Question 1162620
.
Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%.
The total income from all 3 investments is $1600. The income from the 5% and 6% investments is the same 
as the income from the 8% investment. Find the amount invested at each rate.
~~~~~~~~~~~~~~



<pre>
In this problem, the key phrase is that saying  



        The income from the 5% and 6% investments is the same as the income from the 8% investment.



It means that the income from 8% investment is half of the total interest of $1600, i.e.  $800.


Next, since the interest of the 8% investment is $800, it means that the amount of the 8% investment is  {{{800/0.08}}} = 10000 dollars.


Thus, we just know that the 8% investment was $10000; hence, the parts invested at 5% and 6%, make the rest $25000-$10000 = $15000.



Doing in this way, we just reduced the problem from 3 unknowns to only 2,

and the rest of the problem can be solved using one unknown, only.


Let x be the amount invested at 6%;  then the amount invested at 5% is  (15000-x) dollars.


The total interest equation for these two parts is


    0.05*(15000-x) + 0.06x = 800  dollars     (another half of $1600).


From this equation,


    x = {{{(800-0.05*15000)/(0.06-0.05)}}} = 5000.


Thus we have the <U>ANSWER</U> :  $5000 was invested at 6%;  15000-5000 = 10000 dollars was invested at 5%  and  $10000 was invested at 8%.


<U>CHECK</U>.  0.06*5000 + 0.05*10000 + 0.08*10000 = 1600 dollars,

        which is precisely correct total interest.
</pre>

Solved.


---------------


The major lesson to learn is that the problem can be solved using one unknown ONLY (!)


See the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/equations/Advanced-word-problems-to-solve-by-reduction-to-single-linear-equation.lesson>Advanced word problems to solve using a single linear equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/equations/HOW-TO-algebreze-and-solve-these-problems-using-one-eqn-in-one-unknown.lesson>HOW TO algebreze and solve these problems using one equation in one unknown</A> 

in this site and find there other numerous similar solved problems.