What are Mixed numbers?
A mixed number is a combination of a whole number and a proper fraction.
It usually denotes a number that falls between two whole numbers. A whole number, a numerator and a denominator are combined to create a mixed number. The numerator and denominator are both parts of the mixed number’s proper fraction and is partly a fraction and a whole number.
For example, if there are \(2 \frac{1}{2}\) glasses of milk, then in this, \(2 \frac{1}{2}\) is a mixed number.
Converting Mixed numbers to Decimals
Long division can be used to write fractions and mixed numbers as decimals because any integer can be divided by any nonzero integer. These decimals are rational numbers that will either terminate or repeat.
- A decimal that terminates or ends is known as a terminating or a non-repeating decimal. For example, 1.6, -0.54, 16.245.
- A decimal with a repeating pattern is known as a repeating or a non-terminating decimal.
For example, \(-6.222….= -6.\overline{2}, 0.24242…..= 0.\overline{24}\). We can use bar notation to show which digit or group of digits repeat.
For some cases, we can use another way to convert a mixed number to a decimal. First, express the mixed number as an improper fraction and then convert it to an equivalent fraction whose denominator is equal to a power of 10.
Converting Decimals to Mixed numbers or Fractions
Any terminating decimal can be written as a fraction with a power of 10 as the denominator. You can often simplify the resulting fraction using common factors, that is, removing the common factor from the numerator and denominator.
For example, \(0.78 = \frac{78}{100} = \frac{78 \div\ 2}{100 \div\ 2} = \frac{39}{50}\) (or) \(0.78 = \frac{78}{100} = \frac{39 \times 2}{50 \times 2} = \frac{39}{50}\)
This example represents two ways to simplify the fraction. We can use any of the two ways to simplify fractions.
Solved Mixed Numbers Examples
Example 1: Write \(-3 \frac{1}{4}\) as a decimal.
Solution: We have to notice that \(-3 \frac{1}{4} = – \frac{13}{4}\)
Method 1: Use long division to divide 13 by 4.
Divide 13 by 4.
Therefore the remainder is 0. So, it is a terminating decimal.
So, \(- 3 \frac{1}{4} = -3.25\).
Method 2: Use equivalent fractions.
\(\frac{13}{4} = \frac{1\ \times25}{4\ \times\ 25} = \frac{25}{100}\)
So, \(- 3 \frac{1}{4} = -3 \frac{25}{100}\)
\(= -3.25\).
Example 2: Write \(\frac{6}{11}\) as a decimal.
Solution: Use the long division method.
Dividing 6 by 11.
Here the remainder, 5, repeats, so it is a repeating decimal.
So, \(\frac{6}{11} = -0.5\overline{45}\)
Example 3: Write -0.36 as a fraction in its simplest form.
Solution: We have to write the digits after the decimal point in the numerator. The last digit in the given decimal is in the hundredths place. So, use 100 in the denominator.
\(-0.36 = ~-\frac{36}{100}\)
\(=~ – \frac{18 \times\ 2}{50 \times\ 2}\) (Dividing by common factor, 2)
\(=~ – \frac{18}{50}\) (Simplified)
\(=~- \frac{9 \times\ 2}{25 \times\ 2}\) (Dividing by common factor, 2)
\(=~- \frac{9}{25}\) (Simplified)
Therefore, \(-0.36 =~ – \frac{9}{25}\)
Example 4: Write -0.125 as a fraction in its simplest form.
Solution: Write the digits after the decimal point in the numerator and the last digit in the given decimal is in the thousandths place. So, use 1000 in the denominator.
\(-0.125 =- \frac{125}{1000}\)
\( =- \frac{5\ \times\ 25}{40\ \times\ 25}\) (Dividing by the common factor, 25)
\( =- \frac{5}{40}\) (Simplified)
\( =- \frac{5\ \times\ 1}{5\ \times\ 8}\) (Dividing by the common factor, 5)
\( =- \frac{1}{8}\) (Simplified)
Therefore, \(-0.125 =- \frac{5}{40} =- \frac{1}{8}\)
Example 5: Write -10.26 as a fraction in its simplest form.
Solution: The digits of the decimal taken in order will be the numerator, that is, -1026. The last digit in the given decimal is in the hundredths place. So, use 100 in the denominator.
\(-10.26 =- \frac{1026}{100}\)
\( =- \frac{513\ \times\ 2}{50\ \times\ 2}\) (Dividing by common factor, 2)
\( =- \frac{513}{50}\) (Simplified)
Therefore, \(-0.36 =- \frac{513}{50}\)
Example 6: The table shows the elevations of three mountains relative to the mean sea level. Which of the mountains are lower in elevation than the Alaska range? Explain.
Solution: A number line is one way to compare the depths of the mountains. Make each fraction or mixed number a decimal first.
- \(- \frac{15}{10} = -1.5\)
- \(- \frac{3}{11} =\) Using the long division to divide 3 by 11.
Dividing 3 by 11.
Therefore the remainder repeats, so it is a repeating decimal.
So, \(\frac{3}{11} = -0.242424… = 0.\overline{24}\).
\(- 3 \frac{1}{5} = – 3 \frac{2}{10} = -3.2\) (using equivalent fractions)
Now draw the number line and plot all the decimal values on the number line
Both -1.5 and -3.2 are less than -0.27.
So, the elevations of both the Cascade range and the Sierra Nevada are lesser than that of the Alaska range.
Example 7: At a depth of -1.625 feet, a pet rabbit hibernates in the sand and a spotted rabbit hibernates at a depth of \(-9 \frac{1}{2}\) feet. Find out which rabbit digs deeper into the sand to hibernate?
Solution: The pet rabbit hibernates in the sand at a depth of \(=-1.625\) feet
spotted rabbit hibernates at a depth of \(=-9 \frac{1}{2}\) feet \(=- \frac{19}{2}\)
Using the equivalent fractions,
\(\frac{1}{2} = \frac{1\ \times\ 50}{2\ \times\ 50} = \frac{50}{100}\)
So, \(- 9 \frac{1}{2} =- 9 \frac{50}{100} = -9.5\)
So the spotted rabbit hibernates at a depth of = -9.5 feet.
\(-9.5 > -1.625\)
So, the spotted rabbit hibernates deeper in the sand than the pet rabbit.
A mixed number is a combination of a whole number and a proper fraction. It usually denotes a number that falls between two whole numbers.
Mixed numbers are a way of representing improper fractions and vice versa. An improper fraction can be expressed as a sum of a whole number and a proper fraction. This whole number and the proper fraction when combined is known as a mixed number.
An improper fraction is composed of a numerator and a denominator, whereas, a mixed number is composed of a whole number in addition to the numerator and denominator.