Subtracting mixed numbers with borrowing: a math tutorial showing fraction conversion and subtraction steps.

Problem Description:

The task involves subtracting mixed numbers with borrowing. The steps provided in the image are as follows:

1. Convert fractions to their common denominator.
2. Borrow 1 from the whole number of the first fraction.
3. Convert the first mixed number to an improper fraction and subtract.

We need to solve the problem step by step based on these instructions.

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Given Problem:

$$
8 \frac{2}{7} - 5 \frac{4}{5}
$$

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Step-by-Step Solution:

#### Step 1: Convert fractions to their common denominator
The fractions in the mixed numbers are $\frac{2}{7}$ and $\frac{4}{5}$. To subtract them, we need a common denominator.

- The denominators are $7$ and $5$.
- The least common denominator (LCD) is $35$.

Convert each fraction to have the denominator $35$:
$$
\frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35}
$$
$$
\frac{4}{5} = \frac{4 \times 7}{5 \times 7} = \frac{28}{35}
$$

So, the mixed numbers become:
$$
8 \frac{2}{7} = 8 \frac{10}{35}
$$
$$
5 \frac{4}{5} = 5 \frac{28}{35}
$$

Now the problem is:
$$
8 \frac{10}{35} - 5 \frac{28}{35}
$$

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#### Step 2: Borrow 1 from the whole number of the first fraction
In the first mixed number $8 \frac{10}{35}$, the fractional part $\frac{10}{35}$ is smaller than the fractional part $\frac{28}{35}$ in the second mixed number. Therefore, we need to borrow $1$ from the whole number $8$.

- Borrowing $1$ from $8$ leaves $7$ as the whole number.
- Add $1$ (which is $\frac{35}{35}$) to the fractional part $\frac{10}{35}$:
$$
\frac{10}{35} + \frac{35}{35} = \frac{45}{35}
$$

So, $8 \frac{10}{35}$ becomes:
$$
7 \frac{45}{35}
$$

Now the problem is:
$$
7 \frac{45}{35} - 5 \frac{28}{35}
$$

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#### Step 3: Convert the first mixed number to an improper fraction and subtract
First, convert $7 \frac{45}{35}$ to an improper fraction:
$$
7 \frac{45}{35} = \frac{7 \times 35 + 45}{35} = \frac{245 + 45}{35} = \frac{290}{35}
$$

Next, convert $5 \frac{28}{35}$ to an improper fraction:
$$
5 \frac{28}{35} = \frac{5 \times 35 + 28}{35} = \frac{175 + 28}{35} = \frac{203}{35}
$$

Now subtract the two improper fractions:
$$
\frac{290}{35} - \frac{203}{35} = \frac{290 - 203}{35} = \frac{87}{35}
$$

Convert the improper fraction $\frac{87}{35}$ back to a mixed number:
$$
\frac{87}{35} = 2 \frac{17}{35}
$$

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Final Answer:

$$
\boxed{2 \frac{17}{35}}
$$