Converting Between Fractions & Decimals - Sixth Grade Mathematics
Converting smoothly between fractions and decimals enables you to choose the most efficient representation for any mathematical calculation, measurement, or financial problem.
Converting Decimals to Fractions
Converting a terminating decimal into a fraction is straightforward: read the decimal using proper place value, write it as a fraction over a power of 10, and simplify.
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| CONVERTING DECIMALS TO FRACTIONS |
| |
| Step 1: Identify the place value of the last digit. |
| 1 decimal place --> Denominator is 10 |
| 2 decimal places --> Denominator is 100 |
| 3 decimal places --> Denominator is 1,000 |
| Step 2: Write all digits to the right of the decimal point as the |
| numerator, and the power of 10 as the denominator. |
| Step 3: If there is a whole number part, write it as the whole |
| number part of a mixed number. |
| Step 4: Simplify the fraction to simplest form by dividing by GCF. |
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Worked Examples:
Example A: Convert 0.35 to a fraction:
- Two decimal places --> Denominator is 100: 35/100.
- Divide numerator and denominator by GCF (5): (35 / 5) / (100 / 5) = 7/20.
Example B: Convert 4.125 to a mixed number:
- Whole number is 4. Three decimal places --> Denominator is 1,000: 4 and 125/1,000.
- Divide 125 and 1,000 by GCF (125): 125 / 125 = 1, 1000 / 125 = 8.
- Result: 4 and 1/8.
Converting Fractions to Decimals
There are two primary methods for converting fractions into decimals: the Power of Ten Scaling Method and the Long Division Method.
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| TWO PATHWAYS TO CONVERT FRACTION TO DECIMAL |
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| Method 1: Power | Scale denominator to | 3/20 = (3 * 5) / (20 * 5) |
| of Ten Scaling | 10, 100, or 1,000 via | = 15/100 = 0.15 |
| (Fast Mental) | equivalent fraction | Fast, no long division! |
+-------------------+-----------------------+-----------------------------+
| Method 2: Long | Divide numerator by | 5/8: Divide 5.000 by 8: |
| Division | denominator: N / D | 5 / 8 = 0.625 |
| (Universal) | (Add decimal & zeros) | Works for ANY fraction! |
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Method 1: The Power of Ten Scaling Technique
If the denominator of a fraction is a factor of 10, 100, or 1,000 (such as 2, 4, 5, 8, 20, 25, or 50), you can convert it mentally without long division: For 7/25: multiply top and bottom by 4 to get 28/100 = 0.28. For 3/5: multiply top and bottom by 2 to get 6/10 = 0.6. For 7/8: multiply top and bottom by 125 to get 875/1,000 = 0.875.
Method 2: The Universal Long Division Technique
When the denominator cannot easily scale to a power of ten (such as 3, 7, 11, or 12), use long division. Write the numerator inside the division bracket, place a decimal point after it, and append several zeros. Bring the decimal point straight up into the quotient and divide using standard long division.
Convert 3/8 to a decimal using Long Division:
0 . 3 7 5
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8 | 3 . 0 0 0
- 2 . 4 (8 * 3 = 24)
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6 0
- 5 6 (8 * 7 = 56)
-----
4 0
- 4 0 (8 * 5 = 40)
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0 (Remainder 0 -> Terminating decimal!)
Benchmark Fraction-Decimal Equivalencies
Memorizing core benchmark equivalencies builds computational fluency.
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| ESSENTIAL BENCHMARK EQUIVALENCIES |
+-----------------+-------------------+-----------------+-----------------+
| Fraction | Decimal | Fraction | Decimal |
+-----------------+-------------------+-----------------+-----------------+
| 1/2 | 0.5 | 1/8 | 0.125 |
| 1/4 | 0.25 | 3/8 | 0.375 |
| 3/4 | 0.75 | 5/8 | 0.625 |
| 1/5 | 0.2 | 7/8 | 0.875 |
| 2/5 | 0.4 | 1/3 | 0.333... |
| 3/5 | 0.6 | 2/3 | 0.666... |
| 4/5 | 0.8 | 1/10 | 0.1 |
+-----------------+-------------------+-----------------+-----------------+
Exercises
Exercise 1. Convert each decimal into a fraction or mixed number in simplest form: (a) 0.64; (b) 0.075; (c) 5.4.
Exercise 2. Convert each fraction into an equivalent decimal using the scaling method: (a) 9/20; (b) 13/25; (c) 7/50.
Exercise 3. Use long division to convert 5/16 into a decimal. Show all division steps.
Solutions
Solution to Exercise 1. For (a) 0.64: two decimal places means 64/100. Divide top and bottom by 4 to get 16/25. For (b) 0.075: three decimal places means 75/1,000. Divide top and bottom by 25: 75 / 25 = 3, and 1,000 / 25 = 40; the fraction is 3/40. For (c) 5.4: whole number is 5; fraction is 4/10, which reduces by dividing by 2 to 2/5; the mixed number is 5 and 2/5.
Solution to Exercise 2. For (a) 9/20: multiply numerator and denominator by 5: (9 * 5) / (20 * 5) = 45/100 = 0.45. For (b) 13/25: multiply numerator and denominator by 4: (13 * 4) / (25 * 4) = 52/100 = 0.52. For (c) 7/50: multiply numerator and denominator by 2: (7 * 2) / (50 * 2) = 14/100 = 0.14.
Solution to Exercise 3. Set up long division for 5 divided by 16: write 5.0000. 16 into 50 goes 3 times (3 * 16 = 48; remainder 2). Bring down 0 to make 20. 16 into 20 goes 1 time (1 * 16 = 16; remainder 4). Bring down 0 to make 40. 16 into 40 goes 2 times (2 * 16 = 32; remainder 8). Bring down 0 to make 80. 16 into 80 goes exactly 5 times (5 * 16 = 80; remainder 0). The division terminates, yielding the exact decimal 0.3125.