I forgot how to convert a decimal into a fraction, fraction into decimal, fraction into percent, and percent to fraction.

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A mixed number is a combination of a whole number and a proper fraction, and a proper fraction is one where the numerator is less than the denominator. A mixed number can be expressed as an improper fraction, where the numerator is bigger than the denominator. So when I refer to fractions below, I’m including proper and improper fractions.

The conversion of fractions into decimals or percentages can’t always be exact, and it’s possible that decimals can be recurring (non-terminating, but repeating a group of digits in the same order). So, when converting fractions into decimals or percentages the accuracy usually needs to be stated or specified. The only fractions that can be converted into exact decimals are those where the denominator can be factorised entirely into powers of 2 or 5, or both. All other fractions produce recurring decimals and percentages.

Recurring decimals or percentages can always be converted exactly into fractions, but it’s unusual to see percentages expressed as recurring decimals. A percentage can be written as a fraction but is usually written as a decimal, such as 2.5% which is the same as 2½%.

Enough explanation. Now some examples...

Fractions to decimals: ⅖ as a decimal. The denominator is 5 so it can be converted exactly. We simply divide 5 into 2, but we express 2 as a decimal: 2.000... as many zeroes as you like. When we divide by 5, we actually divide 2.0 by 5 to get 0.4. This is the same as dividing 20 by 5 and inserting a decimal point before the 4. The zero in front of the decimal point protects it from being lost and also shows that 2 divided by 5 is zero (because 5 “won’t go” into 2).

Another example: ¹³⁄₈ is an improper fraction representing the mixed number 1⅝. This gives us two ways of converting: divide 8 into 13.00... to get 1.625 (we add as many zeroes as we need to complete the calculation). The other way is to write the whole number part first followed by the decimal point 1. and then divide 8 into 5 to get 0.625, added to 1 is 1.625.

¹⁶⁄₂₅ is 25 divided into 16=¹⁶⁄₅ divided by 5 again: that is, 3.2 divided by 5=0.64.

Now for a recurring decimal. ¹⁄₉ is 9 into 1 which is 0.111... forever. The 1 just goes on repeating.

Here’s another: ²⁄₇ is 7 into 2 which is 0.285714285714... forever. The pattern 285714 just goes on repeating.

Rounding: the above fraction would usually be rounded to a certain number of decimal places. Let’s say 2 decimal places, so the fraction would be written as 0.29.

Decimals to percentages is easy because we just move the decimal point 2 places to the right, inserting zeroes if necessary. So ²⁄₇ would be 28.571428571428...% forever, but rounded to 2 decimal places it would be written 28.57%. A simple decimal like 0.5 would be written 50% representing the fraction ½.

Decimal to fraction: we count how many digits there are after the decimal point and then remove it and divide by 1 followed by as many zeroes as there are digits after the decimal point. Example: 0.25 becomes 25/100 which becomes 1/4 or ¼. Another example: 4.174 becomes 4174/1000=2087/500 or ²⁰⁸⁷⁄₅₀₀. This is the same as the mixed number 4⁸⁷⁄₅₀₀.

Percentages to decimals: we move the decimal point two places to the left and remove the percentage symbol. Example: 48% is 0.48 because, although it’s not written the decimal point is invisibly on the right of a whole number percentage. So 48 is actually 48.0. Then we can go on to convert the decimal to a fraction: 0.48=48/100=12/25 or ¹²⁄₂₅.

Recurring decimals to fractions: we count how many digits are recurring and we divide by the number containing the same number of 9s. Example: 0.027027027... We write 027/999=27/999=3/111=1/37 or ⅟₃₇.

What about 4.6333...? Here we have something in front of the recurring part, so we move it so that the decimal point immediately preceded the recurring part: 46.333... We moved the decimal point one place to the right which is the same as multiplying by 10. Remember that. 0.333... is 3/9=⅓. So we have the mixed number 46⅓. We convert this to an improper fraction: (46×3+1)/3=139/3 and we divide by the 10 we multiplied by before: 139/30=4¹⁹⁄₃₀.


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