I have $95 dollar in hand, One I have deposited on a bank, after one year I get back the initial amount + 15% interest on it. how to calculate the total amount, I have figured out my own solutions and also I get wired solution (that means from my own sense). Could please explain how the and solutions works and where is "#1" is come from.
disclaimer: I'm bad in mathematics, currently learning.
Solutions:
First solution, This I have figured out myself, 95 * (15/100) + 95 = 109.25
Second solution, Elsewhere on the internet, 95 * (1 + 15/100) = 109.2499999.
Note: My main about is, how is #1 is comes in the equation and how the solution become more accurate, basically I don't understand (1 + 15/100) is part on the equation.
*Note: computerized solution (From Erlang).
95 * (15/100) + 95. = 109.25
95 * (1 + 15/100). = 109.24999999999999
We say that multiplication distributes over addition, which means that if you have the sum of two number and multiply the sum by a third number, you can get the same answer by multiplying each of the first two number individually by the third number and then adding the result.
For example, $$ 5 \times (4 + 3) = 5 \times 7 = 35, $$ $$ (5 \times 4) + (5 \times 3) = 20 + 15 = 35. $$
It is the same with your formulas:
$$ 95 \times \left(1 + \frac{15}{100}\right) = \left(95 \times 1\right) + \left(95 \times \left(\frac{15}{100}\right)\right) = 95 + \left(95 \times \left(\frac{15}{100}\right)\right). $$
Also addition is commutative, meaning it does not matter which number comes first when you add two numbers:
$$ 95 + \left(95 \times \left(\frac{15}{100}\right)\right) = \left(95 \times \left(\frac{15}{100}\right)\right) + 95. $$
The "internet" solution was $95 \times \left(1 + \frac{15}{100}\right),$ your solution was $\left(95 \times \left(\frac{15}{100}\right)\right) + 95,$ but they have exactly the same result mathematically.
The result $109.24999999999999$ is an artifact of trying to do these computations in a computer where inexact arithmetic is used. The answer should be $109.25$ but the software represented $1 + \frac{15}{100}$ in an inexact way so that it gets a slightly different answer. The errors introduced by the software depend on which intermediate numbers occur in the calculations, which depends on the order of the operations, so results that would be exactly the same if the arithmetic were perfectly accurate may not be exactly the same in the computer.