Question

What is the final amount if 784 is decreased by 1% followed by a 4% increase?

Answer 1

Answer:

The final amount  if 784 is decreased by 1% followed by a 4% increase is 807.52.

Step-by-step explanation:

General Formula

A way to solve this problem is as follows:

The general formula for this is taking into account that:

$\\ percent\;change = \frac{change}{starting\;point}$

The starting point is number 784.

The most important key is the solution for this answer is that:

$\\ change = starting\;point - x$

Where x is a number that we need to find. Then

$\\ percent\;change = \frac{starting\;point - x}{starting\;point}$ [1]

Is 784 increased or decreased after all?

We also need to evaluate the following: if a quantity decreases a percentage, and then increases in another percentage, what is the final percentage? In this case, we have first that 784 decreases by 1% and then increases by 4%. As a result 784 will +4% - 1% = +3%, that is, 784 will increase in 3%.

Finding the result

If 784 increases by 3%, we have:

$\\ 3\% = \frac{784 - x}{784}$

Which is equal to

$\\ 0.03 = \frac{784 - x}{784}$

Solving for x, we first multiply by 784 to both sides of the previous formula:

$\\ 0.03 * 784 = \frac{784 - x}{784}*784$

$\\ 0.03 * 784 = (784 - x)*\frac{784}{784}$

$\\ 0.03 * 784 = (784 - x)*1$

$\\ 0.03 * 784 = (784 - x)$

Remember that if we add, subtract, multiply, divide... to both sides of the equation, we do not "alter" this equation.

Subtracting 784 to both sides:

$\\ (0.03 * 784) - 784 = (784 - 784 - x)$

$\\ (0.03 * 784) - 784 = (0 - x)$

$\\ (0.03 * 784) - 784 = - x$

$\\ 23.52 - 784 = - x$

$\\ -760.48 = - x$

Multiplying by -1 to both sides:

$\\ -1 * (-760.48) = -1 * (-x)$

We have to remember that:

$\\ +*+ = +; - * - = +; - * + = -; + * - = -$

$\\ 760.48 = x$

Then

$\\ x = 760.48$

Therefore, using formula [1], an increase of 3% is

$\\ 3\% = \frac{784 - 760.48}{784}$

$\\ change = 784 - 760.48$

$\\ change = 23.52$

Since an increase of 3% is 23.52, we have to add this to the starting point 784, and finally the amount is 784 + 23.52 = 807.52.

As a result, the final amount if 784 is decreased by 1% followed by a 4% increase is 807.52.  

In a more terse way of solving this problem, we can say that:

Increase of 4% = 784 * 0.04 = 31.36.

Decrease of 1% = 784 * 0.01 = 7.84.

Difference = 31.36 - 7.84 = 23.52.

Then, we need to add this value to 784, and 784 + 23.52 = 807.52 (the same result).