2.1: Place value and decimals- basic structure

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Problem 40 Without using a calculator:

(a) Work out

(i) 12 345 679 × 9

(ii) 7 × 9 × 11 × 13.

(b) Divide

(i) 123 123 123 by 123

(ii) 111111111 by 111

(iii) 111111111 by 37.

Problem 41 Work out in your head (i) 11² (ii) 11³ (iii) 101².

Problem 42 Try to answer the following questions using only mental arithmetic:

(a) (i) What is the largest and the smallest possible number of digits in the answer when you multiply a 3-digit integer by a 5-digit integer?

(ii) What if we multiply an m-digit integer by an n-digit integer?

(b) (i) How many (base 10) digits are there in the evaluated form of 2²⁰?

(ii) Estimate ${(\frac{1}{2})}^{20}$ to 6 decimal places.

(d) Work out how many zeros there are on the end, and work out the last non-zero digit of (i) 2¹⁵ × 5³ (ii) 20!.

Problem 43 Imagine the sequence of positive integers from 1 to 60 written in a single row as the digits of a very large integer:

1234567891011121314151617181920212223···5960.

You have to cross out 100 of these digits.

(a) Suppose you want to make the remaining number as small as possible. What number is left?

(b) Now suppose that you want to make the remaining number as large as possible. What number is left?