Reverse and combine percent changes

Many SAT questions give you the after number, like a sale price or this year’s population, and ask for the before number. Every percent is a percent of something, called its base. Here, the base is the original, the number you’re looking for. So work backward from the percent that’s left, rather than applying the opposite change to the final amount.

Solution

A $20\%$ discount leaves $80\%$ of the original price. You don’t know the original, so it goes after of. Enter

80% of x = 184

Select the vertical solution line. Its $x$-coordinate is $230$, so the original price was $230, and the answer is C.

Choice B is the trap. Adding $20\%$ of $184 back on gives $220.80, but the discount was $20\%$ of the original price, not of the sale price. Any shortcut that takes a percent of $184 uses the wrong base.

With changes in a row, working out and rounding each middle amount can throw your answer off. Instead, find the base and factor for every step, then enter one equation for the whole chain. When you know the final value and need the original, a nested literal equation keeps the steps in order.

Suppose an amount increases by $18\%$, decreases by $12\%$, and then increases by $5\%$. Its final value is $272.58$. Let $x$ be the original amount, and build the equation from the inside out:

105% of (88% of (118% of x)) = 272.58

The innermost part, 118% of x, happens first. Then comes $88\%$ of that, then $105\%$ of the result. Desmos shows the vertical solution line $x=250$, so the original amount was $250$.

In decimal factors, it’s the same equation, $(1.18)(0.88)(1.05)x=272.58$. The three factors multiply to one overall factor, $1.09032$, so

That overall factor is above $1$, so the amount grew, and an original below the final makes sense.

Related: For more calculator practice, see Solve percent problems in Desmos.

Sometimes the question gives you the overall result and hides the percent. Say an amount goes up by $p\%$, then down by the same $p\%$, and ends $6.25\%$ below where it started. What is $p$?

Put the unknown percent inside each factor. The rise multiplies the amount by $1+\frac{p}{100}$, and the drop multiplies it by $1-\frac{p}{100}$. Ending $6.25\%$ below the start means ending at $100\%-6.25\%=93.75\%$ of it, so

The original amount doesn’t need to appear. With it as $O$, both sides carry it: the left side is multiplied by $O$, and the right side is $0.9375O$. $O$ is positive, so dividing both sides by it leaves the equation above. In general, when the final amount is $k$ times the original, the factors multiply to $k$.

Desmos graphs equations in $x$ and $y$, so type the unknown percent $p$ as $x$. You don’t have to solve for anything first:

(1 + x/100)(1 - x/100) = 0.9375

You’ll see two vertical solution lines, at $x=-25$ and $x=25$. Both values satisfy the equation, because squaring removes the sign. The question’s $p$ is a positive percent, so keep $x=25$. Check: $(1.25)(0.75)=0.9375$.

By hand, use the $(1+r)(1-r)=1-r^2$ pattern you saw with equal percents, with $r=\frac{p}{100}$. The same pair shows up: