Ordering Rational Numbers

Ordering Rational Numbers exercise

• What is the definition of a rational number?

  Hints

  Examples of rational numbers are $3$, $8$, -$2$, $\frac{1}{2}$.

  A rational number terminates, therefore $\sqrt{3}$ is not a rational number because the value of this is $1.732050808...$ which never ends.

  A rational number can be less than $1$. For example -$3$, -$5$ and -$2.3$ are all rational.

  Solution

  A rational number is a number that can be written as a fraction, where the denominator and numerator are integers.

  Integers are whole numbers.

  For example, $0.4$ is rational, as it can be written as $\frac{2}{5}$

• Order rational numbers from least to greatest.

  Hints

  Arrange the amount of pie each person has left, from least to greatest.

  Convert all the fractions to decimals. For example,

  Line the decimals up with the decimal points and compare from left to right. For example, we can see the decimals lined up here. We start from left to right to find the smallest number first.

  Solution

  Results are seen in the table.

  We convert all the numbers to decimals

  $\frac{1}{4}$ = $\frac{25}{100}$ = $0.25$

  $\frac{1}{5}$ = $\frac{20}{100}$ = $0.2$

  Line them up with the decimal points and compare from left to right

  $0.30$

  $0.25$

  $0.19$

  $0.20$

  The zeros help with comparison.

  $0.19$ is the smallest, then $0.2$, then $0.25$, and $0.3$ in $4$th place.

• Arrange the following rational numbers on the spreadsheet from least to greatest.

  Hints

  Convert and order the negative numbers and the positive numbers separately, before putting them together.

  Convert the numbers to decimals and split them into positive and negative.

  Converting to decimals we can change $\frac{-7}{4}$ to a mixed number = $-1$ whole and $\frac{3}{4}$.

  $\frac{3}{4}$ = $\frac{75}{100}$ = $0.75$

  Therefore, $\frac{-7}{4}$ = $-1.75$

  Solution

  Least to greatest:

  • $-2$
  • $ -1.8$
  • $\frac{-7}{4}$
  • $1$
  • $1.5$
  • $\frac{5}{2}$

  Convert all the numbers to decimals and split them into positive and negative.

  Positive Numbers: $1.5$, $1$ and $\frac{5}{2}$ = $2.5$

  Negative Numbers: $-2$, $-1.8$ and $\frac{-7}{4}$ = $-1.75$

• Find the median number of the following list.

  Hints

  The median is the middle number when the values are arranged from least to greatest.

  1) Convert any fractions to decimals.

  2) Arrange the values from least to greatest.

  3) Find the middle number, which is the median.

  We find the median by arranging the data from least to greatest, then choosing the middle value.

  For example, $3$, $5$, $2$, $1$, $8$.

  Arrange from least to greatest $1$, $2$, $3$, $5$, $8$

  $\bf{3}$ is the middle, so it is the median!

  Solution

  Median = $\bf{0.77}$

  Convert the numbers to decimals. then order them from least to greatest.

  $\frac{3}{4}$ =$ 0.75$

  $\frac{7}{10}$ = $0.7$

  $\frac{4}{5}$ = $0.8$

  Line them up and the median is the middle number

  $0.70$, $0.75$, $\bf{0.77}$, $0.80$, $0.90$

• Arrange the following rational numbers from least to greatest.

  Hints

  Remember, $\frac{11}{10}$ = $1.1$, greater than $1$.

  Line the numbers up with the decimal points, and compare from left to right.

  It may help if you draw a number line and position the numbers on there.

  Solution

  $0.93$, $\frac{19}{20}$, $\frac{11}{10}$, $1.2$

  • Convert any fractions to decimals:

  $\frac{19}{20}$ = $\frac{95}{100}$ = $0.95$ (see number line)

  $\frac{11}{10}$ = $\frac{110}{100}$ = $1.1$, (see number line)

  • Line them up with the decimal points and compare from left to right.

  $0.95$

  $1.20$

  $1.10$

  $0.93$

  The zeros help with comparison.

• When $a = 0.2$ order the following from least to greatest.

  Hints

  Substitute the value $0.2$ into each expression.

  For example, $2a = 2 \cdot 0.2 = 0.4$

  Convert the values to decimals for comparison and order the values from least to greatest.

  Solution

  • $a^2$
  • $\frac{a}{2}$
  • $2a$
  • $a + 1$

  ${}$ Evaluate each expression for $a=0.2$.

  $2a = 2\times 0.2 = \bf{0.4}$

  $\frac{a}{2} = \frac{0.2}{2} = \bf{0.1}$

  $a^2 = 0.2 \times 0.2 = \bf{0.04}$

  $a + 1 = 0.2 + 1= \bf{1.2}$