 550 = 87.5

₹ Crores.

Step-by-Step Solution

1

Find missing 2021-22 Education expenditure

580 ₹ Crores

Subtracting known expenditures (680 and 740) from the Education row total (2000 gives

2000 - 1420 = 580

.

2

Compute 2021-22 average department expenditure

550 ₹ Crores

The sum for 2021-22 is 2200 ₹ Crores. Dividing by 4 departments gives

2200 / 4 = 550

.

3

Find missing 2022-23 Agriculture expenditure

480 ₹ Crores

Subtracting known expenditures (450 and 520) from Agriculture row total (1450) gives

1450 - 970 = 480

.

4

Compute 2022-23 average department expenditure

637.5 ₹ Crores

Summing 2022-23 department values (

480 + 680 + 420 + 970 = 2550

) and dividing by 4 gives

2550 / 4 = 637.5

.

5

Subtract 2021-22 average from 2022-23 average

87.5 ₹ Crores

Calculating

637.5 - 550 = 87.5

₹ Crores.

Key Concept

Deriving missing grid values using row totals and computing comparative averages.

Estimated Time:2m 30s

Question 22Question →

A positive integer $N$ when successively divided by $4$ , $5$ , and $6$ leaves remainders of $2$ , $3$ , and $4$ , respectively. What is the sum of the remainders obtained when the smallest such number $N$ is successively divided by $6$ , $5$ , and $4$ , in that order?

Show answer & explanation

Answer: 7

Answer

The sum of the remainders obtained when the smallest such number is successively divided by 6, 5, and 4 is 7.

The smallest positive integer

N

satisfying the given successive division conditions is 94. Successively dividing 94 by 6, 5, and 4 yields quotients of 15, 3, and 0 with remainders 4, 0, and 3, respectively. The sum of these remainders is

4 + 0 + 3 = 7

.

Step-by-Step Solution

1

Formulate the algebraic equations for successive division and calculate the smallest positive value of

N

.

N = 94

By definition of successive division,

N = 4q_1 + 2

,

q_1 = 5q_2 + 3

, and

q_2 = 6q_3 + 4

. Setting the non-negative integer quotient

q_3 = 0

yields

q_2 = 4

,

q_1 = 23

, and

N = 94

.

2

Perform successive division of 94 by the divisors 6, 5, and 4 in sequence.

The sequence of remainders is

4

,

0

, and

3

.

First stage:

94 \div 6 = 15

remainder

4

. Second stage:

15 \div 5 = 3

remainder

0

. Third stage:

3 \div 4 = 0

remainder

3

.

3

Calculate the sum of the three remainders.

4 + 0 + 3 = 7

Adding the individual remainders obtained from each stage of the reverse order division.

Key Concept

Successive Division and Remainder Property

Question 23Question →

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 24Question →

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 25Question →

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 26Question →

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 27Question →

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 28Question →

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 29Question →

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 30Question →

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

Question 31Question →

What is the numerical value of the mathematical expression

80 - [15 + (12 \div 3 \times 2)]

when simplified using the standard BODMAS rule?

Show answer & explanation

Answer: 57

Answer

The simplified numerical value of the given expression is 57.

Evaluating the expression following the strict BODMAS order of operations: first simplify the innermost round brackets by doing division (

12 \div 3 = 4

) then multiplication (

4 \times 2 = 8

). Next, add the numbers inside the square brackets (

15 + 8 = 23

). Finally, subtract 23 from 80 to obtain 57.

Step-by-Step Solution

1

Perform division inside the round brackets.

12 \div 3 = 4

According to the BODMAS rule, division takes precedence over multiplication within brackets.

2

Perform multiplication to complete the round bracket evaluation.

4 \times 2 = 8

Complete the operations contained within the innermost parentheses.

3

Perform addition inside the square brackets.

15 + 8 = 23

Simplify the terms remaining inside the square brackets.

4

Subtract the result of the brackets from the outer value.

80 - 23 = 57

Perform final subtraction to determine the simplified answer.

Key Concept

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Question 32Question →

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

Show answer & explanation

Answer: 23

Answer

The value of

x^2 + y^2

is 23.

By rationalizing the denominators of

x

and

y

, we obtain

x = \frac{5+\sqrt{21}}{2}

and

y = \frac{5-\sqrt{21}}{2}

. Adding these gives

x+y=5

, and multiplying them gives

xy=1

. Using the identity

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

, we calculate

5^2 - 2(1) = 23

.

Step-by-Step Solution

1

Rationalize the expressions for

x

and

y

by multiplying the numerator and denominator by their respective conjugates.

x = \frac{(\sqrt{7}+\sqrt{3})^2}{7-3} = \frac{10 + 2\sqrt{21}}{4} = \frac{5 + \sqrt{21}}{2}

and

y = \frac{(\sqrt{7}-\sqrt{3})^2}{7-3} = \frac{10 - 2\sqrt{21}}{4} = \frac{5 - \sqrt{21}}{2}

Eliminating surds from the denominator simplifies addition and multiplication.

2

Calculate the sum

(x + y)

and the product

(x \cdot y)

of

x

and

y

.

x + y = \frac{5 + \sqrt{21}}{2} + \frac{5 - \sqrt{21}}{2} = 5

and

x \cdot y = \frac{(5 + \sqrt{21})(5 - \sqrt{21})}{4} = \frac{25 - 21}{4} = 1

Simplifying the symmetric expressions reduces computation complexity.

3

Apply the algebraic identity

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

.

x^2 + y^2 = (5)^2 - 2(1) = 25 - 2 = 23

Using the identity avoids evaluating complex squares directly.

Key Concept

Rationalization of surds and application of symmetric algebraic identities

Question 33Question →

A decentralized network consists of $13$ primary verification nodes. A smart contract generates a total of $37^{105}$ encrypted tokens that must be distributed equally among these $13$ nodes. The leftover tokens that cannot be distributed equally are sent to a burn address. How many tokens will be sent to the burn address?

Show answer & explanation

Answer: 8

Answer

8

The correct remainder when

37^{105}

is divided by

13

is

8

. This is found by reducing the base

37

to

-2

modulo

13

, applying Fermat's Little Theorem to reduce the exponent

105

to

9

modulo

12

, and computing

(-2)^9 \pmod{13} = -18 \equiv 8 \pmod{13}

.

Step-by-Step Solution

1

Formulate the problem using modular arithmetic.

Evaluate

37^{105} \pmod{13}

.

Since tokens are distributed equally among 13 nodes, the tokens sent to the burn address represent the remainder when the total is divided by 13.

2

Simplify the base of the exponent.

37 = 13 \times 2 + 11

. Using a negative remainder,

11 \equiv -2 \pmod{13}

.

Working with a smaller absolute base value like -2 makes successive exponentiation much easier than working with 11 or 37.

3

Apply Fermat's Little Theorem.

Since 13 is prime,

(-2)^{12} \equiv 1 \pmod{13}

.

This theorem allows us to significantly reduce large exponents by finding their remainder when divided by

p-1

.

4

Reduce the exponent using the theorem.

105 = 12 \times 8 + 9

. Thus,

(-2)^{105} = ((-2)^{12})^8 \times (-2)^9 \equiv 1^8 \times (-2)^9 \equiv (-2)^9 \pmod{13}

.

The multiples of 12 in the exponent evaluate to 1 modulo 13, leaving only the remainder of the exponent.

5

Calculate the final reduced power.

(-2)^4 = 16 \equiv 3 \pmod{13}

. Therefore,

(-2)^8 = 3^2 = 9 \pmod{13}

. Finally,

(-2)^9 = 9 \times (-2) = -18 \pmod{13}

.

Breaking the calculation into smaller powers prevents the need to calculate very large numbers directly.

6

Convert the negative result to a positive remainder.

-18 = 13 \times (-2) + 8

. The positive remainder is

8

.

Physical quantities, such as the number of leftover tokens, must be represented by a positive remainder.

Key Concept

Modular arithmetic, Fermat's Little Theorem, and negative remainders

Question 34Question →

A water storage tank is currently filled to $0.65$ of its total capacity. If the tank currently contains $130$ liters of water, what is the total capacity of the tank in liters?

Show answer & explanation

Answer: 200

Answer

The total capacity of the tank is 200 liters.

The correct capacity is found by dividing the current volume of water (130 liters) by the decimal that represents the filled portion (0.65). This calculation,

130 \div 0.65

, yields 200 liters.

Step-by-Step Solution

1

Identify the relationship between the filled portion and the total capacity.

Let

C

be the total capacity. We establish the equation:

0.65 \times C = 130

.

Translating the word problem into a mathematical equation allows us to solve for the unknown whole amount.

2

Rearrange the equation to solve for the total capacity.

C = \frac{130}{0.65}

Isolating

C

on one side of the equation gives us the expression needed to find the total capacity.

3

Perform the division by clearing the decimal in the denominator.

C = \frac{13000}{65} = 200

Multiplying the numerator and denominator by 100 eliminates the decimal point, making the division straightforward.

Key Concept

Calculating the total amount when a specific decimal fraction of that amount is known.

Question 35Question →

An event organizer needs to pack $144$ apples, $180$ oranges, and $216$ bananas into identical fruit baskets. Each basket must contain only one type of fruit, and all baskets must contain exactly the same number of fruits. What is the minimum total number of fruit baskets the organizer must prepare?

Show answer & explanation

Answer: 15

Answer

15

To find the minimum number of baskets, we first need to determine the maximum number of fruits that can be placed in each basket. Since the number of fruits must be the same for all baskets and fruits cannot be mixed, this maximum number is the Highest Common Factor (HCF) of 144, 180, and 216. The HCF is 36. Dividing the total count of each fruit by 36 gives 4 apple baskets, 5 orange baskets, and 6 banana baskets. Adding these together yields a total of 15 baskets.

Step-by-Step Solution

1

Determine the mathematical operation required to find the maximum number of fruits per basket.

The problem requires finding the Highest Common Factor (HCF) of 144, 180, and 216.

Because the baskets must be identical in capacity, hold only one type of fruit, and we want the minimum number of total baskets (meaning maximum fruits per basket).

2

Perform prime factorization for each fruit quantity.

144 = 2^4 * 3^2; 180 = 2^2 * 3^2 * 5; 216 = 2^3 * 3^3

Prime factorization is the most reliable method for finding the HCF of three large numbers.

3

Calculate the Highest Common Factor (HCF).

HCF = 2^2 * 3^2 = 4 * 9 = 36.

The HCF is the product of the lowest powers of common prime factors present in all three numbers.

4

Calculate the number of baskets required for each individual fruit type.

Apples: 144 / 36 = 4 baskets. Oranges: 180 / 36 = 5 baskets. Bananas: 216 / 36 = 6 baskets.

Dividing the total quantity of each fruit by the maximum capacity of one basket gives the basket count per fruit.

5

Calculate the total number of baskets.

4 + 5 + 6 = 15 baskets.

The question asks for the minimum total number of fruit baskets to be prepared.

Key Concept

Highest Common Factor (HCF) applied to equitable distribution and grouping

Question 36Question →

In a digital encryption protocol, a security key is generated based on a master integer $N$ . When $N$ is successively divided by $6$ , $8$ , and $9$ , the resulting remainders are $4$ , $3$ , and $7$ , respectively. If $N$ is the smallest possible positive integer that satisfies these conditions, what is the remainder when $N$ is divided by $19$ ?

Show answer & explanation

Answer: 16

Answer

16

By applying the rules of successive division and working backwards from a final quotient of 0, the smallest possible integer

N

is found to be 358. Dividing 358 by 19 yields a quotient of 18 and a remainder of 16.

Step-by-Step Solution

1

Set up equations based on the definition of successive division.

N = 6q_1 + 4

,

q_1 = 8q_2 + 3

, and

q_2 = 9q_3 + 7

Successive division means each division is performed on the quotient of the previous step.

2

Determine the value of the final quotient

q_3

to minimize

N

.

q_3 = 0

The smallest possible positive initial number

N

is obtained when the final successive quotient is zero.

3

Solve for intermediate quotient

q_2

.

q_2 = 9(0) + 7 = 7

Substitute

q_3 = 0

into the equation

q_2 = 9q_3 + 7

.

4

Solve for intermediate quotient

q_1

.

q_1 = 8(7) + 3 = 59

Substitute

q_2 = 7

into the equation

q_1 = 8q_2 + 3

.

5

Calculate the smallest positive integer

N

.

N = 6(59) + 4 = 358

Substitute

q_1 = 59

into the first equation.

6

Divide

N

by

19

to find the final remainder.

358 = 19 \times 18 + 16

. The remainder is

16

.

The problem asks for the remainder when the resulting

N

is divided by

19

.

Key Concept

Successive Division and Remainder Theorem

Estimated Time:1m 30s

Question 37Question →

Compute the exact Highest Common Factor (HCF) for the fractions $\frac{6}{25}$ and $\frac{9}{10}$ . Express your final result as a decimal.

Show answer & explanation

Answer: 0.06

Answer

0.06

The Highest Common Factor (HCF) of a set of fractions is found by dividing the HCF of their numerators by the LCM of their denominators. In this case, HCF(6, 9) is 3, and LCM(25, 10) is 50. This gives the fraction 3/50, which perfectly evaluates to the decimal 0.06.

Step-by-Step Solution

1

Identify the standard rule for calculating the HCF of fractional numbers.

HCF = (HCF of numerators) / (LCM of denominators)

This is the mathematical formula required to find the greatest common divisor of multiple fractions.

2

Find the Highest Common Factor (HCF) of the two numerators, 6 and 9.

HCF(6, 9) = 3

3 is the largest integer that divides both 6 and 9 without leaving a remainder.

3

Find the Least Common Multiple (LCM) of the two denominators, 25 and 10.

LCM(25, 10) = 50

50 is the smallest positive integer that is a multiple of both 25 and 10.

4

Substitute the results into the formula and convert the fraction to a decimal.

3 / 50 = 0.06

Dividing 3 by 50 yields the terminating decimal 0.06, which is the final required format.

Key Concept

HCF and LCM of fractions

Question 38Question →

In a single row of books on a library shelf, an Encyclopedia is $28^{\text{th}}$ from the left end and a Dictionary is $35^{\text{th}}$ from the right end. If there are exactly 12 books placed strictly between them, what is the minimum possible total number of books in this row?

Show answer & explanation

Answer: 49

Answer

The minimum possible total number of books is 49.

The minimum total is found using the overlapping queue formula:

L + R - M - 2

. Substituting the given values yields

28 + 35 - 12 - 2 = 49

.

Step-by-Step Solution

1

Identify the queue configuration for the minimum capacity.

The minimum possible number of books occurs in an overlapping scenario.

In an overlapping scenario, the item counted from the right (Dictionary) is placed to the left of the item counted from the left (Encyclopedia), yielding the smallest overall total length.

2

Apply the minimum capacity formula.

\text{Total} = L + R - M - 2

We add the rank from the left (

L

) and the rank from the right (

R

), then subtract the number of items strictly between them (

M

) and the 2 reference items to correct for double-counting.

3

Substitute the given values and calculate the final count.

\text{Total} = 28 + 35 - 12 - 2 = 49

This yields the exact minimum count of books on the shelf.

Key Concept

Overlapping Minimum Capacity in Ordering and Ranking

Estimated Time:1m 0s

Question 39Question →

A secure digital vault requires a 2-digit numerical access code. The code is determined by calculating the value of the expression $(P + Q) - (R \times S)$ , where the variables are defined based on the mathematical classification of numbers:

* $P$ : The sum of all single-digit prime numbers.
* $Q$ : The product of the smallest positive composite number and the smallest odd prime number.
* $R$ : The total count of whole numbers strictly less than $10$ that are classified as neither prime nor composite.
* $S$ : The smallest positive integer $n$ for which the expression $\sqrt{12n}$ results in a rational number.

What is the final numerical access code?

Show answer & explanation

Answer: 23

Answer

23

The correct calculation evaluates each subset definition perfectly: P = 17 (sum of primes 2, 3, 5, 7), Q = 12 (4 * 3), R = 2 (0 and 1 are neither prime nor composite), and S = 3 (making 12 * 3 = 36 a perfect square). Plugging these into the equation (17 + 12) - (2 * 3) yields exactly 23.

Step-by-Step Solution

1

Identify single-digit primes and sum them to find P.

P = 17

The single-digit primes are 2, 3, 5, and 7. Summing them yields 2 + 3 + 5 + 7 = 17.

2

Identify the smallest composite and smallest odd prime to find Q.

Q = 12

The smallest positive composite number is 4, and the smallest odd prime number is 3. Multiplying them yields 4 * 3 = 12.

3

Count whole numbers less than 10 that are neither prime nor composite to find R.

R = 2

The set of whole numbers strictly less than 10 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Only 0 and 1 fit the classification of being neither prime nor composite.

4

Find the smallest positive integer n making \sqrt{12n} rational.

S = 3

For the square root to be rational, 12n must be a perfect square. Since 12 factors to 2^2 * 3, the smallest integer n to pair the remaining 3 is 3.

5

Evaluate the final expression.

23

(P + Q) - (R * S) = (17 + 12) - (2 * 3) = 29 - 6 = 23.

Key Concept

Classification properties of primes, composites, whole numbers, and rational numbers.

Question 40Question →

A mechanical watch, which gains time at a continuous and uniform rate, is observed to be exactly 5 minutes slow at 8:00 AM on a Sunday. By 8:00 AM on the immediately following Tuesday, the same watch is exactly 7 minutes fast. How many hours after 8:00 AM on that Sunday did the watch display the true, correct time?

Show answer & explanation

Answer: 20

Answer

20

The correct answer is derived by mapping the total error shift over the total elapsed time. The watch gains a total of 12 minutes over a span of 48 hours. This establishes a constant gain rate of 1 minute every 4 hours. Because the watch started exactly 5 minutes slow, it requires

5 \times 4 = 20

hours of elapsed true time for it to catch up and display the correct, synchronized time.

Step-by-Step Solution

1

Calculate the total true time elapsed between the two observations.

48 hours

To find the rate at which the watch gains time, we first need to determine the total duration of the period. From Sunday 8:00 AM to Tuesday 8:00 AM is precisely two full days, which equals 48 hours.

2

Calculate the total amount of time the watch gained over this period.

12 minutes

The watch transitions from being 5 minutes slow (-5) to being 7 minutes fast (+7). The total change in its display relative to true time is

7 - (-5) = 12

minutes.

3

Determine the uniform rate at which the watch gains time.

1 minute gained every 4 hours

By dividing the total elapsed time by the total minutes gained (

48 \div 12

), we discover that the watch gains exactly 1 minute for every 4 hours of true time that pass.

4

Calculate the hours needed to make up the initial 5-minute deficit.

20 hours

In order to display the correct time, the watch must gain precisely the 5 minutes it was initially lagging behind. At a steady rate of 1 minute per 4 hours, this process will take

5 \times 4 = 20

hours.

Key Concept

Uniform time gain and synchronization of faulty clocks

Answer

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