Fraction Decimal and Percentage Conversions
This topic covers the essential skill of converting between fractions, decimals, and percentages, including both terminating and repeating decimals. Fluency in these conversions is critical for comparing values and simplifying problems efficiently without a calculator.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- To compare numbers in different formats, convert them all to a single common format, such as decimals or fractions with a common denominator.
- A fraction will result in a terminating decimal if, in its simplest form, the prime factors of its denominator are only 2s and/or 5s. All other simplified fractions produce recurring decimals.
- The standard method for converting a recurring decimal to a fraction is algebraic: set the decimal equal to x, multiply by powers of 10 to align the repeating part, and subtract the equations to eliminate the infinite tail.
- Percentages are simply fractions with a denominator of 100. To convert a decimal to a percentage, multiply by 100; to convert a percentage to a decimal, divide by 100.
Formulae
(10^k)x - (10^m)x = integer To convert a recurring decimal to a fraction. Let the decimal be 'x'. Choose powers of 10 (k and m) to create two new numbers where the digits after the decimal point are identical, allowing the recurring part to be cancelled out by subtraction.
Definitions
- Terminating Decimal
- A decimal number that has a finite number of digits after the decimal point, such as 0.625.
- Recurring Decimal
- A decimal number with a digit or sequence of digits that repeats infinitely. This is shown with dot notation, e.g., 0.333⋯ is 0.3̇ and 0.142142⋯ is 0.1̇42̇.
Worked example
Arrange the following numbers in ascending order: 4/9, 0.4, 45%, 5/11.
- 1
The most direct way to compare these is to convert all values to decimals.
- 2
Convert the fractions:
4/9 is a known recurring decimal, 0.444⋯
or 0.4̇.
For 5/11, perform division:
5 divided by 11 gives 0.4545⋯
or 0.4̇5̇.
- 3
Convert the percentage:
45% is 45/100, which is 0.45.
- 4
The value 0.4 is already in decimal form.
- 5
Now, list the decimal values:
0.4̇ (0.444⋯), 0.4 (0.400), 0.45 (0.450), and 0.4̇5̇ (0.4545⋯).
- 6
Comparing the decimals:
0.400 < 0.444⋯
< 0.450 < 0.4545⋯
- 7
Therefore, the ascending order using the original numbers is:
0.4, 4/9, 45%, 5/11.
Answer: 0.4, 4/9, 45%, 5/11
Common mistakes
- ×When converting a mixed recurring decimal (e.g., 0.12̇) to a fraction, using an incorrect subtraction. For x=0.12̇, you must calculate 100x - 10x, not 100x - x, to correctly eliminate the recurring part.
- ×Forgetting to simplify a fraction to its lowest terms after converting from a decimal or percentage. For instance, leaving 0.68 as 68/100 instead of simplifying it to 17/25.
- ×Misidentifying the repeating block in a long division, leading to an incorrect recurring decimal representation.
- ×Confusing 0.3̇6̇ (0.3636⋯) with 0.36̇ (0.3666⋯). The placement of the dots is critical.
No-calculator tips
- ✓Memorise common conversions: 1/3=0.3̇, 1/4=0.25, 1/5=0.2, 1/8=0.125, 1/9=0.1̇. These save significant time.
- ✓To change a fraction to a decimal, first check if you can make the denominator a power of 10 (e.g., for 13/20, multiply top and bottom by 5 to get 65/100 = 0.65). This is faster than division.
- ✓A shortcut for converting simple recurring decimals: a number like 0.ababab⋯ is always the fraction ab/99, and 0.abcabc⋯ is abc/999. For example, 0.2̇7̇ = 27/99 = 3/11.