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Solve the following real-world problem: Destiny opens a checking account with $100 and then deposits $27 each week. How many weeks will it take for your account to reach $343? (1 point)
Answers
16.4 weeks
16.4 weeks
12.7 weeks
12.7 weeks
9 weeks
9 weeks
3.16 weeks
To solve this problem, we need to use basic arithmetic. We start with an initial balance of $100 and we want to reach a balance of $343.
Each week, Destiny deposits $27 into the account.
To find the number of weeks it will take for the account to reach $343, we can subtract the initial balance from the desired balance ($343 - $100 = $243).
Then, we divide the desired balance by the weekly deposit amount ($243 / $27 = 9).
Therefore, it will take Destiny 9 weeks to reach a balance of $343 in her account.
So, the correct answer is: 9 weeks.
To solve this problem, we can use the formula for the arithmetic sequence:
\[S_n = \frac{n}{2}(2a + (n-1)d)\]
Where:
\(S_n\) is the sum of the first n terms,
\(a\) is the first term,
and \(d\) is the common difference between successive terms.
In this case, Destiny opens a checking account with $100, which is the first term (a), and deposits $27 each week, which is the common difference (d). We want to find the number of weeks (n) it will take for the account to reach $343.
Let's plug in the values:
\[343 = \frac{n}{2}(2(100) + (n-1)(27))\]
To solve this equation, we can simplify it:
\[343 = (n)(200 + 27n - 27)\]
\[343 = 200n + 27n^2 - 27n\]
\[27n^2 - 27n + 200n - 343 = 0\]
\[27n^2 + 173n - 343 = 0\]
Solving this quadratic equation, we get:
\[n = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\]
Using the quadratic formula, we get:
\[n = \frac{-173 ± \sqrt{173^2 - 4(27)(-343)}}{2(27)}\]
Simplifying further, we have:
\[n = \frac{-173 ± \sqrt{29929 + 36984}}{54}\]
\[n = \frac{-173 ± \sqrt{66913}}{54}\]
Taking the positive square root, we get:
\[n = \frac{-173 + \sqrt{66913}}{54}\]
Using a calculator or approximation, we find that:
\[n \approx 9.0\]
Therefore, it will take approximately 9 weeks for the account to reach $343.
To solve this problem, we need to determine how many weeks it will take for Destiny's account to reach $343.
We know that Destiny starts with $100 and deposits $27 each week. So, we can set up an equation to represent the amount of money in the account after a certain number of weeks:
Total amount = Starting amount + (Deposit amount per week * Number of weeks)
In this case, the total amount we want to reach is $343. Plugging in the given values, the equation becomes:
$343 = $100 + ($27 * Number of weeks)
To solve for the number of weeks, we need to isolate the variable. To do this, we can subtract $100 from both sides of the equation:
$343 - $100 = $27 * Number of weeks
Simplifying the left side of the equation:
$243 = $27 * Number of weeks
Next, we can divide both sides of the equation by $27 to solve for the number of weeks. This will give us:
$243 / $27 = Number of weeks
Simplifying the right side of the equation:
9 = Number of weeks
Therefore, it will take 9 weeks for Destiny's account to reach $343.