Comparing fractions confidently
When denominators differ, rewrite the fractions so they share a denominator, or compare each one to a landmark such as one half.
Same denominator, easy answer
When two fractions already share a denominator, the one with the bigger top number is bigger.
That is only true because the pieces are the same size in both. Five eighths beats three eighths because an eighth is an eighth either way.
Different denominators: make them match
Two thirds and three quarters cannot be compared by looking at the numbers, because thirds and quarters are different sizes.
So cut both into the same size of piece. Twelfths work for both: two thirds becomes eight twelfths, three quarters becomes nine twelfths. Now the tops can be compared, because the pieces finally match.
The one half shortcut
Often you do not need any of that. Compare each fraction to one half first.
A fraction is more than half when the top is more than half of the bottom. Four sevenths is over the line because 4 beats half of 7. Three eighths is under it because 3 is less than half of 8.
One over and one under settles the question in a couple of seconds.
When the shortcut does not settle it
The half test only helps when the two fractions land on opposite sides of the line. If both are over half, or both under, you have learnt something but not enough.
That is when you fall back on a common denominator. The shortcut is a first look, not a replacement.
Same top number, fewer pieces win
There is a second shortcut, and it is even faster than the half test.
When two fractions have the same top number, the one with the smaller bottom number is bigger. Three quarters beats three eighths because you have three pieces either way, and quarters are the larger piece.
It feels backwards the first time. Bigger bottom number, smaller fraction — because the more pieces you cut something into, the less each piece holds.
What this looks like in a test
Selective and OC papers rarely ask you to compare two fractions on their own. They hide it inside something else: which container is fuller, who ran the greater share of a race, which recipe uses more flour.
The work is the same. Name the two fractions, try the half test, try the same-top shortcut, and only reach for a common denominator if neither settles it.
Ordering three or more fractions is the same job done twice. Sort them into over-half and under-half first, and you have usually done most of the work.
Worked examples
Which is larger, five ninths or four sevenths?
Start with the half test. Half of 9 is 4.5 and 5 is above it, so five ninths is just over a half. Half of 7 is 3.5 and 4 is above it, so four sevenths is over a half too. Both over, so the shortcut has not settled it.
Use a common denominator of 63: five ninths is 35 sixty-thirds, four sevenths is 36 sixty-thirds. Four sevenths is larger, but only by one sixty-third.
Answer: Four sevenths is larger
Which is larger, two fifths or five eighths?
The half test does settle this one. Half of 5 is 2.5 and 2 is below it, so two fifths is under a half. Half of 8 is 4 and 5 is above it, so five eighths is over a half. No further working needed.
Answer: Five eighths
Put three quarters, two thirds and five sixths in order, smallest first.
All three are over a half, so go straight to a common denominator. Twelfths suit all of them: three quarters is nine twelfths, two thirds is eight twelfths, five sixths is ten twelfths. In order that is 8, 9, 10 twelfths.
Answer: Two thirds, three quarters, five sixths
Which is larger, three fifths or three eighths?
The tops match, so compare the bottoms. Fifths are bigger pieces than eighths, and there are three of each.
Answer: Three fifths
A jug is two thirds full. A second jug of the same size is five eighths full. Which holds more?
Both are over half, so the half test does not settle it, and the tops do not match either. Use twenty-fourths: two thirds is sixteen twenty-fourths, five eighths is fifteen twenty-fourths.
Answer: The first jug, at two thirds
Order these from smallest to largest: five ninths, three sevenths, seven eighths.
Sort by the half test first. Three sevenths is under half. Five ninths and seven eighths are both over. Then compare the two that are over: seven eighths is nearly a whole, five ninths is barely past half.
Answer: Three sevenths, five ninths, seven eighths
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