정수와 유리수

$\frac{26}{7}$

$\frac{23}{8}$

$\frac{50}{11}$

$\frac{34}{3}$

ⓐ It takes four $\frac{1}{8}$ tiles to exactly cover the $\frac{1}{2}$ tile, so $\frac{4}{8}=\frac{1}{2}.$

ⓑ It takes five $\frac{1}{10}$ tiles to exactly cover the $\frac{1}{2}$ tile, so $\frac{5}{10}=\frac{1}{2}.$

ⓒ It takes six $\frac{1}{12}$ tiles to exactly cover the $\frac{1}{2}$ tile, so $\frac{6}{12}=\frac{1}{2}.$

Suppose you had tiles marked $\frac{1}{20}.$ How many of them would it take to equal $\frac{1}{2}?$ Are you thinking ten tiles? If you are, you’re right, because $\frac{10}{20}=\frac{1}{2}.$

We have shown that $\frac{1}{2},\frac{2}{4},\frac{3}{6},\frac{4}{8},\frac{5}{10},\frac{6}{12},$ and $\frac{10}{20}$ are all equivalent fractions.

2

3

To find a fraction equivalent to $\frac{2}{5},$ we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by $2,3,$ and $5.$

So, $\frac{4}{10},\frac{6}{15},$ and $\frac{10}{25}$ are equivalent to $\frac{2}{5}.$

Correct answers include $\frac{6}{10},\frac{9}{15},$ and $\frac{12}{20}.$

Correct answers include $\frac{8}{10},\frac{12}{15},$ and $\frac{16}{20}.$

To find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in $21.$

Since we can multiply $7$ by $3$ to get $21,$ we can find the equivalent fraction by multiplying both the numerator and denominator by $3.$

$\frac{18}{21}$

$\frac{30}{100}$

Start by locating the proper fraction $\frac{3}{4}.$ It is between $0$ and $1.$ To do this, divide the distance between $0$ and $1$ into four equal parts. Then plot $\frac{3}{4}.$

Next, locate the mixed number $4\frac{1}{5}.$ It is between $4$ and $5$ on the number line. Divide the number line between $4$ and $5$ into five equal parts, and then plot $4\frac{1}{5}$ one-fifth of the way between $4$ and $5$.

Now locate the improper fractions $\frac{4}{3}$ and $\frac{5}{3}$.

It is easier to plot them if we convert them to mixed numbers first.

$$\frac{4}{3}=1\frac{1}{3},\quad \frac{5}{3}=1\frac{2}{3}$$

Divide the distance between $1$ and $2$ into thirds.

Next let us plot $\frac{7}{2}.$ We write it as a mixed number, $\frac{7}{2}=3\frac{1}{2}$. Plot it between $3$ and $4.$

The number line shows all the numbers located on the number line.

Draw a number line. Mark $0$ in the middle and then mark several units to the left and right.

To locate $\frac{1}{4},$ divide the interval between $0$ and $1$ into four equal parts. Each part represents one-quarter of the distance. So plot $\frac{1}{4}$ at the first mark.

To locate $-\frac{1}{4},$ divide the interval between $0$ and $-1$ into four equal parts. Plot $-\frac{1}{4}$ at the first mark to the left of $0.$

Since $1\frac{1}{3}$ is between $1$ and $2,$ divide the interval between $1$ and $2$ into three equal parts. Plot $1\frac{1}{3}$ at the first mark to the right of $1.$ Then since $-1\frac{1}{3}$ is the opposite of $1\frac{1}{3}$ it is between $-1$ and $-2.$ Divide the interval between $-1$ and $-2$ into three equal parts. Plot $-1\frac{1}{3}$ at the first mark to the left of $-1.$

To locate $\frac{5}{2}$ and $-\frac{5}{2},$ it may be helpful to rewrite them as the mixed numbers $2\frac{1}{2}$ and $-2\frac{1}{2}.$

Since $2\frac{1}{2}$ is between $2$ and $3,$ divide the interval between $2$ and $3$ into two equal parts. Plot $\frac{5}{2}$ at the mark. Then since $-2\frac{1}{2}$ is between $-2$ and $-3,$ divide the interval between $-2$ and $-3$ into two equal parts. Plot $-\frac{5}{2}$ at the mark.