40 Fun Facts About Fractions

Which patty size did A&W pitch against McDonald's Quarter Pounder, only for focus groups to judge it smaller?

1/3 lb: the Third Pounder won taste tests, but A&W's owner later wrote that focus groups saw the 3 in 1/3, took it for less meat than the 4 in 1/4, and thought they were being overcharged.

What share of an 18-karat gold ring is actually gold, on a scale where 24 karat means pure?

18-karat gold is 75% gold, because karats count parts per 24 and 18/24 reduces to three quarters. The rest is copper, silver or other metals that make the ring harder.

In which fraction of a dollar did the NYSE quote prices until 1997, a habit from Spanish coins chopped into 'bits'?

Eighths: Spanish dollars were chopped into eight 'bits' to make change, which is why a quarter is still 'two bits', and Wall Street kept its 1/8 tick until sixteenths arrived in 1997 and cents in 2001.

What does a '2×4' board actually measure once it has been dried and planed?

1 1/2 × 3 1/2 in: the name once described the rough-sawn board, before planing and drying trimmed it, and today's mills cut it under the nominal size from the start.

Which Roman punishment killed every tenth man of a unit that had shown cowardice?

Decimation: in Polybius's account the victims were picked by lot and beaten to death by their own comrades, while the survivors were put on barley rations and made to camp outside the defences.

Which church levy took one part in ten of a farmer's produce?

Tithe: it is simply the Old English word for 'tenth', which the church kept using after 'tenth' replaced it in everyday counting, and the dime takes its name from the same idea by way of French.

Which graphic, first drawn by William Playfair in 1801, shows fractions of a whole?

Pie chart: Florence Nightingale later reshaped it into her 'rose' diagram to show how poor sanitation was killing soldiers in the Crimean War, and the French know it as a camembert.

What do British musicians call the note Americans know as a quarter note, worth a quarter of a semibreve?

Crotchet: the name is Old French for 'little hook', from the hooked shape the note had in medieval notation, though today's crotchet is a plain stem and the hooks belong to quavers.

What fraction of an hour is a 15-minute coffee break plus a 10-minute one, the sum 1/4 + 1/6?

5/12: quarters and sixths first line up as twelfths, 3/12 + 2/12, and the clock agrees, since 15 + 10 minutes is 25 of the 60.

Which name does 2 1/2 go by, the form tape measures and recipes prefer to the top-heavy 5/2?

Mixed number: it hides a plus sign, since 2 1/2 means 2 + 1/2, the opposite of algebra, where writing two things side by side means multiplying them.

What is 5/3 called in relation to 3/5, since the two multiply to exactly 1?

The reciprocal flips a fraction over, and dividing by a fraction means you multiply by it instead: 5/3 × 3/5 = 1, so dividing by 3/5 is the same as scaling up by 5/3.

What is 5/6 − 1/4, the difference found by rewriting both fractions over the common denominator 12?

5/6 − 1/4 = 7/12, since 5/6 is 10/12 and 1/4 is 3/12, and 10 − 3 = 7. Subtracting tops and bottoms separately would give the nonsense answer 4/2.

What is 4/5 × 5/6, the product that simplifies the moment the two fives cancel each other?

2/3: going straight across gives 20/30, but cancelling the matching fives first leaves 4/6, and halving both parts finishes the job.

Which unit, written like a % sign with an extra zero on the bottom, often expresses seawater salinity?

Per mille: it counts parts in a thousand, yet oceanographers' practical salinity scale of 1978 works from how well seawater conducts electricity and gives a plain figure with no unit at all.

Which mathematician's 1585 booklet De Thiende told surveyors and tradesmen to calculate in decimal fractions?

Simon Stevin's De Thiende ('The Tenth') won Europe over to decimal fractions, though he did not invent them: al-Uqlidisi had used them in a book written in Damascus in 952.

Which Italian mathematician, taught arithmetic in North Africa as a boy, wrote fractions with the bar a Moroccan scholar had introduced?

Fibonacci: his father acted for Pisa's merchants at the customs house in Bugia, in today's Algeria, and had the boy schooled there, and his Liber Abaci used the fraction bar first attested in al-Hassar's work.

What fraction of a Roman pound was the uncia, a unit whose name survives in both the ounce and the inch?

One-twelfth: Romans counted fractions in twelfths, so the uncia was a twelfth of their foot as well as their pound, and the inch still is one, though the everyday ounce is now a sixteenth of a pound.

What is the bar over the repeating digits of a decimal like 0.1666… called, from the Latin for 'bond' or 'tie'?

Vinculum: the same horizontal 'tie' was once drawn over groups of terms to bind them together, a job parentheses mostly do today, and it survives as the roof of the square-root sign.

Which formal name for the slash in 3/4 recalls the mark's old use as the shilling mark in British prices?

Solidus: it was a Roman gold coin whose name became the 's' for shillings, and the stroke in a price like 7/6 is said to be a stretched long s.

What do older textbooks call the ÷ symbol, a word that also names a dagger mark flagging dubious lines in manuscripts?

Obelus: ancient editors set the mark beside lines they thought spurious, and ÷ had stood for subtraction before a Swiss algebra book of 1659 turned it into the division sign.

How many 1/8-cup scoops of flour fill a 5/8-cup measure, a division done by flipping the divisor to multiply instead?

Five scoops, because 5/8 ÷ 1/8 = 5/8 × 8/1 = 40/8 = 5, since both amounts are counted in the same size of slice.

Which of these denominators gives a decimal that stops rather than repeats, having no prime factor other than 2 and 5?

16: a fraction in lowest terms gives a decimal that stops only when its bottom divides a power of ten, and 10 is just 2 × 5, so 1/16 ends at 0.0625 while 1/12 runs on as 0.08333…

Which shortcut tests whether 3/4 and 9/12 are equal by checking that 3 × 12 matches 4 × 9?

Cross-multiplying: it is shorthand for multiplying both fractions by both bottoms, 4 × 12, which leaves 3 × 12 on one side and 9 × 4 on the other, and both come to 36.

What are fractions like 11/32, with a power of two on the bottom, called, the kind an inch ruler marks?

Dyadic: a computer's binary floating-point numbers are all fractions of this kind, which is why it can store 3/8 exactly but only approximate 1/10.

Which camera setting is written as a fraction like f/8, the lens's focal length divided by eight?

Aperture: f/8 means the opening is one-eighth of the focal length across, so a bigger f-number is a smaller hole, and f/16 lets in a quarter of the light of f/8.

Which of these fractions is equal to 3/5, found by scaling top and bottom by the same factor?

9/15 is 3/5 with both parts tripled, and scaling top and bottom alike never changes a fraction's value, which is why 6/15, with only the top doubled, is a different amount.

What is 28/36 in lowest terms, the form reached once top and bottom are both divided by 4?

28/36 reduces to 7/9 because 4 divides both 28 and 36 and no larger factor does, so 7/9 cannot be simplified any further.

What is 7/8 as a decimal, the figure reached by adding three steps of 0.125 to 0.5?

7/8 = 0.875, since 1/8 is 0.125 and seven of them make 0.875, which is exactly what dividing 7 by 8 on a calculator shows.

A recipe's 3/4 cup of butter is cut to half for a small batch; how much butter goes in?

3/8 cup: halving a fraction doubles its bottom, so 3/4 × 1/2 = 3/8, while doubling the top instead would give 6/4, twice the butter.

What percentage is 7 right answers out of 20, the fraction 7/20 scaled up to a denominator of 100?

7 out of 20 is 35%, because scaling top and bottom by 5 gives 35/100, and 'per cent' literally means 'per hundred'.

Which fraction in lowest terms equals 0.4, the share a '40% off' sale knocks from a price?

0.4 is 2/5, because the decimal means 4/10 and both parts divide by 2. The same 2/5 is 40%, so a 40% discount leaves 3/5 of the price to pay.

A 15% tip is what fraction of the restaurant bill, written in lowest terms?

3/20: 15% means 15 per hundred, and dividing 15/100 top and bottom by 5 gives 3/20, which is why a $3 tip on a $20 bill is exactly 15%.

Which of these fractions is the largest, a comparison easiest made by rewriting each over a common denominator of 20?

7/10 is the largest because over a bottom of 20 the four become 12, 14, 11 and 13, and 14/20 tops the list.

What is 2 3/4 as a top-heavy fraction, found by turning the 2 into quarters and adding the 3 on top?

2 3/4 = 11/4, because two wholes are 8 quarters and three more make 11, which is why a recipe calling for 11/4 cups is really asking for 2 3/4.

What is 25/8 as a whole part plus a fraction, found by dividing 25 by 8 and keeping the remainder on top?

25/8 = 3 1/8, since 8 goes into 25 three times with 1 left over, and that leftover 1 stays as 1/8. Writing it as 2 7/8 would be 23/8 instead.

A runner logs 1 3/4 miles in the morning and 2 1/2 in the evening; what is the day's total?

1 3/4 + 2 1/2 = 4 1/4 miles, because the wholes make 3 and the quarters 3/4 + 2/4 make 5/4, which carries one more whole mile into the total.

A carpenter cuts a 1 5/6-foot piece from a 4-foot board; how much board is left?

2 1/6 feet: think of 4 as 3 6/6, and taking away 1 5/6 leaves 2 1/6; as a check, 2 1/6 + 1 5/6 = 3 6/6, which is the full 4 feet.

How many 3/4-litre bottles can be filled from a 4 1/2-litre cask, a division that flips 3/4 into 4/3?

Six bottles, because 4 1/2 ÷ 3/4 = 9/2 × 4/3 = 36/6 = 6, and a quick check is that four bottles hold 3 litres, so six hold 4 1/2.

A recipe's 3/4 cup of sugar is scaled up 2 1/2 times for a bake sale; how much sugar is needed?

3/4 × 2 1/2 = 1 7/8 cups, since 2 1/2 is 5/2 and 3/4 × 5/2 = 15/8, which is one whole cup and 7/8 more. Adding the two amounts instead would give 3 1/4 cups.

What is the smallest denominator that both 1/10 and 1/15 can be rewritten over before the two are added?

30 is the lowest common denominator of tenths and fifteenths, since 30 is the smallest multiple of both 10 and 15, so 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6.