Addition of Mixed fractions! Subtraction of Mixed fractions!How to add mixed fractions!how to subtract Mixed fractions!

Addition And subtraction of Mixed fractions:

We have learnt About normal addition and subtraction of fractions in our previous blog. Now we learn addition and subtraction of mixed fractions. So let's start.

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Addition of Mixed fractions with same denominators:

Example: Add The mixed fractions 

                      1¾ + 2¾

Solution:

Step 1 - convert each mixed fractions to improper fractions. =7/4 + 11/4.

Step 2 - Add the Numerators and keep denominator same. =18/4.

Step 3 - Convert into simplest Form. =9/2.

Step 4 - Writing as a mixed number.=4½.

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Addition of Mixed fractions with Different denominators:

Example: Add 1¹/7 + 2²/3.

Solution:

Step 1 - Convert each fractions to improper fractions. =8/7 +8/3.

Step 2 - Take the LCM of denominators 7 and 3. The LCM is 21.

Step 3 - Add the Fractions using LCM. 80/21.

Step 4 - Writing as a mixed number. 3¹³/21.

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Subtraction of Mixed fractions with same denominators:

Example: Subtract 6³/7 - 2¹/7.

Solution:

Step 1 - convert each mixed fractions to improper fractions. =45/7 - 15/7.

Step 2 - Subtract the Numerators and keep denominator same. =30/7.

Step 3 - Writing as a mixed number.=4²/7.

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Subtracting From whole Numbers:

Example: Subtract 9 - 2³/8.

Solution:

     

Step 1 - Convert whole number intro fraction. Now we can write as

                        9/1 - 2³/8

Step 2 - Convert mixed fractions to improper fractions.

                        9/1 - 19/8

Step 3 - Find the LCM of denominators 1 and 8. The LCM is 8.

Step 4 - Subtract the Fractions. 53/8.

Step 5 - Convert into simplest form. 6⁵/8.

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Infinite Geometric Series! Common ratio!How do you find the sum of an infinite series using just sigma notation?

Infinite Geometric Series

An infinite geometric series is the sum of an infinite geometric sequence . This series would have no last term. The general form of the infinite geometric series is a1+a1r+a1r2+a1r3+...$a_1+a_{1}r+a_1{r}^2+a_1{r}^3+\mathrm{...}$ , where a1$a_1$ is the first term and r$r$ is the common ratio.

We can find the sum of all finite geometric series. But in the case of an infinite geometric series when the common ratio is greater than one, the terms in the sequence will get larger and larger and if you add the larger numbers, you won't get a final answer. The only possible answer would be infinity. So, we don't deal with the common ratio greater than one for an infinite geometric series.

If the common ratio r$r$ lies between −1$-1$ to 1$1$ , we can have the sum of an infinite geometric series. That is, the sum exits for | r |<1$\left|r\right|<1$ .

The sum S$S$ of an infinite geometric series with −1<r<1$-1<r<1$ is given by the formula,

S=a11−r$S=\frac{a_1}{1-r}$

An infinite series that has a sum is called a convergent series and the sum Sn$S_n$ is called the partial sum of the series.

You can use sigma notation to represent an infinite series.

For example, ∑n=1∞10(12)n−1${\displaystyle \sum _{n=1}^{\infty }10{\left(\frac{1}{2}\right)}^{n-1}}$ is an infinite series. The infinity symbol that placed above the sigma notation indicates that the series is infinite.

To find the sum of the above infinite geometric series, first check if the sum exists by using the value of r$r$ .

Here the value of r$r$ is 12$\frac{1}{2}$ . Since ∣∣12∣∣<1$\left|\frac{1}{2}\right|<1$ , the sum exits.

Now use the formula for the sum of an infinite geometric series.

S=a11−r$S=\frac{a_1}{1-r}$

Substitute 10$10$ for a1$a_1$ and 12$\frac{1}{2}$ for r$r$ .

S=101−12$S=\frac{10}{1-\frac{1}{2}}$

Simplify.

S=10(12)     =20$\begin{array}{l}S=\frac{10}{\left(\frac{1}{2}\right)}\\ =20\end{array}$

$$

S=10(12)     =20