4.2 Percent of a Quantity & Percent Equations

4.2 Percent of a Quantity & Percent Equations\n

Every percent problem involving a single static quantity relates three fundamental components:\n1. The Whole (Base): The total quantity or reference amount (often preceded by the word "of").\n2. The Percent (Rate): The ratio or fractional part per 100 (indicated by % or "percent").\n3. The Part (Amount): The portion of the whole corresponding to the specified percent (often associated with the word "is").\n Understanding how these three elements interact enables you to solve any basic percent problem on the ACCUPLACER, regardless of which value is missing.\n ---\n

1. The Basic Percent Equation\n

The fundamental relationship between Part, Percent, and Whole is expressed by the Percent Equation:\n $\text{Part} = \text{Percent} \times \text{Whole}$\n

[!IMPORTANT]\n> When using the Percent Equation directly, always convert the percent to a decimal or fraction first before performing multiplication or division.\n

Summary of Reorganized Formulas\n

Depending on which quantity is unknown, rearrange the core equation as follows:\n | Target Unknown | Formula | Required Input Format |\n| :--- | :--- | :--- |\n| Part | $\text{Part} = \text{Percent (decimal)} \times \text{Whole}$ | Multiply decimal percent by whole |\n| Whole | $\text{Whole} = \frac{\text{Part}}{\text{Percent (decimal)}}$ | Divide part by decimal percent |\n| Percent | $\text{Percent} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100%$ | Divide part by whole, then multiply by 100% |\n ---\n

2. Solving for the Part\n

When given the percent and the whole, you are calculating a specific portion of the total.\n

Formula\n$\text{Part} = \text{Percent (decimal)} \times \text{Whole}$\n

Worked Example 1: Standard Part Calculation\nProblem: What is $15%$ of $240$?\n- Identify components: $\text{Percent} = 15% = 0.15$, $\text{Whole} = 240$, $\text{Part} = x$.\n- Set up equation: $x = 0.15 \times 240$\n- Calculate:\n $0.15 \times 240 = \frac{15}{100} \times 240 = \frac{3}{20} \times 240 = 3 \times 12 = 36$\n- Answer: $15%$ of $240$ is $36$.\n

Worked Example 2: Percents Greater Than 100%\nProblem: What is $125%$ of $80$?\n- Identify components: $\text{Percent} = 125% = 1.25$, $\text{Whole} = 80$, $\text{Part} = x$.\n- Set up equation: $x = 1.25 \times 80$\n- Calculate:\n $1.25 \times 80 = \frac{5}{4} \times 80 = 5 \times 20 = 100$\n- Answer: $125%$ of $80$ is $100$.\n

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3. Solving for the Whole\n

When given a part and the percent it represents, you are finding the original total.\n

Formula\n$\text{Whole} = \frac{\text{Part}}{\text{Percent (decimal)}}$\n

Worked Example 3: Finding the Total Quantity\nProblem: $45$ is $30%$ of what number?\n- Identify components: $\text{Part} = 45$, $\text{Percent} = 30% = 0.30$, $\text{Whole} = W$.\n- Set up equation: $W = \frac{45}{0.30}$\n- Calculate:\n $W = \frac{45}{0.30} = \frac{450}{3} = 150$\n- Check: $30%$ of $150 = 0.30 \times 150 = 45$. (Correct!)\n- Answer: $45$ is $30%$ of $150$.\n

Worked Example 4: Whole Calculation with Percents Over 100%\nProblem: $84$ is $120%$ of what number?\n- Identify components: $\text{Part} = 84$, $\text{Percent} = 120% = 1.20$, $\text{Whole} = W$.\n- Set up equation: $W = \frac{84}{1.20} = \frac{840}{12} = 70$\n- Answer: $84$ is $120%$ of $70$.\n

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4. Solving for the Percent\n

When given both the part and the whole, you are finding what fraction of the whole the part represents, expressed as a percent.\n

Formula\n$\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\%$\n

Worked Example 5: Finding the Percentage\nProblem: What percent of $75$ is $18$?\n- Identify components: $\text{Part} = 18$, $\text{Whole} = 75$, $\text{Percent} = P$.\n- Set up equation: $P = \frac{18}{75} \times 100%$\n- Simplify the fraction: $\frac{18 \div 3}{75 \div 3} = \frac{6}{25}$\n- Calculate percent:\n $\frac{6}{25} \times 100\% = 6 \times 4\% = 24\%$\n- Answer: $18$ is $24%$ of $75$.\n

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5. The Percent Proportion Framework\n

An alternative to the algebraic equation is the Percent Proportion. Many students find this method easier because it avoids decimal shifts prior to calculation.\n

General Proportion Formula\n$\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} \quad \text{or} \quad \frac{\text{Is}}{\text{Of}} = \frac{\%}{100}$\n

In English word problems:\n- "Is" links to the Part.\n- "Of" links to the Whole.\n- "%" is the Percent number placed over 100.\n

Solving Proportions via Cross-Multiplication\n

To solve $\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100}$:\n1. Cross-multiply: $\text{Part} \times 100 = \text{Whole} \times \text{Percent}$.\n2. Divide by the coefficient of the unknown variable.\n

Comparison Table: Percent Equation vs. Percent Proportion\n

| Problem | Percent Equation Setup | Percent Proportion Setup | Solution |\n| :--- | :--- | :--- | :--- |\n| What is $20%$ of $60$? | $x = 0.20 \times 60$ | $\frac{x}{60} = \frac{20}{100}$ | $x = 12$ |\n| $15$ is $25%$ of what? | $15 = 0.25 \times W$ | $\frac{15}{W} = \frac{25}{100}$ | $W = 60$ |\n| $9$ is what $%$ of $36$? | $9 = P \times 36$ | $\frac{9}{36} = \frac{P}{100}$ | $P = 25%$ |\n ---\n

6. Mental Math & Benchmark Breakdown Strategies\n

You can solve many ACCUPLACER percent problems quickly by breaking the target percent into benchmark components ($10%$, $5%$, $1%$, $50%$, $25%$):\n

• $10%$ of a number: Move the decimal point 1 place left.\n- $1%$ of a number: Move the decimal point 2 places left.\n- $5%$ of a number: Take half of $10%$.\n- $20%$ of a number: Double $10%$.\n- $15%$ of a number: Add $10% + 5%$.\n

Mental Math Example: Find $35%$ of $180$.\n1. Find $10%$ of $180$: $18$\n2. Find $30%$ of $180$: $3 \times 18 = 54$\n3. Find $5%$ of $180$: Half of $18 = 9$\n4. Combine: $35% = 30% + 5% = 54 + 9 = 63$\n

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ACCUPLACER Exam Traps & Common Errors\n

[!WARNING]\n> Trap 1: Misidentifying the Whole\n> In the statement "$18$ is $40%$ of what number?", students often incorrectly calculate $40%$ of $18$ ($7.2$). Remember that the number following "of" is the Whole! Here, $18$ is the Part, so you must divide: $18 \div 0.40 = 45$.\n [!WARNING]\n> Trap 2: Forgetting to Convert Percent to Decimal in Equations\n> Writing $x = 15 \times 240 = 3600$ instead of $x = 0.15 \times 240 = 36$ is a frequent error. Always convert percent to a decimal or use the proportion format $\frac{15}{100}$.\n [!WARNING]\n> Trap 3: Reversing the Division Order When Finding Percent\n> When asked "What percent of $80$ is $20$?", dividing $80 \div 20 = 4$ ($400%$) is incorrect. The formula is $\frac{\text{Part}}{\text{Whole}} = \frac{20}{80} = 0.25 = 25%$.