SAT percentages: multipliers, change, and reverse percent

01Write down what the percentage is of

If 18% of 250 tickets were sold online, the count is $0.18(250)=45$. If those 45 tickets made up 18% of all sales, the total is $45/0.18=250$. Same numbers, different unknown.

Identify the whole before calculating. In "A is 20% greater than B," B is the base, so $A=1.20B$. In "A is 20% of B," $A=0.20B$. The word "greater" changes the equation by a full $B$.

I prefer a multiplier to a separate percent calculation because it keeps the original amount in the work. A 12% increase multiplies by $1.12$; a 12% decrease multiplies by $0.88$. You can still calculate the change separately if that is clearer to you.

02Calculate percent change from the original

A price rises from 80 dollars to 100 dollars. The increase is 20 dollars, and the percentage increase is $\frac{100-80}{80}\cdot 100\%=25\%$. The denominator is the starting price.

If the price falls from 100 dollars to 80 dollars, the decrease is $\frac{100-80}{100}\cdot 100\%=20\%$. The two changes cover the same 20-dollar difference, but they use different bases. That's why a 25% increase can be reversed by a 20% decrease.

Keep the sign if a question asks for a signed percent change: $\frac{\text{new}-\text{old}}{\text{old}}\cdot 100\%$. If it asks for the percentage decrease, give the positive size of the decrease.

03Reverse a discount or increase

After a 30% discount, a jacket costs 56 dollars. The sale price is 70% of the original, so $0.70p=56$ and $p=80$ dollars. Adding 30% to 56 would give 72.80 dollars, which is a different calculation using the discounted price as its base.

A town's population is 2,700 after growing by 8%. Write $1.08P=2700$; the old population is $P=2500$. You don't subtract 8% of 2,700 because the growth was measured from the old population.

For either problem, multiply your recovered original by the given multiplier. That check should return the reported final amount exactly.

04Multiply successive changes

A 200-dollar device gets a 15% discount and then a 10% discount on its reduced price. The final cost is $200(0.85)(0.90)=153$ dollars. The total discount is 47 dollars, or $47/200=23.5\%$, rather than 25%.

The second discount applies to 170 dollars, so it removes 17 dollars. Adding the percentages would incorrectly remove 20 dollars at that step. For any sequence of percent changes, make each change a multiplier and multiply them together.

A 10% increase followed by a 10% decrease produces $1.10(0.90)=0.99$ times the original. Start with 100 and you end at 99. Equal percentages in opposite directions need not cancel.

05Percentage points and percent are different

A survey's approval rate changes from 40% to 50%. That is a rise of 10 percentage points. Relative to the original approval rate, the rise is $\frac{50-40}{40}=0.25$, or 25%.

The SAT can also describe a quantity as "p percent of" another quantity. If $A=1.4B$, then A is 140% of B and 40% greater than B. If $A=0.6B$, A is 60% of B and 40% less than B. Translate the full phrase before selecting an answer.

06Recognize percent models in equations

In $V=1200(0.94{)}^t$, the starting value is 1,200 and the value falls by 6% each time $t$ increases by one. The base $0.94$ is the remaining fraction, not the decay percentage itself.

If the exponent is $t/3$ and $t$ is in months, that 6% decrease happens every three months. Read the exponent's time unit along with the multiplier. Then practice percentages and exponential functions separately in the question bank. Mixing them is useful after you can explain which base each percentage uses.