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© Veneranda Biblioteca Ambrosiana, Milan

Un 1/2 di che numero sarà 1/2? Di: "uno vie 2 fa 2". Parti per l'uno ch'è sopra la riga, ne vien 2, e a questo 2 metti sotto il primo 2 e dira' 2/2, cioè 2 mezzi.
1 1
2 2

0
2 2/21
1

3/4 di che numero fia e 4/5? Tanto è a dire: "dammi e 3/4 di 4/5", quanto è a dire "3/4 di che numero fia e 4/5?" E tutto si fa per una regola.

3 4 12
4 5 16

b

3 3/ 4 6/8
15

4 4

¾

a

Il 4 di b che discende in a, è posto in tal sito sol per denominatore delle figure di sopra la riga, che si debbono intendere essere 3 e 3/4 di quarti e non d'interi, che si de' po' fare di tutto il quesito sedicesimi, che son quarti di quarti; e ritroverai star tal regola nel suo vero essere

Colonna sinistra.

Figura difinita di sotto.

3 4

4 5

3 3 ¾ cioè 12/152

15

4 4
Di' così: "3 via 5 fa 15". Partilo pel 4 che sta sopra il 5, ne verrà 3 e 3/4. E questo 3 s'intende non 3 unità, ma 3 di que' quarti primi, e pe' segno di com'è el 4 che sta sotto il 3, mettera'lo sotto il 3 e 3/4 e dirai poi che ne venga (3 e ¾ )/4 d'un d'essi quarti; onde riducerai esso 3 e ¾ a minor rotto, e dira' " 3 vie 4 fa 12"; aggiugnili il 3 che sta sopra la riga e di': "15/4 di quarti, cioè 15/16". Ora riduci tutta la tua proposta a essi sedicesimi e arai il tuo quisito essere 12/15, che vedi che 12 è veramente e 4/5 di 15. E ritornando a quarti, e 12/15 sarà 3/4, e 15/16 essere tre quarti interi e 3/4 di quarto. E sta bene.

1/2 of what number will be 1/2? Say: 'one times 2 makes 2.' Divide by the one that is above the line, and 2 comes out; and under this 2 put the first 2, and it will read 2/2, that is 2 halves. [1 1 / 2 2 ... 2 2/2 ... 1] 3/4 of what number is 4/5? It is the same to say 'give me 3/4 of 4/5' as to say '3/4 of what number is 4/5?'; and it is all done by one rule. [3 4 12 / 4 5 16; b; 3 3/4 6/8; 15; 4 4; three-quarters; a] The 4 of b that descends into a is placed in that spot only as the denominator of the figures above the line, which are to be understood as 3 and 3/4 of quarters and not of whole units; for the whole problem must then be made into sixteenths, which are quarters of quarters, and you will find this rule to stand in its true being.

Left column.

Figure defined below. [3 4 / 4 5] 3 3 3/4, that is 12/15; 15; 4 4. Say thus: '3 times 5 makes 15.' Divide it by the 4 that stands above the 5, and 3 and 3/4 will come out. And this 3 is understood not as 3 units but as 3 of those first quarters; and as a sign of it, like the 4 that stands under the 3, you will put it under the 3 and 3/4 and then say that there comes (3 3/⁴/4 of one of those quarters; whence you will reduce that 3 3/4 to a smaller fraction and say '3 times 4 makes 12'; add to it the 3 that stands above the line and say '15/4 of quarters, that is 15/16.' Now reduce your whole proposition to those sixteenths and you will have your result to be 12/15, and you see that 12 is truly 4/5 of 15. And returning to quarters, 12/15 will be 3/4, and 15/16 to be three whole quarters and 3/4 of a quarter. And it comes out well.

Solving 'a fraction of what number' problems

Worked examples reducing quarters and fifths to sixteenths

Arithmetic & Proportion

Two worked arithmetic problems of the form 'a fraction of what number equals a given fraction', accompanied by small tables of numbers. The first asks '1/2 of what number is 1/2?'; the second, '3/4 of what number is 4/5?', is solved in detail by turning quarters and fifths into sixteenths (quarters of quarters) and verifying that 12/15 indeed equals 4/5. The notes explain the role of each denominator written below the line and check the answer by converting back to quarters.

On this page

'1/2 of what number is 1/2?'

The first problem multiplies one by two to get two, divides by the numerator above the line, and arrives at 2/2, that is two halves. A second worked case, '3/4 of what number is 4/5?', is set up with the tables 3 4 12 and 4 5 16, noting that the descending 4 serves only as denominator so the figures are read as quarters, then reduced to sixteenths.

'3/4 of what number is 4/5?', reduced to sixteenths

Saying '3 times 5 makes 15' and dividing by the 4 above the 5 yields 3 and 3/4, understood as three of the first quarters. Reducing 3 3/4 gives 15/4 of quarters, that is 15/16; reducing the whole proposition to sixteenths gives the result 12/15, and it is checked that 12 is truly 4/5 of 15.

Reader's Edition — described with AI from the e-Leo transcription.