Converting Between Percents & Decimals - Sixth Grade Mathematics
Converting between percents and decimals involves systematic decimal point relocation based on multiplying or dividing by powers of ten.
The Directional Logic of Decimal Shifts
Because percent means "per 100", transitioning between percents and decimals always involves a shift of exactly two place value positions.
+-------------------------------------------------------------------------+
| PERCENT-DECIMAL SHIFT DIRECTIONAL RULES |
+-------------------+-----------------------+-----------------------------+
| Conversion Task | Mathematical Action | Decimal Point Movement |
+-------------------+-----------------------+-----------------------------+
| Percent to | DIVIDE by 100 | Shift decimal point 2 places|
| Decimal | (Drop the % sign) | to the LEFT (<--) |
+-------------------+-----------------------+-----------------------------+
| Decimal to | MULTIPLY by 100 | Shift decimal point 2 places|
| Percent | (Attach the % sign) | to the RIGHT (-->) |
+-------------------+-----------------------+-----------------------------+
Visualizing the Two-Place Shift:
Percent to Decimal: 4 5 . % --> 0 . 4 5
<-- 2 places left
Decimal to Percent: 0 . 4 5 --> 4 5 . %
2 places right -->
Converting Percents to Decimals
To convert any percent to a decimal: drop the % symbol and move the decimal point two places to the left. If the percent is a whole number, its decimal point is situated at its immediate right.
Examples of Percent to Decimal Conversion:
- Standard Two-Digit: 58% --> 58.0% --> 0.58
- Single-Digit: 6% --> 06.0% --> 0.06 (Prefix leading zero!)
- Decimal Percent: 3.4% --> 03.4% --> 0.034 (Prefix leading zero!)
- Greater than 100%: 245% --> 245.0% --> 2.45
- Microscopic Percent: 0.8% --> 00.8% --> 0.008
The Single-Digit Percent Trap
The most frequent error in percent arithmetic is converting single-digit percents: writing 7% as 0.7 instead of 0.07. Notice that 0.7 represents 7 tenths (70%), not 7 hundredths (7%). Always prefix a leading zero when shifting two places left past a single digit: 7% = 0.07.
Converting Decimals to Percents
To convert any decimal to a percent: move the decimal point two places to the right and attach the % symbol.
Examples of Decimal to Percent Conversion:
- Standard Hundredths: 0.82 --> Shift 2 places right --> 82%
- Tenths: 0.9 --> Shift 2 places right --> 90% (Annex zero!)
- Thousandths: 0.375 --> Shift 2 places right --> 37.5%
- Whole + Decimal: 1.65 --> Shift 2 places right --> 165%
- Pure Integer: 3 --> Shift 2 places right --> 300%
Annexing Zeros When Moving Right
When converting a decimal with only one place value digit (such as 0.4), shifting two places to the right requires annexing a placeholder zero: 0.4 becomes 40%. Converting 2.5 becomes 250%.
Exercises
Exercise 1. Convert each percent to an equivalent decimal: (a) 72%; (b) 4%; (c) 18.5%; (d) 115%; (e) 0.6%.
Exercise 2. Convert each decimal to an equivalent percent: (a) 0.38; (b) 0.05; (c) 0.7; (d) 2.48; (e) 0.0045.
Exercise 3. A store offers a discount described on one sign as "Save 0.15 of the original price" and on another sign as "Save 15%." Are these two promotional discounts mathematically identical? Explain why.
Solutions
Solution to Exercise 1. Drop the % sign and shift the decimal point 2 places to the left: For (a) 72%: 0.72. For (b) 4%: prefix a zero: 0.04. For (c) 18.5%: 0.185. For (d) 115%: 1.15. For (e) 0.6%: prefix zeros: 0.006.
Solution to Exercise 2. Shift the decimal point 2 places to the right and attach the % symbol: For (a) 0.38: 38%. For (b) 0.05: 5%. For (c) 0.7: annex a zero to move 2 places: 70%. For (d) 2.48: 248%. For (e) 0.0045: 0.45%.
Solution to Exercise 3. Yes, the two promotional discounts are mathematically identical. To convert the decimal 0.15 to a percent, multiply by 100 and attach the percent symbol: 0.15 * 100 = 15%. Conversely, converting 15% to a decimal gives 15 / 100 = 0.15. Both statements represent the exact same savings of 15 hundredths (15 cents on every dollar) of the original price.