Mastering Percent Composition & Word Problems
You use percentages every day:
In chemistry, percentages help us describe composition (what things are made of) and proportions (how much of something is present).
Percent literally means "out of 100"
50% = 50 out of 100 = \(\frac{50}{100}\) = 0.50
25% = 25 out of 100 = \(\frac{25}{100}\) = 0.25
100% = the whole thing = 1.00
\[\text{Decimal} = \frac{\text{Percent}}{100}\]
\[\text{Percent} = \text{Decimal} \times 100\]
Examples:
90% → \(\frac{90}{100}\) = 0.90
0.2095 → 0.2095 × 100 = 20.95%
This is the most common type! You're finding what PART of something is made of a specific material.
\[\text{Part} = \frac{\text{Percent}}{100} \times \text{Whole}\]
Problem: Certain coins minted prior to 1965 were made primarily of silver. If a quarter from this era has a mass of 5.670 grams and is 90% silver, how much silver is in one quarter?
Step 1: Identify the WHOLE
The whole = total mass of quarter = 5.670 g
Step 2: Identify the PART
The part = mass of silver (what we're finding)
Step 3: Convert % to decimal
90% = \(\frac{90}{100}\) = 0.90
Step 4: Calculate
Mass of silver = 0.90 × 5.670 g = 5.103 g
Answer: 5.10 g silver (3 sig figs)
Problem: If silver is valued at $0.34 per gram, and 560,390,585 quarters were minted in 1964 (each 90% silver, mass 5.670 g), find the total value of silver.
Breaking it down:
Step 1: Silver per quarter (from Example 1)
5.103 g silver/quarter
Step 2: Total silver in all quarters
\[560,390,585 \text{ quarters} \times 5.103 \text{ g/quarter} = 2.86 \times 10^9 \text{ g}\]
Step 3: Convert to dollars
\[2.86 \times 10^9 \text{ g} \times \frac{\$0.34}{1 \text{ g}} = 9.7 \times 10^8 \text{ dollars}\]
Answer: $970,000,000 (nearly a billion dollars!)
Air isn't pure oxygen - it's a mixture! Same with many liquids. Percentages tell us how much of each component is present.
Problem: The earth's atmosphere is 20.95% oxygen. The average male inhales 500 mL of air during a breath. If he breathes 11 times per minute, how many liters of oxygen are inhaled during this time?
Step 1: Total air inhaled per minute
\[500 \text{ mL/breath} \times 11 \text{ breaths/min} = 5500 \text{ mL air/min}\]
Step 2: Convert % to decimal
20.95% = 0.2095
Step 3: Find oxygen volume
\[\text{O}_2 \text{ volume} = 0.2095 \times 5500 \text{ mL} = 1152.25 \text{ mL O}_2\]
Step 4: Convert to liters
\[1152.25 \text{ mL} \times \frac{1 \text{ L}}{1000 \text{ mL}} = 1.15225 \text{ L}\]
Answer: 1.15 L oxygen (3 sig figs)
Your body is made of different elements in different percentages. These problems combine percentages with mole conversions!
Problem: Roughly one in every trillion (1 × 10¹²) atoms of carbon in the body is radioactive carbon-14. If your body is 18% carbon, determine the number of radioactive carbon atoms in the body of a 75.0 kg person. (12.0 g of carbon contains 6.022 × 10²³ atoms)
This is a MULTI-STEP journey! Let's map it out:
Step 1: Total mass of carbon in body
Body is 18% carbon, so convert % to decimal:
\[75.0 \text{ kg} \times 0.18 = 13.5 \text{ kg carbon}\]
Convert to grams:
\[13.5 \text{ kg} \times 1000 \text{ g/kg} = 13,500 \text{ g carbon}\]
Step 2: Convert grams of carbon to atoms
\[13,500 \cancel{\text{ g C}} \times \frac{6.022 \times 10^{23} \text{ atoms}}{12.0 \cancel{\text{ g C}}} = 6.775 \times 10^{26} \text{ C atoms}\]
Step 3: Find radioactive C-14 atoms
One in every 10¹² is C-14:
\[\frac{6.775 \times 10^{26}}{1 \times 10^{12}} = 6.775 \times 10^{14} \text{ C-14 atoms}\]
Answer: 6.77 × 10¹⁴ radioactive carbon atoms
kg person → kg carbon (%) → g carbon → atoms carbon → C-14 atoms (fraction)
BEFORE conversions: When the percentage applies to the original units
Example: "75 kg person is 18% carbon" → Find kg of carbon FIRST
AFTER conversions: When the percentage applies to final units
Example: "What % of 500 g is 125 g?" → Convert everything to same units FIRST
Problem: A bronze statue is 88% copper and weighs 45.0 pounds. How many copper atoms does it contain?
Path: lb total → g total → g copper (%) → moles copper → atoms copper
Step 1: Convert pounds to grams
\[45.0 \text{ lb} \times \frac{454 \text{ g}}{1 \text{ lb}} = 20,430 \text{ g total}\]
Step 2: Find mass of copper (use % now!)
\[0.88 \times 20,430 \text{ g} = 17,978.4 \text{ g Cu}\]
Step 3: Convert to moles
\[\frac{17,978.4 \text{ g}}{63.55 \text{ g/mol}} = 282.9 \text{ mol Cu}\]
Step 4: Convert to atoms
\[282.9 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} = 1.70 \times 10^{26} \text{ atoms}\]
Answer: 1.70 × 10²⁶ copper atoms
Using % instead of decimal
Wrong: 5.670 × 90 = 510.3
Right: 5.670 × 0.90 = 5.103
Applying % to wrong quantity
Wrong: 0.2095 × 500 mL (one breath)
Right: 0.2095 × 5500 mL (total air)
Wrong order of operations
Always find the TOTAL first, THEN apply %
Forgetting sig figs
90% has 2 sig figs!
90.% or 90.0% has 3 sig figs
\[\text{Part} = \frac{\text{Percent}}{100} \times \text{Whole}\]
\[\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\]
\[\text{Whole} = \frac{\text{Part}}{\text{Percent/100}}\]
| Percent | Decimal | Fraction |
|---|---|---|
| 100% | 1.00 | 1/1 (whole) |
| 50% | 0.50 | 1/2 |
| 25% | 0.25 | 1/4 |
| 20% | 0.20 | 1/5 |
| 10% | 0.10 | 1/10 |
| 1% | 0.01 | 1/100 |