Lottery Winnings Linda won 200,000 in a state lottery. She...

Lottery Winnings Linda won $\$ 200,000$ in a state lottery. She first paid income tax of $30 \%$ on the winnings. She invested some of the rest at $1.5 \%$ and some at $4 \%$ earning $\$ 4350$ interest per year. How much did she invest at each rate?

Key Concepts

Mixture Problems

Mixture problems focus on combining elements with different characteristics, such as percentages, concentrations, or rates, to yield a desired overall effect. This approach is used in scenarios where distinct portions of a total amount must be allocated to achieve a specific weighted result.

Systems of Linear Equations

This concept involves forming and solving multiple equations that represent the relationships between different variables in a problem. It is particularly useful when there are several unknowns that need to be determined simultaneously by leveraging the constraints provided within the problem.

Percentage Calculations

This concept involves applying a percentage to an initial amount to determine a portion of that value, whether it is a deduction (like a tax) or an increment. It is essential for adjusting the original amount to reflect changes due to percentage-based operations, forming the basis for subsequent calculations.

Interest Calculations

Interest calculations involve determining the amount earned or paid over time based on a principal sum and a given interest rate. This concept is fundamental in finance and investment, as it allows one to compute earnings or costs under various rates and time periods.

Transcript

00:01 So we're told that this person wins $200 ,000 in the lottery, and the first thing that she does with the money is she has to pay income taxes.

00:09 So we're going to account for that by having the total amount of money that she won minus however much she has to pay for income taxes.

00:19 So we're told that it's 30 % in income tax.

00:22 So we have 0 .3 times 200 ,000.

00:27 And this is equal to 0 .3 times 200 ,000 is 60 ,000 and 200 ,000 minus 60 ,000 is 140 ,000.

00:37 So she has $140 ,000 left over, and we're told that she invests it in two different accounts, one at 1 .5 % interest and one at 4 ,000, or sorry, at 4 % interest.

00:50 And we're going to assume that she invests the rest of her money in these two accounts.

00:56 So what we're going to say is that 0 .15, which is the interest rate of the first account times x plus 0 .04 times x is equal to 4 ,350.

01:20 And actually, this shouldn't be an x, this should be a y.

01:24 So we're told that 0 .015x plus 0 .04y is equal to 4 ,350.

01:31 And we know this because we're told that she made $4 ,350 on her interest every year.

01:38 So we have this one equation, but we have two variables, so we need to set up another equation.

01:44 And so our other equation is just going to be x plus y is equal to 140 ,000, as we are assuming that she is going to be investing her whole account or the rest of her money.

01:57 And now that we have these two equations, we can just solve for one of our variables in one equation and then plug that into the other equation...

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