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This page sets out the method for fractions, decimals & percentages (FDP) in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Fractions, decimals & percentages (FDP) topic page.
When you see a fraction whose denominator is not 10, 100 or 1000, divide the numerator by the denominator.
When you see fractions, decimals and percentages in the same list, turn everything into decimals.
Percent means per hundred.
1/4 of 100 is 25, so 1/4 = 25%.
25% is 25 hundredths, which is 0.25.
The first decimal place is tenths. The second is hundredths.
1/2 = 0.5 = 50%.
1/5 = 0.2 = 20%, and 1/4 = 0.25 = 25%.
Divide the numerator by the denominator to get the decimal, or make the denominator 100.
3/4 = 75/100 = 0.75.
20% as a fraction is 20/100, which simplifies to 1/5.
Divide the top and bottom by 20.
0.4, 2/5 and 40% are the same size.
Convert them to the same form before you compare.
Read a decimal by its last place. 0.45 ends in the hundredths column, so it is 45 hundredths: 45/100. One decimal place is tenths, so 0.7 = 7/10. Three decimal places are thousandths, so 0.125 = 125/1000. The denominator is 10, 100 or 1000, matching the number of decimal places.
Simplify by dividing the top and the bottom by the same number. 45/100 divides by 5 to give 9/20. Nine and 20 share no common factor greater than 1, so 9/20 is simplest. Dividing only the top, and leaving 9/100, changes the value. Both numbers have to shrink together.
A percentage is already out of 100. 15% = 15/100. Divide by 5 to get 3/20. 8% is 8/100, which divides by 4 to give 2/25. Check a simplification by multiplying back in your head: 3 × 5 = 15 and 20 × 5 = 100, so 3/20 really is 15 hundredths.
Method: 1. Count the decimal places to choose the denominator: 10, 100 or 1000. 2. Put the digits after the point in the numerator. 3. Cancel a common factor if the question asks for the simplest form.
If the denominator is 10, 100 or 1000, write the numerator after the decimal point with the right number of places. 7/10 = 0.7. 75/100 = 0.75. 125/1000 = 0.125. Put a zero in an empty place so the last digit sits in the correct column: 7/100 = 0.07, not 0.7.
If the denominator is not 10, 100 or 1000, make it one of those when you can. 3/4: multiply top and bottom by 25 to get 75/100 = 0.75. 9/20: multiply by 5 to get 45/100 = 0.45. 1/8: the denominator does not jump to 10 or 100 in one easy step, but 1000 ÷ 8 = 125, so 1/8 = 125/1000 = 0.125.
You can also divide the numerator by the denominator. For 3/8, divide 3.000 by 8. Eight into 30 goes 3 times with remainder 6, eight into 60 goes 7 times with remainder 4, and eight into 40 goes 5 exactly. The decimal is 0.375. It terminates, so you stop. 8 × 0.375 = 3 is the check.
Method: 1. If the denominator is 10, 100 or 1000, write the numerator after the point. 2. Otherwise divide the numerator by the denominator. 3. Put a zero in any empty decimal place so the last digit sits in the right column.
To change a decimal into a percentage, multiply by 100. 0.45 × 100 = 45, so 0.45 = 45%. 0.08 × 100 = 8, so 0.08 = 8%. The digits move two places. Multiplying by 10 only moves them one place and gives 4.5 from 0.45, which is not the percentage.
To change a percentage into a decimal, divide by 100. 68% = 68 ÷ 100 = 0.68. 8% = 0.08. Dividing by 1000 instead puts in an extra zero and turns 68% into 0.068, which is 6.8%. The two zeros in 100 match the two decimal places you move.
A fraction, a decimal and a percentage line up once they are all hundredths. 7/20 = 35/100 = 0.35 = 35%. If two of them match and the third does not, the third conversion is the one to redo. Equality is a useful check: 0.35 and 7/20 are the same size, so neither is larger.
Method: 1. Divide a percentage by 100 to get the decimal. 2. Multiply a decimal by 100 to get the percentage. 3. Move the digits two places, and keep a zero if a column would otherwise be empty.
Convert every value to a decimal before you compare. For 7/10, 0.66, 3/4 and 68%: the fraction with denominator 10 is 0.7, the percentage is 0.68, and 3/4 = 0.75. The list in decimals is 0.70, 0.66, 0.75 and 0.68.
Compare hundredths: 66, 68, 70, 75. Smallest to largest is 0.66, 0.68, 0.70, 0.75. Write the originals back in that order: 0.66, 68%, 7/10, 3/4. Do not leave the answer as the decimals unless the question used only decimals.
Padding with a zero can make the columns easier. 0.7 is 0.70. It is larger than 0.68 because 70 hundredths is more than 68 hundredths. It is smaller than 0.75. A percentage such as 68% is not automatically the largest just because 68 looks bigger than 7 or 3. Those 68 hundredths sit between 0.66 and 0.70.
Method: 1. Convert every fraction and percentage into a decimal. 2. Compare those decimals from the left. 3. Write the original values back in the order you found.
Every fraction is a division: the top divided by the bottom. For a friendly denominator you can scale to a power of ten, so 7/20 = 35/100 = 0.35. When that does not work, use short division with extra zeros. For 5/12 divide 5.000 by 12: 50 ÷ 12 = 4 remainder 2, then 20 ÷ 12 = 1 remainder 8, and 80 ÷ 12 = 6 remainder 8.
The remainder 8 has appeared twice, so the sixes repeat: 5/12 = 0.41666ldots, written 0.41dot6. If the question says "correct to 3 decimal places", stop at the fourth digit and round, which gives 0.417. On a calculator paper, type the fraction as a division and copy the display. Always check that the decimal is sensible: 5/12 is just under a half, and 0.417 is just under 0.5.
Method: 1. Try to scale the denominator to 10, 100 or 1000 first. 2. Otherwise divide the top by the bottom with short division, adding zeros after the point. 3. Stop when the remainder is zero or the digits repeat. 4. Round if the question asks for decimal places, and check the size against a half.
Tip: Learn these: 1/8 = 0.125, 3/8 = 0.375, 1/5 = 0.2, 1/3 = 0.333ldots
To write one amount as a percentage of another, make a fraction, then convert. 33 out of 40 is 33/40. Multiply the top and the bottom by 2.5 to get 82.5/100, or divide 33 ÷ 40 = 0.825 and move the decimal two places to get 82.5%. The word "out of" tells you which number is the whole and goes on the bottom.
A fraction can often be changed to a percentage without a calculator by finding an equivalent fraction with denominator 100. For 14/40, simplify to 7/20, then multiply the top and bottom by 5 to get 35/100 = 35%. Questions usually ask for the fraction "in its simplest form" and the percentage separately, so keep both. A percentage can be more than 100% if the first amount is bigger than the second.
Method: 1. Write the part over the whole as a fraction. 2. Convert to a fraction out of 100, or divide to get a decimal. 3. Multiply the decimal by 100 to get the percentage. 4. Check that the percentage is sensible: a little more than half should be a little over 50%.
When a list has fractions, decimals and percentages, choose one form for all of them. Decimals are usually the best because you can line up the digits. Convert 3/8 to 0.375, 36% to 0.36 and 7/20 to 0.35. The decimal 0.4 is already in the right form, so write it as 0.400 to compare.
Now order by comparing the tenths, then the hundredths, then the thousandths. The order is 0.350, 0.360, 0.375, 0.400. Write the final answer using the original numbers from the question, not the decimals: 7/20, 36%, 3/8, 0.4. A table with the original and the decimal side by side is a neat way to show your working and earn the method mark.
Method: 1. Pick one form, usually decimals. 2. Convert every number and write the results next to the originals. 3. Give each decimal the same number of decimal places. 4. Order the decimals and then write the originals in that order.
Tip: If two numbers are equal in decimal form, such as 0.5 and 50%, they are tied. Read the question to see how it wants ties shown.
Test results are often given in different forms: 33/40 in one test and 78% in another. To compare, convert both to a percentage. 33 ÷ 40 × 100 = 82.5%, which is more than 78%. So the first test was better. The question says "give a reason" or "you must show your working", so write both percentages and the comparison in a sentence.
A claim such as "0.4 is bigger than 3/8 because 4 is bigger than 3" is a typical wrong argument. Check it by converting: 3/8 = 0.375 and 0.4 is greater. In this case the claim happens to be right, but the reason is wrong, and the reasoning mark is lost. Always compare the converted values. Percentages above 100% and decimals with a lot of digits are not errors, they just need care.
Method: 1. Convert both amounts to the same form. 2. Compare the converted values. 3. Write a sentence that names the larger amount and quotes both values. 4. Check that the conclusion answers the exact question that was asked.
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