HCF and LCM Alıştırma Soruları — State PSC Exam — Sayfa 2

What is the Highest Common Factor (HCF) of the fractions $\frac{2}{3}$ , $\frac{4}{9}$ , and $\frac{8}{15}$ ?

\frac{8}{3}

\frac{2}{45}

\frac{2}{3}

\frac{8}{45}

Cevabı ve açıklamayı göster

Cevap: $\frac{2}{45}$

Cevap

The correct answer is

\frac{2}{45}

.

The HCF of a set of fractions is found by dividing the HCF of the numerators by the LCM of the denominators. The numerators are 2, 4, and 8, and their HCF is 2. The denominators are 3, 9, and 15, and their LCM is 45. Combining these yields the correct value of

\frac{2}{45}

.

Adım Adım Çözüm

1

Identify the formula for finding the HCF of fractions.

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

This standard formula is required to correctly solve for the highest common factor of any set of fractions.

2

Extract the numerators and find their Highest Common Factor (HCF).

The numerators are 2, 4, and 8. Their HCF is 2.

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

3

Extract the denominators and find their Least Common Multiple (LCM).

The denominators are 3, 9, and 15. Their LCM is 45.

45 is the smallest positive integer that is a multiple of 3, 9, and 15.

4

Apply the calculated values to the fraction HCF formula.

The final fraction is

\frac{2}{45}

.

Substituting the HCF of the numerators (2) and the LCM of the denominators (45) into the formula gives the final answer.

Anahtar Kavram

The HCF of two or more fractions is calculated by dividing the HCF of their numerators by the LCM of their denominators.

Soru 22Soru →

An artisan is preparing metal rods for a custom fence. Two existing rods, which measure $\frac{15}{4}$ meters and $\frac{25}{8}$ meters in length, must be cut into identical smaller pieces of the maximum possible length without wasting any material. What should be the length of each piece?

\frac{75}{4}

meters

\frac{5}{4}

meters

\frac{5}{8}

meters

\frac{75}{8}

meters

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{8}$ meters

Cevap

\frac{5}{8}

meters

The maximum possible length of each piece is found by calculating the HCF of the two fractional lengths. The correct formula is to divide the HCF of the numerators (15 and 25) by the LCM of the denominators (4 and 8). The HCF of 15 and 25 is 5, and the LCM of 4 and 8 is 8, resulting in exactly

\frac{5}{8}

meters.

Adım Adım Çözüm

1

Identify the mathematical operation required for the scenario.

Finding the maximum possible identical length from two given lengths requires calculating their Highest Common Factor (HCF).

The pieces must be of equal length and as large as possible without leaving any remainder.

2

State the formula for finding the HCF of fractions.

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

This is the standard rule for determining the greatest common divisor of rational numbers.

3

Calculate the HCF of the numerators (15 and 25).

The factors of 15 are 1, 3, 5, 15. The factors of 25 are 1, 5, 25. The highest common factor is 5.

The numerator of our final answer must be the HCF of the original numerators.

4

Calculate the LCM of the denominators (4 and 8).

The multiples of 8 (8, 16, 24...) are already divisible by 4. Thus, the least common multiple is 8.

The denominator of our final answer must be the LCM of the original denominators.

5

Construct the final fraction.

\frac{5}{8}

meters.

Combining the results from the previous steps yields the correct maximum length.

Anahtar Kavram

Highest Common Factor (HCF) of fractions

Soru 23Soru →

A logistics manager is packing identical relief kits into crates. When she attempts to pack them in equal batches of $16$ , $24$ , $30$ , or $36$ kits per crate, there are always exactly $8$ kits left over. However, if she packs them in batches of exactly $19$ kits per crate, there are zero kits left over. What is the least possible total number of relief kits she could be packing?

Cevabı ve açıklamayı göster

Cevap: 2888

Cevap

The least possible total number of relief kits is 2888.

The correct answer is derived by finding the general form of a number that leaves a remainder of 8 when divided by 16, 24, 30, and 36. This form is

720k + 8

. By applying the final condition that the total number must be exactly divisible by 19, we find the smallest valid multiplier is

k=4

. Substituting this back gives

720(4) + 8 = 2888

.

Adım Adım Çözüm

1

Establish the relationship for the total number of kits based on the remainders.

The number of kits,

N

, leaves a remainder of

8

when divided by

16

,

24

,

30

, and

36

. Thus,

N = \text{LCM}(16, 24, 30, 36) \times k + 8

.

Any number that leaves the same remainder when divided by multiple divisors can be expressed as a multiple of their least common multiple plus that remainder.

2

Calculate the least common multiple (LCM) of

16

,

24

,

30

, and

36

.

The prime factorizations are

16 = 2^4

,

24 = 2^3 \times 3

,

30 = 2 \times 3 \times 5

, and

36 = 2^2 \times 3^2

. The LCM is the product of the highest powers:

2^4 \times 3^2 \times 5 = 720

.

The LCM is required to find the base repeating cycle for the division condition.

3

Formulate the exact divisibility condition.

Substitute the LCM into the equation to get

N = 720k + 8

. The problem states

N

is exactly divisible by

19

, so

(720k + 8) \equiv 0 \pmod{19}

.

This applies the second constraint of the problem to find the specific multiplier

k

.

4

Simplify the modular arithmetic equation to solve for

k

.

Divide

720

by

19

to find the remainder:

720 = 19 \times 37 + 17

. Substitute

17

for

720

to get

(17k + 8) \equiv 0 \pmod{19}

. This can be written as

(-2k + 8) \equiv 0 \pmod{19}

.

Simplifying large numbers using modulo properties makes finding the integer

k

manageable.

5

Find the smallest positive integer

k

that satisfies the equation.

Solving

-2k + 8 = 0

yields

2k = 8

, which means

k = 4

. Checking:

17(4) + 8 = 68 + 8 = 76

, and

76 \div 19 = 4

, which is exactly divisible.

Finding the smallest valid

k

ensures we calculate the least possible total number of kits.

6

Calculate the final total number of kits.

N = 720(4) + 8 = 2880 + 8 = 2888

.

Substituting

k = 4

back into the original formula for

N

provides the final numerical answer.

Anahtar Kavram

Finding a specific numerical value based on multiple simultaneous divisibility and remainder conditions using Least Common Multiple (LCM) and modular arithmetic.

Soru 24Soru →

A botanist is preparing nutrient solutions. She has three different liquid nutrient extracts measuring $\frac{24}{5}$ liters, $\frac{32}{3}$ liters, and $\frac{40}{7}$ liters. She wants to distribute them entirely into identical small sample vials of maximum possible capacity, such that no extract is left over and the extracts are not mixed. What should be the maximum capacity of each sample vial?

\frac{8}{105}

liters

\frac{1}{480}

liters

480

liters

8

liters

Cevabı ve açıklamayı göster

Cevap: $\frac{8}{105}$ liters

Cevap

The correct maximum capacity is

\frac{8}{105}

liters.

To find the maximum identical capacity that can exactly measure out each of the given quantities without mixing, we must calculate the Highest Common Factor (HCF) of the given fractions. By applying the formula for the HCF of fractions, we divide the HCF of the numerators (8) by the Least Common Multiple (LCM) of the denominators (105), resulting in

\frac{8}{105}

liters.

Adım Adım Çözüm

1

Identify the mathematical operation required.

We need to find the Highest Common Factor (HCF) of the three fractions to determine the maximum equal capacity.

The vials must have the maximum possible capacity and perfectly measure out all three volumes without mixing them or leaving any remainder.

2

Recall the formula for the HCF of fractions.

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

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

3

Calculate the HCF of the numerators: 24, 32, and 40.

The prime factorizations are

24 = 2^3 \times 3

,

32 = 2^5

, and

40 = 2^3 \times 5

. The highest common factor is

2^3 = 8

.

The HCF of the numerators forms the numerator of our final answer.

4

Calculate the LCM of the denominators: 5, 3, and 7.

Since 5, 3, and 7 are all prime numbers, their LCM is their product:

5 \times 3 \times 7 = 105

.

The LCM of the denominators forms the denominator of our final answer.

5

Construct the final fraction.

The capacity is

\frac{8}{105}

liters.

Dividing the HCF of the numerators by the LCM of the denominators gives the final HCF of the original fractions.

Anahtar Kavram

HCF and LCM of fractions

Soru 25Soru →

Calculate the lowest common multiple (LCM) of the fractions $\frac{3}{4}$ , $\frac{9}{10}$ , and $\frac{15}{16}$ . Express your final answer as a decimal.

Cevabı ve açıklamayı göster

Cevap: 22.5

Cevap

22.5

The lowest common multiple (LCM) of a set of fractions is found by dividing the LCM of their numerators by the highest common factor (HCF) of their denominators. For the numerators

3

,

9

, and

15

, the LCM is

45

. For the denominators

4

,

10

, and

16

, the HCF is

2

. Thus, the LCM of the fractions is

\frac{45}{2}

, which equals

22.5

in decimal form.

Adım Adım Çözüm

1

Recall the mathematical formula for finding the LCM of multiple fractions.

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

This formula is the standard method to determine the lowest common multiple when dealing with fractional values.

2

Identify the numerators and calculate their lowest common multiple (LCM).

The numerators are

3

,

9

, and

15

. Their LCM is

45

(since

45

is the smallest number perfectly divisible by

3

,

9

, and

15

).

The numerator of the final fraction requires the LCM of all the given numerators.

3

Identify the denominators and calculate their highest common factor (HCF).

The denominators are

4

,

10

, and

16

. Their HCF is

2

(since

2

is the largest number that divides

4

,

10

, and

16

without a remainder).

The denominator of the final fraction requires the HCF of all the given denominators.

4

Apply the calculated values to the fraction LCM formula.

\text{LCM} = \frac{45}{2}

Combining the results from the previous steps yields the LCM in fractional form.

5

Convert the resulting fraction into a decimal format.

\frac{45}{2} = 22.5

The question explicitly requires the final answer to be expressed as a decimal.

Anahtar Kavram

Calculating the LCM of fractions using the specific formula: LCM of numerators divided by the HCF of denominators.

Soru 26Soru →

The Highest Common Factor (HCF) and Least Common Multiple (LCM) of two positive three-digit integers $P$ and $Q$ (where $P > Q$ ) are $24$ and $1080$ , respectively. If the difference between the two numbers is $96$ , what is the sum of the two numbers ( $P + Q$ )?

Cevabı ve açıklamayı göster

Cevap: 336

Cevap

The sum of the two numbers P and Q is 336.

By representing

P = 24a

and

Q = 24b

with

\gcd(a, b) = 1

, the relation

\text{HCF} \times \text{LCM} = P \times Q

yields

a \times b = 45

. The coprime factor pairs of

45

are

(45, 1)

and

(9, 5)

. The pair

(45, 1)

gives

1080

and

24

, which are not both three-digit numbers. The pair

(9, 5)

gives

P = 216

and

Q = 120

, both of which are three-digit numbers with a difference of

96

. The sum of these two numbers is

216 + 120 = 336

.

Adım Adım Çözüm

1

Express the two numbers in terms of their HCF and coprime factors

Let

P = 24a

and

Q = 24b

where

\gcd(a, b) = 1

and

a > b

.

Any two numbers sharing an HCF of

h

can be represented as

h \cdot a

and

h \cdot b

where

a

and

b

have no common prime factors.

2

Relate the product of the coprime factors to the LCM and HCF

a \times b = \frac{\text{LCM}}{\text{HCF}} = \frac{1080}{24} = 45

.

The product of two numbers equals the product of their HCF and LCM:

(24a)(24b) = 24 \times 1080

.

3

Identify all coprime factor pairs of 45

The coprime factor pairs

(a, b)

with

a > b

are

(45, 1)

and

(9, 5)

.

Factor pairs such as

(15, 3)

are invalid because

\gcd(15, 3) = 3 \neq 1

.

4

Apply the three-digit integer and difference constraints to select the valid pair

For

(a, b) = (9, 5)

,

P = 24 \times 9 = 216

and

Q = 24 \times 5 = 120

. Difference =

216 - 120 = 96

.

The pair

(45, 1)

yields

P = 1080

(four digits) and

Q = 24

(two digits), failing the three-digit criteria.

5

Calculate the sum of

P

and

Q

P + Q = 216 + 120 = 336

.

The question asks specifically for the sum

P + Q

.

Anahtar Kavram

Properties of HCF and LCM including HCF x LCM = Product of Numbers and Prime Factor Coprimality

Soru 27Soru →

Consider the three fractions $\frac{12}{5}$ , $\frac{24}{7}$ , and $\frac{36}{11}$ . What is the exact value of their Least Common Multiple (LCM)?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

The Least Common Multiple (LCM) of the given fractions is 72.

To find the Least Common Multiple (LCM) of a set of fractions, we must divide the LCM of their numerators by the HCF of their denominators. The numerators (

12

,

24

, and

36

) have an LCM of

72

. The denominators (

5

,

7

, and

11

) share no common factors other than

1

, making their HCF

1

. Thus, the final LCM is

72 / 1 = 72

.

Adım Adım Çözüm

1

State the standard formula for finding the LCM of multiple fractions.

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

This mathematical property governs how multiples apply to rational numbers.

2

Calculate the LCM of the numerators.

The LCM of

12

,

24

, and

36

is

72

.

72

is the smallest integer perfectly divisible by all three numerators.

3

Calculate the HCF of the denominators.

The HCF of

5

,

7

, and

11

is

1

.

The numbers

5

,

7

, and

11

are prime and share no common factors other than

1

.

4

Compute the final fraction LCM.

72 \div 1 = 72

.

Dividing the computed numerator LCM by the denominator HCF yields the answer.

Anahtar Kavram

Calculating the LCM of fractions using the specific formula relating numerators and denominators.

Soru 28Soru →

Determine the exact decimal value of the Least Common Multiple (LCM) for the two fractions $\frac{3}{5}$ and $\frac{9}{10}$ .

Cevabı ve açıklamayı göster

Cevap: 1.8

Cevap

1.8

To find the LCM of fractions, we use the standard formula: LCM(numerators) / HCF(denominators). The numerators are 3 and 9, and their LCM is 9. The denominators are 5 and 10, and their HCF is 5. Dividing the LCM of the numerators by the HCF of the denominators gives 9/5. Converting this fraction to a decimal yields exactly 1.8.

Adım Adım Çözüm

1

Identify the correct formula for finding the LCM of fractions.

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

This is the fundamental rule for calculating the least common multiple when dealing with rational numbers.

2

Calculate the Least Common Multiple of the numerators.

The numerators are 3 and 9. Their LCM is 9.

9 is the smallest integer that is perfectly divisible by both 3 and 9.

3

Calculate the Highest Common Factor of the denominators.

The denominators are 5 and 10. Their HCF is 5.

5 is the largest integer that perfectly divides both 5 and 10 without leaving a remainder.

4

Apply the formula and convert the resulting fraction to a decimal.

LCM = 9 / 5 = 1.8

The question specifically asks for the exact decimal value, so the fraction 9/5 must be converted to 1.8.

Anahtar Kavram

The Least Common Multiple (LCM) of a set of fractions is found by dividing the LCM of their numerators by the HCF of their denominators.

Soru 29Soru →

The product of two positive integers is $2160$ and their Highest Common Factor (HCF) is $12$ . What is their Least Common Multiple (LCM)?

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

180

For any two positive integers, the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM) is always equal to the product of the numbers themselves. Given that the product is

2160

and the HCF is

12

, the LCM is calculated by rearranging the formula to

\text{LCM} = \frac{\text{Product}}{\text{HCF}}

. Substituting the values gives

2160 \div 12 = 180

.

Adım Adım Çözüm

1

Identify the fundamental formula connecting HCF, LCM, and the product of two positive integers.

\text{HCF} \times \text{LCM} = \text{Product}

This relationship allows you to find one missing value when the other two are known.

2

Substitute the known values from the problem into the equation.

12 \times \text{LCM} = 2160

The problem explicitly states that the product is

2160

and the HCF is

12

.

3

Solve for the LCM by dividing both sides of the equation by

12

.

\text{LCM} = 180

Isolating the LCM variable provides the final requested value.

Anahtar Kavram

Relationship between the product of two numbers and their HCF and LCM

Soru 30Soru →

Two automated watering sprinklers in a greenhouse operate on continuous cycles. Sprinkler X activates every $\frac{12}{5}$ minutes, and Sprinkler Y activates every $\frac{16}{15}$ minutes. If both sprinklers activate simultaneously at a given moment, what is the minimum time interval, in minutes, before they activate together again?

\frac{48}{5}

\frac{4}{15}

\frac{16}{5}

\frac{4}{5}

Cevabı ve açıklamayı göster

Cevap: $\frac{48}{5}$

Cevap

The minimum time interval is

\frac{48}{5}

minutes.

To find when two periodic events will happen together again, we must calculate the Least Common Multiple (LCM) of their periods. For fractions, the LCM is found by dividing the LCM of the numerators (12 and 16, which is 48) by the HCF of the denominators (5 and 15, which is 5). This results in

\frac{48}{5}

.

Adım Adım Çözüm

1

Identify that the problem requires finding the Least Common Multiple (LCM) of the two time intervals to determine when the periodic events will align again.

Calculate the LCM of

\frac{12}{5}

and

\frac{16}{15}

.

Simultaneous events occurring at regular intervals coincide at the lowest common multiple of their respective periods.

2

Apply the formula for the LCM of fractions.

Formula:

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

.

This is the standard mathematical rule for finding the LCM of fractional values.

3

Find the LCM of the numerators (12 and 16) and the HCF of the denominators (5 and 15).

LCM(12, 16) = 48. HCF(5, 15) = 5.

The smallest positive number divisible by both 12 and 16 is 48. The largest positive number that divides both 5 and 15 is 5.

4

Substitute the calculated values back into the fraction formula.

\frac{48}{5}

Combining the calculated numerator and denominator yields the final time interval.

Anahtar Kavram

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

Soru 31Soru →

In a chemical manufacturing plant, three automated valves release specific additives into a continuous mixing tank. Valve A opens every $\frac{9}{2}$ minutes, Valve B opens every $\frac{15}{4}$ minutes, and Valve C opens every $\frac{45}{8}$ minutes. The system is programmed to record a 'synchronization event' whenever all three valves open at the exact same instant.

If the mixing process runs continuously for exactly $12$ hours, how many synchronization events will be recorded during this period? (Assume a synchronization event is recorded at the very beginning of the process, which counts as the first event).

129

32

271

33

Cevabı ve açıklamayı göster

Cevap: $33$

Cevap

The correct answer is

33

synchronization events.

The synchronization interval is the LCM of the fractions

\frac{9}{2}

,

\frac{15}{4}

, and

\frac{45}{8}

. Using the formula

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

, we get

\frac{\text{LCM}(9, 15, 45)}{\text{HCF}(2, 4, 8)} = \frac{45}{2} = 22.5

minutes. Over a

12

-hour period (

720

minutes), there are precisely

\frac{720}{22.5} = 32

complete intervals. Since a synchronization event is recorded at the very beginning of the process, the total number of recorded events is

32 + 1 = 33

.

Adım Adım Çözüm

1

Identify the mathematical concept required to find the synchronization interval.

The shortest time interval for all valves to open together is the Least Common Multiple (LCM) of their individual time cycles.

Synchronization occurs at common multiples of the cycle times, and the next event always occurs at the least common multiple.

2

Calculate the LCM of the fractions

\frac{9}{2}

,

\frac{15}{4}

, and

\frac{45}{8}

.

\text{LCM of fractions} = \frac{\text{LCM}(9, 15, 45)}{\text{HCF}(2, 4, 8)}

.

The standard formula for the LCM of fractions requires finding the LCM of the numerators divided by the HCF of the denominators.

3

Compute the LCM of numerators and HCF of denominators.

\text{LCM}(9, 15, 45) = 45

and

\text{HCF}(2, 4, 8) = 2

. Thus, the synchronization interval is

\frac{45}{2} = 22.5

minutes.

This establishes that every

22.5

minutes, all three valves will open simultaneously.

4

Calculate the total time in minutes for the

12

-hour shift.

12 \text{ hours} \times 60 \text{ minutes/hour} = 720 \text{ minutes}

.

Units must be consistent to accurately determine the number of intervals.

5

Determine the number of full intervals and total synchronization events.

\text{Number of intervals} = \frac{720}{22.5} = 32

. Total events

= 32 + 1 = 33

.

The division yields the number of periods between events. We add

1

to account for the initial synchronization event explicitly mentioned in the problem (the fencepost principle).

Anahtar Kavram

Calculating the Least Common Multiple (LCM) of fractions and applying it to a temporal boundary condition (fencepost counting).

Tahmini Süre:2m 0s

Soru 32Soru →

A municipal election committee is distributing ballots to various polling stations. When they pack the ballots in bundles of $45$ , $54$ , or $72$ , they find that they are always left with $38$ , $47$ , and $65$ unbundled ballots, respectively. If the total number of ballots printed is the largest possible 4-digit number satisfying these conditions, what is the exact total number of ballots?

Cevabı ve açıklamayı göster

Cevap: 9713

Cevap

9713

The problem describes a scenario where the difference between each divisor (

45

,

54

,

72

) and its respective remainder (

38

,

47

,

65

) is exactly

7

. This means that if

7

more ballots were added, the total would be perfectly divisible by all three numbers. Therefore, the required total is exactly

7

less than a common multiple of these divisors. The LCM of

45

,

54

, and

72

is

1080

. The largest

4

-digit multiple of

1080

is

9720

(

1080 \times 9

). Subtracting the constant difference of

7

from

9720

gives the final answer of

9713

.

Adım Adım Çözüm

1

Calculate the difference between each bundle size and its corresponding remainder.

45 - 38 = 7

,

54 - 47 = 7

, and

72 - 65 = 7

.

To identify if there is a constant difference, which allows the use of the LCM minus constant method.

2

Determine the Least Common Multiple (LCM) of the bundle sizes

45

,

54

, and

72

.

\text{LCM}(45, 54, 72) = 1080

.

The LCM represents the smallest bundle size that perfectly divides by all three numbers. Prime factorizations:

45 = 3^2 \times 5

,

54 = 2 \times 3^3

,

72 = 2^3 \times 3^2

. LCM

= 2^3 \times 3^3 \times 5 = 1080

.

3

Find the largest 4-digit multiple of the LCM.

1080 \times 9 = 9720

.

The problem asks for the largest 4-digit number. Dividing

9999

by

1080

yields

9.258...

, so the largest integer multiplier is

9

.

4

Subtract the constant difference from this largest multiple.

9720 - 7 = 9713

.

Since each division left a remainder that was

7

short of a full bundle, subtracting

7

from a perfect multiple satisfies all three remainder conditions.

Anahtar Kavram

Solving simultaneous remainder problems where the difference between divisors and remainders is constant, by utilizing the LCM and scaling to a specific boundary range.

Soru 33Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 73

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:2m 0s

Soru 34Soru →

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

Cevabı ve açıklamayı göster

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

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Tahmini Süre:1m 30s

Soru 35Soru →

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.

Cevabı ve açıklamayı göster

Cevap: 21.6

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Lowest Common Multiple (LCM) of Fractions

Soru 36Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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.

HCF and LCM

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