Fractions and division of natural numbers (continued)

### 1.1. Knowledge to remember

a) Example 1: There are 2 oranges, divide each into 4 equal parts. Van ate 1 orange and \(\frac{1}{4}\) orange. Write the fraction of the number of oranges Van ate.

We see :

Eating 1 orange means eating 4 servings or \(\frac{4}{4}\) oranges ; eat more \(\frac{1}{4}\) more oranges, ie eat 1 more portion, so Van ate all 5 portions or \(\frac{5}{4}\) oranges.

b) Example 2: Divide 5 oranges equally among 4 people. Find each person’s oranges.

We can do the following: Divide the orange into 4 equal parts. Give each person a portion, i.e. \(\frac{4}{4}\) of each orange in turn. After 5 such divisions, each person gets 5 portions or \(\frac{5}{4}\) oranges.

So : \(5:4 = \frac{5}{4}\) (orange).

c) *Comment :*

- The result of division of a natural number by a (non-zero) natural number can be written as a fraction, for example: \(5:4 = \frac{5}{4}\).

\(\frac{5}{4}\) orange consists of 1 orange and \(\frac{1}{4}\) oranges, so \(\frac{5}{4}\) oranges more than 1 orange.

We write: \(\frac{5}{4} > 1\).

- The fraction \(\frac{5}{4}\) has a numerator greater than the denominator, which is greater than 1.

- The fraction \(\frac{4}{4}\) has a numerator equal to the denominator, which is 1.

We write: \(\frac{4}{4} = 1\).

- The fraction \(\frac{1}{4}\) has a numerator less than the denominator, which is less than 1.

We write: \(\frac{1}{4} < 1\).

### 1.2. Solve the textbook exercises page 110

Lesson 1: Write the quotient of each of the following division as a fraction

9: 7 ; 8: 5 ; 19 : 11 ; 3: 3 ; 2 : 15.

__Solution guide:__

- The result of dividing a natural number by a (non-zero) natural number can be written as a fraction, for example \(5:4 = \frac{5}{4}\).

\(9:7 = \frac{9}{7}\) \(8:5 = \frac{8}{5}\)

\(19:11 = \frac{{19}}{{11}}\) \(3:3 = \frac{3}{3}\) \(2:15 = \frac{2}{{15) }}\)

Lesson 2: There are two fractions 7676 and 712712, which one represents the colored part of figure 1 ? Which fraction represents the colored part of figure 2 ?

a)

b)

__Solution guide:__

- Observe the figure to find the fraction indicating the colored part of each shape.

The fraction \(\frac{7}{6}\) indicates the shaded part of figure 1.

The fraction \(\frac{7}{{12}}\) indicates the shaded part in part 2.

Lesson 3:

In fractions : \(\frac{3}{4} & ;\frac{9}{{14}};\frac{7}{5};\frac{6}{{10}};\frac {{19}}{{17}};\frac{{24}}{{24}}\)

a) Which fraction is less than 1?

b) Which fraction is equal to 1?

c) Which fraction is greater than 1?

__Solution guide:__

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

a) \(\frac{3}{4} < 1\) ; \(\frac{9}{{14}} < 1\); \(\frac{6}{{10}} < 1\)

b) \(\frac{{24}}{{24}} = 1\)

c) \(\frac{7}{5} > 1\) ; \(\frac{{19}}{{17}} > 1\)

### 1.3. Solve the exercises Textbook Practice pages 110, 111

**Lesson 1: **Read quantity measurements: \(\frac{1}{2}kg;\frac{5}{8}m;\frac{{19}}{{12}}\) hours; \(\frac{6}{{100}}m\) .

__Solution guide:__

- When reading fractions, we read the numerator first, the dash reads as “part”, then read the denominator; If there is a unit of measure, then we read the name of the unit of measure.

\(\frac{1}{2}kg\) reads as: one-half of a kilogram ;

\(\frac{5}{8}m\) reads as: five eighths of a meter ;

\(\frac{{19}}{{12}}\) the reading time is: nineteen twelve o’clock ;

\(\frac{6}{{100}}m\) reads as: six hundredths of a meter.

**Lesson 2: **Write fractions: one-quarter, six-tenths; eighteen out of eighty-five; seventy-two parts one hundred.

__Solution guide:__

- When reading fractions, we read the numerator first, the dash read as “part”, then read the denominator. From there, we can write a fraction based on how it is read.

A quarter is written as : \(\frac{1}{4}\) ;

Six tenths are written as : \(\frac{6}{{10}}\) ;

Eighteen out of eighty-five is written as : \(\frac{{18}}{{85}}\) ;

Seventy-two parts per hundred is written as : \(\frac{{72}}{{100}}\).

**Lesson 3: **Write each of the following natural numbers as a fraction with a denominator of 1.

\(8;\,\,\,\,\,\,14;\,\,\,\,\,\,\,\,32;\,\,\,\,\,\,\, \,\,0;\,\,\,\,\,\,\,1\)

__Solution guide:__

- The result of dividing a natural number by a (non-zero) natural number can be written as a fraction, for example \(5:4 = \frac{5}{4}\).

\(8 = \frac{8}{1}\) ; \(14 = \frac{{14}}{1}\) ;

\(32 = \frac{{32}}{1}\) ; \(0 = \frac{0}{1}\) ; \(1 = \frac{1}{1}\).

**Lesson 4: **Write a fraction

a) Less than 11 ; b) Equals 11 ; c) Greater than 11 .

__Solution guide:__

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

The following fractions can be selected:

a) \(\frac{3}{5}\) b) \(\frac{4}{4}\) c) \(\frac{7}{3}\).

**Lesson 5: **Each of the following lines is divided into equal lengths. Write in the dot according to the pattern

*Template: *

\(AI = \frac{1}{3}AB\)

\(IB = \frac{2}{3}AB\)

*Attention : *Writing \(AI = \frac{1}{3}AB\) is a shorthand for : The length of line segment AI is equal to \(\frac{1}{3}\) the length of line AB.

a)

CP = … CD

PD = … CD

b)

MO = … MN

ON = … MN

__Solution guide:__

- Observe the sample example to find fractions that fit the given line segments.

Students write in the marks as follows:

a) CP = \(\frac{3}{4}\) CD b) OM = \(\frac{2}{5}\) MN

PD = \(\frac{1}{4}\) CD ON = \(\frac{3}{5}\) MN

.

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