Solve for x: (3.75 x 98) + 4x = 3.8 x (98 + x).

Step-by-step solution

1. 1

   Read what the equation says. A quantity of $98$ units at grade $3.75$ is blended with $x$ units at grade $4$, and the result must average $3.8$ across all $98+x$ units. Because $3.8$ lies much closer to $3.75$ than to $4$, the answer should be a fairly small $x$ — a useful expectation to hold before computing.

2. 2

   Evaluate the constants and expand. On the left, $3.75\times98=367.5$. On the right, distribute:

   $3.8\times(98+x)=3.8\times98+3.8x=372.4+3.8x.$

   Check the two products: $3.75\times98=3.75\times100-3.75\times2=375-7.5=367.5$ ✓, and $3.8\times98=380-7.6=372.4$ ✓.

3. 3

   Collect the $x$ terms. The equation is $367.5+4x=372.4+3.8x$. Subtracting $3.8x$ from both sides:

   $367.5+0.2x=372.4.$

   The coefficient $0.2=4-3.8$ is the gap between the new material's grade and the target grade — that small number is why $x$ ends up much larger than the constant difference.

4. 4

   Isolate and solve. Subtracting $367.5$:

   $0.2x=4.9\;\Longrightarrow\;x=\frac{4.9}{0.2}=24.5.$

   Dividing by $0.2$ is multiplying by $5$, so $4.9\times5=24.5$ — no long division required.

5. 5

   Check both sides of the original equation. Left: $367.5+4(24.5)=367.5+98=465.5$. Right: $3.8\times(98+24.5)=3.8\times122.5=465.5$ ✓. The blend contains $24.5$ units of the richer material against $98$ of the poorer, which matches the expectation from step 1 that $x$ would be a modest fraction of $98$.