connections between fractions and decimals

do you know that, SOMETIMES when we have a problem in fractions, it is a good idea to try to change it into decimals to make things easier? also, if we have problem doing decimals, we could make things easier by changing it into fractions? yes, they are really connected, or in other words, interchangeable.

say, 1/2 + 1/4, how do you approach this question? we need the HCF of the denominator, etc...
hey, what if we change them into decimals? 1/2 is 0.5 and 1/4 is 0.25, so the answer is 0.75. wahh...so quick, isn't it?

in order to do this properly and efficiently, you need to have a very good number sense. say, 1/2 is half, so half of a number is 0.5 (everyone knows that). a quarter, 1/4 is 0.25, three-quarter, 3/4 is 0.75 etc. yes, a lot of them to be memorized. however, it makes life easier. really... here i list down some of the decimals that could be crucial in ANY calculations.

1/2 = 0.5
1/4 = 0.25
3/4 = 0.75
1/5 = 0.2

for example, take a look at this very famous question.
0.1 x 0.1
the most common mistake that students did was when they gave 0.1 as the answer. why this answer? because (most probably), they did the multiplication "directly", i.e. 0 x 0 = 1 and 1 x 1 = 1, so the answer is also 0.1.

better way?
0.1 x 0.1 can also be expressed as fraction as
1/10 x 1/10
now multiply numerators and denominators, we will get
1/100
which if we convert into decimals, we will get
0.01.

the same goes with the question 0.2 x 0.3, in which the answer is 0.06, not 0.6.

converting these numbers from one form to another is very useful, especially when we are dealing with multiplication (as shown above) and division. another example?

0.8 divide by 0.2.

some teacher taught the moving the decimal trick, but i really don't emphasize that trick to my students. we will discuss that later. now, lets convert these decimals into fraction, then we get

8/10 divide 2/10, and we can change divide to multiplication, and at the same time taking the reciprocal of the fraction on the right, to get

8/10 x 10/2

now, we cancel the 10s, and divide 8 with 2 to get 4 as our answer.

sounds tedious huh? well, first impression is not always true. again, i emphasize, this technique works very well for majority of the questions, NOT all questions. say we cannot do
1/3 + 1/2
by converting them into decimals, since 1/3 is a recurring decimal of 0.3333333...
I always emphasize this; use our own logics.

by the way, before i end this topic, i remember giving out these questions to my tutorial students:
(i) 0.2 x 0.8
(ii) 1.2 x 0.4
this "average" student tried to point out that my technique is useless, since he can calculate the first question directly, i.e. 0x0 =0 and 2x8 = 16, so the answer is 0.16. i just smiled to him, and asked him to do the second question on the whiteboard, and show his technique to the rest of his friends. proudly, he said this.
" this is easy...1x0 = 0 and 2x4 = 8, so the answer is 0.8. "

for readers, what is wrong here?
what should be the answer?

believe me, since that night, this student never argued with my technique, anymore... :-)