Let's break down the solution step by step:
Step 1: Identify Essential Monthly Expenses
Sammy's essential expenses consist of fixed and variable expenses:
- Essential Fixed (Monthly): \[tex]$1,292
- Essential Variable (Monthly): \$[/tex]420
Step 2: Calculate Total Monthly Essential Expenses
To find the total essential monthly expenses, we sum the fixed and variable expenses:
[tex]\[ \text{Total Essential Monthly Expenses} = \$1,292 + \$420 = \$1,712 \][/tex]
Step 3: Determine Emergency Savings Goal
Sammy wants to save 3 times his essential monthly expenses as part of his emergency savings:
[tex]\[ \text{Emergency Savings Goal} = 3 \times \$1,712 = \$5,136 \][/tex]
Step 4: Calculate Monthly Contribution to Emergency Savings
Sammy makes an annual emergency savings contribution of \[tex]$210. To find the monthly savings, we convert this annual amount to a monthly amount:
\[ \text{Monthly Savings} = \frac{\$[/tex]210}{12} = \[tex]$17.5 \]
Step 5: Determine the Time to Reach the Savings Goal
To find out how many months it will take Sammy to reach his savings goal, we divide the emergency savings goal by his monthly savings:
\[ \text{Months to Goal} = \frac{\$[/tex]5,136}{\[tex]$17.5} \approx 293.49 \text{ months} \]
Step 6: Convert Time to Reach the Goal into Years and Months
To simplify the understanding of time, let’s convert the months into years and remaining months:
\[ \text{Years to Goal} = \left\lfloor \frac{293.49}{12} \right\rfloor = 24 \text{ years} \]
\[ \text{Remaining Months to Goal} = 293.49 \mod 12 \approx 5.49 \text{ months} \]
Summary:
1. Sammy's total essential monthly expenses: \$[/tex]1,712.
2. Sammy's emergency savings goal: \[tex]$5,136.
3. Monthly savings contribution: \$[/tex]17.5.
4. Time to achieve the emergency savings goal: approximately 293.49 months.
5. Converted into years and months, Sammy would need 24 years and about 5.49 months to reach his emergency savings goal.
Therefore, Sammy needs to save \[tex]$5,136 for his emergency fund, and it will take him approximately 24 years and 5.5 months to achieve this goal based on his current monthly savings rate of \$[/tex]17.5.
