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Dados los racionales: [tex]r=\frac{2}{13}[/tex], [tex]s=\frac{3}{19}[/tex], [tex]t=\frac{6}{41}[/tex], y [tex]p=\frac{3}{20}[/tex], entonces se cumple que

A) [tex]r \ \textless \ p \ \textless \ s \ \textless \ t[/tex]

B) [tex]t \ \textless \ r \ \textless \ p \ \textless \ s[/tex]

C) [tex]p \ \textless \ r \ \textless \ t \ \textless \ s[/tex]

D) [tex]t \ \textless \ p \ \textless \ r \ \textless \ s[/tex]

Sagot :

GinnyAnsweridn
Para resolver este problema, empezamos identificando los valores de los números racionales dados:

- [tex]\( r = \frac{2}{13} \)[/tex]
- [tex]\( s = \frac{3}{19} \)[/tex]
- [tex]\( t = \frac{6}{41} \)[/tex]
- [tex]\( p = \frac{3}{20} \)[/tex]

Debemos ordenar estos racionales de menor a mayor.

1. Comparación de [tex]\( r \)[/tex] y [tex]\( p \)[/tex]:
[tex]\( r = \frac{2}{13} \approx 0.1538 \)[/tex]
[tex]\( p = \frac{3}{20} = 0.15 \)[/tex]

Comparamos:
[tex]\( \frac{2}{13} > \frac{3}{20} \)[/tex], ya que [tex]\( 0.1538 > 0.15 \)[/tex].

Esto significa que [tex]\( p < r \)[/tex].

2. Comparación de [tex]\( s \)[/tex] y [tex]\( p \)[/tex]:
[tex]\( s = \frac{3}{19} \approx 0.1579 \)[/tex]
[tex]\( p = \frac{3}{20} = 0.15 \)[/tex]

Comparamos:
[tex]\( \frac{3}{19} > \frac{3}{20} \)[/tex], ya que [tex]\( 0.1579 > 0.15 \)[/tex].

Esto significa que [tex]\( p < s \)[/tex].

3. Comparación de [tex]\( t \)[/tex] y [tex]\( p \)[/tex]:
[tex]\( t = \frac{6}{41} \approx 0.1463 \)[/tex]
[tex]\( p = \frac{3}{20} = 0.15 \)[/tex]

Comparamos:
[tex]\( \frac{6}{41} < \frac{3}{20} \)[/tex], ya que [tex]\( 0.1463 < 0.15 \)[/tex].

Esto significa que [tex]\( t < p \)[/tex].

4. Comparación de [tex]\( t \)[/tex] y [tex]\( r \)[/tex]:
[tex]\( t = \frac{6}{41} \approx 0.1463 \)[/tex]
[tex]\( r = \frac{2}{13} \approx 0.1538 \)[/tex]

Comparamos:
[tex]\( \frac{6}{41} < \frac{2}{13} \)[/tex], ya que [tex]\( 0.1463 < 0.1538 \)[/tex].

Esto significa que [tex]\( t < r \)[/tex].

5. Comparación de [tex]\( t \)[/tex] y [tex]\( s \)[/tex]:
[tex]\( t = \frac{6}{41} \approx 0.1463 \)[/tex]
[tex]\( s = \frac{3}{19} \approx 0.1579 \)[/tex]

Comparamos:
[tex]\( \frac{6}{41} < \frac{3}{19} \)[/tex], ya que [tex]\( 0.1463 < 0.1579 \)[/tex].

Esto significa que [tex]\( t < s \)[/tex].

6. Comparación de [tex]\( r \)[/tex] y [tex]\( s \)[/tex]:
[tex]\( r = \frac{2}{13} \approx 0.1538 \)[/tex]
[tex]\( s = \frac{3}{19} \approx 0.1579 \)[/tex]

Comparamos:
[tex]\( \frac{2}{13} < \frac{3}{19} \)[/tex], ya que [tex]\( 0.1538 < 0.1579 \)[/tex].

Esto significa que [tex]\( r < s \)[/tex].

Combinando todas las comparaciones, el orden correcto de los racionales de menor a mayor es:
[tex]\[ t < p < r < s \][/tex]

Por lo tanto, la opción correcta es:
D) [tex]\( t < p < r < s \)[/tex]

La respuesta es opción D.
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