Question 817797: Sam invested $1950, part of it at 6% and the rest at 8% yearly interest. the yearly income on the 8% investment was $6 more than twice the income from 6% investment. How much did he invest at each rate?
Answer by FightinBlueHens(27)   (Show Source):
You can put this solution on YOUR website! Sam divided the his money into 2 parts. Together they equal $1,950, so if one part is x, and the other part is y, then x + y = 1,950.
Because you have 2 variables, you need 2 equations to solve the problem. Above you have the first equation, but you can't solve it on it's own, so you need the second one to help you.
We know that the first one has an 8% yield, the second one has a 6% yeild, and the yield of the first one $6 more than twice the profit of the second amount, so you need to represent in an equation.
Use 1.08 and 1.06 for the yield because recieves interest, but also keeps the amount he started out with, so the equation will be 1.08x = 2(1.06y) + 6
First distribute: 1.08x = 2.12y + 6
Write these steps out on your paper and write the values as decimals, mixed fractions or imporper fractions so you don't get confused. I will show you all 3.
Then you want to get 1 variable on one side, so I'll do this for x: x = (2.12y + 6)/1.08
2.12/1.8 = 1.9629... but it's not an irrational number, it is equal to 1 + 26/27 (1 and 26/27) which is 53/27
6/1.08 = 5.555555... or 5 + 5/9 (5 and 5/9) or 49/9
x = (53/27)y + 49/9 = 53y/27 + 5.55 OR 53y/27 + 49/9
Substitute this value for x in the first equation (x + y = 1950): 53y/27 + 5.55 + y = 1950
Remember: y = 1y, and any number diveded by itself equals 1
multpily y by 27/27 to add this to the 54y/27, so this will be: 53y/27 + 27y/27
Add the neumerators (the denomeinator stays the same): 53 + 27 = 80,
so 53y/27 + 27y/27 = 80y/27
Now you have: 80y/27 + 5.55 = 1950
Subtract 5.55 from both sides: 80y/27 = 1944.44
Multiply both sides by 27 to cancel out the denominator and you get: 27(80y/27)= 1944.44*27
which equals: 80y = 52,500
Divide both sides by 80 to cancel out the 80 on the left hand side and to solve for y: y = 52,500/80
which makes y equal to 656.25
This is how much he invested in the 6% yield option, so to figure out the other part of his investment, subtract this amount from the original amount ($1950): 1950 - 656.25 = 1,293.75
Check your work: substitute x and why in the second equation to make sure that they work:
1.08x = 2(1.06y) + 6
1.08(1,293.75) = 2(1.06* 656.25) + 6
1,397.25 = 2(1.06*656.25) + 6
1,397.25 = 2(695.625) = 6
1,397.25 = 1,391.25 + 6
1,397.25 = 1,397.25
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