SOLUTION: Nicole invested a certain amount of money for one year and earned $78 in interest. Jon invested $260 more at an interest rate that was 1% less than Nicole, but earned the same amou

Question 1034821: Nicole invested a certain amount of money for one year and earned $78 in interest. Jon invested $260 more at an interest rate that was 1% less than Nicole, but earned the same amount of interest. Find Nicole's principal and interest rate.
please explain.
Found 2 solutions by Theo, ankor@dixie-net.com:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let x = the amount she invested.
let r = the rate of interest she invested at.
since her interest earned was 78, her equation becomes:
x*r = 78

jon invested 260 more at an interest rate that was 1% less.
1% is equal to .1
jon's equation becomes:
(x+260) * (r-.1) = 78

you have 2 equations that need to be solved simultaneously.
they are:

x*r = 78
(x+260) * (r-.1) = 78

this system can be solved in a number of ways.
one way to solve it is as follows.

since both expressions on the left side of each equation are equal to 78, you can those expressions equal to each other to get:

x*r = (x+260) * (r-.1)

if you expand this equation, you get:

x*r = x*r - .1*x + 260*r - .1*260

simplify this further to get:

x*r = x*r - .1*x + 260*r - 26

if you subtract x*r from both sides of the equation, they cancel out and you are left with:

0 = -.1*x + 260*r - 26

this is the same as -.1*x + 260*r - 26 = 0

you want to reduce the number of variables in one equation to one so it can be solved.

go back to your original equations.
they are:
x*r = 78
(x+260) * (r-.1) = 78

from the first of these equations, solve for x to get x = 78/r

in your equation of -.1*x + 260*r - 26 = 0, replace x with 78/r to get:

-.1*78/r + 260*r - 26 = 0

simplify this to get:

-7.8/r + 260*r - 26 = 0

multiply both sides of this equation by r to get:

-7.8 + 260*r^2 - 26*r = 0

order the terms of this equation in descending order of degree to get:

260*r^2 - 26*r - 7.8 = 0

use the quadratic formula to solve this to get:

r = 0.2302775637732 or x = -0.1302775637732

your calculator might not give you as many as these digits, but mine did so i used what i had. if your calculator gives you less, then use that. most calculator will take you out to about 8 or 9 digits accuracy. this calculator was an online one that gave me more.

it's not possible to get a positive interest if the interest rate is negative, so the solution has to be r = 0.2302775637732

since r = 0.2302775637732, then (r-.1) must be equal to 0.1302775637732

from the first equation of x*r = 78, you get:
x * 0.2302775637732 = 78
solve for x to get x = 78 / 0.2302775637732 = 338.72166581
that's nicole's principal.

x+260 = 598.72166581
598.72166581 * 0.1302775637732 = 78
that's jon's principal multiplied by jon's interest rate taken from the formula:
(x+260) * (r-.1) = 78.
this calculation confirms that nicole's principal and nicole's interest rate are correct because joh's principal is nocile's principal plus 260 and jon's interest rate is nicole's interest rate - .1.

your solution is:

nicole's principal is 338.72166581 dollars.
nicole's interest rate is 0.2302775637732

lots of digits here to maintain the accuracy.
the concept is fairly simple.
the execution is a bit complicated.

the concept is as stated above.
x * r = 78
(x + 260) * (r - .1) = 78

i set the expressions on the left side of each of those equaitons equal to each other
that's one way to solve it, but you could also have solved by the normal method of substitution.

the equations are still:
x * r = 78
(x + 260) * (r - .1) = 78

in the first equation, solve for x to get x = 78/r

simplify the second equation to get:

x*r - .1*x + 260*r - .1*260 = 78

simplify further to get:

x*r - .1*x + 260*r - 26 = 78

replace x in the second equation with 78/r derived from the first equation to get:

78/r*r - .1*78/r + 260*r - 26 = 78

simplify to get:

78 - 7.8/r + 260*r - 26 = 78

multiply both sides of the equation by r to get:

78*r - 7.8 + 260*r^2 - 26*r = 78*r

subtract 78*r from both sides of the equation to get:

-7.8 + 260*r^2 - 26*r = 0

reorder terms in descending order of degree to get:

260*r^2 - 26*r - 7.8 = 0

this is exactly the same equation we got when we used the other method.
you get the same answer as you should regardless of the method you choose as long as you execute that method correctly.

doing this manually is a chore.
using a calculator helps.
being able to store intermediate results in memory and then reusing those results later on reduces the manual burden involved.
you should not round until you get to the final result, otherwise you can lose accuracy.
hopefully this helps you understand the concept and a couple of ways to execute on the methods involved.
don't sweat the large number of decimal digits i used.
it's because the calculator i used allowed that.
your calculator will more then likely display less.

Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Nicole invested a certain amount of money for one year and earned $78 in interest. Jon invested $260 more at an interest rate that was 1% less than Nicole, but earned the same amount of interest. Find Nicole's principal and interest rate.
:
let n = amt invested by N
J invested $260 more, therefore
(n+260) = amt invested by J
:
then
= N's interest rate in decimal form
J interest rate was 1% less, therefore
- .01 = J's interest rate
:
j's interest rate * j's amt = 78
(-.01)(n+260) = 78
FOIL
78 + - .01n - 2.6 = 78
subtract 78 from both sides
- .01n - 2.6 = 0
multiply by n, form quadratic equation
20280 - .01n^2 - 2.6n = 0
Multiply by -100
n^2 + 260n - 2028000 = 0
you can use the quadratic formula but this will factor
(n+1560)(n-1300) = 0
the positive solution
n = $1300 is N's investment
Find the interest rate
= .06 or 6% int
:
:
Confirm this by finding J's amt and int
1300 + 260 = $1560
= .05 or 5%, on 1 percent less

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