Question 1132304: A certain amount of money was loaned at m per cent simple interest and twice as much at n per cent, interest on both d dollars per year. Find total amt. loaned.
Interest = principal * rate * time
Principal = I /rt
Rate = I / pt
Not sure how to solve. Non-homework.
Found 2 solutions by addingup, Theo:
Answer by addingup(3677)   (Show Source):
You can put this solution on YOUR website! a certain amount was loaned at m%. Let "certain amount" be x. Then we have that x amount was loaned at m, or xm
Then, twice as much was at n%, so this amount = 2xn
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xm + 2xn = d <-- this, as the problem says, is the interest earned in one year.
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OK, so we know that the total amount is x + 2x, where x pays m% and 2x pays n%.
x + 2x = 3x
x/3x = 1/3x this means that m interest came from 1/3 of the money
2x/3x = 2/3x this means that n interest came from 2/3 of the money
d/(1/3xm + 2/3(2x)n) = total amount you invested.
NOTE: Sometimes to make it easier to understand it's better to use some numbers. Let's do it quickly:
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Money invested:
x = 4, thus 2x = 8 (and the total invested is 4 + 8 = 12, but save this. We are going to calculate it)
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Interest
m = 10% and n = 20%
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4(0.1) + 8(0.20) = 2 this is the total interest earned.
Now, we said 1/3 of the interest is from x and 2/3 of the interest comes from 2x
0.10*1/3 + 0.20*2/3 = 0.03333 + 0.13333 = 0.16667
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Total interest earned was 2 and 2/0.16667 = 12 which is the amount you invested in this example.
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Happy learning
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! what i get is as follows.
if the percent interest rate is m, then the interest rate is m / 100.
if the percent interest rate is n, then the interest rate is n / 100.
i = p * r * t.
t = 1 because we're dealing with 1 year only.
that makes i = p * r.
for the m percent rate, the formula becomes i1 = p * m / 100
for the n percent rate, the formula becomes i2 = p * 2 * n / 100
d is the total interest and is therefore equal to i1 + i2 which makes it equal to p * m / 100 + p * 2 * n / 100.
since the denominator is the same, the formula becomes d = (p * m + p * 2 * n) / 100
you can factor out the p to get d = p * (m + 2 * n) / 100
if you solve for p, you get p = 100 * d / (m + 2 * n)
the total amount loaned would be p + 2 * p = 3 * p
since p = 100 * d / (m + 2 * n), then 3 * p = 3 * 100 * d / (m + 2 * n).
that makes 3 * p = 300 * d / (m + 2 * n).
how do you know this formula is good?
you need to put in some randomly generated numbers and see if they come out as you would expect.
i chose p = 1000 and m = 10% and n = 20%.
per the formula of d = p * (m + 2 * n) / 100, you get:
d = 1000 * (10 + 2 * 20) / 100 = 1000 * 50 / 100 = 1000 * .5 = 500
d = 500 = total interest from both investments.
the total amount loaned should be equal to p + 2 * p = 3 * p = 3000.
given d = 500 and m = 10% and n = 20%, and given the formula of 3 * p = 300 * d / (m + 2 * n), you get:
3 * p = 300 * d / (m + 2 * n) becomes 3 * p = 300 * 500 / (10 + 2 * 20) which becomes 3 * p = 150000 / 50 which becomes 3 * p = 3000.
that checks out ok, so the solution looks good.
the formulas check out with the logic.
your solution is that the total amount loaned is equal to 3 * p which is equal to 300 * d / (m + 2 * n).
if you know d and m and n, you can find 3 * p per the formula.
3 * p is the total amount loaned.
if t is not equal to 1, then it becomes patt of the equation.
the interest per year will be the same because you are dealing with simple interest which is calculated off the initial investment only.
instead of i = p * r, you get i = p * r * t.
let me know if you have any questions regarding this.
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