Question: Describe and correct any error a student may have made when solving the equation $0.15(y - 0.2) = 2 - 0.5(1 - y)$.
$0.15(y - 0.2) = 2 - 0.5(1 - y)$
$0.15y - 0.3 = 2 - 0.5 + 0.5y$
$0.15y - 0.3 = 1.5 + 0.5y$
$100(0.15y - 0.3) = 100(1.5 + 0.5y)$
$15y - 30 = 150 + 50y$
$15y - 30 - 15y - 150 = 150 + 50y - 15y - 150$
$-180 = 35y$
$180 \over 35 = y$
The student first made an error when writing $0.15y - 0.3 = 1.5 + 0.5y$. The expression on the left side of that equation should be $2$. The correct answer is $y = 3$.
Simplify your answers.
1. **State the problem:** Solve the equation $$0.15(y - 0.2) = 2 - 0.5(1 - y)$$ and identify any errors in the student's solution.
2. **Distribute terms:**
$$0.15(y - 0.2) = 0.15y - 0.15 \times 0.2 = 0.15y - 0.03$$
$$-0.5(1 - y) = -0.5 + 0.5y$$
So the equation becomes:
$$0.15y - 0.03 = 2 - 0.5 + 0.5y$$
3. **Simplify the right side:**
$$2 - 0.5 = 1.5$$
So:
$$0.15y - 0.03 = 1.5 + 0.5y$$
4. **Identify the student's error:**
The student wrote the left side as $$0.15y - 0.3$$ instead of $$0.15y - 0.03$$. This is a mistake in multiplying $$0.15 \times 0.2$$.
5. **Correct the equation:**
$$0.15y - 0.03 = 1.5 + 0.5y$$
6. **Eliminate decimals by multiplying both sides by 100:**
$$100(0.15y - 0.03) = 100(1.5 + 0.5y)$$
$$15y - 3 = 150 + 50y$$
7. **Bring all terms to one side:**
$$15y - 3 - 15y - 150 = 150 + 50y - 15y - 150$$
$$-153 = 35y$$
8. **Solve for $$y$$:**
$$y = \frac{-153}{35} = -\frac{153}{35}$$
9. **Check the sign and calculation:**
Since the right side was $$1.5 + 0.5y$$, subtracting $$0.5y$$ from both sides:
$$0.15y - 0.5y = 1.5 + 0.03$$
$$-0.35y = 1.53$$
$$y = \frac{1.53}{-0.35} = -4.3714$$
10. **Re-examine the original problem:**
The student's original error was in the multiplication step.
**Correct solution steps:**
$$0.15(y - 0.2) = 2 - 0.5(1 - y)$$
$$0.15y - 0.03 = 2 - 0.5 + 0.5y$$
$$0.15y - 0.03 = 1.5 + 0.5y$$
Subtract $$0.5y$$ from both sides:
$$0.15y - 0.5y - 0.03 = 1.5$$
$$-0.35y - 0.03 = 1.5$$
Add $$0.03$$ to both sides:
$$-0.35y = 1.53$$
Divide both sides by $$-0.35$$:
$$y = \frac{1.53}{-0.35} = -4.3714$$
**Final answer:** $$y \approx -4.37$$
**Summary:** The student's error was incorrectly calculating $$0.15 \times 0.2$$ as $$0.3$$ instead of $$0.03$$, which led to an incorrect equation and solution. The correct solution is $$y \approx -4.37$$, not $$3$$.