Percents and percent change — Digital SAT Math practice
Percent of, percent change, successive changes.
Key terms
- Percent: A ratio expressed as a fraction of 100, representing parts per hundred.
- Base: The original reference amount used to calculate percentages or changes.
- Factor: The multiplier applied to the base when increasing or decreasing by a percent.
- Successive Change: Multiple percentage adjustments applied one after another.
Common traps
- Adding successive percentages directly instead of multiplying their decimal factors.
- Using the wrong base amount when calculating percent more than or less than another value.
- Forgetting to convert the percentage to a decimal before applying it in calculations.
Key ideas to know
What does p% of N mean?
Define the mathematical meaning of p% of N using a formula.
p% of N equals (p divided by 100) multiplied by N. The percent symbol literally means per hundred, so you convert it to a decimal or fraction first.
Watch out: Students often forget to divide by 100 and just multiply p by N, or they misplace the decimal point when converting.
Calculating percent change
Write the formula for percent change given an old value and a new value.
Percent change equals (new value minus old value) divided by old value, multiplied by 100. Always divide by the original amount.
Watch out: Dividing by the new value instead of the old value flips the base and produces an incorrect percentage.
Handling successive percent changes
How do you combine two successive percent increases or decreases into a single multiplier?
Convert each change to a decimal factor (1 plus the increase rate, or 1 minus the decrease rate) and multiply them together. Do not add the percentages.
Watch out: Adding the percentages together ignores compounding and gives the wrong final amount because each step applies to a different base.
Base matters in comparisons
A student writes: If item A costs $80 and item B costs $100, then B is 25% more than A, so A must be 25% less than B. Is this reasoning correct?
The first part is correct ($20 difference over $80 base equals 25%), but the second part is wrong. To find how much less A is than B, you must use B as the base: $20 divided by $100 equals 20%.
Watch out: Students often assume percent differences work both ways symmetrically, but changing the base changes the resulting percentage.
Sample questions
Question 1
A town's population increases from 250 to 300. What is the percent increase?
- 50%
- 25%
- 16.7%
- 20%
Show answer and explanation
Answer: 20%
Increase = 50; 50/250 = 0.20 = 20%.
Question 2
The price of an item decreased from $50 to $30. What was the percent decrease?
- 67%
- 40%
- 35%
- 45%
Show answer and explanation
Answer: 40%
Change = 20; percent = 20/50 × 100% = 40%.
Question 3
In a class, 60% of the students passed a test. If 18 students passed, how many students are in the class?
- 25
- 36
- 28
- 30
Show answer and explanation
Answer: 30
18 is 60% of the class, so the class size is 18/0.60 = 30.