Evaluate 5 3/7 - 9 1/5 + (-3 13/35).

Step-by-step solution

1. 1

   Convert every mixed number to an improper fraction. Mixed-number borrowing across three terms with different denominators is error-prone; improper fractions remove the issue entirely:

   $5\frac{3}{7}=\frac{5\cdot 7+3}{7}=\frac{38}{7},\qquad 9\frac{1}{5}=\frac{46}{5},\qquad 3\frac{13}{35}=\frac{118}{35}$

2. 2

   Resolve the double sign. Adding a negative is subtracting, so

   $\frac{38}{7}-\frac{46}{5}+\left(-\frac{118}{35}\right)=\frac{38}{7}-\frac{46}{5}-\frac{118}{35}$

3. 3

   Rewrite over the least common denominator. $\operatorname{lcm}(7,5,35)=35$:

   $\frac{38}{7}=\frac{190}{35},\qquad \frac{46}{5}=\frac{322}{35},\qquad \frac{118}{35}=\frac{118}{35}$

4. 4

   Combine the numerators in one line.

   $\frac{190-322-118}{35}=\frac{-250}{35}$

5. 5

   Reduce. $\gcd(250,35)=5$, so

   $\frac{-250}{35}=-\frac{50}{7}$

6. 6

   Convert back to a mixed number and sanity-check the sign. $50=7\cdot 7+1$, so

   $-\frac{50}{7}=-7\frac{1}{7}\approx -7.142857$

   The estimate confirms it: $5.43-9.20-3.37\approx -7.14$. Note the whole part and the fraction part both carry the minus sign — $-7\frac17$ means $-\left(7+\frac17\right)$, not $-7+\frac17$.