ireneportillo1211
Answered

Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Helene invested a total of $1,000 in two simple-interest bank accounts. One account paid 5% annual interest; the other paid 8% annual interest. The total amount of interest she earned after 1 year was $65. Find the amount invested in each account.

Sagot :

a  =  total amount invested at 5%

b = total amount invested at 8%

we know 5% was earned in interest from "a", so that'd be (5/100) * a or 0.05a.

we know 8% was earned in interest from "b", so that'd be (8/100) * a or 0.08b.

we also know that the total for both is 1000, a + b = 100.

[tex]\begin{cases} a + b = 1000\\ 0.05a+0.08b=65 \end{cases}\qquad \qquad \stackrel{\textit{since we know}}{a+b=1000}\implies b = 1000-a \\\\\\ \stackrel{\textit{substituting on the 2nd equation}}{0.05a+0.08(100-a)=65}\implies 0.05a+80-0.08a=65 \\\\\\ -0.03a+80=65\implies -0.03a=-15\implies a=\cfrac{-15}{-0.03}\implies \boxed{a = 500} \\\\\\ \stackrel{\textit{we know that}}{b = 1000-a}\implies b=1000-500\implies \boxed{b = 500}[/tex]

Step-by-step explanation:

Let  [tex]\textbf{\textit{x}}[/tex] be the amount invested into the account with a 5% annual interest and [tex]\textit{\textbf{y}}[/tex] be the amount invested into the account with an 8% annual interest,

It is stated that Helene invested a total of $1000, which can be written mathematically as

                                         [tex]x \ + \ y \ = \ 1000 \ \ \ \text{----- \ (1)}[/tex]  .

Subsequently, the total amount of interest after 1 year is $65. This statement can be changed into the equation below.

                                      [tex]0.05x \ + \ 0.08y \ = \ 65 \ \ \text{----- \ (2)}[/tex]  .

First, divide each term in equation (2) by 0.05, yields the expression

                                        [tex]x \ + \ 1.6y \ = \ 1300 \ \ \text{----- \ (3)}[/tex]   .

Then, let equation (1) be subtracted from equation (3) term-by-term,

                                [tex](x \ + \ 1.6y ) \ - \ (x \ + \ y) \ = \ 1300 \ - \ 1000 \\ \\ (x \ - \ x) \ + \ (1.6y \ - \ y) \ = \ 300 \\ \\ \-\hspace{3.15cm} 0.6y \ = \ 300 \\ \\ \-\hspace{3.38cm} \displaystyle\frac{3}{5}y \ = \ 300 \\ \\ \-\hspace{3.65cm} y \ = \ 500[/tex].

Therefore, the amount invested in each account is $500.

We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.