Simplification Part 3 : Decimal To Fractions & Their Operations

Hello friends,

Today we are going to learn 3^rd part of simplification, in which we are going to continue decimal to fractions section.

In last post, we got what are the different types of decimals & what makes them different. Also n/(3,4,5,6,7,8,9,&11). If you didn’t get that you can click on link to continue, don’t bother they are part of mind tricks so it’ll take just 5 minutes.

So, After learning types of decimals we must know how to convert them into fractions. Since if you convert decimals to fractions, you can cancel those fractions with multiplicands.

Example: 22  x 0.34 = 22 X 34/99 = 2 x 34/9 = 7.5

I have short cut for this also, 22 x .34 = 11 x 2 x .34 = 11 x .68 = 6 | 6+8 | 8 = 7.48…This it trick of multiplication with 11. Stay tuned to learn this in upcoming posts.

So let’s learn rules first,

1.   As we can see in above example, 0.34 = 34/100. We are all agree with that. So just convert 100 to 99. To make it approximate & easy for calculation, but in case of recurring only.

2.   Proof,

0.34 = 0.343434…………………………………..Eq.1

Mutiplying both sides by 100, since we want to make recurring part to full part(Real PaRt)

100 x 0.34  = 0.3434 X 100

100 x 0.34 = 34.3434……………………………Eq.2

Subtracting 1 from 2

99 x .034 = 34

0.34 = 34/99

3. Rule for Mixed decimals. Example 0.62323 = 0.623

0.623 = 0.62323 …………………………………..Eq.1

Taking Non-Recurring part before decimal point, by multiplying with 10

10 x 0.623 = 6.2323 …………………………………..Eq.2

Here we’ll make 623 as real part, & for that we need to multiply with 100 to original decimal. So after multiplying Eq.1 with 100 we get

100 x 0.623 = 623.2323 …………………………………..Eq.3

Now Eq.3 - Eq.2

990 x 0.623 = 623 – 6

Therefore 0.623 = 617/999

In short,

1.   Nr = ( All decimals – Non-recurring part ) = 623 – 6

2.   Dr. = Recurring decimals times 9 | Non-recurring times 0. Meaning Recrring decimal’s count( 2, 3) = 2 & non-recurring decimal’s count ( 6 )= 1, So 2times 9 then(|) 1 times 0 = 990 as Dr(Denominator)

Examples of Both types:

1.   0.563 = 3 – 56 / 900 = -53/990…Hey but our decimal is not negative, then result is wrong. Yes. This is as follows,

0.563 = 0.5633 = 0.56333 (Now looks OK for Subtraction) = (Also) 0.563333 (Not Ok for subtraction)

Remember, Take any number & subtract since after all we are going to get approximate number due to 990. Since Recurring decimals can never be expressed as real fractions. Tell me, How One can express partial decimal into correct decimal? We have 0.563 to convert to fraction. Bar that I have places below 3(Wrongly, for my convenience) can not tell how many 3s are there after 0.563. How can I find real & correct fraction then?

Also I we would have provided to convert 0.5633 to fraction in which nothing is recurring & decimal itself is terminating, then we can say, 0.5633 = 5633/10000 is real & correct fraction( Don’t believe then use calculator & check).

Ans. is 0.563= 333-56/900 = 277/900

2.   0.0055 = As a tricky Question we have given ans is  also tricky that is 5-5/9000 = 0/9000 This is wrong since we cannot divide or make parts of Number 0.

Taking next form of 5, Therefore 0.0055 =   55-5/9000 = 50/9000( = 5/900, But I don’t know 5/900 is correct or not), So Semifinal ans is 50/9000

3.   0.842111 = 1111-842/999000 = 269/999000

4.   0.9999 = 0.9 = 9/10(Wrong?)

0.9999 = 9999/10000( Correct, 0.9999 is somehow pure decimal, that’s why !!!)

    5. Last two, 1.5 = 1 + 0.5 = 1 + 5/10 = 1½  & 1.3 = 1+ 3/9  = 1+ 1/3 = 4/3

I thing, there is no need to explain pure decimals to fraction. Skipping that & moving on with next topic.

Suppose we square some number then that square is always greater that that number. Example 2² = 4 , 11² = 121, here 11 < 121 & 4 > 2. But in case of fractions it not correct. Look, ½ ²= ¼ , here ½ > ¼

Also 3/2 ² = 9/4,

Now 3/2 = 1.5 & 9/4 = 2.25, here 3/2 < 2.25, meaning if Numerator is greater then rule of square is follow, but if Nr is less than Dr, it doesn’t.

Adding Decimals:

Let’s try to add following decimals

0.35 + 0.3 + 1.5 = 35/10 + 3/9 + 3/2 = 7/2 + 1/3 + 3/2

= 1 ½ + 3 ½ + 1 ½ = (1+3+1) + ( ½  + ½ + ½ )  = 4 + 3/2 = 11/2 = 5.5

Multiplying :

0.03 x 0.6 = 3 x 6 = 18

Then count no.of decimal. They are 3, So 0.018 is ans

Example 2 : 0.03 x 100 =

1. Count no.of zeros in 100, they are 2.

2. Move Nr’s decimal point 2 positions back, so

0.03 => 00.3 => 003 => 3

Dividing:

0.356/ 100 =

1. Count no.of zeros in 100, they are 2.

2. Move Nr’s decimal point 2 positions forward

0.356 => 0.0356 =>0.00356