A specialized textile loom is weaving a highly detailed traditional tapestry. Different sp… — State PSC Exam

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?

$210$ mmCevap
B
$\frac{7}{60}$ mm
C
$3.5$ mm
D
$7$ mm

Cevap

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.

Adım Adım Çözüm

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.

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.

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.

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.

Anahtar Kavram

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

Tahmini Süre:1m 30s

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