Evaluate 5 3/7 - 9 1/3 + (-3 13/35), giving the answer as a mixed number.

Step-by-step solution

1. 1

   Convert every mixed number to an improper fraction. Multiply the whole part by the denominator and add the numerator:

   $5\tfrac37=\frac{5\cdot7+3}{7}=\frac{38}{7},\quad 9\tfrac13=\frac{9\cdot3+1}{3}=\frac{28}{3},\quad 3\tfrac{13}{35}=\frac{3\cdot35+13}{35}=\frac{118}{35}.$

   Mixed numbers are hard to subtract directly because borrowing across the whole and fractional parts goes wrong easily; improper fractions remove that risk entirely.

2. 2

   Handle the "plus a negative". Adding $-3\tfrac{13}{35}$ is the same as subtracting $3\tfrac{13}{35}$, so the expression is

   $\frac{38}{7}-\frac{28}{3}-\frac{118}{35}.$

   All three terms now carry explicit signs and can be combined in one pass.

3. 3

   Find the least common denominator. With denominators $7=7$, $3=3$ and $35=5\cdot7$, the LCM is $3\cdot5\cdot7=105$. Converting:

   $\frac{38}{7}=\frac{570}{105},\qquad \frac{28}{3}=\frac{980}{105},\qquad \frac{118}{35}=\frac{354}{105}.$

   Check each multiplier: $\times15$, $\times35$, $\times3$ respectively.

4. 4

   Combine the numerators. Over the single denominator:

   $\frac{570-980-354}{105}=\frac{-764}{105}.$

   Running the subtraction in order: $570-980=-410$, then $-410-354=-764$. The result is negative, which the estimate $5.4-9.3-3.4\approx-7.3$ already predicted.

5. 5

   Convert back to a mixed number and confirm. Dividing $764$ by $105$ gives $7$ with remainder $29$ (since $7\times105=735$ and $764-735=29$), so

   $\frac{-764}{105}=-7\tfrac{29}{105}\approx-7.2762.$

   Since $\gcd(764,105)=1$ the fraction is already in lowest terms, and the decimal matches the estimate ✓.