3.1 Percents: Conversions & Applications
Key Takeaways
- Percent means per 100; the % symbol is shorthand for /100, so 25% = 25/100 = 0.25
- To convert percent to decimal, divide by 100 (move the decimal two places left); to convert decimal to percent, multiply by 100
- "Of" always means multiply: 30% of 80 = 0.30 × 80 = 24
- For percent change, divide by the ORIGINAL value (the starting number), never the new value
- 0.5% is not 5% — 0.5% = 0.005; the leading zero changes the value by a factor of ten
What Does "Percent" Mean?
The word percent comes from the Latin per centum, meaning "per hundred." A percent is a ratio with a denominator of 100. The % symbol is shorthand for "/100", so 25% literally means 25/100, or 25 out of every 100.
On the NEX, percents show up in medication concentrations ("a 5% dextrose solution"), supply-order discounts, sales tax, and percent-change questions about intake and output. The arithmetic is Grade 8–10 level; the wording is where people slip.
Converting Between Percent, Fraction, and Decimal
Three forms, one value. Move between them with simple rules.
Percent → Decimal: Drop the % symbol and divide by 100 (move the decimal two places left).
- 45% → 0.45
- 8% → 0.08
- 120% → 1.20
Decimal → Percent: Multiply by 100 (move the decimal two places right), add %.
- 0.3 → 30%
- 0.075 → 7.5%
- 1.4 → 140%
Percent → Fraction: Put the number (without the %) over 100, then simplify.
- 45% = 45/100 = 9/20
- 8% = 8/100 = 2/25
- 125% = 125/100 = 5/4
Fraction → Percent: Convert the fraction to a decimal by dividing, then move the decimal two places right.
- 3/4 = 0.75 = 75%
- 5/8 = 0.625 = 62.5%
Trap: 0.5% is not the same as 5%. 0.5% = 0.005. The leading zero matters.
Finding a Percent of a Number
"Of" means multiply. Always.
What is 30% of 80? 30% of 80 = 0.30 × 80 = 24.
Step-by-step:
- Convert the percent to a decimal: 30% → 0.30
- Multiply: 0.30 × 80 = 24
What is 6% of $240? (a tax-style problem) 0.06 × 240 = 14.40, so the tax is $14.40.
This is the same operation behind medication concentrations: "5% dextrose" means 5 g of dextrose per 100 mL of solution. If you have 250 mL, the dextrose mass is (5/100) × 250 = 0.05 × 250 = 12.5 g.
Finding What Percent One Number Is of Another
Set up a fraction: part ÷ whole, then convert to a percent.
18 is what percent of 72? 18 ÷ 72 = 0.25 → 25%.
A patient consumed 720 mL of a 1-liter order. What percent is that? First make units match: 1 L = 1000 mL. 720 ÷ 1000 = 0.72 = 72%.
Key: identify the whole (the number after "of") and the part (the number before "of" or after "is").
Percent Increase and Percent Decrease (Basics)
The formula, which we'll use in depth in a later chapter:
percent change = (new − original) / original × 100
A supply order rises from $200 to $230. Percent increase? (230 − 200) / 200 = 30/200 = 0.15 = 15% increase.
A patient's heart rate drops from 90 to 72. Percent decrease? (72 − 90) / 90 = −18/90 = −0.20 = 20% decrease. Report it as a positive "20% decrease."
Always divide by the original (starting) value, not the new value. That single step is where most percent-change points are lost.
Real-World Applications
Discount on a nurse kit: A $60 stethoscope is on sale at 25% off. Discount = 0.25 × 60 = $15. Sale price = 60 − 15 = $45. Or in one step: pay 75% of 60 → 0.75 × 60 = $45.
Total with tax: A $240 order with 6% sales tax. Tax = 0.06 × 240 = $14.40. Total = 240 + 14.40 = $254.40. One-step version: 1.06 × 240 = $254.40.
Concentration by percent: A 0.9% saline ("normal saline") solution has 0.9 g NaCl per 100 mL. In 1 L (1000 mL): (0.9/100) × 1000 = 9 g of NaCl per liter.
Common Traps to Avoid
- Forgetting to convert the percent to a decimal before multiplying. 30% × 80 is not 30 × 80. Convert first: 0.30 × 80 = 24.
- Treating "of" as anything other than multiplication. "Of" = ×.
- Misidentifying the "whole" in "percent of" questions. The whole follows "of."
- Confusing 0.5% with 5%. 0.5% = 0.005; 5% = 0.05. A factor of ten.
- Adding percents that should be applied sequentially. A 10% discount then 10% tax is not "no change." Apply each to the running total.
More Worked Examples
A hospital budget was $1.2 million and rises 8%. What is the new budget? Increase = 0.08 × 1,200,000 = $96,000. New budget = 1,200,000 + 96,000 = $1,296,000. One-step form: 1.08 × 1,200,000 = 1,296,000.
A 0.25% bupivacaine solution is used for a nerve block. How much bupivacaine is in 200 mL? 0.25% means 0.25 g per 100 mL. So mass = (0.25/100) × 200 = 0.0025 × 200 = 0.5 g. Notice the percent was tiny (0.25%), so converting to a decimal gave 0.0025 — easy to drop a zero if you rush.
A nurse's salary rises from $58,000 to $62,140. What is the percent increase? Change = 62,140 − 58,000 = 4,140. Divide by the original: 4,140 / 58,000 = 0.0714 ≈ 7.14%. If the question rounds to the nearest whole percent, report 7%.
A patient's oral intake was 1,800 mL yesterday and 1,350 mL today. What is the percent decrease? Change = 1,350 − 1,800 = −450. Divide by original (1,800): −450 / 1,800 = −0.25 = 25% decrease.
Quick Mental-Math Shortcuts
- 10% of any number: move the decimal one place left (10% of 240 = 24).
- 1% of any number: move the decimal two places left (1% of 240 = 2.4).
- 5% = half of 10% (5% of 240 = 12).
- 20% = double 10% (20% of 240 = 48). These shortcuts let you sanity-check calculator output in seconds.
What is 30% of 80?
A patient consumed 720 mL of a 1-liter order. What percent of the order was consumed?