Transforming Terminating Decimals to Fractions and Vice Versa

Transforming Terminating Decimals to Fractions and Vice Versa exercise

• Write the decimals of the recipe as fractions.

  Hints

  Simplify fractions by first determining the largest number that divides into the numerator and the denominator evenly.

  Solution

  Jim's measuring cup gives all of the measurements in fractions. But many of the ingredients in his recipe are written as decimals of cup measurements.

  So Jim has to convert all the decimals in the recipe into fractions.

  Example:

  The decimal $0.75$ goes out to the hundredths place. We can read this as $75$ hundredths. In a fraction, $75$ is the numerator and $100$ is the denominator.

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

  Now the fraction can be simplified. Therefore, Jim has to determine the Greatest Common Factor of the numbers. That is, the largest number that divides both the numerator and the denominator evenly. The Greatest Common Factor of $75$ and $100$ is $25$. So Jim has to divide $75$ and $100$ by $25$.

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

  As you can see, $0.75$ is $\frac{3}{4}$ in fraction form.

• Find the correct decimals for each fraction.

  Hints

  Long division ends when $0$ is our remainder, or if we get a repeating number (or sequence of numbers). If we didn't stop, we'd be dividing FOREVER! (i.e. $\frac13 = 0.33333333...$)

  Solution

  Baking sometimes demands converting fractions into decimals or vice versa. Jim's recipe is written in decimals of cup measurements, but his cup just has fractions. What can he do?

  Fortunately, Jim knows that a fraction is just an unsolved division problem. That means that he can treat the fraction $\frac12$ as if he were dividing 2 into 1: $\frac12 = 1 \div 2$. After executing the long division, that leaves him with $1 \div 2 = 0.5$.

  The second measurement he has to convert is $\frac14$. For getting a decimal he just follows the same steps including the long division. $ 1 \div4 $ equals the decimal $0.25$.

• Convert each fraction into a decimal.

  Hints

  When transforming a fraction into a decimal, you divide the nominator by the denominator.

  Solution

  You can transform fractions into decimals. Fractions are just unsolved division problems which can be calculated by long division. Let's take a look at the following fractions.

  • $\frac18 = 1 \div 8 = 0.125$
  • $\frac38 = 3 \times \frac18 = 3 \times 0.125 = 0.375$
  • $\frac34= 3 \div 4 = 0.75$
  • $\frac78 = 7 \times \frac18 =7 \times 0.125 = 0.875$
  • $\frac9{10} = 9 \div 10 = 0.9$

• Identify where Sue succeeded in converting decimals into fractions.

  Hints

  You can convert a decimal to a fraction by putting the digits behind the decimal point in the nominator of the fraction. The number of digits behind the decimal point indicates the number of zeros behind the $1$ in the denominator.

  The decimal $0.7$ has one digit behind the decimal point. So the fraction is $\frac7{10}$.

  Solution

  It's no surprise this cake tastes disgusting. Let's take a look at the mistakes Sue made as well as the right conversions converting decimals into fractions.

  1. $0.6$ cups of flour is $\frac35$ths of a cup. That's right. You can calculate it like this: $0.6 = \frac{6}{10}= \frac{6\div2}{10\div2} = \frac35$.
  2. $0.45$ cups of butter were required. Let's convert it into fraction form: $0.45=\frac{45}{100} = \frac{45\div5}{100\div5} = \frac{9}{20}$. So Sue should have used $\frac9{20}$ cups of butter instead of $\frac{6}{20}$.
  3. $0.225$ cups of sugar are indicated in the recipe. What is the right fraction? $0.225 = \frac{225}{1000} = \frac{225\div25}{1000\div25} \div = \frac9{40}$. Sue has put $\frac{27}{40}$ cups of sugar into the cake. That is three times more sugar than requested.
  4. $0.3$ cups of apple sauce are $\frac3{10}$.

  The number of eggs and apples needed for the recipe are written in integers. So we don't have to convert any units.

  Now everything is totally clear and Jim and Sue know why their cake was a little too sugary and dry. Oh well, if at first you don't succeed...try again!

• Decide if the given numbers are fractions or decimals.

  Hints

  A fraction always has a nominator and a denominator.

  A decimal can have a thousandths place, a fraction cannot.

  Looking for decimals? Look for the decimal point.

  Solution

  Knowing different ways to represent numbers is important, too. You can often write a decimal as a fraction, and vice versa. Decimals look like this: $0.125$, $0.25$, $0.5$, $1.5$.

  We can transform them in fractions, which look like this: $\frac18$, $\frac14$.

• Find the numbers that represent the same value.

  Hints

  You can divide integers using long division.

  Solution

  Transforming fractions in decimals you can take a fraktion as a unsolved division. You simply divide the nominator by the denominator. Let's do this for all the fractions:

  • $\frac1{25} = 1 \div 25 = 0.04$
  • $\frac{12}{15} = 12 \div 15 = \frac45 = 0.8$
  • $\frac68 = 6 \div 8 = \frac34 = 0.75$
  • $\frac25 = 2 \div 5 = 0.4$