4.3 Percentages, Ratios, and Rapid Proportions

4.3 Percentages, Ratios, and Rapid Proportions

Wonderlic SLE Fast Fact: Percentages, discounts, markups, and multi-part ratios are staples of the SLE's simple-math items. Wonderlic does not publish how many appear on a form. Candidates who set up formal single-variable or two-variable algebraic equations (x, y) frequently run out of time. Success on cognitive assessments depends on deploying intuitive mental arithmetic models: the 10% and 1% Building Block Method for percentages, and the Total Shares Framework for ratios.

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The 10% and 1% Mental Building Block Method

Traditional percentage computation requires converting the percent to a decimal and performing multi-digit multiplication (e.g., finding $15\%$ of 80 by computing $80 \times 0.15 = 12.00$). While accurate, setting up decimal multiplication on scratch paper consumes 15 to 20 seconds.

The Building Block Method relies on the fact that finding $10\%$ and $1\%$ of any number requires nothing more than shifting the decimal point:

• $10\%$ of N: Shift the decimal point one position to the left ($N \div 10$).
• $1\%$ of N: Shift the decimal point two positions to the left ($N \div 100$).

Once $10\%$ and $1\%$ are established as mental anchor points, any target percentage can be constructed using basic addition, doubling, or halving:

Handling Odd and Precise Percentages with the 1% Tool

When an exam question tests non-round percentages like $18\%$ or $7\%$, integrate the $1\%$ unit block:

• Calculate 18% of 250:
  1. $20\% = 10\% \times 2 = 25 \times 2 = 50$.
  2. $1\% = 2.50 \implies 2\% = 5.00$.
  3. $18\% = 20\% - 2\% = 50 - 5 =$ 45.
• Calculate 7% of 600:
  1. $1\% \text{ of } 600 = 6$.
  2. $7\% = 7 \times 6 =$ 42. Mental execution time: under 3 seconds.

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Percent Increase and Decrease: The Base Denominator Rule

Questions asking for percentage change evaluate whether examinees correctly identify the base value (the starting point). The formula for percent change is:

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

The Denominator Rule: The denominator of a percent change calculation is always the initial, chronological baseline before the change occurred. Never place the new, modified value in the denominator unless the problem explicitly asks for the change relative to the final value.

Step-by-Step Percent Change Execution

• Problem 1 (Percent Increase): An employee's hourly wage increases from $\$20$ to $\$25$. What is the percentage increase?

  1. Find the difference: $\$25 - \$20 = \$5$.
  2. Identify the original baseline: $\$20$.
  3. Form the fraction: $\frac{5}{20} = \frac{1}{4}$.
  4. Convert to percentage: $\frac{1}{4} = \mathbf{25\%}$. (Trap: Placing the new wage in the denominator yields $\frac{5}{25} = 20\%$, which is incorrect).

• Problem 2 (Percent Decrease): A store reduces the price of a television from $\$400$ to $\$300$. What is the percentage discount?

  1. Find the difference: $\$400 - \$300 = \$100$.
  2. Identify the original baseline: $\$400$.
  3. Form the fraction: $\frac{100}{400} = \frac{1}{4}$.
  4. Convert to percentage: $\frac{1}{4} = \mathbf{25\%}$. (Trap: Using the sale price yields $\frac{100}{300} = 33.3\%$, a common distractor).

The Asymmetry of Percentage Fluctuations

A common cognitive misconception is assuming percentage gains and losses are symmetrical. They are not:

• If an investment loses $50\%$, it drops from $\$100$ to $\$50$. To return to $\$100$, it must gain $\$50$ on a base of $\$50$—a 100% gain!
• If a price increases by $20\%$ and then decreases by $20\%$:
  1. Start at $\$100 \rightarrow$ after $+20\% = \$120$.
  2. Decrease $\$120$ by $20\%$: $10\% = 12 \implies 20\% = 24$.
  3. $\$120 - 24 = \$96$.
  4. Result: A 4% net loss, not a return to $\$100$. Rule: A sequential increase and decrease of $x\%$ always produces a net decrease of $\left(\frac{x}{10}\right)^2\%$. For $20\%$, $(20/10)^2 = 2^2 = 4\%$ loss.

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The "Reverse Percentage" Trap

A classic trap in percentage word problems is the reverse percentage problem.

The Classic Flawed Intuition

Consider this standard item:

"A jacket is sold on sale for $\$80$ after a $20\%$ discount was applied. What was the original retail price?"

A common rushed approach goes like this:

$$\text{Flawed calculation: } 20\% \text{ of } \$80 = \$16 \implies \$80 + \$16 = \$96$$

This is completely incorrect. The $20\%$ discount was subtracted from the original price, not the sale price. Because the original price was higher than $\$80$, $20\%$ of that original price must be greater than $\$16$.

The Correct Multiplier Protocol

To solve reverse percentage problems, set up the relationship using decimal multipliers:

1. Let the original price equal $100\%$ (or 1.00).
2. After a $20\%$ discount, the sale price represents: $$100\% - 20\% = 80\% = 0.80 \text{ of the original price}$$
3. Express mathematically: $$\text{Original} \times 0.80 = \$80$$
4. Solve for the original: $$\text{Original} = \frac{\$80}{0.80} = \frac{800}{8} = \mathbf{\$100}$$

Summary of Reverse Percentage Multipliers

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Ratio Parts and the Total Shares Framework

Ratio problems describe how a total quantity is partitioned among two or more entities. Writing system-of-equations algebra (2x + 3x + 5x = 600) is unnecessary and slow. The Total Shares Method solves these questions in three visual steps:

$$\text{Total Shares} = \text{Sum of all ratio components}$$ $$\text{Value of 1 Share} = \frac{\text{Total Quantity}}{\text{Total Shares}}$$ $$\text{Individual Allocation} = \text{Ratio Component} \times \text{Value of 1 Share}$$

Standard Three-Part Allocation

• Problem: Divide $\$72,000$ among three departments in the ratio 3 : 4 : 5. How much does each department receive?
  1. Sum the shares: 3 + 4 + 5 = 12 total shares.
  2. Calculate single-share value: $\$72,000 \div 12 = \$6,000$ per share.
  3. Distribute shares:
     • Dept 1 (3 shares): $3 \times \$6,000 = \$18,000$.
     • Dept 2 (4 shares): $4 \times \$6,000 = \$24,000$.
     • Dept 3 (5 shares): $5 \times \$6,000 = \$30,000$.
  4. Verify total: 18,000 + 24,000 + 30,000 = 72,000.

The Shortcut for Part-to-Part Differences

Often, a question does not ask for all individual shares, but rather: "How much more did the largest share receive than the smallest share?"

• Slow method: Compute both shares individually ($\$30,000$ and $\$18,000$), then subtract: $\$30,000 - \$18,000 = \$12,000$.
• Fast method: Subtract the ratio parts directly before multiplying!
  • Difference in parts: 5 - 3 = 2 shares.
  • Multiply by single-share value: $2 \times \$6,000 = \mathbf{\$12,000}$. Saves 8 seconds of redundant computation.

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Unit Rates and Proportional Scaling

A unit rate expresses the quantity of one variable associated with a single unit of another variable (e.g., miles per hour, widgets per worker, cost per ounce).

Proportional Scaling Shortcuts

When scaling rates, look for horizontal or vertical integer scaling factors before cross-multiplying:

• Problem: If 8 pounds of coffee cost $\$52$, how much do 12 pounds cost?

  • Set up proportion: $\frac{\$52}{8 \text{ lbs}} = \frac{\$X}{12 \text{ lbs}}$.
  • Mental scaling: Recognize that 12 is 1.5 times 8 (8 + 4).
  • Therefore, the cost is 1.5 times $\$52$: $$\$52 + \text{half of } \$52 = \$52 + \$26 = \mathbf{\$78}$$

  No long division or multi-step scratchwork required.

• Problem: If 15 workers can pack 450 boxes in 3 hours, how many boxes can 10 workers pack in 4 hours?

  1. Determine unit rate per worker per hour: $$\text{Rate} = \frac{450 \text{ boxes}}{15 \text{ workers} \times 3 \text{ hours}} = \frac{450}{45} = 10 \text{ boxes/worker-hour}$$
  2. Compute new production: $$10 \text{ workers} \times 4 \text{ hours} \times 10 \text{ boxes/worker-hour} = \mathbf{400 \text{ boxes}}$$