un atleta debe recorrer 30 km en sus ultimos 5 dias de preparacion para una competencia. el primer dia recorre frac{23}{2} km, el segundo frac{15}{4} km, el tercer dia frac{16}{3} km y el cuarto dia frac{35}{6} km. ¿cuanto km recorrió el atleta en estos cuatro dias?

The total distance the athlete traveled over the four days is the sum of the four given fractions.

To add fractions, you must first rewrite them to have a common denominator.

For the fractions, $\frac{23}{2},\frac{15}{4},\frac{16}{3},$ and $\frac{35}{6}$, the denominators are 2, 4, 3, and 6.

To find a common denominator, you then need to find the least common multiple of 2, 3, 4, and 6.

Since 2 and 3 are factors of 6, the lowest common multiply of 2, 3, 4, and 6 is the lowest common multiple of 4 and 6.

The first few multiples of 4 are 4, 8, 12, and 16, The first few multiples of 6 are 6, 12, 18, and 24.

The least common multiple of 4 and 6 is then 12.

You then need to rewrite the fractions to have a denominator of 12:

$\frac{23}{2}=23\times \frac{6}{2}\times 6=\frac{138}{12}$

$\frac{15}{4}=15\times \frac{3}{4}\times 3=\frac{45}{12}$

$\frac{16}{3}=16\times \frac{4}{3}\times 4=\frac{64}{12}$

$\frac{35}{6}=35\times \frac{2}{6}\times 12=\frac{70}{12}$

Now that the fractions have a common denominator, you can add them to find the total distance that the athlete traveled.

To add the fractions, add the numerators and leave the denominator the same:

$\frac{138}{12}+\frac{45}{12}+\frac{64}{12}+\frac{70}{12}=\frac{138+45+64+70}{12}=\frac{317}{12}$

To convert an improper fraction to a mixed number, determine how many times the denominator goes into the numerator. 12 goes into 317 twenty-six times with a remainder of 5 since $317=12\times 26+5.$

Therefore $\frac{317}{12}=26\times \frac{5}{12}.$

The athlete then traveled a total distance of $26\times \frac{5}{12}$ km ​ over the four days.