Simplify fractions

### 1.1. Knowledge to remember

a) Given the fraction \(\frac{{10}}{{15}}\). Find fractions with \(\frac{{10}}{{15}}\) but with smaller numerator and denominator.

We can do the following:

We see that 10 and 15 are both divisible by 5. According to the basic properties of fractions, we have:

\(\frac{{10}}{{15}} = \frac{{10:5}}{{15:5}} = \frac{2}{3}\)

So: \(\frac{{10}}{{15}} = \frac{2}{3}\).

*Comment :*

- The numerator and denominator of the fraction \(\frac{2}{3}\) are both less than the numerator and denominator of the fraction \(\frac{{10}}{{15}}\).
- The two fractions \(\frac{2}{3}\) and \(\frac{{10}}{{15}}\) are equal.

We say that: The fraction \(\frac{{10}}{{15}}\) has been reduced to the fraction \(\frac{2}{3}\).

It is possible to reduce the fraction to get a fraction with a smaller numerator and denominator while the new fraction is still equal to the given fraction.

b) *How to shorten fractions*

__ Example 1:__ Simplify fractions \(\frac{6}{8}\).

We see: 6 and 8 are divisible by 2, so:

\(\frac{6}{8} = \frac{{6:2}}{{8:2}} = \frac{3}{4}\)

3 and 4 are not divisible by any natural number greater than 1, so the fraction \(\frac{3}{4}\) can no longer be shortened. We say that : fraction \(\frac{3}{4}\) is the simplest fraction and the fraction \(\frac{6}{8}\) has been reduced to the simplest fraction \(\frac{3}{4}\).

__ Example 2:__ Simplify the fraction \(\frac{{18}}{{54}}\).

We see: 18 and 54 are both divisible by 2, so

\(\frac{{18}}{{54}} = \frac{{18:2}}{{54:2}} = \frac{9}{{27}}\)

9 and 27 are both divisible by 9, so :

\(\frac{9}{{27}} = \frac{{9:9}}{{27:9}} = \frac{1}{3}\)

1 and 3 are not divisible by any natural number greater than 1, so 3434 is a minimal fraction.

So: \(\frac{{18}}{{54}} = \frac{1}{3}\).

*When reducing fractions, you can do the following: *

*Consider whether the numerator and denominator are divisible by any natural number greater than 1.**Divide the numerator and denominator by that number.*

*Keep doing this until you get the simplest fraction. *

### 1.2. Textbook exercise solution page 114

**Lesson 1: **Simplify fractions

a) \(\frac{4}{6};\frac{{12}}{8} & ;\frac{{15}}{{25}};\frac{{11}}{{22}} ;\frac{{36}}{{10}};\frac{{75}}{{36}}\)

b) \(\frac{5}{{10}};\frac{{12}}{{36}};\frac{9}{{72}};\frac{{75}}{{300} };\frac{{15}}{{35}};\frac{4}{{100}}\)

**Solution guide:**

How to shorten fractions:

- Consider whether the numerator and denominator are divisible by any natural number greater than 1.
- Divide the numerator and denominator by that number.

Keep doing this until you get a minimal fraction (a fraction that can’t be reduced anymore).

a)

\(\begin{array}{l}

\frac{4}{6} = \frac{{4:2}}{{6:2}} = \frac{2}{3}\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\frac{{12}}{8} = \frac{{12:4}}{{8:4}} = \frac{3}{ 2}\\

\frac{{15}}{{25}} = \frac{{15:5}}{{25:5}} = \frac{3}{5}\,\,\,\,\,\, \,\,\,\,\,\,\frac{{11}}{{22}} = \frac{{11:11}}{{22:11}} = \frac{1}{2} \\

\frac{{36}}{{10}} = \frac{{36:2}}{{10:2}} = \frac{{18}}{5}\,\,\,\,\, \,\,\,\,\frac{{75}}{{36}} = \frac{{75:3}}{{36:3}} = \frac{{25}}{{12}}

\end{array}\)

b)

\(\begin{array}{l}

\frac{5}{{10}} = \frac{{5:5}}{{10:5}} = \frac{1}{2}\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\frac{{12}}{{36}} = \frac{{12:12}}{{36:12}} = \frac {1}{3}\\

\frac{9}{{72}} = \frac{{9:9}}{{72:9}} = \frac{1}{8}\,\,\,\,\,\,\, \,\,\,\,\,\,\,\frac{{75}}{{300}} = \frac{{75:25}}{{300:25}} = \frac{3}{ {12}} = \frac{{3:3}}{{12:3}} = \frac{1}{4}\\

\frac{{15}}{{35}} = \frac{{15:5}}{{35:5}} = \frac{3}{7}\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\frac{4}{{100}} = \frac{{4:4}}{{100:4}} = \frac{1 }{{25}}

\end{array}\)

**Lesson 2: **In fractions: \(\frac{1}{3};\frac{4}{7};\frac{8}{{12}};\frac{{30}}{{36}};\ frac{{72}}{{73}}\)

a) Which fraction is minimal? Why ?

b) Which fraction can be reduced? Simplify that fraction.

__Solution guide:__

How to shorten fractions:

- Consider whether the numerator and denominator are divisible by any natural number greater than 1.
- Divide the numerator and denominator by that number.

Keep doing this until you get a minimal fraction (a fraction that can’t be reduced anymore).

a) The simplest fractions are : \(\frac{1}{3};\frac{4}{7};\frac{{72}}{{73}}\)

Because the numerator and denominator of each of these fractions are not divisible by any other natural number other than 1.

b) The reduced fractions are : \(\frac{8}{{12}};\frac{{30}}{{36}}\).

\(\frac{8}{{12}} = \frac{{8:4}}{{12:4}} = \frac{2}{3};\,\,\,\,\,\ ,\,\,\,\,\,\,\,\,\frac{{30}}{{36}} = \frac{{30:6}}{{36:6}} = \frac{ 5}{6}\)

**Lesson 3: **Write the correct number in the blank

__Solution guide:__

How to shorten fractions:

- Consider whether the numerator and denominator are divisible by any natural number greater than 1.

- Divide the numerator and denominator by that number.

Keep doing this until you get a minimal fraction (a fraction that can’t be reduced anymore).

\(\begin{array}{l}

\frac{{54}}{{72}} = \frac{{54:2}}{{72:2}} = \frac{{27}}{{36}};\\

\frac{{27}}{{36}} = \frac{{27:3}}{{36:3}} = \frac{9}{{12}};\\

\frac{9}{{12}} = \frac{{9:3}}{{12:3}} = \frac{3}{4}.

\end{array}\)

So we have the following result:

### 1.3. Solve the exercises Textbook Practice page 114

**Lesson 1: **Simplify fractions: \(\frac{{14}}{{28}};\frac{{25}}{{50}};\frac{{48}}{{30}};\frac{) {81}}{{54}}\)

__Solution guide:__

How to shorten fractions:

- Consider whether the numerator and denominator are divisible by any natural number greater than 1.
- Divide the numerator and denominator by that number.

Keep doing this until you get a minimal fraction (a fraction that can’t be reduced anymore).

\(\begin{array}{l}

\frac{{14}}{{28}} = \frac{{14:14}}{{28:14}} = \frac{1}{2};\,\,\,\,\,\ ,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{25}}{{50}} = \frac{{25:25}}{ {50:50}} = \frac{1}{2};\\

\frac{{48}}{{30}} = \frac{{48:6}}{{30:6}} = \frac{8}{5};\,\,\,\,\,\ ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{81}}{{54}} = \frac{{ 81:9}}{{54:9}} = \frac{9}{6} = \frac{{9:3}}{{6:3}} = \frac{3}{2}.

\end{array}\)

**Lesson 2: **Which of the following fractions is equal to \(\frac{2}{3}\) ?

\(\frac{{20}}{{30}};\frac{8}{9};\frac{8}{{12}}\)

__Solution guide:__

- Reduce fractions to minimal fractions (if possible). Fractions with the same minimum fraction are equal.

\(\frac{{20}}{{30}} = \frac{{20:10}}{{30:10}} = \frac{2}{3};\,\,\,\,\ ,\,\,\,\,\,\,\,\,\,\frac{8}{{12}} = \frac{{8:4}}{{12:4}} = \frac{ 2}{3}\)

\(\frac{8}{9}\) is the simplest fraction.

So there are 2 fractions equal to \(\frac{2}{3}\) which is \(\frac{{20}}{{30}};\frac{8}{{12}}\).

**Lesson 3: **Which of the following fractions is equal to \(\frac{{25}}{{100}}\) ?

\(\frac{{50}}{{150}} ;\frac{5}{{20}};\frac{8}{{32}}\)

__Solution guide:__

- Reduce fractions to minimal fractions (if possible). Fractions with the same minimum fraction are equal.

We have : \(\frac{{25}}{{100}} = \frac{{25:25}}{{100:25}} = \frac{1}{4}\)

\(\frac{{50}}{{150}} & = \frac{{50:50}}{{150:50}} = \frac{1}{3};\,\,\,\, \,\,\,\,\,\,\,\,\frac{5}{{20}} = \frac{{5:5}}{{20:5}} = \frac{1}{ 4};\,\,\,\,\,\,\,\,\,\,\,\frac{8}{{32}} = \frac{{8:8}}{{32:8 }} = \frac{1}{4}\)

So the fractions equal to \(\frac{{25}}{{100}}\) are \(\frac{5}{{20}};\frac{8}{{32}}\).

**Lesson 4:** Calculate (according to form)

a) \(\frac{{2 \times 3 \times 5}}{{3 \times 5 \times 7}}\); b) \(\frac{{8 \times 7 \times 5}}{{11 \times 8 \times 7}}\); c) \(\frac{{19 \times 2 \times 5}}{{19 \times 3 \times 5}}\);

*Template:* a)

*Attention :* In the above sample, we have divided the mental product above and the product below the dash by 3, and then divided it together by 5.

__Solution guide:__

- Mentally divide the product above and below the dash by the common factors.

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