Improper Fraction to Mixed Number
- Introduction
- Converting an Improper Fraction to a Mixed Number
- Converting Mixed Number to Improper Fraction
- Addition of Mixed Number and Improper Fraction
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
An improper fraction and a mixed number mathematically mean the same thing, but the way they are represented is different. The value of an improper fraction or a mixed number is always greater than `1`. Improper fraction to mixed number conversion helps us to express any given improper fraction into its equivalent mixed number. This conversion involves dividing the numerator by the denominator. In this article, we will discuss the process of transforming an improper fraction into a mixed number.
Converting an Improper Fraction to a Mixed Number
An improper fraction is a fraction in which the numerator is greater than or equal to the denominator. For instance, `\frac{11}{3}` is an improper fraction. A mixed fraction, or mixed number, combines a whole number and a proper fraction. For example, `4 \frac{1}{5}` is a mixed number, where `4` is the whole number and `\frac{1}{5}` is the proper fraction.
To convert an improper fractions to its equivalent mixed number, divide the numerator by the denominator. The resulting quotient becomes the whole number part of the mixed number, the remainder becomes the new numerator, and the denominator remains unchanged.
Example: Convert improper fraction `\frac{11}{4}` into its equivalent mixed number.
Solution:
Step `1`: Divide the numerator of the improper fraction by the denominator.
Step `2`: Note down the quotient as the whole number part of the mixed number. The remainder becomes the new numerator, while the denominator remains the same.
In simpler terms, we can write the mixed number as:
When dividing `11` by `4`, the quotient is `2` with a remainder of `31`. Therefore, `\frac{11}{4}` can be expressed as the mixed number `2\frac{3}{4}`.
Converting Mixed Number to Improper Fraction
Now, lets look at the reverse process as in converting a given mixed number into its equivalent improper fraction. Converting a mixed number into an improper fraction involves multiplying the denominator by the whole number and then adding the product to the numerator. This process combines the whole number and the proper fraction into a single fraction where the numerator is greater than or equal to the denominator.
Example: Convert the mixed number `4 \frac{3}{7}` into an improper fraction.
Solution:
To convert the mixed number `4 \frac{3}{7}` to an improper fraction, we follow certain steps.
Step `1`: First let’s multiply the whole number \( 4 \), by the denominator \( 7 \).
\(4 \times 7 = 28\)
Step `2`: Then, we add the numerator \( 3 \), to the product \( 28 \) to give the new numerator.
\(28 + 3 = 31\)
Step `3`: The denominator \( 7 \) remains the same.
Therefore, `4 \frac{3}{7}` can be expressed as the improper fraction `\frac{31}{7}`.
Addition of Mixed Number and Improper Fraction
Combining an improper fraction with a mixed number is straightforward, as it involves converting the mixed number into an improper fraction. We will cover two scenarios to illustrate this process.
- When denominators are same
- When denominators are different
When denominators are same
If two fractions have the same denominators, they are referred to as like fractions. When the denominators are identical, the numerators can be added together while the denominator remains unchanged.
Example: Add `\frac{7}{8}` and `1 \frac{3}{8}`. Express the sum as a mixed number.
Solution:
Step `1`: First let’s convert mixed fraction into improper fraction.
`1 \frac{3}{8} = \frac{(8 \times 1) + 3}{8}`
`= \frac{8 + 3}{8}`
`= \frac{11}{8}`
Therefore, `1 \frac{3}{8}` can be expressed as the improper fraction `\frac{11}{8}`.
Step `2`: Now, let’s add both the fractions.
`\frac{7}{8} + \frac{11}{8} = \frac{7 + 11}{8}`
`= \frac{18}{8}`
`= \frac{9}{4}`
Step `3`: Changing improper fractions to mixed numbers.
`\frac{9}{4} = 2 \frac{1}{4}`
When denominators are different
If the denominators of two fractions are different, they are referred to as unlike fractions. In this case, we begin by converting them to a common denominator using the least common multiple (LCM) method.
Example: Add `\frac{3}{4}` and `2 \frac{2}{3}`
Solution:
Step `1`: First let’s convert mixed fraction into improper fraction.
`2 \frac{2}{3} = \frac{(3 \times 2) + 2}{3}`
`= \frac{6 + 2}{3}`
`= \frac{8}{3}`
Therefore, `2 \frac{2}{3}` can be expressed as the improper fraction `\frac{8}{3}`.
Step `2`: Make the denominators the same.
LCM `1(4, 3) = 121`
`\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}`
`\frac{8}{3} =\frac{8 \times 4}{3 \times 4} = \frac{32}{12}`
`\frac{3}{4} + 2\frac{2}{3} = \frac{9}{12} + \frac{32}{12}`
`= \frac{9 + 32}{12}`
`= \frac{41}{12}`
Step `3`: Finally we will convert the improper fraction to a mixed number.
`\frac{41}{12} = 3 \frac{5}{12}`
Solved Examples
Example `1`: Convert improper fraction `\frac{17}{5}` into a mixed number.
Solution:
While converting improper fractions to mixed numbers, the quotient is the whole number part of the mixed number. The remainder becomes the new numerator, while the denominator remains the same.
When dividing `17` by `5`, the quotient is `3` with a remainder of `2`.
Therefore, `\frac{17}{5}` can be expressed as the mixed number `3\frac{2}{5}`.
Example `2`: Convert the mixed number `4 \frac{5}{8}` into an improper fraction.
Solution:
To convert the mixed number `4 \frac{5}{8}` to an improper fraction, we follow certain steps.
Step `1`: First let’s multiply the whole number \( 4 \), by the denominator \( 8 \).
\(4 \times 8 = 32\)
Step `2`: Then, we add the numerator \( 5 \), to the product \( 32 \) to give the new numerator.
\(32 + 5 = 37\)
Step `3`: The denominator \( 8 \) remains the same.
Therefore, `4 \frac{5}{8}` can be expressed as the improper fraction `\frac{37}{8}`.
Example `3`: Subtract `\frac{5}{6}` from `3 \frac{2}{3}`.
Solution:
Step `1`: First let’s convert mixed fraction into improper fraction.
`3 \frac{2}{3} = \frac{(3 \times 3) + 2}{3}`
`= \frac{9 + 2}{3}`
`= \frac{11}{3}`
Therefore, `3 \frac{2}{3}` can be expressed as the improper fraction `\frac{11}{3}`.
Step `2`: Make the denominators the same.
LCM `(6, 3) = 6`
`\frac{5}{6} = \frac{5 \times 1}{6 \times 1} = \frac{5}{6}`
`\frac{11}{3} = \frac{11 \times 2}{3 \times 2} = \frac{22}{6}`
`3\frac{2}{3} - \frac{5}{6} = \frac{22}{6} - \frac{5}{6}`
`= \frac{22 - 5}{6}`
`= \frac{17}{6}`
Step `3`: Finally we will convert the improper fraction to a mixed number.
`\frac{17}{6} = 2 \frac{5}{6}`
Practice Problems
Q`1`. Calculate `\frac{3}{4} + 2 \frac{1}{2}`.
- `\frac{11}{4}`
- `\frac{9}{4}`
- `\frac{13}{4}`
- `\frac{7}{4}`
Answer: c
Q`2`. Simplify `5 \frac{2}{3} + \frac{4}{9}`.
- `6 \frac{1}{9}`
- `5 \frac{7}{9}`
- `6 \frac{2}{9}`
- `5 \frac{5}{9}`
Answer: a
Q`3`. Convert `\frac{23}{6}` as a mixed number.
- `3 \frac{5}{6}`
- `3 \frac{4}{6}`
- `4 \frac{5}{6}`
- `4 \frac{4}{6}`
Answer: a
Q`4`. Convert `5 \frac{4}{7}` to an improper fraction.
- `\frac{37}{7}`
- `\frac{38}{7}`
- `\frac{40}{7}`
- `\frac{39}{7}`
Answer: d
Q`5`. Convert `\frac{31}{7}` as a mixed number.
- `4 \frac{3}{7}`
- `4 \frac{4}{7}`
- `5 \frac{3}{7}`
- `5 \frac{4}{7}`
Answer: a
Frequently Asked Questions
Q`1`. How do I convert an improper fraction to a mixed number?
Answer: Divide the numerator by the denominator, use the quotient as the whole number, and the remainder as the new numerator over the original denominator.
Q`2`. What's the difference between an improper fraction and a mixed number?
Answer: An improper fraction has a numerator larger than or equal to its denominator, while a mixed number consists of a whole number and a proper fraction.
Q`3`. Can a mixed number be converted back to an improper fraction?
Answer: Yes, by multiplying the whole number by the denominator, adding the numerator, and placing the result over the denominator.
Q`4`. Why are mixed numbers useful in real-life situations?
Answer: They help represent quantities more intuitively, especially in measurements, where whole numbers represent units and fractions represent parts of a unit.
Q`5`. How can I simplify mixed numbers or improper fractions?
Answer: For mixed numbers, simplify the fraction part separately, and for improper fractions, divide the numerator and denominator by their greatest common divisor.