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Question

Samantha invests a total of $28,500 in two accounts. The first account earned a rate of return of 11% (after one year). However, the second account suffered a 10% LOSS in the same time period. At the end of one year, the total amount of money gained was $510.00. How much was invested into each account?

$_____ was invested in the account that gained 11%

$_____ was invested in the account that lost 10%

Answer 1

Let X = Account 1

Let Y = Account 2

We know then from the first sentence that X + Y = 28500 (Equation 1).

If Account 1 gained 11%, it could be said to be at 1.11 times its original value. Account 2 lost 10%, so it is at .9 times its original value. If the total gain was known to be $510, 28500+510 being 29010, we could create a second equation. 1.11X + .9Y = 29010 (Equation 2).

We could solve these using a variety of methods, substitution and systems of equations being the most popular. I've seen both taught extensively as both should be known, however knowing when to use the best tool is important too and where systems of equations works best for large matrix and whole numbers, this decimal laden problem begs to be solved using substitution in my opinion. Many ways to do it but find the method that is fastest and most error-free for you on any given problem type, you will find problems like this at all stages of your math career and other STEM disciplines if your education takes you that way.

Substitution:

To turn a number like .9 into 1, simply divide it by .9. However, be sure not to change the equation, so do the same to both sides of the entire equation.

Equation 2 when both sides are divided by .9 becomes:

1.2333 (repeating) X + Y = 32233.33 (repeating).

Simplifying, Y= 32233.33 - 1.23X.

Plugging into Equation 1 gives X + (32233.33 - 1.23X) = 28500.

Simplifying again, -.23X = -3733.33.

Solving: X = 16,000.00.

If Account 1 which gained 11% (X) was $16,000 originally.

Then Account 2 which lost 10% (Y) had the remaining $12,500 at the start (Found by taking Equation 1, substituting X back in solving for Y).

General Advice on This One:

As with all word problems good starting places are reading carefully and turning what you can see into equations first. From there the solution strategy usually jumps out. In this problem it's important to be able to turn percentage gain and lost into simple decimals instantly. 39% loss (1-.39)x becomes .61x instantly. A gain of 7% becomes 1.07y. Being able to do such conversions quickly is a stopping point for some students - wasn't sure if it was for you but definitely a skill to know if don't!

It reduces the number of operations you need to perform and also simplifies expressions. Without simple expressions it's hard to see what to do. And on non-calculator portions of tests like the SAT the less chance for arithmetic mistakes the better.

Good Luck!