Mastering Percentages: Your Secret Weapon for the SAT
Welcome! If you’ve ever gone shopping during a 20% off sale or wondered how much to tip at a restaurant, you’re already using percentages. In the SAT Problem-Solving and Data Analysis section, percentages are a huge deal. They help us compare different sets of data and understand how things change over time.
Don't worry if Math usually feels like a foreign language. We’re going to break this down into simple, bite-sized pieces that will help you tackle these questions with confidence!
1. What is a Percent, Anyway?
The word percent literally means "per 100." Think of it like a dollar: there are 100 cents in 1 dollar. So, 1 cent is 1 percent of a dollar.
The Golden Rule: To use a percentage in a math problem, you almost always need to turn it into a decimal first.
How to convert: Just move the decimal point two places to the left.
Example: \( 25\% \) becomes \( 0.25 \)
Example: \( 5\% \) becomes \( 0.05 \) (Don't forget that extra zero!)
Quick Tip: The "Is over Of" Trick
When you see a word problem, use this simple formula to set it up:
\( \frac{\text{is}}{\text{of}} = \frac{\%}{100} \)
Example: What is 15% of 80?
\( \frac{x}{80} = \frac{15}{100} \)
Cross-multiply and solve: \( 100x = 1200 \), so \( x = 12 \).
Key Takeaway: Percentages are just fractions out of 100. Always convert them to decimals before multiplying!
2. Percent Change: Increases and Decreases
The SAT loves to ask how much something grew or shrank. This is called Percent Change. Whether it's a population growing or a shirt going on sale, the formula is always the same:
\( \text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100 \)
An Easy Way to Remember:
Think "Difference over Original."
1. Find the difference (subtract the numbers).
2. Divide by the starting (original) number.
3. Multiply by 100 to get the percentage.
Common Mistake to Avoid: Students often divide by the new number by accident. Always divide by the original amount (where you started)!
Did you know?
If you get a positive answer, it’s a percent increase. If you get a negative answer, it’s a percent decrease.
Key Takeaway: Always put the "starting" number on the bottom of your fraction.
3. The "Multiplier" Method (Speed Strategy)
On the SAT, time is your most precious resource. Instead of doing two steps (finding the percent and then adding/subtracting it), you can do it in one step using a multiplier.
For Percent Increases:
Add the percent to 100%, then convert to a decimal.
Example: A 20% increase means you now have 120% of the original.
Multiplier: \( 1.20 \)
For Percent Decreases:
Subtract the percent from 100%, then convert to a decimal.
Example: A 30% discount means you are only paying for 70% of the item.
Multiplier: \( 0.70 \)
Example Problem: A $50 jacket is on sale for 20% off. What is the new price?
Step 1: 100% - 20% = 80% (which is 0.80).
Step 2: \( 50 \times 0.80 = 40 \).
The jacket is $40!
Key Takeaway: Multipliers are faster and lead to fewer mistakes. Increase = \( (1 + \text{decimal}) \). Decrease = \( (1 - \text{decimal}) \).
4. Successive Percent Changes
This is a classic SAT "trap" question. It happens when a value changes, and then that new value changes again.
Example: A stock price goes up 10%, and then the next day it drops 10%. Is it back to its original price?
Answer: No!
Why? Because the 10% drop happens to the new, higher price, not the original starting price.
How to solve these:
1. Start with a "fake" number like 100 if no starting number is given.
2. Apply the first change.
3. Apply the second change to your result from Step 2.
Step-by-Step with the Stock Example:
1. Start with \( \$100 \).
2. Up 10%: \( 100 \times 1.10 = 110 \).
3. Down 10%: \( 110 \times 0.90 = 99 \).
The price is now \( \$99 \), not \( \$100 \)!
Key Takeaway: Never just add or subtract the percentages together. Apply them one at a time!
5. Percentages in Data Tables
Since this is the Data Analysis section, you will often see percentages inside tables. The trick here is to read the question very carefully to identify your "Whole" (the denominator).
Look for keywords like "of the females" or "of the people who chose Option A."
- If it says "What percent of the total...", your denominator is the Grand Total.
- If it says "What percent of the males...", your denominator is only the Total for Males.
Quick Review Box:
- Percent to Decimal: Move decimal 2 spots left.
- Finding Percent Change: \( \frac{\text{New} - \text{Old}}{\text{Old}} \).
- Speed Trick: Use multipliers (\( 1.15 \) for 15% increase).
- Successive Changes: Do them one by one; don't add them up!
Final Encouragement: Percentages might seem tricky because they involve moving decimals around, but they follow very logical rules. Practice using the multiplier method—it will make you much faster on test day! You've got this!