Investment problem

Question

Joy invests a total of $10,500 in two accounts. The first account earned a rate of return of 5% (after a year). However, the second account suffered a 2% loss in the same time period. At the end of one year, the total amount of money gained was $315.00. How much was invested into each account?

Answer 1

x = money invested in account 1

y = money invested in account 2

Gain of 5% = + 0.05 = interest rate of account 1

Loss of 2% = -0.02 = interest rate of account 2

(to find a rate from a percent, you divide the percent by 100)

We will make a system of equations, with one showing the total money invested and the other showing the total interest earned or lost.

Equation 1: The total amount of money invested was 10,500

x + y = 10,500

You can imagine there are coefficients of 1 in front of the x and y.

1x + 1y = 10,500

Equation 2: The total interest gained from both account was 315

To find the interest gained by each account, we will multiply the amount invested by the interest rate. These together need to add up to a total of 315 gained in interest.

0.05x - 0.02y = 315

*notice how 0.02y is subtracting —> that is because we are losing interest*

Now we have a system of equations

1x + 1y = 10,500

0.05x - 0.02y = 315

To solve this system, we want to make one pair of x’s or y’s be equal and opposite coefficients. We can see that one of the y’s is negative and the other is positive —> it would be good to match the y’s because they are already opposites of each other. Now we just need to make the coefficients the same value.

I am going to turn 0.02 into 1 by multiplying some number to the entire bottom equation.

0.02 * ? = 1 divide by 0.02 on both sides

? = 50

We need to multiply everything in the second equation by 50 to turn 0.02 into 1

0.05x * 50 - 0.02y*50 = 315*50

2.5x - 1y = 15,750

Now let’s show our first equation with our new second equation

1x + 1y = 10,500

2.5x - 1y = 15,750

If we add the columns down, the +1y and -1y cancel out and we can solve for x. Add the x terms down and the totals down.

3.5x = 26,250 divide by 3.5 on both sides

x = 7,500

We invested $7,500 into the first account, which had 5% gain

Plug back into the first equation to find y

x + y = 10,500

7,500 + y = 10,500 subtract 7,500 from both sides

y = 3,000

We invested $3,000 into the second account, which had 2% loss

Answers: x = 7500 y = 3000

Answer 2

Whenever you deal with percent changes, use multiplication:

Let x be the amount in the 5% return account: end of year, she has 1.05x

A 2% loss is a multiplier of .98, so the other account, at the end of the year, has (10500-x)(.98)

The final equation is that the two accounts gained 315 by the end of the year.

So, 1.05x + (10500-x)(.98) -10500 = 315

Solve for x.by grouping x terms and isolating x.