Seahorse Series · Quiz 9 · Comparing and Ordering Fractions

The Guide

Before you begin

This quiz is about telling which of two fractions is larger, and putting several fractions in order. The method that works for whole numbers (count the digits, then compare from the left, as in Quiz 2) is useless here. Fractions carry two numbers doing two different jobs, and the numbers can pull in opposite directions, so a fraction built from bigger numbers is often the smaller amount. This quiz hands you three tools that settle any comparison. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.

Bigger bottom, smaller piece

This is the whole difficulty in one line, and it comes straight from Quiz 8: the denominator names the size of the piece, and the more pieces you cut a whole into, the smaller each one has to be. Share a pizza among 3 people and you eat well; share it among 8 and you don't. So 13 is larger than 18, even though 8 is larger than 3. The bottom number runs backward.

Drawn to scale. This is called a fraction wall, and it is worth more than any rule on this page.

The two signs, before we use them

Comparisons get written with two symbols, and both are wedges. The wide open side always faces the larger amount, and the narrow point pokes the smaller one. So 3 < 8 and 8 > 3 say one fact from its two sides. Reading them aloud helps: put the left-hand number first and say is less than or is greater than.

Two wedges, one rule. The point marks the smaller side every time.

One warning particular to fractions: the sign points at the smaller amount, never at the smaller-looking numbers. In 13 > 18 the wedge opens toward the side carrying the smaller digit, and it is still correct, because thirds are the larger pieces. Decide which amount is bigger first; only then write the sign.

Tool one: same bottom, count the tops

When the pieces are the same size, comparing is as easy as whole numbers: 37 and 57 are both sevenths, so five of them beats three. Nothing subtle here at all.

Tool two: same top, read the bottoms backward

When you hold the same number of pieces, the bigger fraction is the one with the smaller denominator, because its pieces are bigger: 23 beats 29. Two large slices beat two small ones. Read the wall again if this feels backward, because it is exactly backward, and on purpose.

Tool three: make the bottoms match

When neither number matches, give both fractions the same denominator and fall back to tool one. Any common multiple (a number that both bottoms divide into evenly) works; the smallest one keeps the numbers friendly, and Quiz 7 shows how to find it. To compare 38 and 25, both convert to fortieths: 1540 and 1640. So 25 wins, by one fortieth.

Drawn to scale. The eye can barely call this one, which is exactly why the tools exist.

The shortcut, and what it really is

Cross-multiplying is tool three with the writing left out. For 38 against 25: multiply up the diagonals, 3 × 5 = 15 and 2 × 8 = 16, and the larger answer sits above the larger fraction. Those two answers are the numerators you would have written over 40 anyway. Use it once you can say what it means; a shortcut you can't explain will eventually take you somewhere you didn't intend to go.

Benchmarks: half, and one

Often you don't need an exact comparison, only a verdict, and two landmarks do most of the work. Is it more or less than a half? Double the top: if it passes the bottom, the fraction beats a half. 715 gives 14, short of 15, so it is just under a half. Is it more or less than one? If the top is bigger, the fraction is bigger than a whole. Sorting a list into "under a half" and "over a half" before doing any arithmetic is the fastest honest work in fractions.

Why the old method breaks: in Quiz 2 every digit meant more, because place value only ever climbs. A fraction is a division waiting to happen, so its two numbers push in opposite directions: the top pushes up, the bottom pushes down. Comparing them by size alone is like judging a paycheck by the number of hours worked, without asking the rate.

Worked Examples

Three to study before you start

Example 1 · Same top, backward bottoms

Which is larger, 34 or 38? Three pieces either way, so the size of the piece decides: fourths are bigger than eighths, so 34 wins. Find both on the wall and the answer is visible without a word of arithmetic.

Example 2 · Matching the bottoms

Which is larger, 56 or 79? Neither number matches, so find a common multiple of 6 and 9: eighteen. Then 56 = 1518 and 79 = 1418, so 56 is larger. Cross-multiplying says the same thing faster: 5 × 9 = 45 beats 7 × 6 = 42.

Example 3 · Ordering a whole list

Put 23, 512, 34, 12 in order, least to greatest. Benchmark first: 5/12 is under a half, and the other three are a half or more, so 5/12 goes first in the order. Now give the rest a common denominator of twelve: 8/12, 9/12, 6/12. The order is 5/12, 1/2, 2/3, 3/4. One denominator serves the whole list; there is no need to compare them in pairs.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Write your answers on the lines. Questions 3 and 7 are multiple choice: choose the one best answer.

Write < or > between them: 37 ___ 57
Write < or > between them: 23 ___ 29
Which of these is the largest?
- A) 1/8
- B) 1/3
- C) 1/5
- D) 1/12
Which is larger, 38 or 25? Show the common denominator you used.
Is 715 more or less than a half? Give your reason in a few words.
Put these in order from least to greatest: 23 · 512 · 34 · 12

▶ VideoA video rest stop, for anyone who has kept going this far.

Take two minutes to watch a seahorse father give birth, to some two thousand at once.

Andre says 59 is greater than 56, because 9 is greater than 6. What went wrong?
- A) Nothing; he is correct.
- B) He compared the bottoms as if they were counts, but a bigger bottom means smaller pieces, so 5/6 is the larger.
- C) He should have compared the tops, which are equal, so the fractions are equal.
- D) He needed to add the top and bottom of each first.
On the line below, which letter marks 23?
Two crews are painting identical halls. One crew has finished 58 of its hall, the other 35 of its own. Which crew has done more?
Brian needs the larger of two wrenches: 3/8 inch or 7/16 inch. Which should he reach for?

Answer Key

Check your work

Open the key: after you've finished all ten

1 3/7 < 5/7

Same size pieces, so just count them: five sevenths beats three. This is the one case where whole-number instinct works perfectly, which is why it comes first.

2 2/3 > 2/9

Same count, so the size of the piece decides: thirds are far bigger than ninths. Two large slices beat two small ones. If you wrote the other sign, the bottoms were read as counts rather than as sizes.

3 B: 1/3.

All four hold one piece, so the smallest denominator gives the biggest piece. Read the wall from top to bottom: as the bottom number climbs, the piece shrinks. D) 1/12 is the smallest of the four, and it looks the biggest only if the fraction is misread as a whole number.

4 2/5 is larger; in fortieths, 15/40 against 16/40.

This is the quiz's centerpiece, because every instinct says otherwise: 3/8 shows the bigger numbers and is the smaller amount. Cross-multiplying agrees, 3 × 5 = 15 against 2 × 8 = 16. Any common denominator earns the credit; 40 simply keeps it tidy.

5 Less than a half: doubling 7 gives 14, which falls short of 15.

Half of 15 would be 7½, and 7 doesn't reach it. Any clear version of the reason counts, including "7 is less than half of 15." Benchmarking answers in seconds what a common denominator would take a minute to settle.

6 5/12, 1/2, 2/3, 3/4

In twelfths: 5/12, 6/12, 8/12, 9/12. Give the whole list one denominator rather than comparing in pairs, and the order simply appears. The benchmark shortcut sets 5/12 aside immediately as the only one under a half.

7 B.

5/6 is the larger: in eighteenths, 15/18 against 10/18. Andre's mistake is the exact error this quiz exists to catch, and it is so natural that almost everyone makes it once. C) mistakes equal tops for equal fractions, though the tops only tell how many pieces, never how big. D) adds two numbers that measure different things, which is like adding a price to a quantity.

8 R

The line is cut into six equal steps, so each is a sixth, and 2/3 is another name for 4/6: the fourth step. R stands there. Q is 2/6, which is 1/3, and it is the trap for anyone who reads "2/3" as "the second mark." Count the spaces, not the marks, and convert to the line's own denominator first.

9 The first crew, at 5/8.

In fortieths again: 25/40 against 24/40. A single fortieth of a hall separates them. The useful lesson: because the halls are identical, the two fractions can be compared, and a difference this narrow is invisible to the eye but can be settled on paper. Had the halls been different sizes, the fractions could not be compared at all: same whole, or no contest.

10 The 7/16 inch wrench.

Sixteenths and eighths share a denominator easily, since 3/8 is 6/16, and 7/16 is one sixteenth larger. This is the most practical fraction skill on the whole page: every wrench, drill bit, and socket in a toolbox is labeled in halves, quarters, eighths, sixteenths, and reaching for the wrong one costs a trip back to the truck. Tradespeople convert everything to sixteenths and compare the tops.

Reading your results

Questions	The skill they test	If they gave trouble
1	Same denominator	Same pieces, so count them; whole-number instinct is safe here.
2, 3, 7	The backward bottoms	Study the fraction wall until the shrinking is obvious to the eye.
4, 9, 10	Common denominators	Quiz 7's multiples; convert both, then compare tops.
5, 6	Benchmarks and ordering	Double the top and test it against the bottom; give a whole list one denominator.
8	Fractions on the line	Count spaces, and rename the fraction in the line's own denominator first.

Eight or more right: the tools are yours — on to Quiz 10. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. Draw the fraction wall by hand with your teacher, one bar at a time, and let your eye learn what the rules are only describing. Then come back to these same ten.

The Company · An Interlude

Fellow travelers

Today the company is made of stone, tile, and bronze. Seahorses have been carried into human ornament for as long as we have decorated anything, and one of them makes the argument of this quiz better than the figures do.

A square mosaic panel: four winged seahorses arranged symmetrically around a turtle at the center, with lamps and shell motifs, in coral red, gold, green, and cream tiles — The masthead field, seen whole: a zodiac panel from the Surrogate's Courthouse in Manhattan, designed by William de Leftwich Dodge around 1905 and now in the public domain; photographed by Rhododendrites and shared on Wikimedia Commons under CC BY-SA 4.0. Four seahorses hold the corners, a turtle holds the center, and the whole thing is built from thousands of small tiles: no single one of them is the picture. That is a fraction, tiled — and the symmetry is worth a minute of anyone's time.

A bronze seahorse sculpture on a small stand against a travertine wall, with water running down beside it — A bronze one, standing where the water falls. Armor plates rendered as overlapping tiles, which is roughly what the living animal is doing too.

A stone fountain with mermaid figures above and seahorses supporting the basin below — And a fountain where the seahorses do structural work, holding up the basin. Supporters again, in the heraldic sense, and this time actually holding something up.

Prints only this gallery, full size. The main print button stays photograph-free to spare your ink.