QuestionSamantha invested a total of $\$ 154,000$ in two accounts. The first account earned $19 \%$ after one year. However, the second account suffered a $15 \%$ loss in the same time. At the end of one year, the total amount of money gained was $\$ 4,780$ . \\square\sigma^{5} $at$ 19 \% $P \$$ \square0^{6} $at$ 15 \%$

Studdy Solution

STEP 1

1. Samantha invested a total of \$154,000 in two accounts.
2. The first account earned 19% interest after one year.
3. The second account suffered a 15% loss after one year.
4. The total amount of money gained from both accounts was \$4,780 after one year.

STEP 2

1. Define variables for the amounts invested in each account.
2. Set up equations based on the given conditions.
3. Solve the system of equations to find the amounts invested in each account.

STEP 3

Let $x$ be the amount invested in the first account (19% gain), and $y$ be the amount invested in the second account (15% loss).

STEP 4

Set up the equation for the total investment: $x + y = 154,000$
Set up the equation for the total gain: $0.19x - 0.15y = 4,780$

STEP 5

Solve the system of equations:
First, solve for $y$ in terms of $x$ from the first equation: $y = 154,000 - x$
Substitute $y = 154,000 - x$ into the second equation: $0.19x - 0.15(154,000 - x) = 4,780$
Simplify and solve for $x$ : $0.19x - 0.15 \times 154,000 + 0.15x = 4,780$ $0.34x - 23,100 = 4,780$ $0.34x = 27,880$ $x = \frac{27,880}{0.34}$ $x = 82,000$
Now, substitute back to find $y$ : $y = 154,000 - 82,000$ $y = 72,000$
The amounts invested in each account are: First account: \$82,000 Second account: \$72,000

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