NCERT Solutions Ganita Prakash (Part 1) Chapter 3 .2 IV. The Roman Numerals — In-text Questions

Book page 603 Updated on2026-09-05

Q1.

Do it yourself now: (b) LXXXVII + LXXVIII

Answer

Pool the symbols of both numerals, then regroup from the smallest upwards — remembering that the Roman exchange rates are not all the same.

LXXXVII = L + X + X + X + V + I + I
LXXVIII = L + X + X + V + I + I + I

Pool: L L X X X X X V V I I I I I

Now exchange, one landmark at a time:

Exchange	Because	Left with
I I I I I → V	5 ones make a five	L L, X X X X X, V V V
V V → X	2 fives make a ten	L L, X X X X X X, V
X X X X X → L	5 tens make a fifty	L L L, X, V
L L → C	2 fifties make a hundred	C, L, X, V

Sum = C + L + X + V = CLXV
Check: 87 + 78 = 165 = 100 + 50 + 10 + 5 ✓

Why it happens: notice how much care this needs. Going from I to V you exchange five; from V to X you exchange two; from X to L again five; from L to C again two. In the Egyptian or Hindu systems every single exchange is “ten of these make one of those”, so the same regrouping can be done without thinking. That single difference is why Roman arithmetic needed an abacus and trained specialists.

Q2.

How will you multiply two numbers given in Roman numerals, without converting them to Hindu numerals? Try to find the product of the following pairs of landmark numbers: V × L, L × D, V × D, VII × IX.

Answer

There is no general shortcut, because the Roman landmark numbers are not the powers of any one number. Each product has to be worked out separately by repeated addition or by the distributive law.

Product	Working	Roman numeral
V × L	5 fifties = L+L+L+L+L = 250 = C+C+L	CCL
L × D	50 five-hundreds = 25,000 = 25 thousands	MMMMMMMMMMMMMMMMMMMMMMMMM (M written 25 times)
V × D	5 five-hundreds = 2500 = M + M + D	MMD
VII × IX	VII × (X − I) = LXX − VII = 70 − 7 = 63 = L + X + III	LXIII

Why it happens: in the Egyptian system, landmark × landmark is always another landmark (10² × 10³ = 10⁵), so a product can be written down the moment you know the two factors. Here, V × L = CCL is not a landmark at all, and L × D = 25,000 has no symbol whatever — you are forced to write M twenty-five times. The Roman landmarks jump ×5, ×2, ×5, ×2, ×5, ×2, so there is no rule to remember and no pattern to exploit.

Tip: VII × IX is easiest done as VII × IX = VII × (X − I). Since a smaller symbol before a larger one already means subtraction in this system, the trick is very natural here.

Q3.

Multiply CCXXXI and MDCCCLII

Answer

First read the two numerals.

CCXXXI = 100 + 100 + 10 + 10 + 10 + 1 = 231
MDCCCLII = 1000 + 500 + 300 + 50 + 2 = 1852

Break the second numeral into its landmark parts and use the distributive law:

231 × 1852 = 231 × (1000 + 800 + 50 + 2)
= 231 × 1000 + 231 × 800 + 231 × 50 + 231 × 2
= 231000 + 184800 + 11550 + 462
= 427812

Check: 231000 + 184800 = 415800; 415800 + 11550 = 427350; 427350 + 462 = 427812 ✓

Now try to write 427812 in Roman numerals. The largest Roman symbol is M = 1000, so you would need 427 M’s followed by DCCCXII. Later Romans got round this with a bar over a numeral to mean “times 1000”:

427812 = 427 thousands + 812
427 = CDXXVII, 812 = DCCCXII
so 427812 = C̄D̄X̄X̄V̄Ī Ī DCCCXII

Why it happens: this is the joke in the cartoon. Being asked to multiply CCXXXI by MDCCCLII really was about as pleasant as fighting a lion — not because the arithmetic is deep, but because the notation gives you no help. In Hindu numerals you would set out 231 × 1852 in columns and be done in a minute, because every carry is “ten of these make one of those”.

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