To determine whether the family will have enough money for their trip at the end of August, we first need to calculate how much they will save over the months leading up to the vacation.
- Starting amount: $120
- Target amount for vacation: $1331
- Monthly deposit growth rate: 20%
Now let’s calculate the deposits for each month from March to August. Since the trip is at the end of August, that gives us 6 months of savings: March, April, May, June, July, and August.
Monthly Deposits Calculation
- March:
- Deposit = $120
- April:
- Deposit = Previous Deposit + 20% of Previous Deposit
- Deposit = $120 + 0.20 × $120 = $120 + $24 = $144
- May:
- Deposit = $144 + 0.20 × $144
- Deposit = $144 + $28.80 = $172.80
- June:
- Deposit = $172.80 + 0.20 × $172.80
- Deposit = $172.80 + $34.56 = $207.36
- July:
- Deposit = $207.36 + 0.20 × $207.36
- Deposit = $207.36 + $41.472 = $248.832
- August:
- Deposit = $248.832 + 0.20 × $248.832
- Deposit = $248.832 + $49.7664 = $298.5984
Total Savings Calculation
Now, let's calculate the total savings by adding up all the monthly deposits:
- Total Savings = Starting amount + March + April + May + June + July + August
- Total Savings = $120 + $120 + $144 + $172.80 + $207.36 + $248.832 + $298.5984
Calculating it step-by-step:
- Total Deposits = $120 + $144 + $172.80 + $207.36 + $248.832 + $298.5984
- Total Deposits = $120 + 144 + 172.8 + 207.36 + 248.832 + 298.5984
- Total Deposits = $1191.5904
Now, adding the starting amount:
Total = $1191.5904 + $120 = $1311.5904
Conclusion
Now we compare the total savings to the cost of the vacation:
- Total Savings = $1311.5904
- Cost of Vacation = $1331
Now, we find out how much more they need:
\[ \text{Additional Amount Needed} = \text{Cost of Vacation} - \text{Total Savings} \] \[ \text{Additional Amount Needed} = 1331 - 1311.5904 \] \[ \text{Additional Amount Needed} ≈ 19.41 \]
Thus, they will not have enough money for their trip, and they will need approximately $19.41 more.