Review Basic properties of fractions

### 1.1. Basic properties of fractions

- If you multiply both the numerator and the denominator of a fraction by the same nonzero natural number, you get a fraction equal to the given fraction.
- If both numerator and denominator of a fraction are divisible by the same non-zero natural number, then a fraction is equal to the given fraction.

**For example:** \(\frac {5 }{6}\) = \(\frac {5 \times 3 }{6 \times 3}\) = \(\frac {15 }{18}\).

**For example:** \(\frac {15: 3 }{18:3}\) = \(\frac {5 }{6}\).

### 1.2. Apply basic properties of fractions

Simplify fractions.

**For example:** \(\frac {90 }{120}\) = \(\frac {90 :10}{120:10}\) = \(\frac {9}{12}\) = \(\frac {9 : 3}{12:3}\) = \(\frac {3}{4}\)

or: \(\frac {90 }{120}\) = \(\frac {90 : 30}{120 : 30}\) = \(\frac {3}{4}\); …

Convert the denominators of the fractions.

**For example:** Reduce the denominators of \(\frac {2}{5}\) and \(\frac {4}{7}\).

Take the product of 5 x 7 = 35 as the common denominator (MSC). We have:

\(\frac {2}{5}\) = \(\frac {2 \times 7}{5\times 7}\) = \(\frac {14}{35}\); \(\frac {4}{7}\) = \(\frac {4\times 5}{7\times 5}\) = \(\frac {20}{35}\).

**For example: **Reduce the denominators of \(\frac {3}{5}\) and \(\frac {9}{10}\).

__Comment:__ 10 : 5 = 2, choose 10 as MSC. We have:

\(\frac {3}{5}\) = \(\frac {3\times 2}{5\times 2}\) = \(\frac {6}{10}\); keep \(\frac {9}{10}\).

### 1.3. Instructions for solving textbook exercises

**Lesson 1 of the textbook page 6:** Simplify fractions:

\(\frac{{15}}{{25}};\,\,\frac{{18}}{{27}};\,\,\frac{{36}}{{64}}\)

**Solution**

\(\begin{array}{l}

\frac{{15}}{{25}} = \frac{{15:5}}{{25:5}} = \frac{3}{5}\\

\frac{{18}}{{27}} = \frac{{18:9}}{{27:9}} = \frac{2}{3}\\

\frac{{36}}{{64}} = \frac{{36:4}}{{64:4}} = \frac{9}{{16}}

\end{array}\)

**Lesson 2 Textbook page 6:** Denominator of fractions

a) \(\frac{2}{3}\) and \(\frac{5}{8}\)

b) \(\frac{1}{4}\) and \(\frac{7}{12}\)

c) \(\frac{5}{6}\) and \(\frac{3}{8}\)

**Solution**

a) MSC: 3 x 8 = 24

\(\begin{array}{l}

\frac{2}{3} = \frac{{2 \times 8}}{{3 \times 8}} = \frac{{16}}{{24}}\\

\frac{5}{8} = \frac{{5 \times 3}}{{8 \times 3}} = \frac{{15}}{{24}}

\end{array}\)

b) MSC: 12

\(\begin{array}{l}

\frac{1}{4} = \frac{{1 \times 3}}{{4 \times 3}} = \frac{3}{{12}}\\

\frac{7}{{12}}

\end{array}\)

c) MSC: 6 x 8 = 48

\(\begin{array}{l}

\frac{5}{6} = \frac{{5 \times 8}}{{6 \times 8}} = \frac{{40}}{{48}}\\

\frac{3}{8} = \frac{{3 \times 6}}{{8 \times 6}} = \frac{{18}}{{48}}

\end{array}\)

**Lesson 3 Textbook page 6: **Find equal fractions in the following fractions:

\(\frac{2}{5};\frac{4}{7};\frac{{12}}{{30}};\frac{{12}}{{21}};\frac{{ 20}}{{35}};\frac{{40}}{{100}}\)

**Solution**

\(\begin{array}{l}

\frac{{12}}{{30}} = \frac{{12:6}}{{30:6}} = \frac{2}{5}\\

\frac{{12}}{{21}} = \frac{{12:3}}{{21:3}} = \frac{4}{7}\\

\frac{{20}}{{35}} = \frac{{20:5}}{{35:5}} = \frac{4}{7}\\

\frac{{40}}{{100}} = \frac{{40:20}}{{100:20}} = \frac{2}{5}

\end{array}\)

So \(\frac{{12}}{{30}} = \frac{{40}}{{100}} = \frac{2}{5};\frac{{12}}{{21}} = \frac{{20}}{{35}} = \frac{4}{7}\)

.

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