10.2 Decimal Operations, Place Values & Fraction-Decimal-Percent Conversion

10.2 Decimal Operations, Place Values & Fraction-Decimal-Percent Conversion

Decimal Positional Notation & Place Value Architecture

The base-10 decimal numbering system is positional: the value of every digit depends strictly on its location relative to the decimal point. Moving one position to the left multiplies place value by 10 (10¹, 10², 10³), while moving one position to the right divides place value by 10 (10⁻¹, 10⁻², 10⁻³):

$\dots \quad 10^2 \quad 10^1 \quad 10^0 \quad . \quad 10^{-1} \quad 10^{-2} \quad 10^{-3} \quad 10^{-4} \quad \dots$ $\dots \text{ [Hundreds] } \text{ [Tens] } \text{ [Ones] } . \text{ [Tenths] } \text{ [Hundredths] } \text{ [Thousandths] } \text{ [Ten-Thousandths] } \dots$

Expanded Form and Positional Value

In the decimal number 47.3825:

• 4 occupies the tens place: 4 × 10 = 40
• 7 occupies the ones place: 7 × 1 = 7
• 3 occupies the tenths place: 3 × 0.1 = 3/10 = 0.3
• 8 occupies the hundredths place: 8 × 0.01 = 8/100 = 0.08
• 2 occupies the thousandths place: 2 × 0.001 = 2/1000 = 0.002
• 5 occupies the ten-thousandths place: 5 × 0.0001 = 5/10000 = 0.0005

Trailing Zeros vs. Interior Placeholders

Understanding the distinction between significant placeholders and extraneous zeros prevents careless calculation errors:

• Interior Placeholder Zeros: In 0.04, the zero holds the tenths place, indicating 4/100. Removing it produces 0.4 = 4/10, distorting the value tenfold.
• Trailing Decimal Zeros: In 0.400, the zeros to the right of the last non-zero digit do not alter numerical value (0.4 = 0.40 = 0.400). Trailing zeros serve as essential alignment padding when performing manual column addition and subtraction.

Standard Decimal Rounding Algorithm

To round a decimal to a target place value:

1. Identify the digit in the specified rounding position.
2. Inspect the single digit immediately to its right (the test digit).
3. If the test digit is 5 or greater (5, 6, 7, 8, 9), increment the rounding digit by 1.
4. If the test digit is 4 or less (0, 1, 2, 3, 4), leave the rounding digit unchanged.
5. Truncate all digits to the right of the specified rounding position.

• Example: Round 18.4738 to the nearest hundredth. The hundredths digit is 7; the test digit to its right is 3 (< 5). Thus, round down to 18.47.

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Decimal Addition and Subtraction Mechanics

Adding and subtracting decimals without a calculator requires one non-negotiable rule: vertical alignment of decimal points.

When numbers are written horizontally (e.g., 14.8 - 6.275), candidates who align the rightmost digits mistakenly subtract thousandths from tenths. Decimal points must line up in a single vertical column, ensuring that tenths combine with tenths, hundredths with hundredths, and ones with ones. Trailing zeros must be inserted to fill empty columns.

Worked Subtraction with Padding and Borrowing

Evaluate: 14.8 - 6.275.

14.800 & \text{(Pad two trailing zeros to match thousandths)} \\ - \phantom{0}6.275 & \text{(Decimal points strictly aligned vertically)} \\ \hline \phantom{0}8.525 & \text{(Subtract with standard column borrowing)} \end{array}$$ ### The Constant Difference Subtraction Shortcut To eliminate cumbersome regrouping across multiple consecutive zeros in problems such as 20.00 - 7.64, subtract 0.01 from both terms before computing. Because the difference between two numbers is invariant under equal shifts (A - B = (A - c) - (B - c)): $$20.00 - 7.64 = (20.00 - 0.01) - (7.64 - 0.01) = 19.99 - 7.63 = 12.36$$ Because 19.99 contains only nines, subtraction proceeds without a single regrouping step. --- ## Decimal Multiplication: Counting Total Decimal Places Multiplying decimals does **not** require aligning decimal points. Column alignment during multiplication wastes scratchpad space and breeds confusion. Instead, follow this three-step algorithm: 1. **Ignore decimal points temporarily** and multiply the numbers as if they were positive whole integers. 2. **Count the total number of decimal places** (digits to the right of the decimal point) across **all factors** combined. 3. **Place the decimal point** in the whole-number product so that it contains that exact total number of decimal places, counting from right to left and prepending leading zeros if necessary. ### Worked Example: Multi-Digit Decimal Product Evaluate: 0.035 × 0.24. - **Step 1: Whole-number multiplication:** Multiply 35 × 24. $$35 \times 20 = 700, \quad 35 \times 4 = 140 \implies 700 + 140 = 840$$ - **Step 2: Sum decimal places across factors:** - 0.035 has 3 decimal places. - 0.24 has 2 decimal places. - Total decimal places = 3 + 2 = 5 places. - **Step 3: Insert decimal point:** The integer product is 840. Counting 5 places to the left requires prepending two leading zeros: $$840 \implies 0.00840 = 0.0084$$ ### Multiplying by Powers of 10 Multiplying a decimal by 10ⁿ shifts the decimal point n positions to the **right**: - 4.275 × 10 = 42.75 - 4.275 × 100 = 427.5 - 4.275 × 1,000 = 4,275 --- ## Long Division with Decimals: Divisor Normalization When dividing by a decimal, long division cannot proceed directly with a decimal divisor. The fundamental property of fractions permits multiplying both numerator (dividend) and denominator (divisor) by the identical power of 10 without altering the quotient: $$\frac{\text{Dividend}}{\text{Divisor}} = \frac{\text{Dividend} \times 10^k}{\text{Divisor} \times 10^k}$$ ### The Decimal Division Protocol 1. Count how many decimal places exist in the **divisor** (let this count be k). 2. Shift the decimal point in the **divisor** k places to the right, transforming it into an integer. 3. Shift the decimal point in the **dividend** the exact same k places to the right (appending trailing zeros if the dividend runs out of digits). 4. Place the decimal point in the quotient space **directly above** the new decimal point in the dividend. 5. Execute standard long division as with whole numbers. ### Worked Example: Decimal Long Division Evaluate: 14.28 ÷ 0.35. - Divisor 0.35 has 2 decimal places. Multiply both divisor and dividend by 10² = 100: $$14.28 \div 0.35 = 1,428 \div 35$$ - Long division execution: - 35 goes into 142 four times: 4 × 35 = 140; remainder is 142 - 140 = 2. - Bring down 8 to form 28. 35 goes into 28 zero times (0 in tens place); remainder is 28. - Place decimal point, append trailing zero to form 280. - 35 goes into 280 exactly eight times: 8 × 35 = 280; remainder is 0. - Result: 40.8. --- ## The Fraction-Decimal-Percent (F-D-P) Conversion Triangle Fluency in converting numbers among fractions, decimals, and percentages is tested extensively on the CBEST. Every rational quantity can be expressed in all three formats: - **Fraction to Decimal:** Divide numerator by denominator using scratchpad division: 3/8 = 3 ÷ 8 = 0.375. - **Decimal to Percent:** Multiply by 100 by shifting the decimal point two places to the right and appending the `%` symbol: 0.375 × 100 = 37.5%. - **Percent to Decimal:** Divide by 100 by shifting the decimal point two places to the left and removing the `%` symbol: 42.5% ÷ 100 = 0.425. - **Percent to Fraction:** Write the percent value over a denominator of 100 and simplify via the GCF: 65% = 65/100 = 13/20. - **Decimal to Fraction:** Read the terminal place value, place the digits over the corresponding power of 10 (10, 100, 1,000), and reduce: 0.064 = 64/1000 = 8/125. --- ## High-Frequency Benchmark Conversions Table Memorizing these benchmark equivalencies eliminates long division on the exam and guarantees immediate recognition of options: | Fraction | Decimal | Exact Percentage | Testing Notes & Relationships | | :--- | :--- | :--- | :--- | | 1/2 | 0.5 | 50% | Core foundation; half of 1 | | 1/3 | 0.333… | 33 1/3% or 33.3% | Repeating 3; 2/3 = 0.666… = 66 2/3% | | 1/4 | 0.25 | 25% | Half of 1/2; 3/4 = 0.75 = 75% | | 1/5 | 0.2 | 20% | Multiples: 2/5=0.4=40%, 3/5=0.6=60%, 4/5=0.8=80% | | 1/6 | 0.166… | 16 2/3% or 16.7% | Half of 1/3; 5/6 = 0.833… = 83 1/3% | | 1/8 | 0.125 | 12.5% | Half of 1/4; high-frequency CBEST trap | | 3/8 | 0.375 | 37.5% | 3 × 0.125; frequently tested in measurement | | 5/8 | 0.625 | 62.5% | 0.500 + 0.125; common survey and statistical rate | | 7/8 | 0.875 | 87.5% | 1.000 - 0.125; high-tier completion benchmark | | 1/10 | 0.1 | 10% | Baseline mental math building block | | 1/12 | 0.0833…| 8 1/3% or 8.3% | Half of 1/6; inches-to-feet conversion factor | --- ## Strategic Elimination and Estimation on the CBEST When a CBEST problem asks you to compare or order fractions and decimals, never execute five separate long divisions. Leverage benchmark anchors: - To evaluate whether 17/35 is greater than 0.50, compare it to the exact half-value: half of 35 is 17.5. Because 17 < 17.5, 17/35 < 0.50. - To evaluate 29/40, note that 30/40 = 3/4 = 0.75. Because 1/40 = 0.025, 29/40 = 0.75 - 0.025 = 0.725.