40 Fun Facts About Fractions

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

A&W's 1/3 lb Third Pounder lost to the Quarter Pounder because diners read 3 as less than 4 and assumed a third was less meat than a quarter. It beat the rival in taste tests and was cheaper by weight.

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 ruled Wall Street because the Spanish dollar was literally cut into eight 'bits', and that 1/8 tick lasted until the New York Stock Exchange went to sixteenths in 1997 and to decimals in 2001.

What does a '2×4' board actually measure in inches, once the rough-sawn plank has been dried and planed smooth?

A '2×4' is really 1 1/2 × 3 1/2 in (38 by 89 mm), because the name describes the rough green board before kiln drying shrinks it and planing takes off the faces. By the same rule a '2×2' ends up 1 1/2 in square.

Which Roman punishment, 'destruction of a tenth' in Latin, killed every tenth man of a unit that had shown cowardice?

Decimation chose its victims by lot, and the nine men spared in each group of ten had to club their condemned comrade to death. The survivors were then put on barley rations and made to camp outside the walls.

Which church levy, from the Old English for 'tenth', took one part in ten of a farmer's produce?

A tithe is a tenth, and Genesis has Abraham paying one to Melchizedek from the spoils of battle, the precedent churches cited for centuries of compulsory giving.

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

The pie chart first appeared in Playfair's Statistical Breviary of 1801, and Florence Nightingale later reshaped it into her polar area diagram to show what was killing soldiers in the military hospital she ran.

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

A crotchet is the British quarter note, half as long as a minim and twice as long as a quaver, with the whole note itself going by the name semibreve.

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

1/4 + 1/6 = 5/12, because quarters and sixths only line up over 12: 3/12 + 2/12. In minutes that is 15 + 10 = 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?

A mixed number pairs a whole number with a proper fraction, and any top-heavy fraction can be turned into one by dividing: 5/2 is two with a half left over.

What is 5/3 called in relation to 3/5, the partner every fraction has whose product with it is 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?

4/5 × 5/6 = 2/3, because going straight across gives 20/30 and both parts divide by 10. Adding tops and bottoms to get 9/11 is the classic mistake.

Which symbol, a % with an extra zero in the divisor, is the usual unit for seawater salinity?

Per mille (‰) means 'in each thousand', so a salinity written as 35‰ is 35 parts of salt in every thousand of seawater, a scale ten times finer than a percentage.

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

Simon Stevin's De Thiende ('the art of tenths') sold Europe on decimal fractions, though he did not invent them: al-Uqlidisi had used them in 952 and al-Kashi systematised them in 1427.

Which 12th-century mathematician first wrote fractions with a bar between top and bottom, as Fibonacci soon did too?

Al-Hassar, working in 12th-century Morocco, gave fractions the bar we still use, and Fibonacci carried the notation into Europe in the 13th century.

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

The uncia was one-twelfth of both the Roman pound and the Roman foot, and it lives on as the ounce, the fluid ounce and the inch.

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

A vinculum is the overline that marks a repetend, and the same horizontal 'chain' was once drawn over groups of terms to tie them together, a job parentheses mostly do today.

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

Solidus is the slash's formal name because pre-decimal Britain used the stroke to write shillings, and by the early 20th century 'shilling mark' and 'solidus' were interchangeable.

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

Obelus was the ancient editors' mark beside lines they thought spurious, and the same name passed to the ÷ symbol once it was borrowed for division.

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?

1/16 = 0.0625 and stops, because 16 is a pure power of 2, while 7, 9 and 12 each carry a factor of 3 or 7 and so their decimals repeat forever.

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

Cross-multiplying works because two fractions are equal exactly when the diagonal products agree, and 3 × 12 and 4 × 9 both come to 36.

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

Dyadic rationals are the halves, quarters and sixteenths of a tape measure, and they are also what a computer's floating-point numbers are built from: an integer times a power of two.

Which camera setting, written like f/8, is literally focal length divided by the width of the hole admitting light?

Aperture is what f/8 sets, and because that figure is focal length over the pupil's diameter, a bigger figure means a smaller hole and less light reaching the sensor.

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?

Half of 3/4 cup is 3/8 cup, because halving a fraction doubles the bottom: 3/4 × 1/2 = 3/8. Tripling the batch instead would call for 2 1/4 cups.

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.

What fraction of a $20 restaurant bill is a $3 tip, a rate a waiter would call 15%?

$3 of $20 is 3/20, and since 3 and 20 share no factor it is already in lowest terms; as a decimal it is 0.15, the familiar 15% tip.

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?

4 − 1 5/6 = 2 1/6 feet, because 4 is 3 6/6, and taking 1 5/6 from that leaves 2 1/6. Subtracting just the whole foot and ignoring the 5/6 gives 3, the usual slip.

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.