The annual interest on a $15,000 $ ⁢ 15,000 investment exceeds the interest earned on an $8000 $ ⁢ 8000 investment by $685 $ ⁢ 685 . The $15,000 $ ⁢ 15,000 is invested at a 0.6% 0.6 % higher rate of interest than the $8000 $ ⁢ 8000 . What is the interest rate of each investment?

The interest rate on the $ 8, 000 investment is 9. 657 % = 9.66%

While the interest rate on the $ 15, 000 investment is 10.257% = 10.26%

Step-by-step explanation:

Generally, the formular for calculating simple interest, I is given by

I = (principal x rate x time) / 100

Let the the annual interest on the $8,000 investment be represented as I. While the rate is represented as R

Since the time is 1year, interest on the $8, 000 can be expressed as

I = ( 8000 x R × 1) ÷ 100

I = 8000R/100

I = 80R. ( eqn 1)

Also, let the annual interest on $15,000 investment be K and the rate be V

Since interest on $15, 000 is greater than that of the $8,000 by $685

K = I + 685

Also rate V is 0.6% higher than R

V = (R + 0.6)%

This the interest on the $15, 000 is expressed as

K = ( 15000 × V × 1) ÷ 100 (eqn 2)

Substitute for K and V in eqn 2

That is

I + 685 = ( 15000 x (R + 0.6) x 1) ÷ 100

I + 685 = 15000 (R + 0.6) ÷ 100

I + 685 = 150 (R + 0.6)

Open the bracket

I + 685 = 150R + 15 × 0.6

I + 685 = 150R + 9.

This is the same as

150R + 9 = I + 685. ( eqn 3)

Substitute eqn 1 into eqn 3

That is put

I = 80R in eqn 3

150R + 9 = 80R + 685

Collect like terms with R to one side

150R - 80 R = 685 - 9

70R = 676

R = 676 ÷ 70

R = 9.657 %

Recall that V = (R + 0.6)%

V = 10.257 %

Thus the interest rate on the $ 8, 000 investment is 9. 657 % = 9.66%

While the interest rate on the $ 15, 000 investment is 10.257% = 10.26%

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).