Rational Numbers on the Number Line

Rational Numbers on the Number Line exercise

• Explain how far Tim is above the water shortly after jumping.

  Hints

  $27'3''$ means $27$ feet and $3$ inches.

  Decimals are divided in tenths and hundredths.

  Solution

  The position of Tim shortly after jumping can be represented by a vertical number line. The surface of the water is our zero point. Above the water, we have the positive numbers. Below the water, we have the negative numbers.

  Tim's height right after jumping is $27'3''$ which means $27$ feet and $3$ inches. Because there are $12$ inches in one foot, we can express his position on the number line. It is $27\frac3{12}$, which can be simplified to $27 \frac14$.

  Alternatively his height can be transformed into decimal form: $27\frac14 = 27.25$.

• Determine the decimals and fractions which equal the given values.

  Hints

  There are $12$ inches in $1$ foot.

  You can simplify fractions by finding the Greatest Common Factor of the numerator and denominator. Then, you divide both numbers by the Greatest Common Factor.

  Fractions can be transformed in decimals by dividing the numerator by the denominator.

  Solution

  As you can see, there are several numbers that describe the same value. Feet can be converted to inches: There are $12$ inches in $1$ foot. So now we can express $6''$ as $\dfrac{6}{12}$. If we want, we can simplify it by converting it to a decimal, too.

• Find the height of each jump in feet and inches.

  Hints

  You can convert a decimal into a fraction, too.

  If we display decimals on the number line, we divide our units into tenths, hundredths and so on.

  Therefore, $5.75$ can be interpreted as $5 \frac{75}{100}$, which is a fraction that can be simplified.

  Solution

  Tommy, Richard, Lisa and Harry are jumping off the different diving boards at their local swimming pool. The height of the diving boards can be written as decimals on the number line as well as in feet and inches. Let's take a look:

  Tommy: $22.25$ ft.

  • $22.25$ ft. can be written as a fraction because decimals are divided in tenths and hundredths. So we have $22 \frac{25}{100}$ ft. Because the Greatest Common Factor of $25$ and $100$ is $25$, we can simplify the fraction to $\frac{1}{4}$. $22 \frac{1}{4} \text{ft.}$. Since one foot consists of $12$ inches, $22 \frac{1}{4} \text{ft.} = 22 \frac{3}{12}$ ft. So $22.25$ ft. is the same as $22' 3''$.

  Richard: $19.75$ ft.

  • $0.75$ ft. is $\frac{3}{4}$ ft. Multiplying the numerator and denominator by $3$ makes the denominator $12$, the number of inches in a foot. The number of inches is in the numerator, making our final conversion $19.75\text{ft.} = 19'9''$.

  Lisa: $15.5$ ft.

  • $0.5$ ft. is $\frac{1}{2}$ ft. Multiplying the numerator and denominator by $6$ makes the denominator $12$, the number of inches in a foot. The number of inches is in the numerator, making our final conversion $15.5\text{ft.} = 15'6''$.

  Harry: $12.5$ ft.

  • Again, $0.5$ ft. is $\frac{1}{2}$ ft. Multiplying the numerator and denominator by $6$ makes the denominator $12$, the number of inches in a foot. The number of inches is in the numerator, making our final conversion $12.5\text{ft.} = 12'6''$.

• Convert the depths to see which diver dove deepest.

  Hints

  Convert the depths to the same unit of measure in order to compare more easily.

  Every depth can be written as a fraction as well as in feet and inches.

  Solution

  We want to find out who dove the deepest. Therefore, we have to compare the different values. It is a good idea to convert all the depths so they're expressed in the same units. In the table you can see all conversions. We have ordered the list beginning with the smallest depth to the greatest.

  1. $6.25$ ft.
  2. $6 \frac4{12}$ ft.
  3. $6'5''$
  4. $6 \frac12 $ ft.

• Identify the steps for converting height from fraction to decimal form.

  Hints

  There are $12$ in $1$ foot.

  So in fraction form, $10$ inches is $\frac{10}{12}$ of one foot.

  The Greatest Common Factor of $10$ and $12$ is $2$. So we can simplify the fraction to $\frac{10\div2}{12\div2} = \frac{5}{6}$.

  Solution

  A distance given in feet and inches can be transformed into a decimal by following these simple steps:

  1. We have to determine the distance in feet and inches.
  2. We know that inches are smaller divisions of a foot. There are $12$ inches in one foot.
  3. This can be represented as a fraction. We put the inches in the numerator and the total number of inches in a foot, $12$, in the denominator.
  4. Finally, we divide the numerator by the denominator, getting the decimal.

  e.g. $\frac{5}{6} = 5 \div 6 = 0.8\overline{3}$

• Find out which distances Marshall ran this week.

  Hints

  You can convert every single value given into a fraction or decimal, showing the distance in miles. Alternatively, you can express miles in feet.

  One helpful conversion you should keep in mind is: $\frac{1}{4}$ mile is $1320$ ft.

  Solution

  Marshall seems to be a really good runner. Let's take a look at the distances he covered this week. To compare the distances Marshall ran, we should convert all the numbers we have been given into one form. Let's begin with his shortest distance.

  Remember, when converting units:

  • $\text{feet} \times \dfrac{\text{miles}}{\text{feet}} = \text{miles}$

  1. The distance of $37488\text{ ft.}$ can be transformed like this:

  $\dfrac{37488\text{ ft.}}{5280\text{ ft.}} = 7.1 \text{ mi.}$

  $~$

  2. The next distance is $7.2$ miles, which is equal to:

  $\begin{array}{rcl} 7 \text{ mi.} + 0.2 \text{ mi.} &=&\\ 7 \text{ mi.} + \frac{2}{10} \text{ mi.} &=&\\ 7 \times 5280 \text{ ft.} + \frac{5280 \text{ ft.} \times 2}{10} &=&\\ 36,960 \text{ ft.} + 1,056 \text{ ft.} &=& 38,016 \text{ ft.} \end{array}$

  $~$

  3. The distance $7 \frac{10}{20}$ miles ($7.5$ miles in decimal form) can be converted to feet as follows:

  $\begin{array}{rcl} 7 \text{ mi.} + \frac{10}{20} \text{ mi.} &=&\\ 7 \text{ mi.} + \frac12 \text{ mi.} &=&\\ 7 \times 5280 \text{ ft.} + \frac{5280 \text{ ft.}}{2} &=&\\ 36,960 \text{ ft.} + 2640 \text{ ft.} &=& 39600 \text{ ft.} \end{array}$

  $~$

  4. The distance $7 \frac{3}{5}$ miles can be transformed the same way:

  $\begin{array}{rcl} 7 \text{ mi.}+ \frac{3}{5} \text{ mi.} &=&\\ 7 \times 5280 \text{ ft.} + \frac{5280 \text{ ft.} \times 3}{5} &=&\\ 36,960 \text{ ft.} + 3168 \text{ ft.} &=& 40128 \text{ ft.} \end{array}$

  $~$

  5. The last distance of $41712\text{ ft.}$ can be transformed into miles the following way:

  $41712\text{ ft.} \div 5280 = 7.9 \text{ mi.}$

  $~$

  See? It's much easier to compare distances when they are all in the same unit of measurement. And it`s easier to label numbers on the number line when they are transformed into decimals or mixed fractions.