Phase 00 - Lesson 15

Ratios and Percent

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A percent is just a fraction with the bottom number fixed at 100. "Per cent" literally means "per hundred".

Type: Learn Languages: Python Prerequisites: Fractions and Division (00-14) Time: ~30 minutes

Learning Objectives

Read a ratio as "this many per that many" and turn it into a single number by dividing
Convert any fraction to a percent by multiplying the decimal by 100
Explain what "5 percent" and "97.8 percent recovery" mean in NeuroGrid terms
Avoid the common direction error (which number goes on the bottom)

The Concept

You now know that a fraction is a division (lesson 00-14). A ratio and a percent are both just fractions in disguise.

A ratio is a comparison by division

A ratio compares two amounts using the word "per": miles per hour, dollars per item, weights per byte. You turn a ratio into one number by dividing:

352 km in 5 hours   ->   352 / 5 = 70.4 km per hour

The word "per" is your signal to divide. The thing before "per" goes on top, the thing after "per" goes on the bottom.

A percent is a fraction out of 100

"Per cent" means "per hundred". A percent is a fraction whose bottom is always 100:

5%   =   5/100   =   0.05
50%  =  50/100   =   0.5
100% = 100/100   =   1.0

To go from a fraction to a percent: do the division to get the decimal, then multiply by 100.

17.5 / 352 = 0.0497...   ->   x 100   =   about 5%

To go the other way (percent to a plain number): divide by 100. 5% = 5/100 = 0.05.

Worked example: accuracy recovery

NeuroGrid's full-precision teacher model scores 0.8474 on a legal task. After squeezing it down to 1.58 bits, the ternary student scores 0.829. How much of the teacher's quality did we keep? That is a ratio of student over teacher:

0.829 / 0.8474 = 0.9783...

Multiply by 100: about 97.8%. We kept 97.8 percent of the quality while cutting the storage to roughly an eighth. The whole pitch of the project lives in that one percentage.

A second example, the size side: a tiny adapter of 17.5 units of something inside a 352-unit model is:

17.5 / 352 = 0.0497   ->   about 5%

So the adapter is about 5 percent of the model.

Active recall

Produce the answer. Easiest first.

50% as a plain decimal = ?
3/4 as a percent = ?
0.829 / 0.8474 as a percent (round to one decimal) = ?

Answers: 0.5; 75% (0.75 x 100); about 97.8%.

Misconception callout

The trap is putting the numbers in the wrong order. "Recovery" is student divided by teacher, not teacher divided by student. If you compute 0.8474 / 0.829 you get 1.022, which would claim the student is better than the teacher. Always ask: which is the whole (the bottom) and which is the part (the top)? The part you kept goes on top.

Build It

python phases/00-setup-and-tooling/15-ratios-and-percent/code/percent.py

Why this matters for AI

Almost every result in this curriculum is reported as a percent or a ratio: accuracy recovery, compression ratio, GPU utilization, tokens per second, fraction of parameters fine-tuned. "We recovered 97.8 percent at one eighth the size" is the kind of sentence that decides whether a model ships. It is one division and one multiply by 100.