3.2 Decimals, Percents & Conversions

3.2 Decimals, Percents & Conversions

Quick Summary: Rational numbers can be expressed interchangeably as fractions, decimals, or percentages. Understanding place value, distinguishing between terminating and repeating decimals, and mastering mental estimation without a calculator are key requirements of the Ontario Mathematics Proficiency Test (MPT). This section details exact conversion algorithms, non-calculator percent decomposition strategies, and multi-step percent change calculations.

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1. Structure of Decimals: Terminating vs. Repeating Decimals

A decimal representation is a positional system based on powers of 10. Every rational number $\frac{a}{b}$ in lowest terms can be converted to a decimal via long division, yielding one of two forms:

1. Terminating Decimals: The long division process terminates with a remainder of 0. A fraction $\frac{a}{b}$ (in lowest terms) produces a terminating decimal if and only if the prime factorization of the denominator ( b ) contains no prime factors other than 2 and 5 (i.e., $b = 2^m \cdot 5^n$). Examples: $\frac{3}{8} = \frac{3}{2^3} = 0.375$; $\frac{7}{20} = \frac{7}{2^2 \cdot 5} = 0.35$.

2. Repeating (Periodic) Decimals: The long division produces a repeating sequence of digits (the repetend). A fraction $\frac{a}{b}$ (in lowest terms) produces a repeating decimal if and only if the prime factorization of ( b ) contains at least one prime factor other than 2 or 5. Examples: $\frac{1}{3} = 0.333\dots = 0.\overline{3}$; $\frac{5}{11} = 0.4545\dots = 0.\overline{45}$.

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2. Algebraic Method: Converting Repeating Decimals to Fractions

To convert a repeating decimal to an exact fraction in lowest terms, use the following algebraic algorithm:

1. Set variable ( x ) equal to the repeating decimal.
2. Multiply ( x ) by $10^k$, where ( k ) is the number of digits in the non-repeating part past the decimal point.
3. Multiply ( x ) by $10^{k+p}$, where ( p ) is the length of the repeating period.
4. Subtract the equation in Step 2 from the equation in Step 3 to eliminate the infinite repeating tail.
5. Solve for ( x ) and reduce the fraction to lowest terms.

Detailed Example

Convert $0.2\overline{45}$ to a simplified fraction:

• Let $x = 0.2454545\dots$
• Multiply by 10 (1 non-repeating decimal place): $10x = 2.454545\dots$
• Multiply by 1000 (1 non-repeating + 2 repeating places): $1000x = 245.454545\dots$
• Subtract: $1000x - 10x = 245.454545\dots - 2.454545\dots$

$$990x = 243 \implies x = \frac{243}{990}$$

• Simplify by dividing numerator and denominator by $\operatorname{gcd}(243, 990) = 9$:

$$x = \frac{243 \div 9}{990 \div 9} = \frac{27}{110}$$

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3. Benchmark Conversion Table

Candidates should memorize key fraction-decimal-percent equivalencies for high-speed estimation on the MPT:

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4. Percent of a Quantity & Non-Calculator Estimation Strategies

"Percent" means "per hundred" ($P\% = \frac{P}{100}$). To calculate $P\%$ of a quantity ( N ), evaluate $\frac{P}{100} \times N$.

Non-Calculator Mental Decomposition Techniques

On non-calculator sections of the MPT, decompose target percentages into standard benchmark building blocks:

• $10\%$ of ( N ): Shift decimal point 1 place left ($N \div 10$).
• $1\%$ of ( N ): Shift decimal point 2 places left ($N \div 100$).
• $5\%$ of ( N ): Take half of $10\%$ of ( N ).
• $15\%$ of ( N ): Compute $10\% + 5\%$.
• $25\%$ of ( N ): Divide ( N ) by 4.
• $75\%$ of ( N ): Compute $50\% + 25\%$ or $N - 25\%$.

Example: Compute $17.5\%$ of $\text{\textdollar}480$ mentally:

• $10\% \text{ of } 480 = 48$
• $5\% \text{ of } 480 = 24$
• $2.5\% \text{ of } 480 = 12$
• $17.5\% = 48 + 24 + 12 = \text{\textdollar}84$.

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5. Percent Increase, Percent Decrease, and Multi-Stage Changes

Percent Change Formula

Percent change expresses the relative change between an original base value and a new value:

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

Critical Warning: The denominator is always the original (initial) base value, never the new value!

Multiplicative Scale Factors

• Percent Increase of $r\%$: $\text{New Value} = \text{Original Value} \times (1 + r)$.
• Percent Decrease of $r\%$: $\text{New Value} = \text{Original Value} \times (1 - r)$.

Sequential / Multi-Stage Percent Changes

When multiple percent changes occur in sequence, scale factors multiply. Percentage values cannot be added or subtracted directly.

Example: An item undergoes a $20\%$ discount followed by Ontario's $13\%$ Harmonized Sales Tax (HST).

• Discount factor: ( 1 - 0.20 = 0.80 )
• Tax factor: ( 1 + 0.13 = 1.13 )
• Combined factor: $0.80 \times 1.13 = 0.904$
• Net effect: $(0.904 - 1) \times 100\% = -9.6\%$ (a net discount of $9.6\%$, not $20\% - 13\% = 7\%$).

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6. Step-by-Step Worked Problems

Worked Problem 1: Converting a Complex Repeating Decimal to Lowest Terms Fraction

Problem: Express the repeating decimal $0.15\overline{4}$ (where only the digit 4 repeats) as a fraction in lowest terms.

Step-by-Step Solution:

1. Define the Variable Let $x = 0.154444\dots$

2. Multiply by Power of 10 to Move Non-Repeating Digits Past Decimal

   • Non-repeating part has 2 digits (15), so multiply by $10^2 = 100$: $$100x = 15.4444\dots$$

3. Multiply by Power of 10 to Include One Repeating Period

   • Repeating block has 1 digit (4), so multiply by $10^{2+1} = 1000$: $$1000x = 154.4444\dots$$

4. Subtract to Eliminate Repeating Decimal Tail

   $$1000x - 100x = 154.4444\dots - 15.4444\dots$$

   [ 900x = 139 ]

5. Solve for ( x ) and Check for Reducibility

   $$x = \frac{139}{900}$$
   • Check prime factors of 139: $\sqrt{139} \approx 11.79$. Primes to check: 2, 3, 5, 7, 11. 139 is not divisible by any of these, so 139 is prime.
   • Final Answer: $\frac{139}{900}$.

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Worked Problem 2: Sequential Discount and Tax Calculation

Problem: A school board orders science equipment listed at $\text{\textdollar}650.00$ per unit. The supplier provides a $15\%$ educational discount. Following the discount, Ontario's $13\%$ HST is added at checkout. Calculate: (a) the final cost per unit, and (b) the overall net percentage change relative to the original list price of $\text{\textdollar}650.00$.

Step-by-Step Solution:

1. Calculate Discounted Price

   • Educational discount multiplier: ( 1 - 0.15 = 0.85 ).
   $$\text{Discounted Price} = 650.00 \times 0.85 = \$552.50$$

2. Calculate Final Price with Ontario HST (13%)

   • Tax multiplier: ( 1 + 0.13 = 1.13 ).
   $$\text{Final Price} = 552.50 \times 1.13 = \$624.325$$
   • Rounding to the nearest cent gives $\text{\textdollar}624.33$.

3. Calculate Overall Net Percentage Change

   • Combined multiplier: $0.85 \times 1.13 = 0.9605$.
   • Net percentage change: $$(0.9605 - 1) \times 100\% = -3.95\%$$
   • Final Answer: (a) $\text{\textdollar}624.33$ per unit, (b) a net decrease of $3.95\%$ relative to list price.

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Worked Problem 3: Percent Change & School Demographics Word Problem

Problem: In 2024, a high school enrolled 480 students. In 2025, enrollment increased by $12.5\%$. In 2026, enrollment decreased by $10\%$ relative to the 2025 enrollment level. Calculate the number of students enrolled in 2026 and the net percentage change from 2024 to 2026.

Step-by-Step Solution:

1. Calculate 2025 Enrollment

   • $12.5\% = \frac{1}{8}$.
   • Increase: $480 \times \frac{1}{8} = 60$ students.
   • 2025 Enrollment: ( 480 + 60 = 540 ) students.

2. Calculate 2026 Enrollment

   • Decrease of $10\%$ on 540 students: $540 \times 0.10 = 54$ students.
   • 2026 Enrollment: ( 540 - 54 = 486 ) students.

3. Calculate Net Percentage Change (2024 to 2026)

   $$\text{Net Change} = 486 - 480 = +6 \text{ students}$$ $$\text{Net Percent Change} = \frac{+6}{480} \times 100\% = \frac{1}{80} \times 100\% = 1.25\%$$
   • Final Answer: 486 students in 2026, representing an overall $1.25\%$ increase over 2024.