The same sum, worked out two ways

You add 27 and 15 two ways. Then you count every mark each way writes. The columns way uses our digits in their places. The symbols way uses Roman symbols.

A mark is one symbol or digit written. A carry is the 1 moved to the next column.

In the symbol way, each number is written as usual first. Then the symbols of both numbers are put into one pile. The pile uses no pairs, so 9 is VIIII there. A trade swaps a group of symbols for one bigger symbol. Every 5 I's trade for 1 V. Every 2 V's trade for 1 X. Each trade writes the whole pile again. Every symbol in it counts.

For 27 + 15, columns write 27, 15, a carry and 42. That is 2 + 2 + 1 + 2 = 7 marks.

For 27 + 15, symbols write XXVII, XV, XXXVVII, XXXXII and XLII. The last is the tidy answer. That is 5 + 2 + 7 + 6 + 4 = 24 marks.

Rule: Columns cost about a mark per digit. For these sums, symbols cost at least three times as many.

Try 8 + 9. By columns that is 5 marks: 8, 9, a carry and 17. By symbols, write VIII, IX, VVIIIIIII, VVVII and XVII. That is 4 + 2 + 9 + 5 + 4 = 24 marks.

Try the lab to finish this lesson.