Transforming Simple Repeating Decimals to Fractions and Vice Versa

Transforming Simple Repeating Decimals to Fractions and Vice Versa exercise

• Explain how to divide 100 dollars evenly between three people.

  Hints

  Divide $9$ by $3$:

  $\frac93=\frac{3}{1}=3$

  Subtract $9$ from $10$ to get $10-9=1$

  • one $\$100$ bill is equal to ten $\$10$ bills
  • one $\$10$ bill is equal to ten $\$1$ bills
  • one $\$1$ bill is equal to ten dimes
  • one dime is equal to ten pennies

  Solution

  How can we divide $\$100$ evenly amongst the three band members?

  First, we exchange the $\$100$ bills for ten $\$10$ bills.

  Each band member gets three $\$10$ bills, but one $\$10$ bill is left.

  Next, we have to exchange the last $\$10$ bill for ten $\$1$ bills.

  Again, everyone gets three $\$1$ bills and, again, one $\$1$ bill is left.

  In the same manner, we can proceed until only one penny is left.

  We can describe this situation with a fraction $ \frac13$, which can be written as $1\div3$. This is the part of a penny each member is entitled to.

• Define a repeating decimal.

  Hints

  $1\div 3=0.3333333........$ is an infinite but periodic decimal.

  $0.\bar3$ is a repeating decimal.

  Solution

  Dividing $1$ by $3$ gives us $0.3333...$ with infinite $3$s behind the decimal point. Numbers with one or more repeating decimals are infinite, or without end, and are called repeating decimals.

  The number of repeated decimal places can vary:

  $\begin{array}{lcl} 0.\bar2 &&1 \text{ repeating number after the decimal}\\ 0.\overline{36} &&2 \text{ repeating numbers after the decimal}\\ 0.\overline{459} &&3 \text{ repeating numbers after the decimal} \end{array}$

  All these numbers are written with a horizontal bar above the decimals that are repeated.

  For example $0.\overline{459}=0.459459459459.....$.

  Repeating decimals are infinite.

• Decide how much money each band member should receive.

  Hints

  First, take a look at the amount of money they will be paid at each location. Then, see how you can divide that amount amongst the $3$ boys.

  For example, if the boys got paid $\$110$ for a concert in Plano, Texas, the equation would look like this:

  • $\dfrac{\text{total money}}{\text{number of boys}}=\dfrac{\$110}{3}$

  $\begin{array}{rcl} \dfrac{\$110}{3}&=&\$36\frac23\\ \end{array}$

  Solution

  Lucky for them, the Musical Triplets are not only excellent singers, they are also very talented mathematicians, so they are able to calculate how much each band member will receive for each gig.

  • $\dfrac{\text{total money}}{\text{number of boys}}$

  Boston

  $\begin{array}{rcl} \dfrac{\$160}{3}&=&\$53\frac13\\ \end{array}$

  Since we know that $1\div3=0.\overline{3}$, we can say that each band member will be paid $\$53.\bar3$.

  Chicago

  $\begin{array}{rcl} \dfrac{\$200}{3}&=&\$66\frac23\\ \end{array}$

  Since $2\div3=0.\overline{6}$, each band member will be paid $\$66.\bar6$.

  Memphis

  $\begin{array}{rcl} \dfrac{\$205}{3}&=&\$68\frac13\\ \end{array}$

  Each band member will be paid $\$68.\overline{3}$.

  Atlanta

  $\begin{array}{rcl} \dfrac{\$130}{3}&=&\$43\frac13\\ \end{array}$

  Each band member will be paid $\$43.\overline{3}$.

• Write repeating decimals as fractions.

  Hints

  Simplify the fraction by dividing the numerator and the denominator by their GCF (Greatest Common Factor).

  Solution

  You can write a repeating decimal as a fraction as follows:

  $0.\overline{123}=\frac{123}{999}$

  1. We place the repeated numbers in the numerator.
  2. We place as many $9$s in the denominator as the number of digits that repeat in the numerator.

  Now, we should try and simplify the fraction by dividing the nominator and denominator by the Greatest Common Factor (GCF):

  $\frac{123}{999}=\frac{123\div3}{999\div 3}=\frac{41}{333}$

  $~$

  1. We can write the repeating decimal $\mathbf{0.\overline{45}}$ as a simplified fraction:

  $\begin{align*} 0.\overline{45}&=\frac{45}{99}\\ &=\frac{45\div 9}{99\div 9}\\ &=\frac{5}{11}\\ \end{align*}$

  2.We can write the repeating decimal $\mathbf{0.\overline{324}}$ as a simplified fraction:

  $\begin{align*} 0.\overline{324}&=\frac{324}{999}\\ &=\frac{324\div 27}{999\div 27}\\ &=\frac{12}{37}\\ \end{align*}$

  3.We can write the repeating decimal $\mathbf{0.\overline{18}}$ as a simplified fraction:

  $\begin{align*} 0.\overline{18}&=\frac{18}{99}\\ &=\frac{18\div 9}{99\div 9}\\ &=\frac{2}{11}\\ \end{align*}$

  4.We can write the repeating decimal $\mathbf{0.\overline{132}}$ as a simplified fraction:

  $\begin{align*} 0.\overline{132}&=\frac{132}{999}\\ &=\frac{132\div 3}{999\div 3}\\ &=\frac{44}{333} \end{align*}$

• Convert the fraction $\frac13$ into a decimal by using long division.

  Hints

  You have to do the same steps over and over again.

  To show that a decimal is a repeating decimal, place a horizontal line over the repeating numbers.

  Solution

  You already know that a fraction bar indicates a division.

  So we can say that $\frac13=1\div3$.

  But how can you divide $1$ by $3$? On the right, you can see the problem worked out.

  If we use long division, we will always have a remainder of $1$. We can repeat this process over and over and OVER again.

  $1\div 3=0.33333......$.

  We call this a repeating decimal and indicate the repeating portion by placing a horizontal line over the numbers:

  • $0.\overline3$
  • $0.\overline{623}$
  • $0.\overline{1025}$

  ...and so on.

• Determine the repeating decimal equivalent for each given fraction and vice versa.

  Hints

  The factors

  • of $99$ are: $1$, $3$, $9$, $11$, $33$, and $99$
  • of $999$ are: $1$, $3$, $9$, $27$, $37$, $111$, $333$, and $999$

  Is $37$ a factor of $99$?

  Is $37$ a factor of $999$?

  You can write every repeating decimal in fraction form in which the denominator has as many $9$s as the number of digits that repeat in the numerator.

  For example $0.\overline{1234}$ would look like this as a fraction:

  • $\dfrac{1234}{9999}$

  Solution

  When transforming fractions into repeating decimals...

  • You can use long division
  • Or you can multiply the fraction so that the denominator is $9...9$

  $~$

  For the fraction $\frac{46}{111}$:

  If we multiply both the numerator and the denominator by $9$ we get:

  $\begin{array}{rcl} \frac{46}{111}&=&\frac{46\times 9}{111\times 9}\\ &=&\frac{414}{999}\\ &=&0.\overline{414} \end{array}$

  $~$

  For the fraction $\frac{4}{27}$:

  We multiply both the numerator and the denominator by $37$ to get:

  $\begin{array}{rcl} \frac{4}{27}&=&\frac{4\times 37}{27\times 37}\\ &=&\frac{148}{999}\\ &=&0.\overline{148} \end{array}$

  $~$

  To transform repeating decimals into fractions:

  • write all repeating numbers in the numerator
  • in the denominator, place as many $9$s as there are repeating digits
  • reduce the fraction by dividing the numerator and denominator by the GCF

  $~$

  For the repeating decimal $0.\overline{81}$:

  $\begin{array}{rcl} 0.\overline{81}&=&\frac{81}{99}\\ &=&\frac{81\div9}{99\div 9}\\ &=&\frac{9}{11} \end{array}$

  $~$

  For the repeating decimal $0.\overline{63}$:

  $\begin{array}{rcl} 0.\overline{63}&=&\frac{63}{99}\\ &=&\frac{63\div9}{99\div 9}\\ &=&\frac{7}{11} \end{array}$