Solve each problem. See Example 4. Zhu inherited $200,000 from...

Question

Solve each problem. See Example 4. Zhu inherited $200,000 from her grandmother. She first gave 30% to her favorite charity. 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?

Accepted Answer

Hello, today we're going to be solving the following word problem. So what we are told is that Steven wins the lottery for $500,000, of the total amount was deducted in taxation. Then he invested some of the remaining amount at a 5% and a 3% interest rate. We want to find the amount invested at each rate if the interest earned per year is $17,000. So before we begin this problem, we want to first figure out what was the remaining total after the 20% deduction? And in order to do that, let's first figure out how much was deducted from $500,000. So we are told that 20% was deducted from 500,000. So to figure out that 20%, we're gonna multiply 500,000 by 20%. Now, normally we don't like to work with percentages. So we want to convert this percentage into a decimal. And in order to do that, we're gonna divide 20% by 100 doing this is going to leave us with 500,000 times 0. And 500,000 times 0.2 is going to give us 100,000. So this amount represents the total amount of money that was deducted from the initial winning. So in order to figure out the remaining amount, We're going to take $500,000 And subtract the 20% amount which was $100,000 and that is going to leave us with $400,000. So this is going to be the amount of money that Stephen was left with after the 20% deduction. Now the word problem states that he takes this $400,000 and deposits it into two accounts one at a 5% interest rate and the other at a 3% interest rate. So for the 5% interest rate, we don't know how much money he invested into this. So we're going to represent the unknown amount with the variable x. and for the 3% interest rate, We want to go ahead and figure out what was the remaining amount after the initial deposit from the $400,000? And we can represent that by 400, -1. So now that we have the scenario set up, we can go ahead and set this equal to $17, because we know that this is going to be the interest earned per year. So setting up the equation is going to be the amount of money that was invested at the 5% interest rate Plus the amount of money that was invested in the 3% interest rate, which is 400,000 -1 times 3%. And that is going to equal to $17,000. And so now we just need to go ahead and simplify the equation in order to solve for X. Now again, we don't want to work with percentages. So we want to convert the percentages into a decimal by dividing each one by 100. And that is going to allow us to rewrite the equation as X times 0.5 plus the quantity 400, -X times 0.03. And that is going to equal to 17,000. On the left hand side, x times 0.05 is going to give us 0. x. And in order to simplify this seek this quantity here, we're going to distribute 0.3 to 400, minus X And 400,000 times 0. is going to give us 12,000 -0. x and that is going to equal to 17,000. Now, we can go ahead and combine like terms by combining the 0.5 X with the 0.3 X and 0.5 minus 0.3 is going to leave us with 0.2 X plus 12,000 is equal to 17,000. Next, in order to Sulfur X, we're going to subtract 12,000 to both sides. And that's going to leave us with 0.02, x equal to 5000. And finally, in order to solve for X, we're going to divide both sides of the equation by 0.02, Again, divide both sides by 0.02. And that's going to leave us with X is equal to 250,000. So now we know the amount for X and we can go ahead and replace every X in the given scenarios, we set up with what we currently solved. So the 5% interest had an initial deposit of 250,000 because we let X equal 250,000. And for the 3%, we can go ahead and replace X with 250,000 and that's going to give us 400, -250000 And that is going to equal to $150,000. So in order to summarize what we just solved in the 5% interest rate, Stephen deposited 250,000 of the remaining $400,000. And in the 3% interest rate, he deposited 150,000 from the $400,000. And with that being said, the answer to this problem is going to be a so I hope this video helps you in understanding how to approach this problem and I'll go ahead and see you all in the next video.

Solve each problem. See Example 4. Zhu inherited \$200,000 from her grandmother. She first gave 30% to her favorite charity. 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

Percentage Calculation

Systems of Linear Equations

Simple Interest Calculation

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