All practice questions

Question 521Question →

An environmental agency uses three autonomous drones to patrol a protected reserve. The drones fly on continuous looping routes. Drone X completes one full route in $\frac{42}{15}$ hours, Drone Y completes a route in $\frac{35}{20}$ hours, and Drone Z completes a route in $\frac{63}{30}$ hours.

If all three drones depart simultaneously from the base station, after how many hours will they all meet at the base station again for the first time?

Show answer & explanation

Answer: 42

Answer

42

The drones will meet again at a time that is a common multiple of all their individual cycle times. The very first time this happens is represented by the Least Common Multiple (LCM). When calculating the LCM of fractions, it is mathematically required to reduce them to their simplest terms first:

\frac{14}{5}

,

\frac{7}{4}

, and

\frac{21}{10}

. Using the correct formula,

\frac{\text{LCM}(14, 7, 21)}{\text{HCF}(5, 4, 10)}

, we get

\frac{42}{1} = 42

hours.

Step-by-Step Solution

1

Identify the mathematical operation required.

The convergence time is the Least Common Multiple (LCM) of the three cycle times.

The drones will meet again at a time that is a common multiple of all their individual route completion times.

2

Simplify the given fractions to their lowest terms.

\frac{42}{15} \rightarrow \frac{14}{5}

;

\frac{35}{20} \rightarrow \frac{7}{4}

;

\frac{63}{30} \rightarrow \frac{21}{10}

The formula for the LCM of fractions requires all fractions to be in their simplest form to yield the correct result.

3

Calculate the LCM of the simplified numerators.

\text{LCM}(14, 7, 21) = 42

The numerator of the resulting fraction must be divisible by all original numerators.

4

Calculate the HCF of the simplified denominators.

\text{HCF}(5, 4, 10) = 1

The denominator of the resulting fraction must evenly divide all original denominators.

5

Compute the final fraction.

\frac{42}{1} = 42

hours

Applying the formula

\frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}

gives the exact time of the next simultaneous meeting.

Key Concept

Calculating the Least Common Multiple (LCM) of fractions, emphasizing the critical prerequisite of simplifying the fractions first.

Estimated Time:2m 30s

Question 522Question →

Consider the following statements regarding the properties of numbers:

Statement 1: The integer $0$ is neither positive nor negative, but it is classified as an even rational number.
Statement 2: The addition of any rational number and any irrational number always results in an irrational number.
Statement 3: The fraction $\frac{22}{7}$ is an irrational number because it represents the exact value of $\pi$ .

Which of the statements given above is/are mathematically correct?

1 and 2 only

1 and 3 only

2 and 3 only

1, 2, and 3

Show answer & explanation

Answer: 1 and 2 only

Answer

The correct answer includes Statement 1 and Statement 2 only.

The correct answer correctly identifies that Statement 1 and Statement 2 are mathematically sound truths, while Statement 3 is fundamentally flawed. Statement 1 holds because

0

satisfies the definition of an even number (

2n

) and a rational number (

\frac{p}{q}

). Statement 2 holds due to mathematical contradiction proofs regarding sums. Statement 3 is false because

\frac{22}{7}

is a rational number by definition, despite being an approximation for the irrational

\pi

.

Step-by-Step Solution

1

Evaluate the mathematical properties of

0

in Statement 1.

Statement 1 is valid.

Zero has no sign (neither positive nor negative). It is an even integer because it can be expressed as

2 \times 0

. It is a rational number because it can be written as

\frac{0}{1}

.

2

Analyze the closure property of addition for Statement 2.

Statement 2 is valid.

If a rational number

r

is added to an irrational number

x

, the sum must be irrational. (If

r + x = q

where

q

is rational, then

x = q - r

, meaning

x

would be rational, which contradicts the premise).

3

Examine the classification of

\frac{22}{7}

in Statement 3.

Statement 3 is invalid.

The fraction

\frac{22}{7}

is the ratio of two integers, which strictly defines it as a rational number. It is merely a common numerical approximation for

\pi

, not the exact irrational value of

\pi

itself.

Key Concept

Core properties of rational/irrational numbers and fundamental definitions of integers like zero.

Question 523Question →

An industrial chemical plant stores a specialized solvent in three large cylindrical vats. The first vat contains $53.\overline{3}$ liters, the second contains $71.\overline{1}$ liters, and the third contains $26.\overline{6}$ liters of the solvent. The plant manager wants to completely transfer the solvent from all three vats into identical, smaller drums such that every drum is completely filled, no solvent is left over in any vat, and solvents from different vats are not mixed. What should be the maximum possible capacity of each drum?

8.\overline{8}

liters

26.\overline{6}

liters

71.\overline{1}

liters

213.\overline{3}

liters

Show answer & explanation

Answer: $8.\overline{8}$ liters

Answer

The maximum possible capacity of each drum is

8.\overline{8}

liters.

To find the maximum identical capacity that leaves no remainder, we must calculate the Highest Common Factor (HCF) of the three volumes. By converting the recurring decimals to fractions (

\frac{160}{3}

,

\frac{640}{9}

,

\frac{80}{3}

) and applying the fraction HCF formula (

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

), we obtain

\frac{80}{9}

, which perfectly translates to

8.\overline{8}

liters.

Step-by-Step Solution

1

Convert the given recurring decimals representing the solvent volumes into their simplest fractional forms.

53.\overline{3} = 53 + \frac{3}{9} = \frac{160}{3}

,

71.\overline{1} = 71 + \frac{1}{9} = \frac{640}{9}

, and

26.\overline{6} = 26 + \frac{6}{9} = \frac{80}{3}

.

Fractional forms are required to accurately and properly compute the highest common factor (HCF) of non-integer values.

2

Identify the mathematical operation required based on the physical constraints described in the problem.

We must calculate the Highest Common Factor (HCF) of the three volumes:

\frac{160}{3}

,

\frac{640}{9}

, and

\frac{80}{3}

.

The solvent must be divided equally without any remainders, meaning the drum size must be a common factor of all three initial volumes, and the problem asks for the 'maximum possible capacity'.

3

Apply the standard formula for finding the HCF of multiple fractions.

The formula is:

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

.

This formula ensures the resulting fraction will evenly divide each of the original fractions without leaving a remainder.

4

Calculate the HCF of the numerators (

160

,

640

,

80

) and the LCM of the denominators (

3

,

9

,

3

).

\text{HCF}(160, 640, 80) = 80

, and

\text{LCM}(3, 9, 3) = 9

. Thus, the HCF of the fractions is

\frac{80}{9}

.

80

is the largest integer dividing

160

,

640

, and

80

.

9

is the smallest integer divisible by

3

,

9

, and

3

.

5

Convert the resulting fraction back into a recurring decimal.

\frac{80}{9} = 8 + \frac{8}{9} = 8.\overline{8}

liters.

The final calculated capacity should match the formatting style of the given options.

Key Concept

Calculating the Highest Common Factor (HCF) of recurring decimals by converting them to fractions and using the fraction HCF rule.

Estimated Time:2m 30s

Question 524Question →

A boutique chocolatier is preparing special gift assortments. They have three large blocks of premium cocoa weighing $\frac{21}{2}$ kg, $\frac{35}{4}$ kg, and $\frac{49}{8}$ kg. They must mold these blocks into the largest possible identical solid chocolate bars such that no cocoa is left over from any of the three blocks. What should be the exact weight of each chocolate bar?

\frac{7}{8}

kg

\frac{7}{2}

kg

\frac{735}{2}

kg

7

kg

Show answer & explanation

Answer: $\frac{7}{8}$ kg

Answer

The exact weight of each chocolate bar should be

\frac{7}{8}

kg.

To find the maximum identical weight that can perfectly divide all three cocoa blocks, we must calculate the Highest Common Factor (HCF) of the three fractions. The formula dictates finding the HCF of the numerators (21, 35, 49), which is 7, and dividing it by the LCM of the denominators (2, 4, 8), which is 8. This results in the correct weight of 7/8 kg.

Step-by-Step Solution

1

Identify the mathematical operation required.

The problem asks for the 'largest possible identical' divisions of the blocks with no remainders, meaning we must find the Highest Common Factor (HCF) of the given fractional weights.

HCF provides the maximum uniform size that can perfectly divide a given set of quantities.

2

Recall the formula for finding the HCF of fractions.

HCF of fractions = (HCF of Numerators) / (LCM of Denominators).

This is the standard mathematical rule for calculating the greatest common divisor for fractional values.

3

Calculate the HCF of the numerators: 21, 35, and 49.

The highest common factor for 21, 35, and 49 is 7.

21 = 3 × 7; 35 = 5 × 7; 49 = 7 × 7. The greatest shared prime factor is 7.

4

Calculate the LCM of the denominators: 2, 4, and 8.

The lowest common multiple for 2, 4, and 8 is 8.

8 is a multiple of both 2 and 4, making it the smallest common denominator.

5

Apply the calculated values to the fraction formula.

\frac{7}{8}

kg.

Combining the calculated numerator HCF and denominator LCM yields the final maximum weight.

Key Concept

HCF of Fractions

Question 525Question →

Eight colleagues—K, L, M, N, O, P, Q, and R—are sitting around a square table. Four of them sit in the middle of the sides and face the center of the table, while the other four sit at the four corners and face outside (away from the center). They are seated according to the following conditions:

- M sits in the middle of one of the sides.
- K sits third to the left of M.
- Only two people sit between K and R.
- O sits to the immediate right of R.
- P sits second to the left of K.
- N is an immediate neighbor of O.
- Q sits to the immediate right of P.

Based on the given seating arrangement, which of the following statements are correct? (Select all that apply)

Select all that apply

L sits to the immediate right of M.

R sits to the immediate left of O.

O sits second to the left of K.

Exactly three people sit between K and L when counted from the right of K.

Show answer & explanation

Answer: L sits to the immediate right of M.; Exactly three people sit between K and L when counted from the right of K.

Answer

The correct statements are that L sits to the immediate right of M, and exactly three people sit between K and L when counted from the right of K.

Based on the seating conditions, M is at a middle position (facing center). K is 3rd to the left (clockwise), placing K at a corner facing outside. R is separated from K by two people, meaning R is at the opposite middle position (facing center). O is to the immediate right of R (counter-clockwise), placing O at a corner facing outside. P is 2nd to the left of K (counter-clockwise), so P is at a corner. N is O's neighbor, leaving only one adjacent spot. Q is immediate right of P (clockwise). L takes the last spot. In this final layout, L is immediately counter-clockwise from M, which is M's right side, making the first statement correct. Counting clockwise from K (K's right side), we pass N, O, and R before reaching L, making the statement about three people between K and L correct.

Step-by-Step Solution

1

Determine M and K's positions.

M is at a middle position (facing center). K is 3rd to the left of M. Since M faces the center, left is clockwise, placing K at a corner (facing outside).

M is explicitly given as sitting in the middle. The direction rules (center = left is clockwise) determine K's placement.

2

Determine R and O's positions.

R must be at the opposite middle position (facing center). O is to the immediate right of R, placing O at the adjacent corner counter-clockwise (facing outside).

Only two people sit between K and R, so R is opposite M. R faces the center, so its right is counter-clockwise.

3

Place P, N, Q, and L.

P is 2nd to the left of K (counter-clockwise), N is the only available neighbor of O, Q is to the immediate right of P (clockwise), and L takes the final remaining position.

Following the remaining clues step-by-step while strictly adhering to the left/right rules for inward vs. outward facing positions ensures a unique arrangement.

4

Evaluate the given statements based on the final arrangement.

The statements about L's position relative to M, and the number of people between K and L (counting from K's right) are correct.

By applying the left/right rules to the completed seating layout, we can confirm the validity of each statement.

Key Concept

Polygonal Seating Arrangement with Mixed Facing Directions

Estimated Time:2m 0s

Question 526Question →

When evaluating the fundamental categories and properties of the real number system, which of the following statements are mathematically correct?

Select all that apply

The square root of any prime number inherently belongs to the set of irrational numbers.

Zero is classified as a positive integer that is neither odd nor even.

Every integer can be expressed as a rational number in the form

\frac{p}{q}

, where

q

is a non-zero integer.

The mathematical constant

\pi

is exactly equivalent to the ratio

\frac{22}{7}

, meaning it belongs to the rational numbers.

Show answer & explanation

Answer: The square root of any prime number inherently belongs to the set of irrational numbers.; Every integer can be expressed as a rational number in the form $\frac{p}{q}$ , where $q$ is a non-zero integer.

Answer

The mathematically correct statements are that the square root of any prime number is irrational, and every integer can be expressed as a rational number.

The correct statements accurately identify that prime numbers lack perfect square roots, making their roots inherently irrational, and that all integers are essentially a subset of rational numbers since any integer can be expressed as a fraction with a denominator of one.

Step-by-Step Solution

1

Analyze the statement regarding prime numbers and irrationality.

Since prime numbers only have two distinct positive divisors, they are never perfect squares. Therefore, their square roots cannot be simplified into rational fractions, making the statement correct.

To verify the relationship between primes and algebraic irrationality.

2

Evaluate the properties of zero given in the second statement.

Zero is determined to be neutral (neither positive nor negative) and even (since

0 = 2 \times 0

). Therefore, the statement claiming it is positive and neither odd nor even is false.

To check for misconceptions regarding the classification and parity of zero.

3

Assess the definition of rational numbers applied to integers.

An integer

z

can always be written as

\frac{z}{1}

. Since

1

is a non-zero integer, this satisfies the

\frac{p}{q}

definition of rational numbers, making the statement correct.

To confirm the subset relationship between integers and rational numbers.

4

Examine the classification of

\pi

.

While

\frac{22}{7}

is a common approximation,

\pi

cannot be written as a simple fraction of integers. It is irrational, making the statement false.

To differentiate between an exact mathematical constant and its heuristic approximation.

Key Concept

Classification of Numbers and Fundamental Properties

Question 527Question →

A large agricultural cooperative is dividing a massive tract of land for different crops. They allocate $0.\overline{428571}$ of the total land to cultivate sunflowers, $0.1\overline{6}$ of the total land to cultivate maize, and $0.0\overline{5}$ of the total land to cultivate organic vegetables. The remaining land, which measures exactly $110$ hectares, is preserved as a wildlife reserve. What is the total area of the tract of land, in hectares?

Show answer & explanation

Answer: 315

Answer

315

The total area is found by properly converting all recurring decimals into exact fractions (

3/7

,

1/6

, and

1/18

), summing them to find the total allocated land (

41/63

), determining the remaining land fraction (

22/63

), and setting it equal to the given

110

hectares. Solving for the whole yields exactly

315

hectares.

Step-by-Step Solution

1

Convert the pure recurring decimal

0.\overline{428571}

into a simplified fraction.

The fraction is

\frac{3}{7}

.

Recognizing that

\frac{1}{7} = 0.\overline{142857}

, we can multiply by

3

to get

0.\overline{428571}

. Alternatively, using the algebraic method:

\frac{428571}{999999} = \frac{3}{7}

.

2

Convert the mixed recurring decimals

0.1\overline{6}

and

0.0\overline{5}

into fractions.

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

and

0.0\overline{5} = \frac{5-0}{90} = \frac{5}{90} = \frac{1}{18}

.

To operate with mixed recurring decimals, subtract the non-repeating part from the entire number, and place it over a denominator consisting of 9s (for repeating digits) followed by 0s (for non-repeating digits after the decimal point).

3

Calculate the total fraction of land allocated to the three crops.

\frac{3}{7} + \frac{1}{6} + \frac{1}{18} = \frac{3}{7} + \frac{3+1}{18} = \frac{3}{7} + \frac{4}{18} = \frac{3}{7} + \frac{2}{9} = \frac{27+14}{63} = \frac{41}{63}

.

Finding a common denominator (

63

) allows us to sum the individual crop fractions to determine the total proportion of cultivated land.

4

Determine the fraction representing the wildlife reserve and calculate the total land area.

Reserve fraction =

1 - \frac{41}{63} = \frac{22}{63}

. Total Area =

110 \times \frac{63}{22} = 315

hectares.

The unallocated fraction represents the reserve area. Setting this fraction of the total area (

T

) equal to

110

hectares (

\frac{22}{63} \times T = 110

) gives the final answer.

Key Concept

Fractions and Decimals

Estimated Time:2m 30s

Question 528Question →

A pharmaceutical laboratory uses three automated dispensers to add chemical reagents into a continuous reaction vessel. Dispenser X adds a drop every $\frac{14}{9}$ seconds, Dispenser Y every $\frac{35}{6}$ seconds, and Dispenser Z every $\frac{49}{12}$ seconds. If all three dispensers release a drop simultaneously, what is the minimum time interval (in seconds) until they all release a drop together again?

\frac{36}{7}

\frac{490}{3}

\frac{7}{36}

\frac{245}{18}

Show answer & explanation

Answer: $\frac{490}{3}$

Answer

The minimum time interval is

\frac{490}{3}

seconds.

The correct answer is found by applying the LCM formula for fractions. Since the events repeat over time, their next simultaneous occurrence happens at a time that is a common multiple of all three periods. By calculating the LCM of the numerators (

14, 35, 49 \rightarrow 490

) and dividing it by the HCF of the denominators (

9, 6, 12 \rightarrow 3

), we get

\frac{490}{3}

seconds.

Step-by-Step Solution

1

Identify the mathematical operation required.

We need to find the Lowest Common Multiple (LCM) of the three fractional time intervals.

Simultaneous periodic events repeat exactly at the LCM of their individual periods.

2

State the formula for the LCM of fractions.

LCM of fractions =

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

.

This standard formula ensures the resulting value is a multiple of each original fraction.

3

Calculate the LCM of the numerators.

The numerators are

14

,

35

, and

49

. Their prime factorizations are

14 = 2 \times 7

,

35 = 5 \times 7

, and

49 = 7^2

. The LCM is

2 \times 5 \times 7^2 = 490

.

The LCM must include the highest power of all prime factors present in the numbers.

4

Calculate the HCF of the denominators.

The denominators are

9

,

6

, and

12

. Their prime factorizations are

9 = 3^2

,

6 = 2 \times 3

, and

12 = 2^2 \times 3

. The highest common factor is

3

.

The HCF is the largest positive integer that divides each of the numbers without leaving a remainder.

5

Apply the formula to find the final LCM.

LCM =

\frac{490}{3}

.

Substituting the calculated numerator LCM and denominator HCF into the fraction formula.

Key Concept

Lowest Common Multiple (LCM) of fractions for periodic events

Estimated Time:1m 30s

Question 529Question →

Five delegates—Alice, Ben, Clara, David, and Emma—are sitting in a single straight row facing North during a conference.

Read the following conditions carefully:
1. David is sitting at the extreme left end of the row.
2. Ben is sitting second to the right of Alice.
3. Clara is an immediate neighbor of Ben, but does not sit next to Alice.

Arrange the delegates in their correct seating order from the extreme left end to the extreme right end of the row.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from left to right is: David, Alice, Emma, Ben, Clara.

The arrangement must logically begin with David at position 1. Alice must sit at position 2 and Ben at position 4; otherwise, Clara would be forced to sit next to Alice (if Alice were at 3 and Ben at 5). Consequently, Clara takes position 5, leaving position 3 for Emma.

Step-by-Step Solution

1

Place David according to the first condition.

David is at position 1. The row configuration from left to right is: David, _, _, _, _.

The condition explicitly states David is at the extreme left end.

2

Determine the possible positions for Alice and Ben.

Since Ben is second to the right of Alice, the possible position pairs for (Alice, Ben) are (2, 4) or (3, 5).

This maintains exactly one empty seat between Alice and Ben to her right.

3

Evaluate the (3, 5) position pair using Clara's constraint.

If Alice is at 3 and Ben is at 5, Clara must be at 4 to be an immediate neighbor of Ben. However, this places Clara next to Alice, which violates the third condition.

Testing hypotheses against negative constraints is necessary to eliminate invalid scenarios.

4

Place Alice, Ben, and Clara in the only valid remaining configuration.

Alice is securely at position 2, and Ben is at position 4. Clara must sit at position 5 to be next to Ben but not next to Alice.

This is the only configuration that satisfies both the 'second to the right' rule and the negative neighbor constraint.

5

Place Emma in the final remaining seat.

Emma sits at position 3. The final order is David, Alice, Emma, Ben, Clara.

There is only one person left (Emma) and one seat available (position 3).

Key Concept

Linear Seating Arrangement with negative constraints

Question 530Question →

A city's public transport network features three distinct tram lines that operate on continuous circular routes departing from a central station. Tram Line 1 completes its route every $\frac{45}{4}$ minutes. Tram Line 2 completes its route every $\frac{25}{2}$ minutes, and Tram Line 3 takes $\frac{75}{8}$ minutes per loop. If all three trams depart from the central station simultaneously, how many minutes will it take for them to depart together again for the first time?

Show answer & explanation

Answer: 112.5

Answer

It will take 112.5 minutes for all three trams to depart together again.

The correct answer is found by taking the Least Common Multiple of the fractional times. By finding the LCM of the numerators (225) and dividing it by the Highest Common Factor of the denominators (2), we get 225/2, which evaluates to exactly 112.5 minutes.

Step-by-Step Solution

1

Determine the mathematical operation required to find when the events will synchronize.

Identify the need to calculate the Least Common Multiple (LCM) of the fractions

\frac{45}{4}

,

\frac{25}{2}

, and

\frac{75}{8}

.

The trams will meet again at a time that is a common multiple of their individual loop durations. The 'first time' indicates the least common multiple is needed.

2

Apply the rule for calculating the LCM of fractional values.

Use the formula:

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

.

To synchronize fractional frequencies, the numerators must reach a common multiple while strictly maintaining the largest common baseline unit defined by the denominators.

3

Calculate the LCM of the numerators: 45, 25, and 75.

The LCM of 45, 25, and 75 is 225.

Prime factorization:

45 = 3^2 \times 5

;

25 = 5^2

;

75 = 3 \times 5^2

. Taking the highest powers gives

3^2 \times 5^2 = 9 \times 25 = 225

.

4

Calculate the HCF of the denominators: 4, 2, and 8.

The HCF of 4, 2, and 8 is 2.

2 is the largest integer that can divide 4, 2, and 8 without leaving a remainder.

5

Compute the final synchronized time.

Divide the LCM of numerators by the HCF of denominators:

\frac{225}{2} = 112.5

.

Applying the values to the fraction LCM formula yields the exact time in minutes.

Key Concept

Calculating the Least Common Multiple (LCM) for fractions to solve simultaneous event problems.

Question 531Question →

Consider the numerical expression $E = 8^{52} - 3^{27}$ . If $u$ represents the unit digit of the positive integer $E$ , which of the following correctly classifies the number $u$ ?

It is an odd composite number.

It is an even composite number.

It is a negative integer.

It is a unique number that is neither prime nor composite.

Show answer & explanation

Answer: It is an odd composite number.

Answer

The calculated unit digit is 9, which is classified as an odd composite number.

Evaluating the expression requires finding the unit digits of both exponential terms. The unit digit of

8^{52}

is 6, and the unit digit of

3^{27}

is 7. When subtracting 7 from 6 in a larger positive number, borrowing from the tens place results in

16 - 7 = 9

. The number 9 is odd and has three distinct factors (1, 3, 9), making it an odd composite number.

Step-by-Step Solution

1

Determine the unit digit of

8^{52}

.

The unit digit is 6.

The cyclicity pattern for powers of 8 is 4 (ending in 8, 4, 2, 6). Since 52 is a perfect multiple of 4 (remainder is 0), the unit digit matches the 4th power in the cycle, which is 6.

2

Determine the unit digit of

3^{27}

.

The unit digit is 7.

The cyclicity pattern for powers of 3 is 4 (ending in 3, 9, 7, 1). Dividing the exponent 27 by 4 leaves a remainder of 3. Therefore, the unit digit matches the 3rd power in the cycle (

3^3

), which ends in 7.

3

Calculate the unit digit

u

of the difference

E

.

u = 9

Subtracting the unit digits gives

6 - 7

. Because

E

is a positive integer, we must borrow 10 from the next higher place value in the base-10 system, yielding

16 - 7 = 9

.

4

Classify the resulting number 9.

9 is an odd composite number.

The integer 9 cannot be evenly divided by 2 (making it odd) and has positive divisors other than 1 and itself (1, 3, and 9), which classifies it as composite.

Key Concept

Integration of power cyclicity rules and fundamental number classification.

Question 532Question →

A metal fabrication company has three long copper rods measuring $\frac{54}{5}$ meters, $\frac{81}{10}$ meters, and $\frac{108}{25}$ meters. The company needs to cut all three rods into smaller segments of equal length such that no copper material is left over. To minimize the total number of segments, what is the maximum possible length of each individual segment?

\frac{162}{25}

meters

\frac{324}{5}

meters

\frac{27}{50}

meters

\frac{27}{5}

meters

Show answer & explanation

Answer: $\frac{27}{50}$ meters

Answer

The maximum possible length of each segment is 27/50 meters.

The maximum possible length for the equal segments is found by calculating the HCF of the given fractions. Using the mathematical rule for fractions, HCF = HCF(numerators) / LCM(denominators). The HCF of 54, 81, and 108 is 27. The LCM of 5, 10, and 25 is 50. Therefore, the maximum length is 27/50 meters.

Step-by-Step Solution

1

Identify the mathematical operation required to find the maximum possible equal length.

The problem requires finding the Highest Common Factor (HCF) of the three fractional lengths.

The rods must be cut into equal pieces without wastage, and the length of each piece must be maximized to minimize the number of segments.

2

Recall the formula for finding the HCF of fractions.

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

This formula is necessary to accurately compute the greatest common divisor of non-integer values.

3

Calculate the HCF of the numerators: 54, 81, and 108.

The factors give 54 = 2 × 27, 81 = 3 × 27, and 108 = 4 × 27. The HCF is 27.

We need the greatest integer that divides all the numerators evenly.

4

Calculate the LCM of the denominators: 5, 10, and 25.

The multiples of 25 are 25, 50, 75, etc. Since 50 is divisible by both 5 and 10, the LCM is 50.

We need the smallest common multiple for the denominators to complete the fraction formula.

5

Construct the final fraction.

The maximum length is 27/50 meters.

Dividing the computed numerator HCF by the denominator LCM gives the correct value.

Key Concept

HCF of Fractions

Question 533Question →

Seven diplomats—Alan, Boris, Chloe, David, Elena, Felix, and Grace—are sitting around a circular table facing the center.

Read the following conditions carefully:
1. Alan sits third to the right of Boris.
2. Chloe sits second to the left of Alan.
3. David sits immediately between Elena and Felix.
4. Elena is not an immediate neighbor of Boris.

Based on the above seating arrangement, who sits second to the right of Grace?

Boris

Felix

Elena

Alan

Show answer & explanation

Answer: Elena

Answer

Elena sits second to the right of Grace.

Following the conditions step-by-step: If we place Boris at position 1, Alan goes to position 4 (third to the right, anti-clockwise). Chloe goes to position 2 (second to the left of Alan, clockwise). The three consecutive seats for Elena, David, and Felix must be 5, 6, and 7, meaning David is at 6. Grace takes the only empty seat at 3. Because Elena cannot be next to Boris (position 1), she must take position 5, leaving Felix at position 7. The complete anti-clockwise order is Boris, Chloe, Grace, Alan, Elena, David, Felix. Grace is at position 3; the second person to her right is at position 5, which is occupied by Elena.

Step-by-Step Solution

1

Place Boris and determine Alan's position using the first condition.

Assume Boris is at position 1. Since they face the center, 'right' is anti-clockwise. Alan is third to the right, placing him at position 4.

Fixing a reference point is the standard method for solving circular arrangements.

2

Place Chloe using the second condition.

Chloe sits second to the left (clockwise) of Alan (position 4). Thus, Chloe is placed at position 2.

Chloe's position is strictly relative to Alan's fixed position.

3

Identify the seating block for David, Elena, and Felix.

David sits immediately between Elena and Felix, requiring three consecutive empty seats. The currently occupied seats are 1, 2, and 4. The only consecutive empty seats are 5, 6, and 7. Therefore, David must sit exactly in the middle of them, at position 6.

This is the only way to satisfy the requirement of three people sitting together without splitting the group.

4

Place Grace in the arrangement.

With positions 1, 2, 4, 5, 6, and 7 accounted for, position 3 is the only remaining seat. Grace must sit at position 3.

By the process of elimination, the last remaining person takes the last available seat.

5

Determine the exact positions of Elena and Felix using the final condition.

Elena and Felix occupy positions 5 and 7 in some order. Boris is at position 1, making positions 2 and 7 his immediate neighbors. Since Elena cannot be an immediate neighbor of Boris, she cannot sit at position 7. Therefore, Elena is at position 5, and Felix is at position 7.

This fully resolves all ambiguities in the arrangement while satisfying the negative constraint.

6

Answer the specific question asked in the stem.

Grace is at position 3. Facing the center, the second seat to her right (anti-clockwise) is position 5. Elena occupies position 5.

Applying the final relative directional check on the completed diagram.

Key Concept

Solving complex circular seating arrangements by logically deducing relative positions and applying negative constraints.

Question 534Question →

A numerical analysis task requires evaluating four specific values to classify them into their correct number sets. The values are defined as follows:

- $K = 18 - 6 \div 2 + 1$
- $L =$ The unit digit of $8^{32}$
- $M =$ The remainder when $-23$ is divided by $5$
- $N =$ The Highest Common Factor (HCF) of $\frac{3}{4}$ and $\frac{9}{10}$

Based on the correct mathematical evaluation of these expressions, which of the following statements regarding their classification are mathematically correct?

Select all that apply

The value of

K

is a perfect square, and

M

is the only prime number among the integer results.

The value of

N

is a rational fraction strictly between

0

and

1

, and

L

is an even composite number.

The value of

K

is a prime number, and

N

is an improper fraction greater than

1

.

The value of

L

is a positive odd integer, and

M

is a negative integer.

Show answer & explanation

Answer: The value of $K$ is a perfect square, and $M$ is the only prime number among the integer results.; The value of $N$ is a rational fraction strictly between $0$ and $1$ , and $L$ is an even composite number.

Answer

The mathematically correct statements are that K is a perfect square, M is the only prime among the integer results, N is a rational fraction between 0 and 1, and L is an even composite number.

Based on rigorous mathematical evaluation, the true values are

K = 16

,

L = 6

,

M = 2

, and

N = \frac{3}{20}

.

K

(

16

) is a perfect square, and

M

(

2

) is the only prime among the integer results (

16, 6, 2

). Furthermore,

N

(

0.15

) is a rational fraction strictly between

0

and

1

, and

L

(

6

) is an even composite number. Therefore, these descriptive classifications perfectly match the evaluated properties.

Step-by-Step Solution

1

Evaluate expression K using proper BODMAS rules.

K = 16

Division must be performed before addition and subtraction.

K = 18 - (6 \div 2) + 1 = 18 - 3 + 1 = 16

, which classifies as a perfect square.

2

Determine the unit digit of L based on cyclicity.

L = 6

The unit digit of powers of

8

follows a 4-step cycle (

8, 4, 2, 6

). Since the exponent

32

is perfectly divisible by

4

, the unit digit is the 4th in the cycle, which is

6

(an even composite number).

3

Calculate the mathematically correct positive remainder for M.

M = 2

By the formal division algorithm,

-23 = 5 \times (-5) + 2

. Remainder must be non-negative, so the remainder is

2

, which is an even prime number.

4

Compute the HCF of the given fractions for N.

N = \frac{3}{20}

The HCF of fractions is computed as

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

.

\text{HCF}(3,9) = 3

and

\text{LCM}(4,10) = 20

. Thus

N = \frac{3}{20} = 0.15

, a rational number strictly between

0

and

1

.

5

Cross-reference the correctly evaluated numbers against the provided statements.

The statements categorizing

K

as a perfect square,

M

as the only prime among integers,

N

between

0

and

1

, and

L

as an even composite are correct.

The integer results are

16, 6

, and

2

, where exactly one (

2

) is prime. The other statements rely on distinct computational and conceptual errors.

Key Concept

Applying fundamental arithmetic rules and modular arithmetic to properly classify numbers into distinct mathematical sets.

Estimated Time:1m 30s

Question 535Question →

An event organizer is arranging chairs for a large conference. When the chairs are arranged in rows of $18$ , $24$ , or $36$ , there are always exactly $5$ chairs left over. However, when the chairs are arranged in rows of $13$ , all chairs are perfectly accommodated with none left over. What is the minimum possible total number of chairs the organizer has?

Show answer & explanation

Answer: 221

Answer

The minimum possible total number of chairs is 221.

The correct answer is derived by first establishing that any number leaving a remainder of

5

when divided by

18

,

24

, and

36

must be of the form

72k + 5

, where

72

is the LCM of the divisors. By systematically checking values of

k

, we find that

k=3

is the smallest integer that makes the expression

(72k + 5)

perfectly divisible by

13

, resulting in

72(3) + 5 = 221

.

Step-by-Step Solution

1

Set up the conditions for the total number of chairs mathematically.

Let the total number of chairs be

N

. We are given

N \equiv 5 \pmod{18}

,

N \equiv 5 \pmod{24}

,

N \equiv 5 \pmod{36}

, and

N \equiv 0 \pmod{13}

.

Translating the word problem into modular arithmetic helps systematically apply the concepts of LCM and divisibility.

2

Find the Least Common Multiple (LCM) of the first set of divisors.

The divisors are

18

,

24

, and

36

. Their prime factorizations are

18 = 2 \times 3^2

,

24 = 2^3 \times 3

, and

36 = 2^2 \times 3^2

. The LCM is

2^3 \times 3^2 = 8 \times 9 = 72

.

Any number that leaves the same remainder when divided by multiple divisors must be a multiple of their LCM plus that remainder.

3

Express

N

using the LCM and the common remainder.

Since

N

leaves a remainder of

5

when divided by

18

,

24

, or

36

, we can write

N = 72k + 5

, where

k

is a non-negative integer (

k = 0, 1, 2, \dots

).

This general formula captures all possible numbers of chairs that satisfy the first condition.

4

Apply the final divisibility condition to find

k

.

We require

N

to be perfectly divisible by

13

, meaning

72k + 5 \equiv 0 \pmod{13}

.

This guarantees the solution satisfies the second condition where arranging chairs in rows of 13 leaves no remainder.

5

Simplify the congruence modulo 13 and solve for

k

.

Divide

72

by

13

to find the remainder:

72 = 13 \times 5 + 7

. So,

72k \equiv 7k \pmod{13}

. The equation becomes

7k + 5 \equiv 0 \pmod{13}

. Testing values for

k

: if

k=1

,

7(1)+5=12

(not divisible); if

k=2

,

7(2)+5=19

(not divisible); if

k=3

,

7(3)+5=26

(divisible by

13

, since

26 = 13 \times 2

). The smallest valid

k

is

3

.

Finding the smallest non-negative integer

k

ensures we find the minimum possible number of chairs.

6

Calculate the final value of

N

.

N = 72(3) + 5 = 216 + 5 = 221

.

Substituting

k=3

back into our general formula gives the final answer.

Key Concept

Solving simultaneous remainder and divisibility conditions using the Least Common Multiple (LCM).

Question 536Question →

Eight students—P, Q, R, S, T, U, V, and W—are seated in two parallel rows containing four people each, such that there is an equal distance between adjacent persons.

In Row 1, P, Q, R, and S are seated and all of them are facing South.
In Row 2, T, U, V, and W are seated and all of them are facing North.
In the given seating arrangement, each member seated in a row faces another member of the other row.

- Q sits second to the left of S.
- The person facing Q sits to the immediate right of W.
- P sits at one of the extreme ends of the row.
- V faces the person who sits to the immediate right of R.
- T sits at an extreme end of the row, and does not face P.

Based on the given arrangement, which of the following statements are correct? (Select all that apply)

Select all that apply

U sits at an extreme end of the row and faces P.

W sits second to the left of U.

R faces the person who sits to the immediate left of T.

P sits to the immediate left of S.

Show answer & explanation

Answer: U sits at an extreme end of the row and faces P.; R faces the person who sits to the immediate left of T.

Answer

The correct statements are that U sits at an extreme end facing P, and R faces the person who sits to the immediate left of T.

Based on the final arrangement, U is at the leftmost extreme end of Row 2 and faces P at the leftmost extreme end of Row 1, making the statement about U correct. T is at the rightmost extreme end of Row 2 facing North, meaning W is to T's immediate left; R sits directly opposite W, making the statement about R correct.

Step-by-Step Solution

1

Establish the left and right directions for both rows based on their facing directions.

Row 1 (South): Left is rightward, Right is leftward (from the observer's view). Row 2 (North): Left is leftward, Right is rightward.

Correctly orienting directions is critical before placing individuals in a parallel arrangement.

2

Place S and Q using the clue: 'Q sits second to the left of S'.

Since S faces South, 'left' means moving to the observer's right. Possibility 1: S is at position 1 (far left), Q is at position 3. Possibility 2: S is at position 2, Q is at position 4.

This establishes the relative spacing between two individuals in Row 1.

3

Incorporate the clues about W and P to test the possibilities.

Using 'P sits at an extreme end': In Possibility 1, P must be at 4. In Possibility 2, P must be at 1. Using 'The person facing Q sits to the immediate right of W': In Possibility 1, Q(3) faces pos 3. W must be at 2. In Possibility 2, Q(4) faces pos 4. W must be at 3.

Combining extreme end constraints with facing constraints helps fill the grid.

4

Apply the clue regarding V and R: 'V faces the person who sits to the immediate right of R'.

In Possibility 1, the remaining Row 1 person R is at 2. R's right (facing South) is pos 1 (S). V faces S, so V is at 1. In Possibility 2, R is at 3. R's right is pos 2 (S). V faces S, so V is at 2.

This locks in the position of V in Row 2 based on the remaining individuals in Row 1.

5

Resolve the final positions using the clue: 'T sits at an extreme end... and does not face P'.

In Possibility 1, V=1, W=2. T must be at 4 (extreme end). But P is at 4, meaning T faces P, which violates the condition. Thus, Possibility 1 is invalid. In Possibility 2, V=2, W=3. T must be at 1 or 4. If T is at 1, T faces P (at 1), violating the condition. Therefore, T must be at 4. The remaining person U goes to position 1.

This eliminates the invalid scenario and provides the single correct final arrangement.

6

Document the finalized seating grid.

Row 1 (South-facing, left-to-right): P, S, R, Q. Row 2 (North-facing, left-to-right): U, V, W, T.

A complete layout is required to evaluate the given statements in the options.

Key Concept

Applying relative directional reasoning (left/right inversion) in a two-row parallel seating arrangement.

Question 537Question →

A gourmet coffee roaster has three batches of rare coffee beans weighing $\frac{15}{2}$ kg, $\frac{25}{4}$ kg, and $\frac{35}{8}$ kg respectively. The roaster wants to package all the beans into smaller, equal-sized bags such that each bag contains the maximum possible weight of coffee beans without mixing the batches, and no beans are left over. What should be the weight of each bag?

\frac{525}{2}

kg

\frac{525}{8}

kg

\frac{5}{8}

kg

\frac{5}{2}

kg

Show answer & explanation

Answer: $\frac{5}{8}$ kg

Answer

The weight of each bag should be

\frac{5}{8}

kg.

The problem asks for the maximum possible equal capacity for the bags, which requires calculating the Highest Common Factor (HCF) of the given fractional weights. The HCF of a set of fractions is found by calculating the HCF of their numerators (

15

,

25

,

35

) and dividing it by the LCM of their denominators (

2

,

4

,

8

). The HCF of the numerators is

5

, and the LCM of the denominators is

8

. Therefore, the correct weight is

\frac{5}{8}

kg.

Step-by-Step Solution

1

Determine the mathematical operation required.

We need to find the Highest Common Factor (HCF) of the given fractional weights.

The bags must be of equal size, hold the maximum possible weight, and leave no remainder, which matches the definition of HCF.

2

Recall the formula for finding the HCF of fractions.

The formula is: (HCF of numerators) / (LCM of denominators).

This formula ensures the resulting fraction correctly divides all the given fractions into integers.

3

Calculate the HCF of the numerators.

The numerators are

15

,

25

, and

35

. Their HCF is

5

.

5

is the largest integer that divides

15

,

25

, and

35

without leaving a remainder.

4

Calculate the LCM of the denominators.

The denominators are

2

,

4

, and

8

. Their LCM is

8

.

8

is the smallest integer that is a multiple of

2

,

4

, and

8

.

5

Combine the results to find the HCF of the fractions.

The final HCF is

\frac{5}{8}

.

Substituting the calculated HCF and LCM into the fraction formula yields

\frac{5}{8}

.

Key Concept

Finding the Highest Common Factor (HCF) of fractions.

Question 538Question →

Three mathematical values, $X$ , $Y$ , and $Z$ , are defined below:

- $X = \frac{22}{7} - \pi$
- $Y = \text{The positive remainder obtained when } -23 \text{ is divided by } 6$
- $Z = 2 + 18 \div 3 \times 2$

Based on the fundamental properties of numbers, which of the following statements correctly classifies all three values?

Value

X

is an irrational number,

Y

is neither prime nor composite, and

Z

is an even composite number.

Value

X

is an irrational number,

Y

is neither prime nor composite, and

Z

is an odd prime number.

Value

X

is an irrational number,

Y

is an odd prime number, and

Z

is an even composite number.

Value

X

is a rational number,

Y

is neither prime nor composite, and

Z

is an even composite number.

Show answer & explanation

Answer: Value $X$ is an irrational number, $Y$ is neither prime nor composite, and $Z$ is an even composite number.

Answer

Value

X

is an irrational number,

Y

is neither prime nor composite, and

Z

is an even composite number.

First,

X

is the difference between a rational number (

22/7

) and an irrational number (

\pi

), which results in a non-zero irrational number. Second, applying the division algorithm (

a = bq + r

where

0 \le r < b

), we evaluate

-23 = 6 \times (-4) + 1

. Thus, the positive remainder

Y

is 1, which is uniquely defined as a natural number that is neither prime nor composite. Finally, evaluating

Z

requires applying BODMAS rules from left to right for multiplication and division:

18 \div 3 = 6

, then

6 \times 2 = 12

, and

2 + 12 = 14

. The result 14 is an even composite number.

Step-by-Step Solution

1

Evaluate and classify the value of

X = \frac{22}{7} - \pi

.

X

is a non-zero irrational number.

Since

22/7

is a rational approximation and not strictly equal to the irrational number

\pi

, their difference yields a non-zero irrational number.

2

Determine the true positive remainder of

-23 \div 6

to find

Y

.

Y = 1

, which is neither prime nor composite.

Using the division algorithm (

a = bq + r

where

0 \le r < b

), we write

-23 = 6 \times (-4) + 1

. The positive remainder is 1, a natural number defined as neither prime nor composite.

3

Calculate

Z = 2 + 18 \div 3 \times 2

using proper operational precedence.

Z = 14

, which is an even composite number.

According to BODMAS, division and multiplication are evaluated strictly from left to right before addition. Thus,

18 \div 3 = 6

, followed by

6 \times 2 = 12

, and finally

2 + 12 = 14

.

Key Concept

Classification of Real Numbers, Modulo Arithmetic, and Order of Operations

Question 539Question →

Seven researchers—Fiona, George, Harry, Iris, Jack, Kelly, and Liam—are sitting around a circular table for a meeting. Four of them face the center, while the remaining three face outward. They are seated according to the following conditions:

1. Harry faces outward. Jack sits second to the left of Harry.
2. Fiona sits exactly between Harry and George.
3. George sits adjacent to Iris and faces the center.
4. Iris sits second to the right of Jack.
5. Kelly is not an immediate neighbor of Harry and faces the same direction as George.
6. The immediate neighbors of Jack face opposite directions (i.e., one faces the center and the other faces outward).
7. Fiona faces the center.

Based on the given information, who sits third to the left of Kelly?

Fiona

George

Liam

Harry

Show answer & explanation

Answer: Harry

Answer

The correct person sitting third to the left of Kelly is Harry.

Based on the conditions, Kelly must sit in seat 5 facing the center, while Harry sits in seat 1 facing outward. Since Kelly is facing the center, her left direction moves clockwise. Moving three positions clockwise from seat 5 (to 6, then 7, then 1) lands exactly on Harry.

Step-by-Step Solution

1

Set up a 7-seat circular arrangement (numbered 1 to 7 clockwise) and place Harry and Jack.

Harry is at seat 1 facing outward. Jack is at seat 6.

Since Harry faces outward, his 'left' direction is counter-clockwise. Second to the left of seat 1 (counter-clockwise) is seat 7, then seat 6.

2

Place Fiona and George based on relative positions.

Fiona is at seat 2 and George is at seat 3.

Fiona must be exactly between Harry (1) and George. This forces them to occupy consecutive seats (1, 2, 3), placing Fiona at 2 and George at 3. George is given as facing the center. Fiona is given as facing the center.

3

Place Iris and determine Jack's facing direction.

Iris is at seat 4, and Jack faces the center.

George (3) is adjacent to Iris. Since Fiona is already at 2, Iris must be at 4. Iris (4) is second to the right of Jack (6). For seat 4 to be on the right of seat 6 (a counter-clockwise move), Jack must face the center.

4

Place Kelly and Liam, and deduce the remaining directions.

Kelly is at seat 5 facing the center, and Liam is at seat 7 facing outward. Iris faces outward.

Kelly cannot be next to Harry (1), so Kelly cannot take seat 7. Kelly takes seat 5, leaving seat 7 for Liam. Kelly faces the same direction as George (center). Jack's neighbors are Kelly (5) and Liam (7). Since Kelly faces center, Liam must face outward. We now have 4 facing center (Fiona, George, Jack, Kelly), meaning the rest (Harry, Liam, Iris) must face outward.

5

Identify the person sitting third to the left of Kelly.

Harry.

Kelly is at seat 5 facing the center. Left of a center-facing person is clockwise. Tracing three spots clockwise from 5: seat 6 (1st), seat 7 (2nd), and seat 1 (3rd), which is occupied by Harry.

Key Concept

Solving complex circular seating arrangements with mixed facing directions (inward and outward) by tracking relative left-right orientations.

Question 540Question →

Consider the following statements regarding the 'Mahtari Vandan Yojana' implemented by the Government of Chhattisgarh:

1. It provides direct financial assistance of ₹1,000 per month (₹12,000 annually) to eligible beneficiaries through Direct Benefit Transfer (DBT).
2. Unmarried women aged 18 years and above residing in the state are the primary target beneficiaries.
3. The Department of Women and Child Development serves as the nodal administrative department for its execution.

Which of the statements given above is/are correct?

1 and 2 only

2 and 3 only

1 and 3 only

1, 2, and 3

Show answer & explanation

Answer: 1 and 3 only

Answer

1 and 3 only

The combination indicating statements 1 and 3 only is correct because the Mahtari Vandan Yojana provides ₹1,000 per month via Direct Benefit Transfer and is administered by the Department of Women and Child Development of Chhattisgarh. Statement 2 is incorrect because the scheme restricts eligibility to married women aged 21 and above, rather than unmarried women aged 18.

Step-by-Step Solution

1

Evaluate statement 1 regarding financial quantum and disbursement mode

Statement 1 is correct. The Mahtari Vandan Yojana provides ₹1,000 per month (total ₹12,000 per annum) directly credited to the beneficiary's Aadhaar-linked bank account via DBT.

Verifying the core financial entitlement parameter of the state welfare scheme.

2

Evaluate statement 2 regarding beneficiary target demographic and age criteria

Statement 2 is incorrect. The scheme is designed exclusively for married women who are domicile residents of Chhattisgarh, having attained the age of 21 years as of January 1 of the application year. Widows, divorced, and abandoned women are also eligible, but unmarried women are not covered.

Checking eligibility criteria against official state policy guidelines.

3

Evaluate statement 3 regarding the nodal implementing department

Statement 3 is correct. The Department of Women and Child Development, Government of Chhattisgarh, is the nodal authority responsible for planning, verifying, and executing the scheme.

Identifying the administrative machinery governing the flagship initiative.

4

Synthesize the findings to choose the correct combination

Statements 1 and 3 are correct, while statement 2 is incorrect.

Matching the evaluated statements with the appropriate answer option.

Key Concept

Eligibility conditions, financial transfers, and administrative mechanisms of state flagship women empowerment schemes

Estimated Time:1m 15s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept