Percentages Word Problems Study Guide

Master the Strategy for Chemistry Calculations

The Universal Percentage Strategy

Ask yourself these questions EVERY time:

  1. What is the WHOLE? (What does 100% represent?)
  2. What is the PART? (What am I finding?)
  3. What's the percentage? (Convert to decimal!)
  4. Do I multiply or divide? (Usually multiply: Part = % × Whole)

Problem Type 1: Finding the Mass of a Component

Example: A quarter from 1964 is made of 90% silver and has a mass of 5.670 g. What is the mass of silver in the quarter?

Key Question: "90% of WHAT equals the silver mass?"

1Identify the WHOLE

The WHOLE is the total mass of the quarter = 5.670 g

Ask: "What does 100% represent here?" → The entire quarter

2Identify the PART

The PART is what we're finding = mass of silver

Ask: "What portion am I calculating?" → The silver portion

3Convert percentage to decimal

\[90\% = \frac{90}{100} = 0.90\]

4Calculate: Part = Decimal × Whole

\[\text{Mass of silver} = 0.90 \times 5.670 \text{ g} = 5.10 \text{ g}\]

Practice Template for Type 1:

The WHOLE (100%) = __________

The PART (what I'm finding) = __________

Percentage as decimal = __________

Calculation: __________ × __________ = __________

Problem Type 2: Value Calculations (Two-Step Problems)

Example: If silver is worth $31.02 per troy ounce, and there are 31.1 g in 1 troy ounce, what is the value of silver in all 1964 quarters ever minted (1,950,000,000 quarters)?

Key Question: "What do I need to find FIRST before I can find the final answer?"

1Break into sub-problems

Sub-problem A: Total mass of silver in all quarters

Sub-problem B: Convert grams to troy ounces

Sub-problem C: Calculate dollar value

2Solve Sub-problem A: Total silver mass

From Type 1, we know each quarter has 5.10 g of silver

\[\text{Total silver} = 5.10 \text{ g/quarter} \times 1,950,000,000 \text{ quarters}\]

\[= 9,945,000,000 \text{ g} = 9.945 \times 10^9 \text{ g}\]

3Solve Sub-problem B: Convert to troy ounces

\[9.945 \times 10^9 \cancel{\text{ g}} \times \frac{1 \text{ troy oz}}{31.1 \cancel{\text{ g}}} = 3.20 \times 10^8 \text{ troy oz}\]

4Solve Sub-problem C: Calculate value

\[3.20 \times 10^8 \cancel{\text{ troy oz}} \times \frac{\$31.02}{1 \cancel{\text{ troy oz}}} = \$9.93 \times 10^9\]

Final Answer: $9.93 billion

Common Mistake: Jumping to the end!

Multi-step problems require patience. Write down each intermediate answer. Don't try to do it all in one calculation!

Problem Type 3: Finding Volume or Amount from Percentage

Example: The atmosphere is 20.95% oxygen. If you breathe 0.5 L of air, what volume of oxygen did you breathe?

Key Question: "20.95% of WHAT volume equals the oxygen I breathed?"

1Identify the WHOLE

The WHOLE is the total volume of air = 0.5 L

Ask: "What does 100% represent?" → All the air breathed

2Identify the PART

The PART is what we're finding = volume of oxygen

3Convert percentage to decimal

\[20.95\% = \frac{20.95}{100} = 0.2095\]

4Calculate: Part = Decimal × Whole

\[\text{Volume of O}_2 = 0.2095 \times 0.5 \text{ L} = 0.10 \text{ L}\]

Key Insight:

Type 3 problems are structurally identical to Type 1 - just with different units! The strategy doesn't change. Mass, volume, number of atoms - the math is the same.

Problem Type 4: Finding a Subset of a Component

Example: Your body contains about 16 kg of carbon, and 18% of that carbon is carbon-12. If you have \(3.0 \times 10^{27}\) carbon atoms total, how many are carbon-12?

Key Question: "18% of WHAT equals the carbon-12 atoms?"

1Identify the WHOLE

The WHOLE is the total carbon atoms = \(3.0 \times 10^{27}\) atoms

Note: The 16 kg is extra information - we already have the atom count!

2Identify the PART

The PART is what we're finding = number of C-12 atoms

3Convert percentage to decimal

\[18\% = \frac{18}{100} = 0.18\]

4Calculate: Part = Decimal × Whole

\[\text{C-12 atoms} = 0.18 \times 3.0 \times 10^{27} = 5.4 \times 10^{26} \text{ atoms}\]

Watch Out: Red Herring Information!

Some problems give you extra information you don't need. The 16 kg of carbon was a distraction - we already had the atom count. Always ask: "What information do I actually need for this calculation?"

Quick Decision Tree

Use this flowchart to decide your approach:

Question 1: Is this a one-step or multi-step problem?

  • One-step: "Find X% of Y" → Use Part = Decimal × Whole
  • Multi-step: Break into sub-problems, solve in order

Question 2: What does 100% represent?

  • The total mass? → Finding mass of component
  • The total volume? → Finding volume of component
  • The total atoms? → Finding number of specific atoms

Question 3: Am I given unnecessary information?

  • Read carefully - some numbers might be distractors!
  • Cross out or ignore information you don't need

Top 5 Mistakes to Avoid

  1. Forgetting to convert percentage to decimal - 90% is NOT 90 in calculations!
  2. Multiplying by the percentage instead of the decimal - Use 0.90, not 90
  3. Confusing the WHOLE and the PART - Always ask "What is 100%?"
  4. Skipping units - Keep track of grams, liters, atoms, etc.
  5. Not showing work - Even if you can do it in your head, write it down!
Percentage Decimal Example Calculation
100% 1.00 100% of 50 g = 1.00 × 50 g = 50 g
90% 0.90 90% of 5.670 g = 0.90 × 5.670 g = 5.10 g
50% 0.50 50% of 100 atoms = 0.50 × 100 = 50 atoms
20.95% 0.2095 20.95% of 0.5 L = 0.2095 × 0.5 L = 0.10475 L
18% 0.18 18% of 3.0 × 10²⁷ = 0.18 × 3.0 × 10²⁷ = 5.4 × 10²⁶
10% 0.10 10% of 200 mL = 0.10 × 200 mL = 20 mL
5% 0.05 5% of 1000 g = 0.05 × 1000 g = 50 g
1% 0.01 1% of 250 atoms = 0.01 × 250 = 2.5 atoms
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