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(iv) XCIX = (100 - 10) + (10 - 1) = 99
(v) CLXVI=l00+50+10+(5+1)=166
(vi) CCXXVI = 200 + 20 + (5 + 1) = 226 2 FACTORS AND MULTIPLES
(vii) CCCXL= 300 + (50 - 10) = 340
(viii) CDXLVI = (500 - 100) + (50 - 10) + (5 + 1) = 446

EXAMPLES. Show that each of the following is meaningless. Give reason in each case. Suppose we are given a numerical expression consisting of numbers with two or more of the fundamental oper-
(i)XXXX (ii) VX (iii) IC (iv) XVV ations. We perform these operations strictly in the following order:
(i)Division (ii) Multiplication (iii) Addition (iv) Subtraction
SOLUTION (i) No symbol is repeated more than three times. EXAMPLE:- Simplify: 18 - 6÷ 2 × 3
XXXX is wrong. SOLUTION: The given expression
(ii) V. L, Dare never subtracted. = 18 - 6 ÷ 2 × 3
VX is wrong. = 18 - 3 × 3 [performing division]
(iii) I can be subtracted from V and X only. = 18 - 9 [performing multiplication]
IC is wrong. = 9 [performing subtraction].
(iv) V, L, D are never repeated. EXAMPLE: Simplify: 15 + 24 ÷ 3 - l × 6
XVV is wrong. SOLUTION: The given expression
JUST TRY: 1 Write the following numbers in Roman numerals. = 15 + 24 ÷ 3 - 1 × 6
a. 57 = 15 + 8 - 1 x 6 [performing division]
b. 89 = 15 + 8 - 6 [performing multiplication]
c. 25 = 23 - 6 [performing addition]
JUST TRY:2 Write the following Roman numerals in Arabic numbers. = 17 [performing subtraction]
a. LXII
b. DCXXLYI
c. CDXXYII USE OF BRACKETS
We know that brackets are used to separate various parts of an expression. There are four kinds of brackets.
1. Write the following numbers in Roman numerals.
a.45 b.69 c.99 d.80 e. 75 (i) Bar, or Vinculum −
2. Write the following Roman numerals as Arabic numbers. (ii) Round brackets, or Small brackets, or Parentheses ( )
a. LXXII b. XLIX c. CCXLIX d. MDCLIV e. XCYIII (iii) Curly brackets, or Braces { }
3. Classify the following as meaningful or meaningless. If meaningful, write them in Hindu-Arabic numerals. (iv) Square brackets, or Big brackets [ ]
a. IXX b. IVC c. LIL d. XLIV e. CCLXIV
4. Put the correct symbol >, < or = in the blank boxes. While simplifying expressions, the terms in brackets are taken as independent units. The order of simplifying
a. XL □ LV b. XLIX □ LXXI c. XLIV □ LXIV the brackets is:
d. XL □ xc e. XCVIII □ LXIII
5. Find the sum of the following Roman numerals and give the answer in both (i) Bar first (ii) Parentheses next (iii) Braces next (iv) Square brackets in the last
Roman numerals and Arabic numbers.
a. LX + XC b. DC+ XLIX c. XCIV + XL For simplifying the expressions involving brackets. we must proceed according to the letters of the word
d. XXXVI + IX e. CLIX + CD ‘VBODMAS’.

Here V. B, 0, D, M, A, S stand for Vinculum, Brackets, Of, Division, Multiplication, Addition and Subtraction
respectively.