5.4 Percent Calculations, Increase, Decrease, and Applications

5.4 Percents, Rate Applications & Number Comparisons

Core Concept: A percent is a ratio expressing a quantity per $100$ parts (from Latin per centum). On the ACCUPLACER, you must seamlessly convert between fractions, decimals, and percentages, calculate rates of increase and decrease, solve multi-step commercial finance applications, and evaluate the relative ordering and absolute values of signed rational numbers.

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1. The Tri-Modal Equivalence: Fractions, Decimals & Percents

Every rational quantity can be expressed in three interchangeable formats:

                    TRI-MODAL CONVERSION PATHWAYS

       [ FRACTION: a/b ] <========================> [ DECIMAL: d ]
              ▲                                            ▲
              │                                            │
              │            [ PERCENT: p% ]                 │
              └───────────────────┴────────────────────────┘

  • Fraction to Decimal: Divide numerator by denominator (a ÷ b).
  • Decimal to Percent:  Multiply by 100 and append % (shift decimal 2 places right).
  • Percent to Decimal:  Divide by 100 and remove % (shift decimal 2 places left).
  • Percent to Fraction: Write over 100 and simplify via GCF.

• Examples:
  • Convert $7.5%$ to decimal: $7.5 \div 100 = \mathbf{0.075}$.
  • Convert $0.625$ to percent: $0.625 \times 100% = \mathbf{62.5%}$.
  • Convert $\frac{7}{8}$ to percent: $7 \div 8 = 0.875 \implies 0.875 \times 100% = \mathbf{87.5%}$.
  • Convert $36%$ to a simplified fraction: $\frac{36}{100} = \frac{36 \div 4}{100 \div 4} = \mathbf{\frac{9}{25}}$.

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2. The Core Percent Equation & Three Question Archetypes

All basic percentage problems relate three quantities via the fundamental formula:

$\text{Part} = \text{Percent (in decimal form)} \times \text{Whole} \quad \Longleftrightarrow \quad \frac{\text{Is}}{\text{Of}} = \frac{P}{100}$

+-------------------------------------------------------------------------------------------------------+
|                                 THE THREE PERCENT QUESTION ARCHETYPES                                 |
+-------------------+-------------------------------+-----------------------+---------------------------+
| Archetype         | Question Phrasing             | Algebraic Setup       | Worked Solution           |
+-------------------+-------------------------------+-----------------------+---------------------------+
| 1. Find the Part  | "What is 18% of 450?"         | Part = 0.18 × 450     | Part = 81                 |
| 2. Find Percent   | "42 is what percent of 280?"  | P = 42 / 280          | P = 0.15 = 15%            |
| 3. Find the Whole | "78 is 65% of what number?"   | Whole = 78 / 0.65     | Whole = 120               |
+-------------------+-------------------------------+-----------------------+---------------------------+

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3. Percent Increase, Percent Decrease & Successive Percentage Traps

The Universal Percent Change Formula

Percent change measures relative variation against the initial (original) baseline value:

$\text{Percent Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\% = \frac{\Delta}{\text{Original}} \times 100\%$

• Critical Exam Rule: The denominator is always the original starting value, never the updated final value.

• Worked Example (Percent Decrease): A jacket originally priced at $$160$ is marked down to $$112$. What is the percent decrease? $\text{Percent Decrease} = \frac{160 - 112}{160} \times 100\% = \frac{48}{160} \times 100\% = 0.30 \times 100\% = \mathbf{30\%}$

The Successive Percent Trap

Percentages cannot be added directly across successive steps because the base changes after each step.

• Question: If a stock increases by $20%$ on Monday and decreases by $20%$ on Tuesday, what is the net percent change from its initial price?
• Solution: Let the initial price be $$100$.
  • After Monday ($+20%$): $$100 \times 1.20 = $120$.
  • After Tuesday ($-20%$): $$120 \times (1 - 0.20) = $120 \times 0.80 = $96$.
  • Net change: $\frac{96 - 100}{100} \times 100% = \mathbf{-4%}$ (a net $4%$ decrease, not $0%$).

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4. Commercial & Financial Arithmetic Applications

+-------------------------------------------------------------------------------------------------------+
|                                 COMMERCIAL ARITHMETIC FORMULA MATRIX                                  |
+-----------------------+---------------------------------------+---------------------------------------+
| Application           | Mathematical Formula                  | Multiplier Format                     |
+-----------------------+---------------------------------------+---------------------------------------+
| Sales Tax & Total     | Total = Base Price + (Price × Tax)   | Total = Base × (1 + r_tax)            |
| Retail Discount       | Sale Price = Retail - (Retail × Disc) | Sale Price = Retail × (1 - r_disc)    |
| Sales Commission      | Commission = Sales Volume × Rate      | Comm = Sales × r_comm                 |
| Simple Interest       | Interest = Principal × Rate × Time    | I = P · r · t   (t in years)          |
| Total Balance (Loans) | Total Accumulated Amount = P + I      | A = P(1 + rt)                         |
+-----------------------+---------------------------------------+---------------------------------------+

• Worked Example (Simple Interest): A student deposits $$2,400$ into a fixed certificate of deposit earning $4.5%$ simple annual interest for a period of $30$ months. What is the total accumulated balance at maturity?
  1. Identify parameters: $P = $2,400$, $r = 0.045$, $t = \frac{30}{12} = 2.5$ years.
  2. Compute simple interest: $I = P \cdot r \cdot t = 2,400 \times 0.045 \times 2.5$ $2,400 \times 0.045 = 108 \implies 108 \times 2.5 = 270$ $I = \$270$
  3. Total accumulated balance: $A = P + I = \$2,400 + \$270 = \mathbf{\$2,670}$

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5. Comparing and Ordering Rational Numbers & Absolute Value

Number Line Geometry & Signed Comparisons

On a real number line, values increase from left to right. For negative numbers, a larger numerical magnitude represents a value further to the left (i.e., smaller): $-0.8 < -0.6$.

  <---|-------|-------|-------|-------|-------|-------|-------|-------|--->
     -1.0    -0.8    -0.6    -0.4    -0.2     0.0    +0.2    +0.4    +0.6

Systematic Protocol for Ordering Mixed Rational Numbers

1. Convert all numbers (fractions, mixed numbers, percentages) into standard decimal format to equal decimal precision.
2. Separate negative values from positive values.
3. Order negative values from most negative (smallest) to least negative.
4. Order positive values from smallest to largest.

• Worked Example: Order from least to greatest: $-\frac{5}{8}, ; -0.65, ; 38%, ; \frac{1}{3}, ; -0.6, ; 0.35$.
  1. Convert all elements to decimals (3 decimal places):
     • $-\frac{5}{8} = -0.625$
     • $-0.65 = -0.650$
     • $38% = +0.380$
     • $\frac{1}{3} = +0.333...$
     • $-0.6 = -0.600$
     • $0.35 = +0.350$
  2. Order negative numbers: $-0.650 < -0.625 < -0.600 \implies -0.65 < -\frac{5}{8} < -0.6$.
  3. Order positive numbers: $+0.333... < +0.350 < +0.380 \implies \frac{1}{3} < 0.35 < 38%$.
  4. Unified ascending order: $\mathbf{-0.65 < -\frac{5}{8} < -0.6 < \frac{1}{3} < 0.35 < 38\%}$

Absolute Value Properties

The absolute value $|x|$ represents the non-negative geometric distance of a number $x$ from $0$ on the number line:

• $|x| = x$ if $x \ge 0$
• $|x| = -x$ if $x < 0$
• $|a - b| = |b - a|$ (distance between points $a$ and $b$)