Compare two fractions with the same denominator

### 1.1. Knowledge to remember

**For example :** Compare two fractions \(\frac{2}{5}\) and \(\frac{3}{5}\).

Draw line segment AB. Divide line segment AB into 5 equal parts. The length of line segment AC is equal to \(\frac{2}{5}\) length of line segment AB. The length of line segment AD is equal to \(\frac{3}{5}\) length of line segment AB.

Looking at the picture we see:

\(\frac{2}{5}\) \(< \frac{3}{5}\) ; \(\frac{3}{5} > \) \(\frac{2}{5}\)

*In two fractions with the same denominator:*

*The fraction with the smaller numerator is smaller.**The fraction with the larger numerator is larger.**If the numerators are equal, then the two fractions are equal.*

### 1.2. Textbook exercise solution page 119

**Lesson 1:** Compare two fractions

\(\frac{3}{7}\) and \( \frac{5}{7}\) b) \(\frac{4}{3}\) and \(\frac{2}{3} \) c) \(\frac{7}{8}\) and \(\frac{5}{8}\) d) \(\frac{2}{{11}}\) and \(\frac {9}{{11}}\)

**Solution guide:**

In two fractions with the same denominator:

- The fraction with the smaller numerator is smaller.
- The fraction with the larger numerator is larger.
- If the numerators are equal, then the two fractions are equal.

a) \(\frac{3}{7} < \frac{5}{7}\) b) \(\frac{4}{3} > \frac{2}{3}\)

c) \(\frac{7}{8} > \frac{5}{8}\) d) \(\frac{2}{{11}} < \frac{9}{{11}}\)

**Lesson 2:**

a) Comment:

\(\frac{2}{5} < \frac{5}{5}\) where \(\frac{5}{5}=1\) should \(\frac{2}{5} < 1\ ).

If the numerator is less than the denominator, the fraction is less than 1.

\(\frac{8}{5} > \frac{5}{5}\) where \(\frac{5}{5}=1\) should \(\frac{8}{5} >1\ ).

b) Compare the following fractions with 1 :

\(\frac{1}{2};\frac{4}{5};\frac{7}{3};\frac{6}{5};\frac{9}{9}\); \(\frac{{12}}{7}\)

__Solution guide:__

- If the numerator is less than the denominator, the fraction is less than 1.
- If the numerator is greater than the denominator, the fraction is greater than 1.
- If the numerator is equal to the denominator, the fraction is equal to 1.

\(\frac{1}{2} < 1\) ; \(\frac{4}{5} < 1\) ; \(\frac{7}{3} > 1\) ;

\(\frac{6}{5} > 1\); \(\frac{9}{9}=1\) ; 1\(\frac{{12}}{7} > 1\).

**Lesson 3: **Write fractions less than 1, whose denominator is 5 and the numerator is not 0

__Solution guide:__

- If the numerator is less than the denominator, the fraction is less than 1.

Fractions less than 1, with a denominator of 5 and a non-zero numerator are:

\(\frac{1}{5};\frac{2}{5};\frac{3}{5};\frac{4}{5}\)

### 1.3. Textbook exercise solution page 120

**Lesson 1:** Compare two fractions

a) \(\frac{3}{5}\) and \(\frac{1}{5}\) b) \(\frac{9}{{10}}\) and \(\frac{{ 11}}{{10}}\)

c) \(\frac{{13}}{{17}}\) and \(\frac{{15}}{{17}}\) d) \(\frac{{25}}{{19} }\) and \(\frac{{22}}{{19}}\)

__Solution guide:__

In two fractions with the same denominator:

- The fraction with the smaller numerator is smaller.
- The fraction with the larger numerator is larger.
- If the numerators are equal, then the two fractions are equal.

a) \(\frac{3}{5} > \frac{1}{5}\) b) \(\frac{9}{{10}} < \frac{{11}}{{10}} \)

c) \(\frac{{13}}{{17}} < \frac{{15}}{{17}}\) d) \(\frac{{25}}{{19}} > \frac {{22}}{{19}}\)

**Lesson 2: **Compare the following fractions with 1

\(\frac{1}{4};\frac{3}{7};\frac{9}{5};\frac{7}{3};\frac{{14}}{{15}} ;\frac{{16}}{{16}};\frac{{14}}{{11}}\)

__Solution guide:__

- If the numerator is less than the denominator, the fraction is less than 1.
- If the numerator is greater than the denominator, the fraction is greater than 1.
- If the numerator is equal to the denominator, the fraction is 1.

\(\frac{1}{4} < 1\) ; \(\frac{3}{7} < 1\) ; \(\frac{9}{5} > 1\) ; \(\frac{7}{3} > 1\) ;

\(\frac{{14}}{{15}} < 1\) ; \(\frac{{16}}{{16}}=1\) ; \(\frac{{14}}{{11}} > 1\)

**Lesson 3: **Write fractions in order from smallest to largest

a) \(\frac{1}{5};\frac{4}{5};\frac{3}{5}\) b) \(\frac{6}{7};\frac{8} {7};\frac{5}{7}\)

c) \(\frac{8}{9};\frac{5}{9};\frac{7}{9}\) d) \(\frac{{12}}{{11}};\ frac{{16}}{{11}};\frac{{10}}{{11}}\)

__Solution guide:__

In two fractions with the same denominator:

- The fraction with the smaller numerator is smaller.
- The fraction with the larger numerator is larger.
- If the numerators are equal, then the two fractions are equal.

a) Since 1 < 3 < 4, we have \(\frac{1}{5} < \frac{3}{5} < \frac{4}{5}\)

So the fractions written in order from smallest to largest are: \(\frac{1}{5};\frac{3}{5};\frac{4}{5}\)

b) Since 5 < 6 < 8 we have \(\frac{5}{7} < \frac{6}{7} < \frac{8}{7}\)

So the fractions written in order from smallest to largest are \(\frac{5}{7};\frac{6}{7};\frac{8}{7}\)

c) Since 5 < 7 < 8 we have \(\frac{5}{9} < \frac{7}{9} < \frac{8}{9}\)

So the fractions written in order from smallest to largest are \(\frac{5}{9};\frac{7}{9};\frac{8}{9}\)

d) Since 10 < 12 < 16, we have \(\frac{{10}}{{11}} < \frac{{12}}{{11}} < \frac{{16}}{{11} }\)

So the fractions written in order from smallest to largest are \(\frac{{10}}{{11}};\frac{{12}}{{11}};\frac{{16}}{{) 11}}\).

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