Linear percentage problems

A percentage change is always taken of some amount, then added or subtracted.
Increasing by a percent means adding that percent of the value; decreasing means subtracting it.
Multiplying by $(1 + \text{rate})$ or $(1 - \text{rate})$ does the same thing in one step.

"$X\%$ more than a value" means you add $X\%$ of that value.
If Model X costs $75\%$ more than Model Y's $100, then Model X costs $100 + 75 = 175$ dollars.
In symbols, $u = c + 0.75c = 1.75c$.

A discount tells you what fraction of the price remains.
A $25\%$ discount means you pay $100\% - 25\% = 75\%$ of the price, so you multiply by $0.75$.
A $25\%$ discount on $80 gives $0.75 \times 80 = 60$ dollars.

Often you are given the value after the change and must find the original.
Let the original be $x$, translate the change into an equation, and solve.
If a value increased $10\%$ to become $110$, then $x + 0.1x = 1.1x = 110$, so $x = \frac{110}{1.1} = 100$.

The same idea works after a decrease.
If a price dropped $20\%$ to $48$, then it kept $80\%$, so $0.8x = 48$ and $x = \frac{48}{0.8} = 60$.
Always start by writing what the original amount times the change factor equals.

A price rose by $20\%$ to become $120$ dollars. What was the original price?
Let the original price be $x$. Then $1.2x = 120$.
So $x = \frac{120}{1.2} = 100$ dollars.

A factory made $4000$ items per month, and the monthly average is to rise by $150\%$. What is the new average?
Add $150\%$ of $4000$, which is $6000$.
So the new average is $4000 + 6000 = 10000$ items.

After a $25\%$ discount, a bag costs $60$ dollars. What was the original price?
Let the original price be $x$. A $25\%$ discount means paying $75\%$, so $0.75x = 60$.
Then $x = \frac{60}{0.75} = 80$ dollars.