11.2 Non-Calculator Benchmark Percentages, Fractional Shortcuts, and Decimal Scaling

The Percentage Commutative Property and Benchmark Building Blocks

Flipping Difficult Percentages ($a\% \text{ of } b = b\% \text{ of } a$)

One of the most powerful time-savers in LANTITE Section 2 is the Percentage Commutative Property. Because finding $a\%$ of $b$ means computing $\frac{a}{100} \times b$, and multiplication can be performed in either order, you can always swap the percentage and the base number:

$$a\% \text{ of } b = \frac{a}{100} \times b = \frac{a \times b}{100} = \frac{b}{100} \times a = b\% \text{ of } a$$

Whenever a question asks for an awkward percentage of a clean benchmark number ($25, 50, 75, 20, 12.5,$ or their multiples of $10$), flip the two numbers immediately:

• $16\% \text{ of } 25$: Flip to $25\% \text{ of } 16 = \frac{1}{4} \times 16 = 4$.
• $48\% \text{ of } 50$: Flip to $50\% \text{ of } 48 = \frac{1}{2} \times 48 = 24$.
• $32\% \text{ of } 75$: Flip to $75\% \text{ of } 32 = \frac{3}{4} \times 32 = 24$.
• $64\% \text{ of } 12.5$: Flip to $12.5\% \text{ of } 64 = \frac{1}{8} \times 64 = 8$.
• $28\% \text{ of } 250$: Rewrite $250$ as $25 \times 10$, or flip to $250\% \text{ of } 28 = 2.5 \times 28 = 70$.

Benchmark Building Blocks and Subtraction from $100\%$

When neither number is a unit-fraction benchmark, build the target percentage from two instant mental anchors—$10\%$ (move the decimal point one place left) and $1\%$ (move the decimal point two places left):

• $5\%$: Half of $10\%$.
• $2.5\%$: Half of $5\%$ (or one-quarter of $10\%$). For example, to find $17.5\%$ of $160, compute $10\% = 16$, $5\% = 8$, and $2.5\% = 4$, then sum: $16 + 8 + 4 = 28$ dollars.
• $0.5\%$: Half of $1\%$. For example, $3.5\%$ of $400 equals $(3 \times 4) + 2 = 14$ dollars.
• Complement Subtraction from $100\%$: For percentages close to $100\%$ (such as $85\%, 90\%, 95\%,$ or $98\%$), find the small missing complement and subtract it from the original total:
  • $95\%$ of $360: Find $5\%$ of $360 (half of $36 is $18) and subtract from $360: $360 - 18 = 342$ dollars.
  • $85\%$ of $240: Find $15\%$ of $240 ($10\% = 24$ plus $5\% = 12$, totaling $36) and subtract: $240 - 36 = 204$ dollars.

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Replacing Awkward Decimals and Percentages with Exact Fractions

Multiplying or dividing by decimals such as $0.125, 0.375, 0.75,$ or $1.5$ using long paper algorithms wastes valuable time and invites column-placement errors. Instead, convert the decimal or percentage into its simplest fraction and use integer division and multiplication:

• $\times 0.25$ or $25\% = \times \frac{1}{4}$: Divide by $4$ (halve twice). Conversely, $\div 0.25 = \times 4$ (because dividing by $\frac{1}{4}$ finds how many quarters fit into the number).
• $\times 0.125$ or $12.5\% = \times \frac{1}{8}$: Divide by $8$ (halve three times). Conversely, $\div 0.125 = \times 8$.
• $\times 0.375$ or $37.5\% = \times \frac{3}{8}$: Divide by $8$ first to keep numbers small, then multiply by $3$. For instance, $37.5\%$ of $64 = (64 \div 8) \times 3 = 8 \times 3 = 24$.
• $\times 0.75$ or $75\% = \times \frac{3}{4}$: Divide by $4$ first, then multiply by $3$. Conversely, $\div 0.75 = \times \frac{4}{3}$ (divide by $3$, then multiply by $4$).
• $\times 0.2$ or $20\% = \times \frac{1}{5}$: Divide by $5$. Conversely, $\div 0.2 = \times 5$.
• $\times 1.5 = \times \frac{3}{2}$: Add half of the number to itself (e.g., $34 \times 1.5 = 34 + 17 = 51$).
• $\div 1.5 = \div \frac{3}{2} = \times \frac{2}{3}$: Divide by $3$ first, then multiply by $2$. For example, $45 \div 1.5 = (45 \div 3) \times 2 = 15 \times 2 = 30$.

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Clearing Decimals in Division and Pre-Simplifying Fractions

Whole-Number Scaling for Decimal Division

Whenever you must divide by a decimal in Section 2, write the division as a fraction $\frac{\text{dividend}}{\text{divisor}}$ and multiply both the numerator and denominator by the same power of $10$ ($10, 100,$ or $1{,}000$) so that the denominator becomes a whole number:

$$3.6 \div 0.04 = \frac{3.6}{0.04} = \frac{3.6 \times 100}{0.04 \times 100} = \frac{360}{4} = 90$$

Critical Place-Value Warning: You must shift the decimal point the exact same number of places in both numbers, governed by the divisor. Because $0.04$ has two decimal places, both $3.6$ and $0.04$ must be multiplied by $100$ (shifting two places right to $360$ and $4$). Shifting $3.6$ only one place to $36$ while shifting $0.04$ two places to $4$ yields $9$, a tenfold place-value error that makes a classic multiple-choice distractor.

Cancelling Common Factors Before Multiplying

In multi-step rate, ratio, or percentage-change calculations—such as finding $\frac{45 \times 28}{35}$ or scaling a recipe for $36$ students from a base of $24$—never multiply the numerator out into a large three- or four-digit integer first. Cancel common factors between any numerator term and any denominator term before multiplying:

$$\frac{45 \times 28}{35} = \frac{45 \times (28 \div 7)}{35 \div 7} = \frac{45 \times 4}{5} = (45 \div 5) \times 4 = 9 \times 4 = 36$$

Worked Section 2 Classroom Examples

Example 1 (Laboratory Solution Division): A science teacher has $4.5\text{ litres}$ of saline solution to dispense into laboratory beakers that each hold $0.075\text{ litres}$. How many beakers can be completely filled?

1. Write as a fraction and clear decimals: Because $0.075$ has three decimal places, multiply both numerator and denominator by $1{,}000$:

$$\frac{4.5}{0.075} = \frac{4.5 \times 1{,}000}{0.075 \times 1{,}000} = \frac{4{,}500}{75}$$

1. Cancel common factors: Divide numerator and denominator by $25$ (using $4{,}500 \div 25 = 45 \times 4 = 180$ and $75 \div 25 = 3$), or note that $450 \div 75 = 6$, so $4{,}500 \div 75 = 60$ beakers. (Alternatively, recognize $0.075 = \frac{3}{40}$, so $4.5 \div \frac{3}{40} = (4.5 \div 3) \times 40 = 1.5 \times 40 = 60$.)

Example 2 (School Survey Benchmarks): In a travel survey of $75$ Year 10 students, $36\%$ walk to school, and $37.5\%$ of the remaining students catch the train. How many students catch the train?

1. Walkers via commutative percentage flip: $36\% \text{ of } 75 = 75\% \text{ of } 36 = \frac{3}{4} \times 36 = 27$ students.
2. Remaining students: $75 - 27 = 48$ students.
3. Train commuters via unit fraction $\frac{3}{8}$: $37.5\% \text{ of } 48 = \frac{3}{8} \times 48 = (48 \div 8) \times 3 = 6 \times 3 = 18$ students.

Non-Calculator Fraction, Decimal, and Percentage Shortcut Reference