Tüm alıştırma soruları

What is the simplified numerical value of the mathematical expression

25 - \left[ 10 + \left\{ 8 \div \left( 6 - \overline{4 - 2} \right) \right\} \right]

when evaluated using the standard BODMAS rule?

13

17

11

7

Cevabı ve açıklamayı göster

Cevap: $13$

Cevap

The correct simplified value of the given expression is 13.

Evaluating the expression step-by-step using VBODMAS rules:
1. Vinculum:

\overline{4 - 2} = 2

2. Round brackets:

6 - 2 = 4

3. Curly brackets:

8 \div 4 = 2

4. Square brackets:

10 + 2 = 12

5. Final subtraction:

25 - 12 = 13

.
Thus, the correct value is 13.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum (bar)

\overline{4 - 2} = 2

According to VBODMAS, operations under a vinculum (bar) have the highest precedence.

2

Simplify the innermost round brackets

6 - 2 = 4

Substitute the vinculum result into the round brackets

(6 - 2)

.

3

Evaluate the division inside the curly brackets

8 \div 4 = 2

Division within the curly braces

\{8 \div 4\}

takes precedence.

4

Evaluate the addition inside the square brackets

10 + 2 = 12

Add the terms inside the square brackets

[10 + 2]

.

5

Perform the final subtraction

25 - 12 = 13

Subtract the bracket result from 25.

Anahtar Kavram

Order of operations following the VBODMAS rule (Vinculum, Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Soru 822Soru →

Consider the positive integer $N = 2^4 \times 3^3 \times 5^2$ . Which of the following statements regarding the properties of the factors and multiples of $N$ is/are correct?

Geçerli olan tümünü seçin

The number of positive factors of

N

that are perfect squares is equal to 12.

The sum of the reciprocals of all positive factors of

N

is equal to

\frac{\sigma(N)}{N}

, where

\sigma(N)

is the sum of all positive factors of

N

.

The number of ordered pairs of positive integers

(x, y)

such that

\text{lcm}(x, y) = N

is equal to 315.

The product of all positive factors of

N

is equal to

N^{60}

.

Cevabı ve açıklamayı göster

Cevap: The number of positive factors of $N$ that are perfect squares is equal to 12.; The sum of the reciprocals of all positive factors of $N$ is equal to $\frac{\sigma(N)}{N}$ , where $\sigma(N)$ is the sum of all positive factors of $N$ .; The number of ordered pairs of positive integers $(x, y)$ such that $\text{lcm}(x, y) = N$ is equal to 315.

Cevap

The statements asserting that the number of perfect square factors is 12, the sum of factor reciprocals equals

\frac{\sigma(N)}{N}

, and the number of ordered pairs

(x, y)

such that

\text{lcm}(x, y) = N

is 315 are all correct.

The statements asserting that

N

has 12 perfect square factors, that the sum of reciprocals of all factors is

\frac{\sigma(N)}{N}

, and that there are 315 ordered pairs

(x, y)

with

\text{lcm}(x, y) = N

are all mathematically correct applications of prime factorization principles.

Adım Adım Çözüm

1

Determine the number of perfect square factors of

N = 2^4 \times 3^3 \times 5^2

.

Any factor

d = 2^a \times 3^b \times 5^c

is a perfect square if

a, b, c

are even. The allowed values are

a \in \{0, 2, 4\}

(3 options),

b \in \{0, 2\}

(2 options), and

c \in \{0, 2\}

(2 options). Total perfect square factors =

3 \times 2 \times 2 = 12

.

Perfect square factors require non-negative even exponents for each prime factor.

2

Evaluate the sum of the reciprocals of all factors of

N

.

As

d

ranges over all divisors of

N

,

\frac{N}{d}

also ranges over all divisors of

N

. Therefore,

\sum_{d|N} \frac{1}{d} = \sum_{d|N} \frac{N/d}{N} = \frac{1}{N} \sum_{d'|N} d' = \frac{\sigma(N)}{N}

.

Pairing each factor

d

with its complementary factor

N/d

converts the reciprocal sum into the total factor sum divided by

N

.

3

Calculate the number of ordered pairs

(x, y)

such that

\text{lcm}(x, y) = N

.

For each prime

p_i^{e_i}

, the maximum exponent in

x

and

y

must be

e_i

. The number of choices for

(a_i, b_i)

such that

\max(a_i, b_i) = e_i

is

2e_i + 1

. For

N = 2^4 \times 3^3 \times 5^2

, total ordered pairs =

(2(4)+1)(2(3)+1)(2(2)+1) = 9 \times 7 \times 5 = 315

.

The exponent of each prime factor in

\text{lcm}(x, y)

is determined independently by

\max(a_i, b_i) = e_i

.

4

Compute the product of all positive factors of

N

.

Total number of factors

D(N) = (4+1)(3+1)(2+1) = 60

. The product of all factors is

N^{D(N)/2} = N^{60/2} = N^{30}

.

Factors can be grouped into

D(N)/2

pairs, each multiplying to

N

.

Anahtar Kavram

Advanced properties of prime factorization including perfect square factor enumeration, reciprocal factor sum identity, LCM pair counting formula, and factor product relations.

Soru 823Soru →

Evaluate the expression given below by applying the standard order of operations (VBODMAS):

2.5 \times \left[ 18 \div \left\{ 4.5 + \frac{1}{2} \text{ of } \left( 7.8 - \overline{3.4 + 1.4} \right) \right\} \right] + 3.5

Which of the following represents the correct simplified result?

11

16.25

6.5

17.25

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The correct simplified result is 11.

Evaluating step-by-step using VBODMAS order yields: Vinculum gives 4.8; Round bracket yields 7.8 - 4.8 = 3.0; 'of' operation gives 0.5 * 3.0 = 1.5; Curly brackets give 4.5 + 1.5 = 6.0; Square bracket gives 18 / 6.0 = 3; Final calculation gives 2.5 * 3 + 3.5 = 7.5 + 3.5 = 11.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum (bar line).

\overline{3.4 + 1.4} = 4.8

According to VBODMAS, the vinculum takes precedence over all standard brackets.

2

Simplify the innermost round brackets.

7.8 - 4.8 = 3.0

Resolve terms within parenthetical brackets next.

3

Execute the 'of' operation inside the curly brackets.

\frac{1}{2} \text{ of } 3.0 = 1.5

The 'of' operator represents multiplication with higher priority than division and standard addition.

4

Complete the addition within the curly brackets.

4.5 + 1.5 = 6.0

Finish evaluating all operations contained within the curly brackets.

5

Perform division inside the square brackets.

18 \div 6.0 = 3

Evaluate the remaining square bracket operation.

6

Perform final multiplication and addition operations from left to right according to operator precedence.

2.5 \times 3 + 3.5 = 7.5 + 3.5 = 11

Multiplication takes precedence over final addition.

Anahtar Kavram

VBODMAS Rule (Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction)

Tahmini Süre:1m 15s

Soru 824Soru →

What is the unit digit of the expression $P = (678^{120} + 359^{45}) \times 243^{37} - 714^{82}$ ?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The unit digit of the expression is 9.

Evaluating each term using cyclicity rules gives

678^{120} \rightarrow 6

,

359^{45} \rightarrow 9

,

243^{37} \rightarrow 3

, and

714^{82} \rightarrow 6

. Combining these gives

(6 + 9) \times 3 - 6 \equiv 15 \times 3 - 6 \equiv 5 \times 3 - 6 \equiv 15 - 6 \equiv 9 \pmod{10}

.

Adım Adım Çözüm

1

Calculate the unit digit of

678^{120}

Unit digit is 6

The cyclicity of 8 is 4 (8, 4, 2, 6). Exponent 120 mod 4 = 0, which corresponds to the 4th term in the cycle.

2

Calculate the unit digit of

359^{45}

Unit digit is 9

The cyclicity of 9 is 2 (9 for odd powers, 1 for even powers). Exponent 45 is odd.

3

Find the unit digit of

(678^{120} + 359^{45})

Unit digit is 5

Sum of unit digits is 6 + 9 = 15, which has unit digit 5.

4

Calculate the unit digit of

243^{37}

Unit digit is 3

The cyclicity of 3 is 4 (3, 9, 7, 1). Exponent 37 mod 4 = 1, so the unit digit is 3^1 = 3.

5

Multiply the sum by

243^{37}

Unit digit is 5

Product of unit digits is 5 * 3 = 15, which has unit digit 5.

6

Calculate the unit digit of

714^{82}

Unit digit is 6

The cyclicity of 4 is 2 (4 for odd powers, 6 for even powers). Exponent 82 is even.

7

Compute the final unit digit of the overall expression

Unit digit is 9

Subtracting unit digits yields 5 - 6. Adding 10 for borrowing gives 15 - 6 = 9.

Anahtar Kavram

Unit Digit and Cyclicity Rules for Exponential Expressions

Tahmini Süre:1m 30s

Soru 825Soru →

Arrange the following real numbers in ascending order (from smallest to largest):

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order is

\frac{23}{40} < 0.\overline{57} < 0.5\overline{7} < \frac{7}{12}

.

Converting all numbers to decimal expansions gives:

\frac{23}{40} = 0.5750

,

0.\overline{57} = 0.575757...

,

0.5\overline{7} = 0.577777...

, and

\frac{7}{12} = 0.583333...

. Comparing these digit by digit confirms the correct ascending sequence is

\frac{23}{40} < 0.\overline{57} < 0.5\overline{7} < \frac{7}{12}

.

Adım Adım Çözüm

1

Convert each fraction and recurring decimal into its expanded decimal representation.

\frac{23}{40} = 0.57500...

,

0.\overline{57} = 0.575757...

,

0.5\overline{7} = 0.577777...

, and

\frac{7}{12} = 0.583333...

Converting all terms into a common decimal format allows direct place-value comparison.

2

Compare the values digit by digit from left to right starting at the tenths place.

All terms start with

0.57

, except

\frac{7}{12}

which is

0.58...

(making

\frac{7}{12}

the largest). For the remaining three terms, compare the thousandths digit:

\frac{23}{40}

has

0

in the ten-thousandths place (

0.5750

),

0.\overline{57}

has

7

(

0.5757

), and

0.5\overline{7}

has

7

in the thousandths place (

0.5777

).

Determining place-value differences establishes the relative magnitudes clearly.

3

Sequence the original numbers according to their evaluated decimal order.

0.5750 < 0.5757... < 0.5777... < 0.5833... \implies \frac{23}{40} < 0.\overline{57} < 0.5\overline{7} < \frac{7}{12}

Re-substituting the original expressions yields the required ascending order sequence.

Anahtar Kavram

Ordering and comparison of rational numbers, terminating fractions, pure recurring decimals, and mixed recurring decimals.

Soru 826Soru →

What is the simplified numerical value of the expression $15 + \left\{ 20 - \left( 8 \div \overline{6 - 2} \right) \right\}$ when evaluated using the standard BODMAS rule?

Cevabı ve açıklamayı göster

Cevap: 33

Cevap

The simplified numerical value of the expression is 33.

Applying VBODMAS order (Vinculum

\rightarrow

Brackets

\rightarrow

Division

\rightarrow

Subtraction

\rightarrow

Addition) yields:

\overline{6-2} = 4

,

8 \div 4 = 2

,

20 - 2 = 18

, and

15 + 18 = 33

.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum (bar)

\overline{6 - 2} = 4

The vinculum (bar line) takes precedence over standard operators.

2

Perform division inside the round brackets

8 \div 4 = 2

Operations inside the innermost round brackets are evaluated next.

3

Perform subtraction inside the curly brackets

20 - 2 = 18

Evaluate the expression enclosed in the curly brackets.

4

Perform final addition outside brackets

15 + 18 = 33

Complete the evaluation by adding the remaining outer terms.

Anahtar Kavram

BODMAS / VBODMAS Rule (Order of Operations with Vinculum and Brackets)

Soru 827Soru →

What is the simplified numerical value of the following mathematical expression when evaluated strictly according to the BODMAS rule?

25\% \text{ of } 120 - \left[ 4 \times \left\{ 15 - \left( 18 \div \overline{7 - 4} \right) \right\} \right] \div 1.5

6

-4

54

16

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

Evaluating the vinculum

\overline{7 - 4}

yields

3

. The round bracket becomes

18 \div 3 = 6

. The curly bracket yields

15 - 6 = 9

. The square bracket evaluates to

4 \times 9 = 36

. Concurrently,

25\%

of

120

equals

30

. Following operator hierarchy, division is executed next:

36 \div 1.5 = 24

. Finally, performing subtraction gives

30 - 24 = 6

.

Adım Adım Çözüm

1

Evaluate the expression under the bar (vinculum)

\overline{7 - 4} = 3

According to BODMAS, vinculum (bar bracket) has the highest priority among all grouping symbols.

2

Evaluate the innermost round bracket

18 \div 3 = 6

Perform division inside the round bracket after resolving the vinculum.

3

Evaluate the curly bracket

15 - 6 = 9

Subtract the result of the round bracket from 15 inside the curly bracket.

4

Evaluate the square bracket and the percentage term

4 \times 9 = 36

and

25\% \text{ of } 120 = \frac{25}{100} \times 120 = 30

Multiply 4 by the curly bracket result, and independently calculate the percentage term.

5

Perform division before final subtraction

36 \div 1.5 = 24

BODMAS mandates performing division before subtraction.

6

Perform the final subtraction

30 - 24 = 6

Subtract the quotient from the percentage value.

Anahtar Kavram

BODMAS Order of Operations with Vinculum and Mixed Numbers

Soru 828Soru →

The table below details the renewable energy generation (in Gigawatt-hours, GWh) across four zones in 2025:

Zone	Solar	Wind	Hydro	Biomass
North	120	150	200	50
South	180	220	160	40
East	90	60	250	80
West	210	190	100	70

What is the total solar energy generation (in GWh) across all four zones combined in 2025?

Cevabı ve açıklamayı göster

Cevap: 600

Cevap

The total solar energy generation across all four zones in 2025 is 600 GWh.

Summing the values in the Solar column (

120 + 180 + 90 + 210

) yields a total of

600

GWh.

Adım Adım Çözüm

1

Locate and extract the values under the 'Solar' column for all four zones.

North = 120, South = 180, East = 90, West = 210.

The question asks specifically for total solar energy generation.

2

Sum the extracted values across all rows.

120 + 180 + 90 + 210 = 600

GWh.

Combining values across all four zones yields the overall total for the solar category.

Anahtar Kavram

Data Extraction and Summation from Tables

Soru 829Soru →

Three quantities are given as $0.\overline{6}$ , $\frac{12}{25}$ , and $1.4$ . What is the exact ratio of the Least Common Multiple (LCM) of these three quantities to their Highest Common Factor (HCF)?

6300

\frac{1}{6300}

420

84

Cevabı ve açıklamayı göster

Cevap: 6300

Cevap

6300

Converting the given values yields the reduced fractions

\frac{2}{3}

,

\frac{12}{25}

, and

\frac{7}{5}

. Applying standard fraction formulas,

\text{LCM} = \frac{\text{LCM}(2,12,7)}{\text{HCF}(3,25,5)} = \frac{84}{1} = 84

and

\text{HCF} = \frac{\text{HCF}(2,12,7)}{\text{LCM}(3,25,5)} = \frac{1}{75}

. Taking the ratio

\frac{\text{LCM}}{\text{HCF}}

gives

84 \div \frac{1}{75} = 6300

.

Adım Adım Çözüm

1

Convert all terms to irreducible fractions

0.\overline{6} = \frac{6}{9} = \frac{2}{3}

,

\frac{12}{25} = \frac{12}{25}

, and

1.4 = \frac{14}{10} = \frac{7}{5}

All numbers must be expressed as simplified fractions

\frac{a}{b}

in lowest terms before applying fraction HCF and LCM formulas.

2

Calculate the Least Common Multiple (LCM) of the fractions

\text{LCM}\left(\frac{2}{3}, \frac{12}{25}, \frac{7}{5}\right) = \frac{\text{LCM}(2, 12, 7)}{\text{HCF}(3, 25, 5)} = \frac{84}{1} = 84

The LCM of a set of fractions is given by

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

.

3

Calculate the Highest Common Factor (HCF) of the fractions

\text{HCF}\left(\frac{2}{3}, \frac{12}{25}, \frac{7}{5}\right) = \frac{\text{HCF}(2, 12, 7)}{\text{LCM}(3, 25, 5)} = \frac{1}{75}

The HCF of a set of fractions is given by

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

.

4

Find the ratio of the LCM to the HCF

\text{Ratio} = \frac{84}{\frac{1}{75}} = 84 \times 75 = 6300

Dividing the computed LCM by the computed HCF yields the required ratio.

Anahtar Kavram

HCF and LCM of Fractions and Recurring Decimals

Soru 830Soru →

A positive integer $N$ has exactly $12$ positive factors. The sum of all distinct prime factors of $N$ is $10$ . If $N$ is divisible by $12$ but not divisible by $9$ , what is the total number of positive factors of $5N$ ?

15

16

18

24

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The total number of positive factors of

5N

is

18

.

The distinct prime factors of

N

that sum to

10

and satisfy divisibility by

12

are

2, 3,

and

5

. Because

N

has

12

factors, its exponents must be

2, 1, 1

. Since

N

is divisible by

12

and not divisible by

9

,

N = 2^2 \times 3^1 \times 5^1 = 60

. Consequently,

5N = 2^2 \times 3^1 \times 5^2

, which possesses

(2+1)(1+1)(2+1) = 18

factors.

Adım Adım Çözüm

1

Determine the set of distinct prime factors of

N

The distinct prime factors of

N

are

2

,

3

, and

5

.

The sum of distinct prime factors is

10

. Possible sets of distinct primes summing to

10

are

\{3, 7\}

and

\{2, 3, 5\}

. Because

N

is divisible by

12

, its prime factorization must include

2

and

3

, forcing the set of prime factors to be

\{2, 3, 5\}

.

2

Determine the exponent structure of

N

The prime factorization of

N

takes the form

p_1^2 \times p_2^1 \times p_3^1

for distinct primes

p_1, p_2, p_3

.

The number of positive factors is given by

(a_1+1)(a_2+1)(a_3+1) = 12

. The only integer factorization of

12

into three factors greater than

1

is

3 \times 2 \times 2

, requiring exponents

2, 1, 1

in some arrangement.

3

Assign exponents to prime factors

2

,

3

, and

5

N = 2^2 \times 3^1 \times 5^1 = 60

.

Since

N

is divisible by

12 = 2^2 \times 3^1

, the exponent of

2

must be at least

2

and the exponent of

3

at least

1

. Since

N

is not divisible by

9 = 3^2

, the exponent of

3

cannot be

2

or greater, so it must be exactly

1

. Thus, the prime

2

takes exponent

2

, leaving prime

5

with exponent

1

.

4

Calculate the prime factorization and factor count of

5N

5N = 2^2 \times 3^1 \times 5^2

, which has

18

positive factors.

Multiplying

N

by

5

increments the exponent of

5

from

1

to

2

. The factor count formula yields

(2+1) \times (1+1) \times (2+1) = 3 \times 2 \times 3 = 18

.

Anahtar Kavram

Factors, Multiples, and Prime Factorization

Soru 831Soru →

The following missing data table presents the annual cargo volume (in Thousand Metric Tonnes, TMT) handled across three terminals (Terminal A, Terminal B, and Terminal C) at a major seaport for three distinct cargo categories: Bulk Cargo, Liquid Cargo, and Containerized Cargo.

Terminal	Bulk Cargo	Liquid Cargo	Containerized Cargo	Total
Terminal A	120	—	150	450
Terminal B	—	210	180	500
Terminal C	160	140	—	420
Total	390	530	450	1370

Based on the table above, what is the volume of Liquid Cargo (in Thousand Metric Tonnes) handled by Terminal A?

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

The volume of Liquid Cargo handled by Terminal A is 180 Thousand Metric Tonnes.

The row total for Terminal A is

450\text{ TMT}

. The known components are Bulk Cargo (

120\text{ TMT}

) and Containerized Cargo (

150\text{ TMT}

). Subtracting these from the total yields

450 - (120 + 150) = 180\text{ TMT}

. Alternatively, using the column total for Liquid Cargo (

530\text{ TMT}

) minus the Liquid Cargo of Terminal B (

210\text{ TMT}

) and Terminal C (

140\text{ TMT}

) gives

530 - 350 = 180\text{ TMT}

.

Adım Adım Çözüm

1

Examine the row for Terminal A to find total cargo and known cargo types

Total cargo for Terminal A =

450\text{ TMT}

, Bulk Cargo =

120\text{ TMT}

, Containerized Cargo =

150\text{ TMT}

.

The sum of all cargo categories for a terminal equals the row total.

2

Calculate the sum of known cargo categories for Terminal A

120 + 150 = 270\text{ TMT}

.

Combining known entries simplifies finding the unknown value.

3

Subtract the sum of known categories from the row total

450 - 270 = 180\text{ TMT}

.

Liquid Cargo for Terminal A is the remaining portion of its total cargo volume.

4

Cross-verify using the column total for Liquid Cargo

Total Liquid Cargo =

530\text{ TMT}

. Liquid Cargo for Terminal B =

210\text{ TMT}

and Terminal C =

140\text{ TMT}

. Terminal A Liquid Cargo =

530 - (210 + 140) = 530 - 350 = 180\text{ TMT}

.

Cross-verification confirms numerical consistency.

Anahtar Kavram

Missing Data Interpretation via Row and Column Equations

Soru 832Soru →

What is the unit digit of the value obtained from the numerical expression $N = (512^{160} + 737^{203}) \times 434^{116} - 618^{140}$ ?

3

2

8

4

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

Evaluating each term using cyclicity rules gives

512^{160} \rightarrow 6

,

737^{203} \rightarrow 3

,

434^{116} \rightarrow 6

, and

618^{140} \rightarrow 6

. The composite expression evaluates to

(6 + 3) \times 6 - 6 = 54 - 6

. Performing the unit digit subtraction

4 - 6

with borrowing yields

14 - 6 = 8

.

Adım Adım Çözüm

1

Determine the unit digits of the terms inside the parentheses:

512^{160}

and

737^{203}

Unit digit of

512^{160}

is

6

; unit digit of

737^{203}

is

3

.

For

512

, the base unit digit is

2

(cyclicity pattern 2, 4, 8, 6). Exponent

160 \div 4

leaves remainder

0

, indicating the

4^{\text{th}}

power in cyclicity:

2^4 = 16 \rightarrow 6

. For

737

, the base unit digit is

7

(cyclicity pattern 7, 9, 3, 1). Exponent

203 \div 4

leaves remainder

3

, indicating

7^3 = 343 \rightarrow 3

.

2

Add the unit digits inside the parentheses and multiply by the unit digit of

434^{116}

Unit digit of the product term is

4

.

Sum inside parentheses:

6 + 3 = 9

. For

434^{116}

, the base unit digit is

4

(cyclicity pattern 4, 6). An even exponent

116

gives

4^2 = 16 \rightarrow 6

. The product is

9 \times 6 = 54 \rightarrow 4

.

3

Subtract the unit digit of

618^{140}

from the product's unit digit

The final unit digit is

8

.

For

618^{140}

, base unit digit is

8

(cyclicity pattern 8, 4, 2, 6). Exponent

140 \div 4

leaves remainder

0

, giving

8^4 = 4096 \rightarrow 6

. Subtracting gives

4 - 6

. Since

4 < 6

, add

10

to borrow from the tens place:

14 - 6 = 8

.

Anahtar Kavram

Cyclicity of Unit Digits and Modular Subtraction

Tahmini Süre:1m 30s

Soru 833Soru →

The table below presents the operational metrics for Agricultural Processing Units across four districts of a state for FY 2025–26:

District	Total Processing Capacity (in '000 MT)	Operational Units Count	Total Raw Produce Received (in '000 MT)	Processing Efficiency Rate (%)
District A	450	18	360	$85\%$
District B	600	24	480	$90\%$
District C	500	20	420	$80\%$
District D	350	14	320	$75\%$

*Note: Processing Efficiency Rate is defined as the percentage of Total Raw Produce Received that is converted into finished products. The remaining portion of received raw produce is lost as processing waste.*

What is the overall average processing waste (in '000 Metric Tonnes) per operational unit across all four districts combined?

Cevabı ve açıklamayı göster

Cevap: 3.5

Cevap

The overall average processing waste per operational unit across all four districts combined is

3.5

thousand Metric Tonnes.

The correct answer is derived by first finding the waste volume for each district based on raw produce received: District A (

360 \times 0.15 = 54

), District B (

480 \times 0.10 = 48

), District C (

420 \times 0.20 = 84

), and District D (

320 \times 0.25 = 80

). Summing these gives a combined total waste of

266

thousand MT. Dividing this total waste by the total count of operational units (

18 + 24 + 20 + 14 = 76

) yields

\frac{266}{76} = 3.5

thousand MT per unit.

Adım Adım Çözüm

1

Calculate the processing waste percentage for each district

District A waste rate =

100\% - 85\% = 15\%

; District B waste rate =

100\% - 90\% = 10\%

; District C waste rate =

100\% - 80\% = 20\%

; District D waste rate =

100\% - 75\% = 25\%

.

Processing waste is the complement of processing efficiency rate.

2

Calculate total processing waste generated in each district in '000 MT

District A waste =

360 \times 0.15 = 54

'000 MT; District B waste =

480 \times 0.10 = 48

'000 MT; District C waste =

420 \times 0.20 = 84

'000 MT; District D waste =

320 \times 0.25 = 80

'000 MT.

Absolute waste is obtained by multiplying Total Raw Produce Received by the waste rate.

3

Calculate total waste and total operational units across all four districts combined

Total Waste =

54 + 48 + 84 + 80 = 266

'000 MT. Total Operational Units =

18 + 24 + 20 + 14 = 76

units.

Aggregating metrics across districts is required to find the combined weighted average.

4

Divide total combined waste by total combined operational units

Average Waste per Unit =

\frac{266}{76} = 3.5

'000 MT per unit.

Overall average metric equals total aggregate waste divided by total aggregate count of units.

Anahtar Kavram

Weighted Average and Percentage Calculation from Data Grids

Tahmini Süre:2m 30s

Soru 834Soru →

Let $N = 2^a \times 3^b \times 7^c$ be a positive integer, where $a$ , $b$ , and $c$ are positive integers. The number of positive factors of $N$ that are divisible by $6$ is $36$ , the number of positive factors of $N$ that are divisible by $14$ is $32$ , and the number of positive factors of $N$ that are divisible by $21$ is $30$ . What is the total number of positive factors of $N$ ?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The total number of positive factors of

N

is 60.

By setting up the system of equations based on exponent constraints for each divisor (

6, 14, 21

), we find

a=4, b=3, c=2

. Substituting these into the total factor formula

(a+1)(b+1)(c+1)

yields

5 \times 4 \times 3 = 60

.

Adım Adım Çözüm

1

Formulate algebraic expressions for the constrained factor counts

ab(c+1) = 36

,

ac(b+1) = 32

, and

bc(a+1) = 30

A factor

2^x \times 3^y \times 7^z

is divisible by

6 = 2^1 \times 3^1

if

x \ge 1

,

y \ge 1

,

z \ge 0

, yielding

a \cdot b \cdot (c+1)

factors. Similarly, divisibility by

14

requires

x \ge 1, z \ge 1

, giving

a \cdot c \cdot (b+1)

, and divisibility by

21

requires

y \ge 1, z \ge 1

, giving

b \cdot c \cdot (a+1)

.

2

Expand equations and substitute

P = abc

ab = 36 - P

,

ac = 32 - P

,

bc = 30 - P

Expanding

ab(c+1) = abc + ab = 36

gives

ab = 36 - P

. Applying the same expansion to the other two equations gives expressions for

ac

and

bc

in terms of

P

.

3

Multiply the three equations together and solve for

P

(ab)(ac)(bc) = (abc)^2 = P^2 \Rightarrow (36-P)(32-P)(30-P) = P^2 \Rightarrow P = 24

Testing positive integer values of

P

that divide into products reveals that for

P = 24

,

(36-24)(32-24)(30-24) = 12 \times 8 \times 6 = 576 = 24^2

.

4

Calculate the values of

ab

,

ac

,

bc

, and solve for

a

,

b

,

c

ab = 12

,

ac = 8

,

bc = 6

, giving

a = 4

,

b = 3

,

c = 2

a^2 = \frac{(ab)(ac)}{bc} = \frac{12 \times 8}{6} = 16 \Rightarrow a = 4

. Similarly,

b^2 = \frac{12 \times 6}{8} = 9 \Rightarrow b = 3

, and

c^2 = \frac{8 \times 6}{12} = 4 \Rightarrow c = 2

.

5

Compute the total number of positive factors of

N

(4+1)(3+1)(2+1) = 5 \times 4 \times 3 = 60

The total number of factors of

N = 2^a \times 3^b \times 7^c

is

(a+1)(b+1)(c+1)

.

Anahtar Kavram

Constrained Factor Counting and Prime Factorization Systems

Soru 835Soru →

If $3^{x + 2} = 81$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

x

is 2.

Rewriting 81 as

3^4

gives

3^{x+2} = 3^4

. Since the bases are equal, setting the exponents equal gives

x + 2 = 4

, which solves to

x = 2

.

Adım Adım Çözüm

1

Express the right-hand side with a base of 3

81 = 3^4

Both sides must have the same base to apply the laws of indices.

2

Equate the exponents

x + 2 = 4

If

a^m = a^n

(where

a > 0

and

a \neq 1

), then

m = n

.

3

Solve for

x

x = 2

Subtracting 2 from both sides isolates

x

.

Anahtar Kavram

Equating exponents of equal bases

Soru 836Soru →

What is the unit digit of the expression $K = (377^{123} + 428^{164}) \times 533^{200} - 619^{86}$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The unit digit of the expression is 8.

Using cyclicity principles for each power term,

377^{123}

yields unit digit 3 (remainder 3),

428^{164}

yields unit digit 6 (remainder 0

\rightarrow

4th power),

533^{200}

yields unit digit 1 (remainder 0

\rightarrow

4th power), and

619^{86}

yields unit digit 1 (even power of 9). Combining these gives

(3 + 6) \times 1 - 1 = 8

.

Adım Adım Çözüm

1

Find the unit digit of

377^{123}

Unit digit is 3

The unit digit of 377 is 7, which follows a cyclicity pattern of 4 (7, 9, 3, 1). Dividing the exponent 123 by 4 leaves a remainder of 3. Thus, the unit digit is given by

7^3 = 343 \rightarrow 3

.

2

Find the unit digit of

428^{164}

Unit digit is 6

The unit digit of 428 is 8, which has a cyclicity of 4 (8, 4, 2, 6). The exponent 164 is divisible by 4 (remainder 0), which corresponds to the 4th position in the cyclicity cycle (

8^4 = 4096 \rightarrow 6

).

3

Evaluate the sum inside the parentheses

(377^{123} + 428^{164})

Unit digit is 9

Adding the individual unit digits gives

3 + 6 = 9

.

4

Find the unit digit of

533^{200}

Unit digit is 1

The unit digit of 533 is 3, which has a cyclicity of 4 (3, 9, 7, 1). Exponent 200 is divisible by 4 (remainder 0), corresponding to the 4th power in the cycle (

3^4 = 81 \rightarrow 1

).

5

Multiply the bracketed sum by

533^{200}

Unit digit is 9

Multiplying the unit digits yields

9 \times 1 = 9

.

6

Find the unit digit of

619^{86}

Unit digit is 1

The unit digit of 619 is 9, which has a cyclicity of 2 (9 for odd powers, 1 for even powers). Since 86 is even, the unit digit is 1.

7

Subtract

619^{86}

from the product

Unit digit is 8

Subtracting the unit digits gives

9 - 1 = 8

.

Anahtar Kavram

Unit Digit and Cyclicity

Soru 837Soru →

A district officer allocates a development fund among four sectors: Health, Education, Infrastructure, and Agriculture. Health receives a fraction equal to $0.1\overline{6}$ of the total fund. Education receives $\frac{3}{8}$ of the remaining fund after the Health allocation. Infrastructure is directly allocated $0.4\overline{3}$ of the original total fund. The remaining amount is given to Agriculture. If the amount allocated to Infrastructure exceeds the amount allocated to Education by ₹ $87,000$ , what is the total development fund in Rupees?

Cevabı ve açıklamayı göster

Cevap: 720000

Cevap

The total development fund is ₹ 720,000.

Converting the recurring decimals gives Health's share as

\frac{1}{6}

and Infrastructure's share as

\frac{13}{30}

. The remaining fund after Health allocation is

\frac{5}{6}

, making Education's share equal to

\frac{3}{8} \times \frac{5}{6} = \frac{5}{16}

of the total. The difference between Infrastructure's and Education's shares is

\frac{13}{30} - \frac{5}{16} = \frac{29}{240}

. Equating

\frac{29}{240}

of the total fund to ₹ 87,000 yields the total fund as ₹ 720,000.

Adım Adım Çözüm

1

Convert the recurring decimals into simplified vulgar fractions.

0.1\overline{6} = \frac{16-1}{90} = \frac{15}{90} = \frac{1}{6}

and

0.4\overline{3} = \frac{43-4}{90} = \frac{39}{90} = \frac{13}{30}

.

Converting mixed recurring decimals to fractions allows exact algebraic calculations.

2

Calculate the fraction of the total fund received by Education.

Remaining fund after Health =

1 - \frac{1}{6} = \frac{5}{6}

. Education's fraction =

\frac{3}{8} \times \frac{5}{6} = \frac{5}{16}

.

Education receives a fraction of the remaining fund, not the original total fund.

3

Find the difference between Infrastructure's fraction and Education's fraction.

\frac{13}{30} - \frac{5}{16} = \frac{104 - 75}{240} = \frac{29}{240}

.

The LCM of denominators 30 and 16 is 240.

4

Set up the linear equation with the given monetary difference to find the total fund

F

.

\frac{29}{240} \times F = 87,000 \implies F = \frac{87,000 \times 240}{29} = 3,000 \times 240 = 720,000

.

Solving for

F

yields the original total fund allocation.

Anahtar Kavram

Converting mixed recurring decimals to vulgar fractions and evaluating multi-step compound fractional parts of a total quantity.

Soru 838Soru →

If $x = \sqrt{6 + 2\sqrt{5}} - \sqrt{6 - 2\sqrt{5}}$ and $\left(\frac{1}{27}\right)^{x - 5} = 9^{y + 1}$ , what is the value of $x + y$ ?

\frac{11}{2}

\frac{9}{2}

-\frac{7}{2}

10

Cevabı ve açıklamayı göster

Cevap: $\frac{11}{2}$

Cevap

The value of

x + y

is

\frac{11}{2}

.

Simplifying

\sqrt{6 + 2\sqrt{5}}

yields

\sqrt{5} + 1

and

\sqrt{6 - 2\sqrt{5}}

yields

\sqrt{5} - 1

. Taking their difference gives

x = 2

. Substituting

x = 2

into the exponential equation gives

(1/27)^{-3} = 27^3 = (3^3)^3 = 3^9

. The right side is

9^{y+1} = (3^2)^{y+1} = 3^{2y+2}

. Equating exponents gives

2y + 2 = 9

, which leads to

y = 7/2

. Adding

x

and

y

gives

2 + 7/2 = 11/2

.

Adım Adım Çözüm

1

Simplify the nested surds for

x

\sqrt{6 + 2\sqrt{5}} = \sqrt{5} + 1

and

\sqrt{6 - 2\sqrt{5}} = \sqrt{5} - 1

For a nested surd

\sqrt{a \pm 2\sqrt{b}}

, express

a

as

m+n

and

b

as

m \cdot n

. Here

5+1=6

and

5 \cdot 1 = 5

.

2

Compute the value of

x

x = (\sqrt{5} + 1) - (\sqrt{5} - 1) = 2

Subtracting the second simplified surd from the first cancels out

\sqrt{5}

and adds the integer terms.

3

Express the exponential equation using a common prime base

3

(3^{-3})^{2 - 5} = (3^2)^{y + 1} \implies (3^{-3})^{-3} = 3^{2y + 2} \implies 3^9 = 3^{2y + 2}

Rewriting

\frac{1}{27} = 3^{-3}

and

9 = 3^2

allows direct comparison of exponents.

4

Solve for

y

and evaluate

x + y

2y + 2 = 9 \implies y = \frac{7}{2}

, so

x + y = 2 + \frac{7}{2} = \frac{11}{2}

Equating exponents with identical bases and adding the resolved values of

x

and

y

.

Anahtar Kavram

Simplification of nested surds

\sqrt{a \pm 2\sqrt{b}}

combined with solving exponential equations using common prime bases.

Tahmini Süre:2m 0s

Soru 839Soru →

What is the simplified value of the expression $(256)^{0.16} \times (256)^{0.09}$ ?

4

16

64

1

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

The simplified value of

(256)^{0.16} \times (256)^{0.09}

is

4

.

According to the basic laws of indices, multiplying terms with equal bases requires adding their exponents:

(256)^{0.16} \times (256)^{0.09} = (256)^{0.16 + 0.09} = (256)^{0.25}

. Expressing

0.25

as

\frac{1}{4}

converts the expression to the fourth root of

256

, which equals

4

because

4^4 = 256

.

Adım Adım Çözüm

1

Apply the law of indices

a^m \times a^n = a^{m+n}

(256)^{0.16 + 0.09} = (256)^{0.25}

When multiplying exponential terms with identical bases, add their powers.

2

Convert the decimal exponent to a simplified fraction

0.25 = \frac{25}{100} = \frac{1}{4}

Converting to a fraction simplifies root evaluation.

3

Express

256

as a base with power

4

256 = 4^4

Writing the base as

4^4

allows straightforward simplification with the exponent

\frac{1}{4}

.

4

Evaluate

(4^4)^{1/4}

4^{4 \times \frac{1}{4}} = 4^1 = 4

Apply the power rule

(a^m)^n = a^{m \times n}

to arrive at the final simplified value.

Anahtar Kavram

Laws of Indices: Product Rule (

a^m \times a^n = a^{m+n}

) and Power of a Power Rule (

(a^m)^n = a^{mn}

)

Soru 840Soru →

A train running at a constant speed crosses a stationary platform of length $200\text{ meters}$ . What is the speed of the train in $\text{km/h}$ ?

Statement I: The train takes $20\text{ seconds}$ to completely cross the platform.
Statement II: The train takes $8\text{ seconds}$ to cross a telegraph pole standing beside the track.

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 alone nor Statement II alone is sufficient, and both statements together are not sufficient to answer the question.

Both Statement I and Statement II together are necessary and sufficient to answer the question.

Cevabı ve açıklamayı göster

Cevap: Both Statement I and Statement II together are necessary and sufficient to answer the question.

Cevap

Both Statement I and Statement II together are necessary and sufficient to answer the question.

Evaluating each statement individually shows that neither Statement I nor Statement II alone provides enough information to determine the speed of the train because the length of the train remains unknown. However, combining both statements yields a system of two independent linear equations with two variables (length and speed of the train), allowing us to solve uniquely for the speed of the train (

60\text{ km/h}

). Therefore, both statements together are necessary and sufficient.

Adım Adım Çözüm

1

Analyze Statement I alone.

Let the length of the train be

L\text{ meters}

and its speed be

v\text{ m/s}

. The equation is

L + 200 = 20v

. Since there are two unknowns (

L

and

v

), Statement I alone is not sufficient.

The length of the train is an unknown variable.

2

Analyze Statement II alone.

Crossing a telegraph pole means distance traveled equals the length of the train. The equation is

L = 8v

. Since there are two unknowns (

L

and

v

), Statement II alone is not sufficient.

The speed cannot be determined without knowing the length of the train.

3

Evaluate Statement I and Statement II together.

Substitute

L = 8v

into the first equation:

8v + 200 = 20v \implies 12v = 200 \implies v = \frac{50}{3}\text{ m/s}

. Converting to

\text{km/h}

:

v = \frac{50}{3} \times \frac{18}{5} = 60\text{ km/h}

. Thus, both statements together give a unique answer.

Two independent equations are sufficient to solve for two unknown variables.

Anahtar Kavram

Data Sufficiency evaluation for relative motion and distance-speed problems using linear equations.

Tahmini Süre:1m 30s

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