ireneportillo1211
Answered

Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

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 appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.