Connecting Fractions, Decimals, and Percents: Fifth Grade Mathematics
Fractions, decimals, and percents are three interchangeable languages that represent the exact same mathematical value.
The Equivalence Triangle
Every rational value can be translated among three forms: Fraction (parts of a whole), Decimal (base-ten place value), and Percent (parts per hundred).
+--------------------------------------------------------------------------+
| THE FRACTION - DECIMAL - PERCENT TRIANGLE |
| |
| [ FRACTION ] |
| a/b |
| / \ |
| Divide numerator by denom/ \ Multiply or divide |
| / \ by powers of 10 |
| v v |
| [ DECIMAL ] <=====> [ PERCENT ] |
| 0.xx Shift xx% |
| 2 places |
+--------------------------------------------------------------------------+
Conversion Reference Table
Memorizing common benchmark equivalencies builds mathematical fluency:
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| COMMON BENCHMARK EQUIVALENCIES |
+---------------------+---------------------+------------------------------+
| Fraction | Decimal | Percent |
+---------------------+---------------------+------------------------------+
| 1/1 (One whole) | 1.0 | 100% |
| 1/2 | 0.50 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.20 | 20% |
| 2/5 | 0.40 | 40% |
| 1/10 | 0.10 | 10% |
| 1/8 | 0.125 | 12.5% |
+---------------------+---------------------+------------------------------+
Conversion Rules
- Decimal to Percent: Multiply by 100 (shift decimal point 2 places RIGHT) and add %: 0.35 -> 35%.
- Percent to Decimal: Divide by 100 (shift decimal point 2 places LEFT) and remove %: 42% -> 0.42.
- Fraction to Percent: Convert to a fraction over 100, or convert to a decimal and shift 2 places right: 3/20 = 15/100 = 15%.
Exercises and Solutions
Exercise 1: Three-Way Conversions. Fill in the missing representations for each row: Row A: Given fraction 3/5; find decimal and percent. Row B: Given decimal 0.08; find fraction in lowest terms and percent. Row C: Given percent 45%; find decimal and fraction in lowest terms.
Solution 1: Row A: 3/5 = (3 * 20)/(5 * 20) = 60/100 = 0.60 decimal = 60%. Row B: 0.08 = 8 hundredths = 8/100. Simplify 8/100 by dividing by 4 to get 2/25 fraction. Percent: shift decimal 2 places right = 8%. Row C: 45% = 0.45 decimal. As a fraction: 45/100. Simplify 45/100 by dividing by 5 to get 9/20 fraction.
Exercise 2: Comparing Mixed Formats. Arrange the following three values in order from least to greatest: 0.65, 3/4, and 60%.
Solution 2: Convert all three numbers to the same format, such as decimals or percents: 0.65 = 65%; 3/4 = 75%; 60% = 60%. Compare the percentage values: 60% < 65% < 75%. Therefore, from least to greatest, the original numbers are: 60%, 0.65, and 3/4.