# What does "reduce by a factor of 10" mean in terms of percentage in the following context?

"Pneumatic tampers and stronger formwork, as used nowadays for instance in Australia, can reduce the labor input by the factor of 10." How much percentage does "by a factor of 10" make? Many thanks for your help.

• Oh, I once reported 1 millimolar as .1 mM. Guess I had reduced the real number by a factor of 10. Mar 13 '15 at 19:58
• There seem to be a lot of wrong answers to this question, so I updated my answer with several examples showing the correct usage of this phrase. Mar 14 '15 at 6:15

Speaking as an engineer, I would interpret "reduce ... by a factor of ten" as meaning "reduce to one tenth of its original value". The new value would be 10% of the old one, which would be a 90% reduction.

Saying "reduce by 90%" sounds more precise and has less emotional impact. "Factor of ten" doesn't have to mean exactly 10.0.

You'll sometimes hear "order of magnitude" used instead of "factor of ten". Technically, they mean almost the same thing, but in casual speech "order[s] of magnitude [more/less]" is used figuratively to mean "a lot".

EDIT: A lot of people seem to be getting this wrong, so I will provide some examples. First, from the Wikipedia article on orders of magnitude:

An order-of-magnitude difference between two values is a factor of 10. For example, the mass of the planet Saturn is 95 times that of Earth, so Saturn is two orders of magnitude more massive than Earth. Order-of-magnitude differences are called decades when measured on a logarithmic scale. [emphasis mine]

From HyperPhysics:

...a commonly used rule of thumb for sound loudness suggests that an offending sound must be reduced by a factor of ten in intensity to be reduced to half as loud.

Here's a more extreme example from a page about gravitational wave detection:

In all, noise is reduced by a factor of ten billion.

Obviously this doesn't mean that the noise was reduced by 0.00000001%, since that would be a ridiculously small difference.

From a page on chemistry:

Serial dilutions involve diluting a stock or standard solution multiple times in a row. Typically, the dilution factor remains constant for each dilution, resulting in an exponential decrease in concentration. For example, a ten-fold serial dilution could result in the following concentrations: 1 M, 0.1 M, 0.01 M, 0.001 M, and so on. As is evidenced in this example, the concentration is reduced by a factor of ten in each step. [emphasis mine]

From a page on software for data backup:

At its worst, we took more than 72 hours for a full backup ... With the Quantum solution, a full backup of our work is now completed in eight to ten hours. The time needed has been reduced by a factor of ten. [emphasis mine]

From a paper on trash reduction at take-out restaurants:

At Homeplate, the trash has been reduced by a factor of ten, meaning the usual fifty pounds of trash coming from the venue from each meal (lunch, dinner) has decreased to a mere five pounds of trash and five pounds of composting.

I found these by Googling "reduced by a factor of ten" and looking for documents that mentioned specific numbers. In no case did I find anything to suggest that reducing by a factor of ten means a 10% reduction. I have never heard such a usage and will be very surprised if I ever do.

• +1 Adam, your answer is correct and I posted my article to make a point- the same one you made which is why a writer would not choose language that is more precise and specific that something that may be loaded with much more than the essential information- including connotation, usage predominance, emotion (as you wrote), etc. This question reminds me of what S I Hayakawa (a semanticist) wrote about the "types" of language.
– Gary
Mar 13 '15 at 22:47
• @Adam Haun : I think I understand now, thanks to your examples :-) I had googled too but didn't find any adequate solution, thank you. I was puzzled because in France we don't use this sort of expression "by a factor of 10", we tend to use percentages, which I find a lot easier to understand! Ah, it will always be the French way of thinking vs the Anglo-Saxon one! LOL Mar 14 '15 at 9:30
• @FrenchMan It is not an Anglo-Saxon specific expression, and actually it is used also in French, especially in a scientific context. It is also used in Italian. Mar 14 '15 at 10:17
• "Obviously this doesn't mean that the noise was reduced by 0.00000001%, since that would be ridiculously small.": yes, it means exactly that. In gravitational wave detectors noise should, and is, so ridiculously small, otherwise they wouldn't be able to detect the gravitational wave signal. Mar 14 '15 at 21:40
• The noise should be small in absolute terms, and the noise-signal ratio might be that small, but a percentage change in noise that small wouldn't count as news. And "reduced by a factor of ten billion" definitely does not mean that. Mar 14 '15 at 21:47

# Original Sentence:

Pneumatic tampers and stronger formwork, as used nowadays for instance in Australia, can reduce the labor input by the factor of 10.

# Meaning of "by a factor of"

The phrase "reduced by a factor of 10" is a mathematical idiom within standard math literacy. It means a reduction to one-tenth of its original value. This usage well-defined:

• factor (American Heritage): A quantity by which a stated quantity is multiplied or divided, so as to indicate an increase or decrease in a measurement: The rate increased by a factor of ten.

• factor (Merriam-Webster): *a quantity by which a given quantity is multiplied or divided in order to indicate a difference in measurement; "costs increased by a factor of 10".

The usage allows some ambiguity: it can mean "exactly 10" or "approximately 10". Lest anyone still has doubt, the following passage proves that the standard idiomatic usage is being used in this case:

Advances in forming technology have increased the efficiency and quality with which rammed earth walls can be built. Today, with the help of front-loading tractors, pneumatic tampers, and well-designed formwork, earth walls can be built in a fraction of the time it used to take. Under good conditions it is possible for a crew of four men to complete 300 square feet of wall per day, compared to the 40 or 50 square feet per day for a four man crew working with hand tampers and baskets.

Source: Building Rammed Earth Homes by David Easton, Mother Earth News April/May 1996. Also found in Google Ebook, The Rammed Earth House, by David Easton, Chelsea Green Publishing, Jun 13, 2007.

An increase from 40 to 300 per day is an increase by a factor of 10 (when exacting accuracy is not important). Conversely, "labor input" is reduced by a factor of 10. To produce 400 the old way took 40 X 10 days, and to produce 400 the new way takes about one day.

In this case, the idiom means "approximately 10".

# A Mathematician Joke

Consider the following joke, which was made for this post:

Two mathematicians were talking over lunch. One said to the other, "I just bought a car at a fraction of it's manufacturing cost!"

The other said, "That's great! What did it cost and how much did you pay for it?"

The first said, "It cost \$20,000 and I paid \$25,000."

The other said, "That's very good! Most cars sell at a larger fraction than that!"

The mathematician is making a mistake in standard English since "at a fraction of the cost" means "substantially less". In this case, the mathematician was thinking of the fraction 5/4. Similarly, anyone who uses "reduce by a factor of 10" to mean "reduce by 10 percent" is making a competence error relative to mathematical literacy.

# Pragmatic Ambiguity

In typical usage, the word factor is used to mean "contributing element" as in, "That was an important factor in my final decision."

One who is unfamiliar with the mathematical idiom will think of the phrase as meaning "reduced by a contributing element of 10", which is ambiguous. Even if they know the mathematical definition of "factor", they still don't recognize the phrase as idiomatic. They see it as straight prose and are faced with an ambiguity of synonyms. Such a person might say, "the phrase requires more context" or "perhaps there's a better way to say it that would be less ambiguous".

To say reduced by a "factor of ten" is misguided, and misguiding. try now saying by 10%. of by 90% (which is what you mean to say.) so using factor is jargon and jargon ism jargonistic, and jargonismal, as well i might "opposing subtract."

And furthermore leaving only 10% of any economy is heinonimously more than jargon. bite me! is better said. i can see the teeming homeless already upon our shores of prosperity and dying as we step over the bodies here in the richest city in AMerd-rica.

i cannot imagine a worse scenario: unless i try to imagine a stubbornly selfish and hoarding Econ 101 90% smaller, even as the poor switch from a threatened social security to the social security of many children to care for them in later years of even more perverse poverty.

the proper response to this crisis, is to tax assets until they are public or communal, local or state. AND to force all stake holders on to the boards of directors, and manage an honest return to democracy by penalty of death if voters do not vote. all profits should return to the people. everything ever made rests upon the shoulders of what went before. we are all obligated to the past performance of humanity... open your heart and hand and get real.