#### To determine

**To express:** The given number as a ratio of integers.

#### Answer

The number 10.135¯ can be expressed as 5017495.

#### Explanation

**Given:**

10.135¯=10.1353535...

**Result used:**

The geometric series ∑n=1∞arn−1 (or) a+ar+ar2+⋯ is convergent if |r|<1 and its sum is a1−r, where *a* is the first term and *r* is the common ratio of the series.

**Calculation:**

Rewrite the number and express 10.135¯ as follows,

10.135¯=10.1+0.035¯=10.1+(0.035+0.00035+⋯)=10.1+(35103+35105+⋯)

10.135¯=10.1+∑n=1∞35102n+1 (1)

Here, ∑n=1∞35102n+1 is geometric series with first term of the series is a=35103 and common ratio is r=1102.

Since |r|<1 and by the Result stated above, the geometric series ∑n=1∞35102n+1 is convergent.

Obtain the sum of the geometric series.

Since a=351000 and r=1100, the series becomes,

∑n=1∞35102n+1=3510001−1100=35100099100=351000×10099=35990

Thus, the series is convergent to the sum ∑n=1∞35102n+1=35990. (2)

Substitute equation (2) in equation (1),

10.135¯=10.1+35990=9999+35990=10,034990=5017495

Therefore, the number 10.135¯ can be expressed as 5017495.