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(c) In 13768, the tens digit is 6 > 5. SOLUTION: 42 estimated to the nearest ten= 40.
:. the required rounded number = 13800. 58 estimated to the nearest ten = 60.
(d) In 1249, the tens digit is 4 < 5. Hence, the required estimation = ( 40 × 60) = 2400.
:. the required rounded number = 1200.
EXAMPLE 10: Estimate the product of 34 and 75.
EXAMPLE 3: Round each of the following numbers to the nearest thousand:
(a) 5486 (b) 6823 (c) 14380 (d) 23659 SOLUTION: 34 estimated to the nearest ten = 30.
75 estimated to the nearest ten = 80.
SOLUTION: (a) In 5486, the hundreds digit is 4 < 5. Hence, the required estimation = (30 × 80) = 2400.
:. the required rounded number = 5000.
(b) In 6823, the hundreds digit is 8 > 5. EXAMPLE 11: Estimate the product of 367 x 231 by rounding off each number to the nearest hundred.
:. the required rounded number = 7000.
(c) In 143801 the hundreds digit is 3 < 5. SOLUTION: 367 estimated to the nearest hundred = 400.
:. the required rounded number = 14000. 231 estimated to the nearest hundred = 200.
(d) In 23659, the hundreds digit is 6 > 5. Hence, the estimated product = 400 × 200 = 80000.
:. the required rounded number = 24000.
EXAMPLE 12: Estimate the product of 183 x 153 by rounding off the first number upwards and the second
EXAMPLE 4: Estimate the sum (64 + 79) to the nearest ten. number downwards.

SOLUTIONS: 64 estimated to the nearest ten = 60. SOLUTION: 183 estimated upwards= 200.
79 estimated to the nearest ten = 80. 153 estimated downwards = 100.
Hence, the required estimation = (60 + 80) = 140. Hence, the estimated product= 200 × 100 = 20000.

EXAMPLE 5: Estimate the sum (267 + 132) to the nearest ten.
ROMAN NUMERALS
SOLUTION: 267 estimated to the nearest ten= 270.
132 estimated to the nearest ten= 130. ROMAN NUMERALS One of the early systems of writing numerals is the system of Roman numerals.
Hence, the required estimation= (270 + 130) = 400.
There are seven basic symbols to write any numeral.
EXAMPLE 6: Estimate the sum (2 7 4 + 143) to the nearest hundred. These symbols are given below.

SOLUTION: 274 estimated to the nearest hundred= 300.
143 estimated to the nearest hundred = 100. Roman numeral I V X L C D M
Hence, the required estimation= (300 + 100) = 400.
Hindu - Arabic numeral 1 5 10 50 100 500 1000
EXAMPLE 7: Estimate the sum (21397 + 2 7807 + 42505) to the nearest thousand.
If a bar is placed over a numeral, it is multiplied by 1000.

SOLUTION: 21397 estimated to the nearest thousand = 21000. Thus, V = 5000 and × = 10000, etc.
27807 estimated to the nearest thousand = 28000. Using these symbols, we may form all Roman numerals by adopting the rules given below.
42505 estimated to the nearest thousand = 43000
Hence, the required estimation= (21000 + 28000 + 43000) = 92000 RULE 1 Repetition of a symbol in a Roman numeral means addition.

EXAMPLE 8: Estimate the difference (673 - 258) to the nearest hundred. CAUTIONS (i) Only I, X, C, M can be repeated.
(ii) V. L and D are never repeated.
SOLUTION: 673 estimated to the nearest hundred = 700. (iii) No symbol in a Roman numeral can be repeated more than 3 times.
258 estimated to the nearest hundred = 300.
Hence, the required estimation= (700 -300) = 400.

EXAMPLE 9: Estimate the product of 42 and 58.