What is a TERMINATING DECIMAL?

Describe the ratio of integers that results in a terminating decimal

A decimal which ends without repeating

e.g. 0.2 , 0.47, 0.375

Some integer

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Some Power of 10 (i.e. some integer that can be expressed in the form  2^x5^y where x and y are integers)

e.g. 0.2 = 2/10 = 1/5   

e.g. 0.375 = 375/1000 = 3/8

When will a decimal terminate and why? 

When will a decimal NOT terminate and why?

If, after being fully reduced, the denominator ONLY has factors of 2 and/or 5, the decimal WILL TERMINATE

If, after being fully reduced, the denominator has  any prime factors OTHER than 2 or 5, the decimal WILL NOT TERMINATE

This is because a terminating decimal ONLY arises as a result of an integer being divided by a power of 10. i.e. the denominator should only have prime factors of 2 and/or 5

When a denominator of a fraction is 9, 99, 999 or another power of 10 minus 1, what's an easy way of determining the REPEATING DIGITS of the decimal equivalent of the fraction?

Look at the numerator... This will give you the repeating digits (perhaps with leading zeroes) if the denominator of the fraction is 1 less than a power of 10.

e.g.

1/11 = 9/99 = 0.09 (rep.)

3/11 = 27/99 = 0.27 (rep.)

 4/9 = 0.4 (rep.)

23/99 = 0.23 (rep.)

List all the properties of a terminating decimal 

A number is increased by 25%... what must you reduce the new number by to get the old number again?

20%... because 5/4 = 125% and 4/5 = 80% (reciprocal of 5/4)... therefore 80% of the new number is the old number, therefore you must reduce the new number by 20% to get this amount...

4/9 = ? = ?repeating
1/11 = 9/99 = ? = ?repeating
3/11 = 27/99 = ? = ?repeating
23/99 = ? = ?repeating

(3/7)^-2 = ? = ? = 49/9

(x/y)^-w = ? = ?