Solve for x: 0.8(x + 100) = 160.

Step-by-step solution

1. 1

   Choose between distributing and dividing. Two routes work: expand to $0.8x+80=160$, or divide both sides by $0.8$ straight away. Dividing is quicker here because $160$ is a convenient multiple of $0.8$, and it keeps the bracket intact.

2. 2

   Divide both sides by 0.8.

   $x+100=\frac{160}{0.8}$

   Because $\tfrac{1}{0.8}=1.25$, dividing by $0.8$ is the same as multiplying by $1.25$:

   $\frac{160}{0.8}=160\times1.25=200$

   so $x+100=200$.

3. 3

   Isolate x. Subtract $100$ from both sides:

   $x=200-100=100$

4. 4

   Cross-check by the other route. Expanding instead: $0.8x+80=160$, so $0.8x=80$ and $x=\tfrac{80}{0.8}=100$ ✓. Both methods land on the same value, which is a genuine independent check rather than a repeat of the same arithmetic.

5. 5

   Verify in the original equation. $0.8(100+100)=0.8\times200=160$ ✓, matching the right side exactly.