Evaluate (2/5) * (-209/7) + 1041 + 4/5.

Step-by-step solution

1. 1

   Do the multiplication first (order of operations). Multiply numerators and denominators straight across, keeping the negative sign:

   $\frac{2}{5}\times\left(-\frac{209}{7}\right)=-\frac{418}{35}$

   Nothing cancels since $\gcd(2,7)=\gcd(209,5)=1$.

2. 2

   Group the two fractional terms before touching 1041. Handling $-\frac{418}{35}+\frac{4}{5}$ on its own keeps the numbers small. Over the denominator $35$, $\frac{4}{5}=\frac{28}{35}$, so

   $-\frac{418}{35}+\frac{28}{35}=-\frac{390}{35}=-\frac{78}{7}$

   (dividing numerator and denominator by $5$).

3. 3

   Write the whole number over the same denominator.

   $1041=\frac{1041\times 7}{7}=\frac{7287}{7}$

4. 4

   Combine into a single fraction.

   $\frac{7287}{7}-\frac{78}{7}=\frac{7209}{7}$

5. 5

   Check whether it reduces, then convert. $7209=3\times 2403=3^{2}\times 801=\ldots$ contains no factor $7$ (indeed $7\times 1029=7203$, remainder $6$), so $\frac{7209}{7}$ is already in lowest terms. As a mixed number it is $1029\frac{6}{7}$, i.e. $x\approx 1029.857143$.

6. 6

   Estimate as a check. $\frac{2}{5}\times(-29.857)\approx -11.94$, and $-11.94+1041+0.8\approx 1029.86$ ✓, matching the exact value.