Math CentralQuandaries & Queries


Question from Kenneth:


Can you explain mathematically how the following scales become inches to feet?
For example, how does the scale of 1:160 become 1/16" = 1'?

(1:160) (1/16" = 1 foot) scale
(1:87) (1/8" = 1 foot) scale

I saw more of these different scale examples at the following:

I thank you for your reply!

Hi Kenneth,

I am confused also. In their examples that say a man is approximately 6 feet tall and using the N scale that 1/16"=1' they get that the scaled man is 3/8" inches tall. I agree, if 1 foot scales to 1/16 inches then 6 feet scales to $6 \times \frac{1}{16} = \frac{6}{16} = \frac{3}{8}$ inches.

The scale is then

\[\frac{1}{16} \mbox{ inches} : 1 \mbox{ foot.}\]

To exxpress both parts of the ratio in the same units write 1 foot as 12 inches to get

\[\frac{1}{16} \mbox{ inches} : 12 \mbox{ inches.}\]

As with fractions you can convert a ratio to an equivalent ratio by mulyiplying moth sides be the same non-zero number. If you multiply bothe sides of the above ratio by 16 then, since $12 \times 16 = 192$ you get a scale of

\[1 : 192.\]

I don't know how they get the scale of $1:160.$



1/16" = 1 ' is 1:(16x12) = 1:192
1/8" = 1' is 1:(8x12) = 1:96

What is happening is that they are equating industry-standard scales (N,HO,...) that have precise ratio equivalents with ratios of "X to one foot" where X is a "round number" length in binary-divided inches. These are approximations on the "a pint's a pound" or "one degree Celsius is two degrees Fahrenheit" level. They figure this is close enough for model making. See the quote on that web page:

"but they actually vary is scale between manufacturers, from 1:20, 1:22, 1:24 & 1:25 (1/2" scale). Because 1/2 Scale or (G) Scale or 1:24 Scale items are hard to find, most hobbiest [sic] and scale modelers will mix and match the use of items across the spectrum of the scale."

Good hunting!

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