Ordering Fractions, Decimals and Percentages Together

Here is a list to put in order: 0.7, three fifths, 65 percent and 0.62. You cannot rank them as they stand, because they are written in three different forms. Is three fifths bigger than 65 percent? Nothing about how they look tells you. It is like being asked to sort distances given in metres, miles and feet — you have to convert to one unit first. The fix is to turn everything into a single form, and decimals are the best choice. Every fraction and every percentage converts to a decimal easily, and decimals line up in place-value columns so comparing them is quick. So 0.7 stays as it is, three fifths becomes 3 divided by 5, which is 0.6, 65 percent becomes 0.65, and 0.62 stays as it is.

Once everything is a decimal you still have to compare them properly, and that means lining the digits up in their place-value columns. Take 0.7, 0.62 and 0.125. Pad them all to three decimal places, so 0.7 becomes 0.700 and 0.62 becomes 0.620. Adding those zeros changes nothing at all. Now compare the tenths column first: 7 beats 6 beats 1, so 0.7 is the largest and 0.125 is the smallest. You only move on to the hundredths column if the tenths tie. This kills the number one error in the topic, which is thinking that more digits means a bigger number. With whole numbers that is true, but after the decimal point length means nothing — 0.125 is the longest and the smallest.

Here is the full routine on the original list. Step one, convert every item to a decimal: 0.7 stays, three fifths becomes 0.6, 65 percent becomes 0.65, and 0.62 stays. Step two, pad them to the same length, giving 0.70, 0.60, 0.65 and 0.62. Step three, sort those decimals: 0.60, 0.62, 0.65, 0.70. Step four, and this is the one people forget, swap back to the forms the question actually used. So the answer is three fifths, 0.62, 65 percent, 0.7. A good habit is to pencil the decimal above each item rather than replacing it, so the original is still visible when you write your answer. And always check the direction — ascending means smallest first, descending means reverse the whole list.

A number line makes the whole thing visual. Take one quarter, 0.4, 55 percent and three quarters. Convert each to a decimal to know where it goes: 0.25, 0.40, 0.55 and 0.75. The line runs from 0 to 1 with tick marks every tenth, so one quarter sits halfway between 0.2 and 0.3, 0.4 lands right on a tick, 55 percent sits halfway between 0.5 and 0.6, and three quarters sits halfway between 0.7 and 0.8. Now the ordering is simply reading left to right, which gives one quarter, 0.4, 55 percent, three quarters. Even when you do not need to draw one, a quick mental number line is the best way to check an ordering you have already written — a value out of sequence jumps out at you.

Look at some lists where the answer is genuinely hard to see. One eighth, 0.15 and 12 percent become 0.125, 0.15 and 0.12, so smallest first gives 12 percent, one eighth, 0.15 — notice that one eighth lands in the middle, which you could never have guessed from how it looks. Two fifths, 0.35 and 38 percent become 0.4, 0.35 and 0.38, so the order is 0.35, 38 percent, two fifths. And 0.5, 45 percent and two fifths become 0.5, 0.45 and 0.4, giving two fifths, 45 percent, 0.5. The last one is different: 0.8, three quarters and 78 percent, largest first. The decimals are 0.8, 0.75 and 0.78, so the answer is 0.8, 78 percent, three quarters. Same working, reversed answer — so underline the direction word before you start.