Problem

Investment problems. An investment club used fun…

11:19

Problem 19 Medium Difficulty

Investment problems.
A salesperson used her 3,500 dollar year-end bonus to purchase some old gold and silver coins. She earned 15% annual simple interest on the gold coins and 12% annual simple interest on the silver coins. If she saw a return on her investment of 480 dollar the first year, how much did she invest in
each type of coin?

Answer

Silver: $\$ 1,500$ gold: $\$ 2,000$

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Precalculus

Algebra

Elementary and Intermediate Algebra

Chapter 2

Equations, Inequalities, and Problem Solving

Section 6

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Video Transcript

this problem reads. A salesperson used her $3500 year end bonus to purchase some old gold and silver coins. She earned 15% annual simple interests on the gold coins and 12% annual simple interest on the silver coins. If she saw return on her investment of $480 the first year, how much did she invest in each type of coin? So I made a chart. Um, that has the equation p principle times rate. Our times tee time is equal to interest. And we know that our total that she earned an interest waas $480. So we're gonna fill that in, and then we're gonna have to fill in the rest of the chart Gold is in the top row and then silvers in the bottom row. So let's say that the amount that's invested in gold was X s. So that means that the amount that she invested in silver must be 3500 minus X, since that's a total that she invested in both queens and then the interest rate on the gold was 15% so as a decimal, that 0.15 the interest rate on the silver Queens was 12%. So that is 0.12 as a decimal. And then, since it's at the end of the year, we're going to say that the time is one. So it's a funder interest. We're gonna multiply our principal times rate times time. So our interest for gold is 0.15 X or is there a 0.15 times X and then for silver? We have 0.12 times 3500 for 3500 minus x. And then to figure out what, um, values were invested in each type of queen we're gonna first, right, an equation. Um, that involves the interest. So we have 0.15 x plus 0.12 times 3500 minus X is equal to 480. And then from here, we're gonna actually multiply each side by 100 because we want to get rid of thes decimals right here. This your 1000.15 and that's your 0.12. So we're gonna multiply each side by 100. So now we have 100 time 0.15 x plus 100 time 0.12 times 3500 minus X is equal to for her 480 times 100 which is 48,000 and then we're gonna have some multiply. So we have 100 times your 1000.15 exes, just 15 x, and then we have 100 times their 1000.12 which is just 12. And then we're gonna have to multiply 12 times 3500 minus X is equal to 48,000. And then now we're going to distribute the 12 to the 3500 then the 12 to the negative X. So we have 15 X plus 12 times 3500 is 42,000 and then we have 12 times negative X, which is negative. 12 X is equal to 48,000 that we're gonna have to combine like terms. So we have five X minus 12 x, which is three X so we have three eggs, plus 42,000 is equal to 48,000 and this is a two step equation. So the first thing that we want to do is move our constant over to the other side. So to get 42,000 to the other side, we're gonna have to subtract 42,000 on each side. So we have 42,000 minus 42,000 which is zero. So then we're left with three ex is equal to 48,000 minus 42,000 which is 6000. And now we have three. X is equal to 6000. We want to get the three to the other side. So we're gonna divide both sides by three. So three divided by three is one. So we're left with X is equal to and then we have 6000 Divide by three, which is 2000. So we have X is equal to 2000. So that means that $2000 was invested in gold earned gold coins and then to figure out how much was invested in silver coins, we have to subtract the 35 100 or 3500 minus the X, which we now know is 2000. So we have 3500 minus 2000. Ceremony zero is 00 minus 005 minus zero is five three minus two is once. That means that $1500 was invested in silver coins and that is our solution

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