Percentages, Percent Change & Percent Error
Read a percent as 'per 100', find a percent of a number, and measure how much a value changed or how far off a measurement was.
A percent is just a number written 'per 100', and once you can read it that way you can compare things that started at different sizes. Percent of a number sizes a part against a whole (how many of 40 cells survived). Percent change tells you how much something grew or shrank (a tumor that shrank 30 percent), which is more honest than a raw difference because it accounts for the starting amount. Percent error tells you how close a measurement came to the true value, so a lab result can be judged, not just reported. Nurses read percent change in a patient's white-blood-cell count to spot an infection. Lab technologists compute percent error to decide whether an instrument passed calibration. Epidemiologists report survival percentages so hospitals of different sizes can be compared fairly, and geneticists report allele percentages to describe a population. Master this one skill and most of the numbers you meet in biology stop being mysterious.
- Common Core · HSN-Q.A.1Use units and quantities to reason through multi-step problems, including expressing and interpreting a part of a whole as a percent.
- Ohio · Ohio HS N.Q.1Reason quantitatively with rates and percents to solve real problems and to interpret results in context.
- NGSS · SEP-4Analyzing and Interpreting Data: use percentages and percent change to summarize and compare data sets.
- NGSS · SEP-5Using Mathematics and Computational Thinking: apply percent error to evaluate the accuracy of a measurement against an accepted value.
- AP · AP Bio SP 6 (Quantitative)Work with quantities, including percentages and percent change, when analyzing and evaluating biological data.
- Convert a fraction to a decimal: A percent is a part out of 100, so students must turn a part-over-whole fraction into a decimal to compute with it.
- Multiply by a decimal: Finding a percent of a number is multiplying by a decimal (0.20 times 45), the core move in every percent problem.
- Subtract, then divide: Percent change and percent error both find a difference first, then divide by a starting or accepted value.
Prerequisites are inferred: pending teacher review.
Re-learn the skill with worked practice and clear examples.
Use one setup for change and error: find the gap first, then divide by the right starting number, then times 100. For percent change, divide the gap by the OLD value. For percent error, take the positive size of the gap and divide by the ACCEPTED value. A negative percent change means a decrease.
A cell count rose from 40 to 50. Using percent change (new minus old, over old, times 100), what is the percent increase?
Reviewed- A.10%
- B.20%
- C.25%
- D.125%
Show the worked solution ▾
Answer: C. 25%
- Step 1: Find the gap: New minus old is 50 minus 40, which is 10.
- Step 2: Divide by the old value and times 100: 10 divided by 40 is 0.25; times 100 is 25%.
Why it's right: The gap of 10 divided by the old value 40 is 0.25, so the increase is 25%.
- A: This is the raw gap of 10, reported as if it were the percent.
- B: This divided the gap by the new value 50 instead of the old value 40.
- D: This divided the new value by the gap, or read the final size as the change.
Aligned to NGSS SEP-4: percent change · reading level ~grade 9
Read the table, then use percent error. A student measures a mass of 9.6 g for a sample whose accepted value is 10.0 g. What is the percent error?
Reviewed| Quantity | Value |
|---|---|
| Measured mass | 9.6 g |
| Accepted mass | 10.0 g |
| Gap (positive size) | 0.4 g |
- A.0.4%
- B.4%
- C.4.2%
- D.40%
Show the worked solution ▾
Answer: B. 4%
- Step 1: Find the positive gap: The size of 9.6 minus 10.0 is 0.4 g (the bars mean take the positive value).
- Step 2: Divide by the accepted value and times 100: 0.4 divided by 10.0 is 0.04; times 100 is 4%.
Why it's right: The gap of 0.4 g divided by the accepted 10.0 g is 0.04, so the percent error is 4%.
- A: This forgot to multiply by 100 (0.04 written as if it were 0.4%).
- C: This divided 0.4 by the measured 9.6 instead of the accepted 10.0.
- D: This divided by 1.0 instead of 10.0 (a decimal-place slip).
Aligned to NGSS SEP-5: percent error · reading level ~grade 9
A tumor measured 80 mm before treatment and 60 mm after. What is the percent change in its size?
Reviewed- A.20% decrease
- B.25% decrease
- C.25% increase
- D.33% decrease
Show the worked solution ▾
Answer: B. 25% decrease
- Step 1: Find the gap: New minus old is 60 minus 80, which is negative 20 (it shrank).
- Step 2: Divide by the old value and times 100: Negative 20 divided by 80 is negative 0.25; times 100 is negative 25%, a 25% decrease.
Why it's right: The gap of negative 20 divided by the old value 80 is negative 0.25, a 25% decrease.
- A: This reported the raw 20 mm gap as a percent instead of dividing by 80.
- C: The value went down, so this is a decrease, not an increase.
- D: This divided the gap of 20 by the new value 60 instead of the old value 80.
Aligned to NGSS SEP-4: percent change (decrease) · reading level ~grade 9
- A nurse computes the percent change in a patient's white-blood-cell count between two blood draws to judge whether an infection is worsening.
- A lab technologist finds the percent error of a pipette against a certified reference mass to decide whether it passed calibration.
- An oncologist reports a tumor's percent decrease after treatment so responses can be compared across patients whose tumors started at different sizes.
Fill these in as you work through the lesson.
- Percent (a part out of 100):
- Percent of a number (change to a decimal, then multiply):
- Percent change (gap over the OLD value):
- Percent error (positive gap over the ACCEPTED value):
For percent change, divide the gap by the value; for percent error, take the size of the gap and divide by the value; then multiply by .
- Write 25% two other ways: as a fraction and as a decimal.
- A count goes from 40 to 50. Which number do you divide the gap of 10 by, and why?
- In percent error, why do you divide by the accepted value instead of the measured value?
A balance reads 21.0 g for a sample whose accepted value is 20.0 g. The gap size is 1.0 g. Divide 1.0 by the accepted value 20.0 to get ____, then multiply by 100 to get ____ percent error.
The vocabulary of this topic, shown in the way you will meet it.