All practice questions

Question 4561Question →

A civil supplies department procures three batches of edible oil weighing $0.\overline{6}$ metric tonnes, $1.\overline{3}$ metric tonnes, and $2.\overline{2}$ metric tonnes. The department intends to repackage the entire procured quantity into smaller, identical standardized containers. To minimize the number of containers used without mixing the batches, what should be the maximum capacity of each standardized container?

6.\overline{6}

metric tonnes

0.\overline{2}

metric tonnes

0.\overline{6}

metric tonnes

0.1

metric tonnes

Show answer & explanation

Answer: $0.\overline{2}$ metric tonnes

Answer

The maximum capacity of each standardized container should be

0.\overline{2}

metric tonnes.

The maximum capacity of the standardized containers is determined by finding the Highest Common Factor (HCF) of the oil volumes. First, the given recurring decimals are converted to fractions:

\frac{2}{3}

,

\frac{4}{3}

, and

\frac{20}{9}

. The HCF of these fractions is found by taking the HCF of the numerators (2, 4, 20), which is 2, and dividing it by the LCM of the denominators (3, 3, 9), which is 9. This calculation yields

\frac{2}{9}

, which is equivalent to the recurring decimal

0.\overline{2}

.

Step-by-Step Solution

1

Convert the recurring decimals representing the batch weights into fractions in their simplest form.

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

,

1.\overline{3} = \frac{13-1}{9} = \frac{12}{9} = \frac{4}{3}

, and

2.\overline{2} = \frac{22-2}{9} = \frac{20}{9}

.

Mathematical operations involving divisors and multiples are most accurate when recurring decimals are expressed exactly as fractions.

2

Identify the mathematical requirement to find the maximum standardized container capacity.

We must determine the Highest Common Factor (HCF) of the three fractional quantities.

To divide all batches completely into identical containers of maximum capacity without leftovers, the container size must be the greatest common divisor of all the given volumes.

3

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

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

.

This formula ensures that the resulting fraction will evenly divide each of the original fractional quantities.

4

Calculate the HCF of the numerators and the LCM of the denominators for

\frac{2}{3}

,

\frac{4}{3}

, and

\frac{20}{9}

.

Numerators (2, 4, 20): The

\text{HCF}

is 2. Denominators (3, 3, 9): The

\text{LCM}

is 9.

These values provide the numerator and denominator for the final maximum capacity.

5

Form the final fraction and convert it back to a recurring decimal to match the given options.

\text{Maximum capacity} = \frac{2}{9} = 0.\overline{2}

metric tonnes.

Converting back to recurring decimal format answers the problem in the original units.

Key Concept

Fractions and Decimals - Converting Recurring Decimals and Finding HCF of Fractions

Question 4562Question →

Two positive integers $A$ and $B$ (both strictly greater than $12$ ) have a Highest Common Factor (HCF) of $12$ and a Least Common Multiple (LCM) of $504$ . Based on this information, which of the following statements are true?

Select all that apply

The sum of the integers

A

and

B

can be

156

.

The positive difference between

A

and

B

can be

132

.

There are exactly four distinct unordered pairs

\{A, B\}

that satisfy all given conditions.

The Least Common Multiple (LCM) of the fractions

\frac{1}{A}

and

\frac{1}{B}

is

\frac{1}{504}

.

Show answer & explanation

Answer: The sum of the integers $A$ and $B$ can be $156$ .; The positive difference between $A$ and $B$ can be $132$ .

Answer

The statements confirming that the sum of the integers can be 156 and that their positive difference can be 132 are both correct.

Based on the prime factorization constraints, the valid unordered pairs

\{A, B\}

are

\{24, 252\}

,

\{36, 168\}

, and

\{72, 84\}

. Because

72 + 84 = 156

and

168 - 36 = 132

, the descriptive statements asserting that the sum can be

156

and the positive difference can be

132

are logically true.

Step-by-Step Solution

1

Use the property that the product of two numbers equals the product of their HCF and LCM.

A \times B = 12 \times 504 = 6048

.

This establishes the fundamental numerical relationship between the two integers.

2

Express

A

and

B

in terms of their HCF to simplify.

Let

A = 12x

and

B = 12y

, where

x

and

y

are coprime integers. Substituting into the product gives

144xy = 6048

, which simplifies to

xy = 42

.

Factoring out the HCF ensures that any pairs found will inherently respect the greatest common divisor requirement.

3

Find all coprime pairs

(x, y)

that multiply to 42 and calculate the corresponding integer pairs

\{A, B\}

.

The coprime pairs for

42

are

(1, 42)

,

(2, 21)

,

(3, 14)

, and

(6, 7)

. The corresponding

\{A, B\}

pairs are

\{12, 504\}

,

\{24, 252\}

,

\{36, 168\}

, and

\{72, 84\}

.

These pairs represent all mathematical possibilities before applying the specific contextual constraints.

4

Apply the problem's contextual condition that both integers must be strictly greater than 12.

The pair

\{12, 504\}

must be excluded. The valid pairs are exactly three:

\{24, 252\}

,

\{36, 168\}

, and

\{72, 84\}

.

Filtering out invalid mathematical solutions ensures strict adherence to the stated facts.

5

Evaluate the given statements against the three valid integer pairs.

The sum

72 + 84 = 156

validates the first statement. The difference

168 - 36 = 132

validates the second. The count of valid pairs is

3

, refuting the third statement. The LCM of

1/A

and

1/B

is

1/12

, refuting the fourth.

Systematically verifying which descriptive statements accurately reflect the discovered pairs.

Key Concept

Applying HCF and LCM properties alongside algebraic constraints to evaluate paired integers.

Estimated Time:2m 0s

Question 4563Question →

A jeweler is cutting equal-length pieces of gold wire from three different spools to make uniform necklace links without any wastage. The three spools contain gold wire of lengths $\frac{27}{2}$ cm, $\frac{45}{4}$ cm, and $\frac{63}{5}$ cm. What is the maximum possible length of each uniform gold wire piece that can be cut? (Provide your answer as a precise decimal)

Show answer & explanation

Answer: 0.45

Answer

0.45

The maximum uniform length is found by computing the Highest Common Factor (HCF) of the fractional wire lengths. Applying the formula yields an HCF of numerators (9) over the LCM of denominators (20), giving

\frac{9}{20}

, which correctly converts to exactly

0.45

cm.

Step-by-Step Solution

1

Determine the mathematical operation required for the scenario.

Calculate the Highest Common Factor (HCF) of the three fractional lengths.

Cutting multiple lengths into the maximum possible equal segments without wastage is defined by the HCF.

2

Apply the rule for finding the HCF of fractions.

HCF = HCF(Numerators) / LCM(Denominators)

This is the standard formula for finding the greatest common divisor of multiple fractional values.

3

Find the HCF of the numerators 27, 45, and 63.

HCF(27, 45, 63) = 9

9 is the largest integer that divides perfectly into 27, 45, and 63.

4

Find the LCM of the denominators 2, 4, and 5.

LCM(2, 4, 5) = 20

20 is the smallest integer that is a multiple of 2, 4, and 5.

5

Combine the results into the final fraction and convert to a decimal.

\frac{9}{20} = 0.45

cm

The question requires the precise decimal representation of the fraction.

Key Concept

Calculating the Highest Common Factor (HCF) of fractions to solve optimization word problems.

Question 4564Question →

A botanical garden is installing a new irrigation system and has three main supply hoses measuring $\frac{44}{5}$ meters, $\frac{77}{4}$ meters, and $\frac{121}{10}$ meters in length. The landscaping team needs to cut all three hoses into smaller, equal-length segments to connect to individual planters. If no material can be wasted, what is the minimum total number of segments that can be produced from all three hoses combined?

Show answer & explanation

Answer: 73

Answer

73 segments

The minimum number of total segments is achieved when each segment is cut to its maximum possible equal length. This maximum length is the HCF of the three fractions (44/5, 77/4, 121/10), which evaluates to 11/20 meters. Dividing the original lengths by this HCF yields exactly 16, 35, and 22 pieces respectively, totaling 73 individual segments.

Step-by-Step Solution

1

Determine the mathematical requirement for minimizing the number of segments.

Identify that the Highest Common Factor (HCF) of the hose lengths must be found.

To get the minimum number of pieces, each piece must be of the maximum possible equal length.

2

Apply the rule for finding the HCF of fractions.

Use the formula:

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

.

This formula allows the calculation of the greatest common divisor for non-integer fractional values.

3

Calculate the HCF of the numerators.

The numerators are

44

,

77

, and

121

. Their HCF is

11

.

The largest integer that perfectly divides

44

,

77

, and

121

is

11

(

44 = 11 \times 4

,

77 = 11 \times 7

,

121 = 11 \times 11

).

4

Calculate the LCM of the denominators.

The denominators are

5

,

4

, and

10

. Their LCM is

20

.

The smallest integer that is a multiple of

5

,

4

, and

10

is

20

.

5

Determine the maximum segment length.

The segment length is

\frac{11}{20}

meters.

Combining the results from the previous steps using the fraction HCF formula yields the length.

6

Calculate the number of segments produced from each hose.

First hose:

\frac{44}{5} \div \frac{11}{20} = 16

. Second hose:

\frac{77}{4} \div \frac{11}{20} = 35

. Third hose:

\frac{121}{10} \div \frac{11}{20} = 22

.

Dividing the total length of each hose by the length of one segment gives the segment count per hose.

7

Sum the segment counts.

16 + 35 + 22 = 73

segments.

The question asks for the minimum total number of segments produced from all three hoses combined.

Key Concept

Calculating the Highest Common Factor (HCF) of fractions and applying it to optimize division in real-world scenarios.

Estimated Time:2m 0s

Question 4565Question →

A cryptography algorithm generates a numerical verification key, $V$ , through a specific two-step mathematical process. First, it determines the value of $K$ , which is strictly defined as the positive remainder when $-26$ is divided by $7$ . Second, it calculates the final key using the expression $V = K \times (18 \div 6 \times 3)$ . Based on the fundamental classification of numbers, which of the following accurately describes the final value of $V$ ?

It is a positive, even, and composite number.

It is a negative, odd integer.

It is a positive, even, and prime number.

It is a positive, odd, and composite number.

Show answer & explanation

Answer: It is a positive, even, and composite number.

Answer

The final value is 18, which is uniquely classified as a positive, even, and composite number.

By applying the formal definition of the modulo operation, the positive remainder of

-26

divided by

7

is found by the equation

-26 = 7 \times (-4) + 2

, giving

K = 2

. Next, using the correct order of operations (left to right for consecutive division and multiplication), the expression evaluates to

(18 \div 6) \times 3 = 3 \times 3 = 9

. Multiplying these results yields

V = 2 \times 9 = 18

. The number 18 is classified as positive (greater than zero), even (divisible by 2), and composite (having factors other than 1 and itself).

Step-by-Step Solution

1

Determine the positive remainder when

-26

is divided by

7

to find the value of

K

.

K = 2

By the formal definition of Euclidean division, the remainder must be non-negative. We must express

-26

as

7 \times (-4) + 2

, which gives a valid positive remainder of

2

.

2

Evaluate the mathematical expression

18 \div 6 \times 3

following the correct order of operations.

The expression evaluates to

9

.

According to the BODMAS rule, division and multiplication share the same level of precedence and must be evaluated strictly from left to right. Therefore,

(18 \div 6) = 3

, and

3 \times 3 = 9

.

3

Calculate the final value of the verification key

V

and classify it.

V = 2 \times 9 = 18

. The number

18

is a positive, even, and composite number.

The number

18

is greater than zero (positive), perfectly divisible by

2

(even), and possesses divisors other than

1

and itself, such as

2, 3,

and

6

(composite).

Key Concept

Classification of Numbers and Fundamental Arithmetic Operations

Question 4566Question →

Six software engineers—Arjun, Brenda, Carlos, Divya, Ethan, and Fatima—are seated around a circular conference table. Some are facing the center while others are facing outward (away from the center).

1. Ethan faces the center and Arjun sits third to his right.
2. Arjun faces outward. Carlos sits second to the left of Arjun.
3. Divya sits to the immediate right of Arjun and faces the same direction as Arjun.
4. Carlos faces the center. Brenda sits to the immediate left of Carlos.
5. The immediate neighbors of Ethan face the same direction as each other.
6. Fatima and Brenda face opposite directions.

Based on the given arrangement, who sits second to the right of Brenda?

Divya

Ethan

Carlos

Fatima

Show answer & explanation

Answer: Divya

Answer

Divya

Based on the step-by-step logical placement, Brenda is seated at position 3 and faces outward. Because she is facing away from the center, her right side is in the counter-clockwise direction. Therefore, the person to her immediate right is at position 4 (Arjun), and the person second to her right is at position 5, which is Divya.

Step-by-Step Solution

1

Place Ethan and determine Arjun's position.

Ethan is at position 1 facing the center. Arjun is at position 4.

Assuming positions 1 to 6 clockwise, Ethan at 1 facing inward means his right is counter-clockwise. Third to the right (6, 5, 4) places Arjun exactly opposite at position 4.

2

Place Carlos relative to Arjun.

Carlos is at position 2 facing the center.

Arjun at position 4 faces outward. His left side is clockwise (towards 3, 2). Second to the left is position 2, so Carlos is placed there. The clue also states Carlos faces the center.

3

Place Divya relative to Arjun.

Divya is at position 5 facing outward.

Divya is to the immediate right of Arjun. Since Arjun faces outward at position 4, his right is counter-clockwise at position 5. Divya faces the same direction as Arjun (outward).

4

Place Brenda relative to Carlos.

Brenda is at position 3.

Carlos faces the center at position 2. His left is clockwise towards position 3, which is where Brenda sits.

5

Place Fatima and determine remaining facing directions.

Fatima is at position 6 (facing the center) and Brenda faces outward.

Fatima takes the last spot at 6. Ethan's neighbors (Carlos at 2 and Fatima at 6) must face the same direction. Carlos faces inward, so Fatima faces inward. Since Brenda and Fatima face opposite directions, Brenda faces outward.

6

Find the person second to the right of Brenda.

The person is Divya.

Brenda is at position 3 facing outward. Her right side points counter-clockwise towards position 4. The first to her right is Arjun (4), and the second to her right is Divya (5).

Key Concept

Determining left and right orientations in circular arrangements with mixed facing directions.

Question 4567Question →

A teacher provides a list of eight numerical values on the board for a classification exercise:

(I) $\sqrt{144}$
(II) $\frac{22}{7}$
(III) $\pi$
(IV) $0.\overline{36}$
(V) $\sqrt{12}$
(VI) $\frac{\sqrt{3}}{\sqrt{27}}$
(VII) $3.14159$
(VIII) $0.121221222...$ (where the number of 2s increases by one each time)

What is the exact count of rational numbers in this list?

Show answer & explanation

Answer: 5

Answer

There are exactly 5 rational numbers in the provided list.

By simplifying each expression and applying the definitions of real numbers, exactly 5 of the 8 values (values I, II, IV, VI, and VII) satisfy the conditions of being a rational number.

Step-by-Step Solution

1

Define the criteria for a rational number.

A rational number is any number that can be expressed as a fraction of two integers (p/q, where q is not zero). This includes all integers, terminating decimals, and repeating decimals.

Establishing the definition is necessary to accurately classify each item.

2

Evaluate the square roots and fractions in the list.

\sqrt{144} = 12

, which is an integer (Rational).

\frac{22}{7}

is a ratio of two integers (Rational).

\sqrt{12} = 2\sqrt{3}

, which contains the root of a non-perfect square (Irrational).

\frac{\sqrt{3}}{\sqrt{27}}

simplifies to

\sqrt{\frac{3}{27}} = \sqrt{\frac{1}{9}} = \frac{1}{3}

(Rational).

Radicals must be simplified to their lowest terms to reveal their true mathematical classification.

3

Evaluate the decimal representations and constants in the list.

0.\overline{36}

is a repeating decimal (Rational).

3.14159

is a terminating decimal (Rational).

\pi

is a transcendental constant with infinite, non-repeating digits (Irrational).

0.121221222...

has a changing pattern, making it non-terminating and non-recurring (Irrational).

Decimals must be analyzed by their termination or repetition properties.

4

Count the total number of items identified as rational.

The rational values are (I), (II), (IV), (VI), and (VII). The total count is 5.

The question asks for the exact numerical count of rational numbers.

Key Concept

Identifying rational and irrational numbers by their fractional, radical, and decimal properties.

Question 4568Question →

In an industrial robotics laboratory, three distinct robotic arms perform repetitive cyclic tasks. Arm X completes one full operational cycle every $\frac{16}{15}$ seconds, Arm Y completes a cycle every $\frac{24}{25}$ seconds, and Arm Z completes a cycle every $\frac{32}{35}$ seconds. If all three robotic arms begin their cycles simultaneously from a synchronized home position, what is the minimum time required for all three arms to return to the home position at the exact same instant?

\frac{8}{525}

seconds

\frac{32}{175}

seconds

\frac{96}{5}

seconds

\frac{8}{5}

seconds

Show answer & explanation

Answer: $\frac{96}{5}$ seconds

Answer

The minimum time required for all three robotic arms to synchronize again is

\frac{96}{5}

seconds.

To determine the next simultaneous occurrence of repeating events, we compute the Lowest Common Multiple (LCM) of their cycle durations. For fractions, the LCM is determined by dividing the LCM of the numerators by the Highest Common Factor (HCF) of the denominators. Thus,

\frac{\text{LCM}(16, 24, 32)}{\text{HCF}(15, 25, 35)} = \frac{96}{5}

.

Step-by-Step Solution

1

Identify the mathematical operation required.

Calculate the Lowest Common Multiple (LCM) of the three cycle times.

To find the first instance when multiple independent repeating events synchronize, we must find the LCM of their individual intervals.

2

State the formula for finding the LCM of fractions.

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

This is the standard algebraic rule for determining the lowest common multiple of rational numbers.

3

Calculate the LCM of the numerators: 16, 24, and 32.

The LCM of 16, 24, and 32 is 96.

The prime factorizations are

16 = 2^4

,

24 = 2^3 \times 3

, and

32 = 2^5

. The highest powers of prime factors are

2^5

and

3^1

, so

2^5 \times 3 = 32 \times 3 = 96

.

4

Calculate the HCF of the denominators: 15, 25, and 35.

The HCF of 15, 25, and 35 is 5.

The prime factorizations are

15 = 3 \times 5

,

25 = 5^2

, and

35 = 5 \times 7

. The only common prime factor is 5.

5

Apply the values to the fraction LCM formula.

\frac{96}{5}

Dividing the computed numerator LCM (96) by the computed denominator HCF (5) yields the final answer.

Key Concept

Calculating the Lowest Common Multiple (LCM) of fractional values to synchronize repeating events.

Estimated Time:1m 30s

Question 4569Question →

An urban traffic control system manages three independent electronic toll gates. Based on their internal sensor loops, Gate A completes its automated scanning cycle every $\frac{12}{5}$ seconds, Gate B every $\frac{18}{25}$ seconds, and Gate C every $\frac{27}{35}$ seconds. If all three gates reset their cycles simultaneously, how many seconds will it take for all three gates to reset simultaneously again? Express your answer as an exact decimal.

Show answer & explanation

Answer: 21.6

Answer

21.6

Simultaneous repeating events require finding the Lowest Common Multiple (LCM) of their individual periods. For fractional periods, the rule is

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

. The LCM of

12

,

18

, and

27

is

108

, and the HCF of

5

,

25

, and

35

is

5

. Therefore, the LCM of the fractions is

\frac{108}{5}

, which evaluates precisely to

21.6

.

Step-by-Step Solution

1

Identify the mathematical operation required.

Calculate the LCM of the fractions

\frac{12}{5}

,

\frac{18}{25}

, and

\frac{27}{35}

.

Simultaneous cyclic events coincide at the Lowest Common Multiple of their individual time intervals.

2

Determine the LCM of the numerators.

The LCM of

12

,

18

, and

27

is

108

.

The formula for the LCM of fractions requires calculating the LCM of their respective numerators.

3

Determine the HCF of the denominators.

The HCF of

5

,

25

, and

35

is

5

.

The formula for the LCM of fractions requires calculating the HCF of their respective denominators.

4

Apply the fraction LCM formula.

\frac{108}{5} = 21.6

Dividing the LCM of numerators by the HCF of denominators yields the final LCM of the given fractions.

Key Concept

Lowest Common Multiple (LCM) of Fractions

Question 4570Question →

A meteorological research station operates three automated data collection buoys in the ocean. The buoys transmit complete environmental telemetry packages to a satellite every $\frac{8}{21}$ hours, $\frac{11}{35}$ hours, and $\frac{15}{49}$ hours, respectively. If all three buoys initiate a transmission simultaneously at a given moment, how many hours will pass before they all initiate a transmission together again?

\frac{1}{735}

\frac{88}{49}

\frac{1320}{7}

\frac{1}{7}

Show answer & explanation

Answer: $\frac{1320}{7}$

Answer

The buoys will transmit together again after

\frac{1320}{7}

hours.

To find the time when all three buoys will transmit together again, we must calculate the Lowest Common Multiple (LCM) of the given time intervals. For fractions, the applicable formula is the LCM of the numerators divided by the highest common factor (HCF) of the denominators. The LCM of the numerators (8, 11, 15) is 1320. The HCF of the denominators (21, 35, 49) is 7. Therefore, the LCM of the fractions is 1320/7.

Step-by-Step Solution

1

Identify that the problem requires finding the Lowest Common Multiple (LCM) of the given fractional time intervals.

The mathematical goal is to compute

\text{LCM}(\frac{8}{21}, \frac{11}{35}, \frac{15}{49})

.

Simultaneous periodic events synchronize again at the least common multiple of their individual periods.

2

Apply the standard formula for the LCM of fractions.

The formula requires calculating

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

.

This is the established algebraic rule to compute the LCM for rational numbers.

3

Calculate the LCM of the numerators (8, 11, 15).

Since 8, 11, and 15 share no common prime factors, their LCM is simply their product:

8 \times 11 \times 15 = 1320

.

To find the smallest whole number that is evenly divisible by all three numerators.

4

Calculate the HCF of the denominators (21, 35, 49).

Breaking them down into prime factors gives

21=3\times7

,

35=5\times7

, and

49=7\times7

. The highest common factor is 7.

To find the largest whole number that divides evenly into all three denominators.

5

Combine the calculated numerator and denominator results.

\frac{1320}{7}

Substituting the calculated values back into the fraction LCM formula yields the final synchronized interval.

Key Concept

The lowest common multiple (LCM) of multiple fractions is calculated as the LCM of their numerators divided by the highest common factor (HCF) of their denominators.

Question 4571Question →

A financial analyst is evaluating the long-term performance of four mutual funds. The average annualized yield rates of the funds are given below. Arrange the funds in descending order based on their yield rates (from highest to lowest).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct descending order is Fund X, Fund Y, Fund W, then Fund Z.

By converting all values to their extended decimal forms, we determine their precise magnitudes: Fund X is

8.166...\%

, Fund Y is

8.155...\%

, Fund W is

8.151...\%

, and Fund Z is

8.142...\%

. Ordering these from highest to lowest yields Fund X, Fund Y, Fund W, then Fund Z.

Step-by-Step Solution

1

Convert Fund W's yield to extended decimal form for precise comparison.

Fund W =

8.\overline{15}\% = 8.1515...\%

Expanding the recurring decimals reveals the sequence of digits beyond the hundredths place.

2

Convert Fund X's yield to decimal form.

Fund X =

8 \frac{1}{6}\% = 8 + 0.1666...\% = 8.1666...\%

Converting the mixed fraction standardizes the format to match the others.

3

Convert Fund Y's yield to extended decimal form.

Fund Y =

8.1\overline{5}\% = 8.1555...\%

Properly expanding the specific repeating digit distinguishes it from Fund W.

4

Convert Fund Z's yield to decimal form.

Fund Z =

\frac{57}{7}\% = 8 \frac{1}{7}\% \approx 8.1428...\%

Division provides the decimal equivalent needed for a direct comparison.

5

Compare the decimal values digit by digit from left to right.

Highest is

8.166...

(Fund X), then

8.155...

(Fund Y), then

8.151...

(Fund W), and lowest is

8.142...

(Fund Z).

Sorting by the tenths, hundredths, and thousandths digits establishes the definitive descending sequence.

Key Concept

Converting mixed fractions, improper fractions, and different notations of recurring decimals into standard decimal form for accurate comparison.

Question 4572Question →

A botanical research institute is preparing identical standardized nutrient doses for a plant growth experiment. They have three concentrated liquid nutrient solutions measuring $\frac{14}{3}$ liters, $\frac{35}{6}$ liters, and $\frac{21}{4}$ liters respectively. The researchers need to distribute these solutions into small testing vials such that every vial contains the exact same maximum possible volume of solution, without mixing the different types of nutrients, and leaving no solution behind in the original containers. What should be the volume capacity of each testing vial?

\frac{7}{12}

liters

\frac{1}{210}

liters

210

liters

7

liters

Show answer & explanation

Answer: $\frac{7}{12}$ liters

Answer

\frac{7}{12}

liters

To find the maximum identical volume that can evenly divide all three solutions without remainders, we must calculate the Highest Common Factor (HCF) of the fractions

\frac{14}{3}

,

\frac{35}{6}

, and

\frac{21}{4}

. Using the rule that the HCF of fractions equals the HCF of their numerators divided by the LCM of their denominators, we evaluate HCF(14, 35, 21) = 7 and LCM(3, 6, 4) = 12. This results in exactly

\frac{7}{12}

liters per vial.

Step-by-Step Solution

1

Identify the mathematical operation required based on the scenario.

The problem asks for the maximum identical volume to evenly distribute three different quantities without remainders. This requires finding the Highest Common Factor (HCF) of the given fractional volumes.

HCF is used when dividing varying quantities into the largest possible equal segments.

2

Recall the formula for calculating the HCF of fractions.

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

This is the standard rule for finding the greatest common divisor of fractional values.

3

Calculate the HCF of the numerators (

14, 35, 21

).

The prime factors are:

14 = 2 \times 7

;

35 = 5 \times 7

;

21 = 3 \times 7

. The highest common factor is

7

.

To form the numerator of our final fraction answer.

4

Calculate the LCM of the denominators (

3, 6, 4

).

The multiples of the denominators meet at

12

. (Since

12

is divisible by

3

,

6

, and

4

).

To form the denominator of our final fraction answer.

5

Substitute the calculated values back into the fraction HCF formula.

\frac{7}{12}

liters.

Provides the final vial capacity.

Key Concept

Highest Common Factor (HCF) of Fractions in Word Problems

Question 4573Question →

A civil engineer is evaluating the thickness of four different structural panels, measured in centimeters (cm). To meet safety standards, the panels must be stacked from the thinnest to the thickest. Arrange the following panel thicknesses in ascending order.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order from thinnest to thickest is: Panel Z, Panel X, Panel W, Panel Y.

The correct sequence is determined by converting all measurements into standard decimal forms. Panel Z is exactly

1.375

cm, Panel X is approximately

1.428

cm, Panel W is

1.444...

cm, and Panel Y is exactly

1.450

cm. Sorting these values in ascending order yields Panel Z, Panel X, Panel W, Panel Y.

Step-by-Step Solution

1

Convert the thickness of Panel Z into a standard decimal format.

Panel Z =

1\frac{3}{8} = 1 + 0.375 = 1.375

cm.

Converting a mixed fraction to a decimal allows for direct magnitude comparison.

2

Convert the thickness of Panel X into a standard decimal format.

Panel X =

\frac{10}{7} \approx 1.42857...

cm.

An improper fraction must be divided out to identify its exact decimal value.

3

Expand the thickness of Panel W to clearly see its decimal value.

Panel W =

1.\overline{4} = 1.4444...

cm.

The overline indicates a recurring decimal that continues infinitely with the digit 4.

4

Compare all four decimal values and arrange them from smallest to largest.

1.375 < 1.42857... < 1.4444... < 1.450

. Therefore, Panel Z < Panel X < Panel W < Panel Y.

The stem requires ordering from thinnest to thickest.

Key Concept

Converting mixed fractions, improper fractions, and recurring decimals into a common decimal format to accurately compare and order them.

Question 4574Question →

A highway construction company is contracted to pave a new road. In the first month, they complete exactly $0.\overline{3}$ of the total project length. In the second month, they pave $0.625$ of the *remaining* length. If there are exactly $21$ kilometers left to pave in the third month to finish the project, what is the total length of the highway in kilometers?

84

80

63

504

Show answer & explanation

Answer: 84

Answer

The total length of the highway is 84 kilometers.

The total length of the highway is 84 km. In the first month,

1/3

is completed, leaving

2/3

. In the second month,

5/8

of this remaining

2/3

is completed, which means

(1 - 5/8) = 3/8

of the remainder is left for the third month. Calculating this relative to the total:

3/8 \times 2/3 = 1/4

. Since

1/4

of the total project equals 21 km, the total length must be

21 \times 4 = 84

km.

Step-by-Step Solution

1

Convert the decimal and recurring decimal into fractions for exact calculation.

0.\overline{3} = \frac{1}{3}

and

0.625 = \frac{625}{1000} = \frac{5}{8}

.

Fractional calculations prevent rounding errors and simplify multi-step multiplication.

2

Determine the fraction of the highway remaining after the first month.

1 - \frac{1}{3} = \frac{2}{3}

of the total length remains.

The second month's progress is based on this remaining amount, not the total.

3

Calculate the fraction of the total project paved in the second month.

\frac{5}{8} \times \frac{2}{3} = \frac{10}{24} = \frac{5}{12}

of the total length.

The second month's work is

0.625

of the remaining

\frac{2}{3}

.

4

Find the final remaining fraction for the third month.

\frac{2}{3} - \frac{5}{12} = \frac{8}{12} - \frac{5}{12} = \frac{3}{12} = \frac{1}{4}

of the total project.

Subtracting the second month's progress from what was left after the first month gives the final remainder.

5

Set the remaining fraction equal to the given remaining distance and solve for the total.

\frac{1}{4} \times \text{Total} = 21 \Rightarrow \text{Total} = 21 \times 4 = 84

km.

This establishes the relationship between the fractional parts and the actual physical distance.

Key Concept

Converting recurring and terminating decimals to fractions and computing successive remainders.

Estimated Time:1m 30s

Question 4575Question →

A specialized aerospace component is manufactured from a custom metal alloy weighing exactly $160$ kg. The alloy's composition by weight is $0.35$ aluminum and $\frac{5}{16}$ magnesium, with the remaining portion consisting entirely of titanium. What is the exact mass of titanium in this component?

Show answer & explanation

Answer: 54

Answer

54

To find the mass of titanium, the total proportions of the other metals must first be unified into a common format (either fractions or decimals). By converting the magnesium's share of

\frac{5}{16}

to the decimal

0.3125

, the combined proportion of aluminum and magnesium is

0.35 + 0.3125 = 0.6625

. The titanium makes up the remaining portion of the whole, which is

1 - 0.6625 = 0.3375

. Finally, multiplying this decimal proportion by the total alloy mass of

160

kg yields exactly

54

kg.

Step-by-Step Solution

1

Convert the fractional part to a decimal.

\frac{5}{16} = 0.3125

Unifying the terms into a single format (decimals) makes addition straightforward.

2

Add the portions of aluminum and magnesium.

0.35 + 0.3125 = 0.6625

This finds the total proportion of the alloy that is NOT titanium.

3

Calculate the proportion of titanium.

1 - 0.6625 = 0.3375

The sum of all material proportions in the whole alloy must equal exactly

1

.

4

Calculate the final mass of titanium.

0.3375 \times 160 = 54

Multiplying the component's proportion by the total mass gives the specific weight of that component.

Key Concept

Converting between fractions and decimals and applying proportional reasoning to find a part of a whole.

Question 4576Question →

Three automated drones patrol a large forest reserve and return to their base station for a rapid data sync at regular intervals. Drone A returns every $3.\overline{3}$ hours, Drone B returns every $2.5$ hours, and Drone C returns every $2.\overline{7}$ hours. If all three drones depart from the base station simultaneously, how many hours will it take for all three to meet at the base station again for a simultaneous data sync?

50

\frac{5}{18}

5

\frac{25}{9}

Show answer & explanation

Answer: $50$

Answer

It will take 50 hours for all three drones to meet simultaneously at the base station.

The time required for all drones to meet again is the LCM of their individual interval times. Converting the intervals to fractions gives 10/3, 5/2, and 25/9. The LCM of fractions is computed by dividing the LCM of the numerators (10, 5, 25) by the HCF of the denominators (3, 2, 9). The LCM of the numerators is 50 and the HCF of the denominators is 1, resulting in exactly 50 hours.

Step-by-Step Solution

1

Convert the recurring and terminating decimals into fractions in their simplest form.

Drone A:

3.\overline{3} = 3 \frac{1}{3} = \frac{10}{3}

. Drone B:

2.5 = \frac{5}{2}

. Drone C:

2.\overline{7} = 2 \frac{7}{9} = \frac{25}{9}

.

Fractional forms are required to accurately compute the lowest common multiple of non-integer time intervals.

2

Identify the mathematical operation needed to find the time of the next simultaneous meeting.

We must find the Least Common Multiple (LCM) of the three fractional time intervals.

The LCM represents the smallest time value that is a perfect multiple of all individual drone intervals.

3

State the formula for the LCM of fractions.

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

.

This is the standard arithmetic rule for finding the LCM of multiple fractions.

4

Calculate the LCM of the numerators.

The numerators are

10, 5, 25

. Their prime factorizations are

10 = 2 \times 5

,

5 = 5

, and

25 = 5^2

. The

\text{LCM}(10, 5, 25) = 2 \times 5^2 = 50

.

The LCM must include the highest power of each prime factor present in the numerators.

5

Calculate the HCF of the denominators.

The denominators are

3, 2, 9

. Since they do not share any common prime factors, their

\text{HCF}(3, 2, 9) = 1

.

The Highest Common Factor is the largest integer that divides all the denominators evenly.

6

Compute the final LCM of the fractions.

\text{LCM} = \frac{50}{1} = 50

hours.

Applying the values from Steps 4 and 5 into the formula from Step 3 yields the final synchronization time.

Key Concept

LCM of Fractions and Recurring Decimals

Estimated Time:2m 0s

Question 4577Question →

During a physical endurance test, four candidates completed different portions of a 10-kilometer cross-country course before a mandatory rest period. The fraction or decimal representing the portion of the course completed by each candidate is given below. Arrange the candidates in ascending order based on the portion of the course they completed (from least to most).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

Candidate Q, Candidate S, Candidate R, Candidate P

To correctly order the candidates, all values must be evaluated in a uniform format. Candidate P is exactly 0.85. Candidate Q is approximately 0.818. Candidate R is approximately 0.844. Candidate S is approximately 0.833. Arranging these numerical values from least to most gives: Q (0.818...), S (0.833...), R (0.844...), P (0.850).

Step-by-Step Solution

1

Convert the portion completed by Candidate P into a standard decimal format.

\frac{17}{20} = \frac{17 \times 5}{20 \times 5} = \frac{85}{100} = 0.85

Converting all values to standard decimals makes it easier to directly compare their magnitudes.

2

Convert the portion completed by Candidate Q into an expanded decimal format.

0.\overline{81} = 0.818181\dots

Expanding the recurring bar notation clarifies the digits beyond the hundredths place.

3

Convert the portion completed by Candidate R into an expanded decimal format.

0.8\overline{4} = 0.844444\dots

Expanding the single recurring digit makes its exact magnitude clear for comparison.

4

Convert the portion completed by Candidate S into a decimal format.

\frac{5}{6} = 5 \div 6 = 0.833333\dots

Performing the division provides the decimal value needed to compare against the others.

5

Compare the resulting decimal values and arrange them in ascending order.

0.8181\dots < 0.8333\dots < 0.8444\dots < 0.8500

Sorting the decimals from smallest to largest yields the order: Candidate Q, Candidate S, Candidate R, Candidate P.

Key Concept

Converting mixed fraction types and recurring decimals into a uniform decimal format to accurately compare their relative magnitudes.

Question 4578Question →

An agricultural research center is preparing experimental test plots. The facility has exactly $2 \frac{5}{8}$ kg, $2 \frac{11}{12}$ kg, and $3 \frac{1}{16}$ kg of three different premium organic soil additives. The lead researcher wants to distribute these additives into smaller, identical sample packets such that no material is left over from any of the three original quantities. To ensure efficiency, the packets must be as large as possible. What should be the exact weight capacity of each sample packet?

\frac{4}{735}

kg

1 \frac{3}{4}

kg

\frac{7}{48}

kg

183 \frac{3}{4}

kg

Show answer & explanation

Answer: $\frac{7}{48}$ kg

Answer

The correct packet capacity is

\frac{7}{48}

kg.

To find the largest possible identical packet size that leaves no remainder, we must calculate the Highest Common Factor (HCF) of the three fractional quantities. After converting the mixed numbers to improper fractions (

\frac{21}{8}

,

\frac{35}{12}

,

\frac{49}{16}

), we apply the required formula: HCF of fractions = (HCF of numerators) / (LCM of denominators). The HCF of the numerators

(21, 35, 49)

is

7

, and the LCM of the denominators

(8, 12, 16)

is

48

. Therefore, the required capacity is

\frac{7}{48}

kg.

Step-by-Step Solution

1

Convert the mixed numbers representing the quantities of additives into improper fractions.

2 \frac{5}{8} = \frac{21}{8}

,

2 \frac{11}{12} = \frac{35}{12}

, and

3 \frac{1}{16} = \frac{49}{16}

.

Calculations involving the HCF and LCM of fractions require the numbers to be in improper fraction format.

2

Identify the mathematical operation required to solve the problem.

Calculate the Highest Common Factor (HCF) of the three fractions.

The problem asks for the largest possible identical packet size that can divide all quantities without a remainder, which maps to finding the HCF.

3

Apply the standard formula for the HCF of fractions.

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

This formula allows the determination of the greatest common divisor for fractional quantities.

4

Calculate the HCF of the numerators and the LCM of the denominators.

The HCF of the numerators

(21, 35, 49)

is

7

. The LCM of the denominators

(8, 12, 16)

is

48

.

Finding these values provides the numerator and denominator for the final fractional answer.

5

Combine the calculated values to form the final fraction.

The maximum identical capacity is

\frac{7}{48}

kg.

Dividing the numerator HCF by the denominator LCM yields the exact required capacity.

Key Concept

Calculating the Highest Common Factor (HCF) of fractions using the formula: HCF of numerators / LCM of denominators.

Estimated Time:1m 30s

Question 4579Question →

A civil contractor is tasked with upgrading a rural highway. In the first month, the crew successfully upgrades $0.275$ of the total highway length. During the second month, they upgrade $\frac{5}{12}$ of the remaining length. If the crew has exactly $10.15$ kilometers left to upgrade in the third month to complete the project, what is the total length of the highway in kilometers?

Show answer & explanation

Answer: 24

Answer

The total length of the highway is 24 kilometers.

The correct total length is calculated by accurately determining the remaining fraction of the highway step-by-step. Converting the decimal 0.275 to a fraction (11/40) allows for clean, exact operations without repeating decimal rounding errors. By tracking the remainder accurately, we establish that the final 10.15 km represents exactly 203/480 of the total length, resulting in a total of 24 km.

Step-by-Step Solution

1

Determine the fraction of the highway remaining after the first month.

\frac{29}{40}

of the total length remains.

The crew upgraded

0.275

of the total. Converting

0.275

to a fraction gives

\frac{275}{1000}

, which simplifies to

\frac{11}{40}

. Subtracting this from the whole gives

1 - \frac{11}{40} = \frac{29}{40}

.

2

Calculate the fraction of the total highway upgraded in the second month.

\frac{29}{96}

of the total length.

The crew upgraded

\frac{5}{12}

of the remaining length. Therefore, multiply the two fractions:

\frac{5}{12} \times \frac{29}{40} = \frac{1 \times 29}{12 \times 8} = \frac{29}{96}

.

3

Calculate the total fraction of the highway remaining for the third month.

\frac{203}{480}

of the total length.

Subtract the second month's progress from the remainder after the first month:

\frac{29}{40} - \frac{29}{96}

. The least common multiple of

40

and

96

is

480

. Converting to common denominators gives

\frac{348}{480} - \frac{145}{480} = \frac{203}{480}

.

4

Set up the final equation and solve for the total highway length.

24

km

The remaining fraction equals the given physical distance of

10.15

km. So,

\frac{203}{480} \times \text{Total} = 10.15

. Solving for Total yields

\frac{10.15 \times 480}{203}

. Since

10.15 \div 203 = 0.05

, the Total is

0.05 \times 480 = 24

km.

Key Concept

Solving sequential parts-of-a-whole word problems by effectively converting between decimals and fractions to find a remaining proportion.

Question 4580Question →

A specialized textile loom is weaving a highly detailed traditional tapestry. Different specialty threads are inserted at specific periodic intervals along the length of the fabric:
- Gold thread is inserted every $\frac{14}{15}$ of a millimeter.
- Silver thread is inserted every $\frac{21}{20}$ of a millimeter.
- Crimson thread is inserted every $\frac{35}{12}$ of a millimeter.

If all three threads are inserted simultaneously at the very beginning of the pattern, at what exact distance from the start will all three specialty threads be inserted at the exact same position again for the first time?

210

mm

\frac{7}{60}

mm

3.5

mm

7

mm

Show answer & explanation

Answer: $210$ mm

Answer

The three specialty threads will be inserted at the exact same position again for the first time at a distance of

210

mm.

The correct answer is found by calculating the Least Common Multiple (LCM) of the three fractions. The formula requires dividing the LCM of the numerators (14, 21, 35) by the HCF of the denominators (15, 20, 12). The LCM of the numerators is 210 and the HCF of the denominators is 1, resulting in exactly 210 mm.

Step-by-Step Solution

1

Identify the mathematical operation required.

The problem asks for the first time three periodic intervals coincide, which requires finding the Least Common Multiple (LCM) of the fractions

\frac{14}{15}

,

\frac{21}{20}

, and

\frac{35}{12}

.

The LCM finds the smallest common multiple of multiple values, representing the first common distance where the intervals align.

2

Calculate the LCM of the numerators.

The numerators are 14, 21, and 35. Their prime factorizations are

14 = 2 \times 7

,

21 = 3 \times 7

, and

35 = 5 \times 7

. The lowest common multiple of these values is

2 \times 3 \times 5 \times 7 = 210

.

The formula for the LCM of fractions dictates that the numerator of the result must be the LCM of the individual numerators.

3

Calculate the HCF (Highest Common Factor) of the denominators.

The denominators are 15, 20, and 12. Their prime factorizations are

15 = 3 \times 5

,

20 = 2^2 \times 5

, and

12 = 2^2 \times 3

. Because there is no single prime factor shared across all three denominators, their HCF is 1.

The formula for the LCM of fractions dictates that the denominator of the result must be the HCF of the individual denominators.

4

Compute the final LCM of the fractions.

\frac{\text{LCM}(14, 21, 35)}{\text{HCF}(15, 20, 12)} = \frac{210}{1} = 210

mm.

Applying the values derived in the previous steps to the fraction LCM formula yields the final coincidence distance.

Key Concept

Calculating the Least Common Multiple (LCM) of fractions using the formula: LCM of fractions = (LCM of numerators) / (HCF of denominators).

Estimated Time:1m 30s

Answer

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