Let's analyze Mara's work step-by-step and identify the error she made.
1. Setup the Equation:
The problem states that the principal (initial amount of the investment) is [tex]$4300, the rate is 0.062 (or 6.2%), and the interest (final amount minus principal) is $[/tex]2666. This leads us to set up the correct equation for simple interest, which is:
[tex]\[
I = P \times r \times t
\][/tex]
Substituting in the given values, the equation becomes:
[tex]\[
2666 = 4300 \times 0.062 \times t
\][/tex]
2. Solve for the Time (t):
To find [tex]\( t \)[/tex], we need to isolate it in the equation. This involves dividing both sides of the equation by [tex]\( 4300 \times 0.062 \)[/tex]:
[tex]\[
t = \frac{2666}{4300 \times 0.062}
\][/tex]
Simplifying inside the denominator first:
[tex]\[
4300 \times 0.062 = 266.6
\][/tex]
Therefore, the equation for [tex]\( t \)[/tex] is:
[tex]\[
t = \frac{2666}{266.6}
\][/tex]
Performing the division:
[tex]\[
t = 10
\][/tex]
So, the correct length of time [tex]\( t \)[/tex] is 10 years.
3. Identify Mara's Error:
Let's revisit the step-by-step process Mara followed and identify where she went wrong.
- Mara correctly started with the setup [tex]\( 2666 = 4300 \times 0.062 \times t \)[/tex]
- Mara correctly simplified [tex]\( 4300 \times 0.062 \)[/tex] to get [tex]\( 266.6 \)[/tex]
- Then, Mara rewrote the equation as [tex]\( 2666 = 266.6 \times t \)[/tex]
- Next, Mara solved for [tex]\( t \)[/tex] by incorrectly writing [tex]\( \frac{266.6}{2666} = t \)[/tex]
[tex]\(\boxed{\text{Here lies her error}}: \text{Mara mistakenly divided } 266.6 \text{ by } 2666, \text{ instead of dividing } 2666 \text{ by } 266.6.\)[/tex]
4. Conclusion:
The correct operation to find [tex]\( t \)[/tex] would be [tex]\( \frac{2666}{266.6} \)[/tex], which gives us [tex]\( t = 10 \)[/tex]. Mara instead calculated [tex]\( \frac{266.6}{2666} \)[/tex], which erroneously led her to [tex]\( t = 0.1\)[/tex].
Therefore, Mara's error was "Mara did not divide correctly."
