Mixed Numbers Calculator
Answer |
\( =5\frac{11}{35}\)
In decimal: 5.3142857142857
Solution breakdown:
An equation with whole numbers and fractions apart
\((4 + 0) + \frac{5}{7} + \frac{9}{15}\)
Solving the whole numbers part:
\(4 + 0 = 4\)
Solving the fraction part of the equation, we need to find the LCD of 5/7 and 9/15
LCD = 105
Solution of the fraction part with equivalent fractions is:
\( \frac{5}{7} + \frac{9}{15} = \frac{75}{105} + \frac{63}{105} = \frac{138}{105}\)
After simplifying 138/105, the result is:
\(\frac{138}{105} = \frac{46}{35} = 1\frac{11}{35}\)
Adding the whole and fraction parts
\((4 + 1) + \frac{11}{35} = 5\frac{11}{35}\)
Mixed Numbers and Operations with Them
A mixed number represents a quantity that combines a whole number with a proper fraction. This mathematical notation expresses values greater than one in a format that's often more intuitive than improper fractions. For instance, 2¾ indicates two complete units plus three-quarters of another unit, making it easier to visualize than the equivalent improper fraction 11/4.
Components of a Mixed Number:
- Whole part: The integer component (e.g., the "2" in 2¾)
- Fractional part: A proper fraction where the numerator is smaller than the denominator (e.g., the "¾" in 2¾)
Converting Mixed Numbers to Improper Fractions
An improper fraction has a numerator equal to or greater than its denominator. To transform a mixed number into this format, multiply the whole number by the denominator, then add the numerator. Conversion formula:
\( a\frac{b}{c} = \frac{a \times c + b}{c} \)Where:
- a - whole number portion
- b - numerator of the fraction
- c - denominator of the fraction
Example. Convert 3⅖ to an improper fraction:
\( 3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5} \)
Converting Improper Fractions to Mixed Numbers
To express an improper fraction as a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, while the remainder forms the new numerator over the original denominator. Conversion formula:
\( \frac{a}{b} = q\frac{r}{b} \)
Where:
- a - original numerator
- b - denominator
- q - quotient (whole number part)
- r - remainder (new numerator)
Example. Convert 23/4 to a mixed number:
23 ÷ 4 = 5 (r3)
\( \frac{23}{4} = 5\frac{3}{4} \)
Addition of Mixed Numbers
When adding mixed numbers, you can either work with them directly or convert them to improper fractions first.
Method 1: Direct Addition
- Add the whole number parts
- Add the fractional parts (find common denominator if needed)
- If the fractional sum is improper, convert it and add to the whole number
- Join the whole part with the fraction part, that gives a sum of two given mixed numbers.
Method 2: Improper Fraction Method
\( a\frac{b}{c} + d\frac{e}{f} = \frac{ac + b}{c} + \frac{df + e}{f} \)
Example: add 2⅓ + 1¾
Using Method 2:
\( 2\frac{1}{3} + 1\frac{3}{4} = \frac{7}{3} + \frac{7}{4} \)
Finding common denominator (LCD). LCD (3, 4) = 12:
\( \frac{7 \times 4}{12} + \frac{7 \times 3}{12} = \frac{28}{12} + \frac{21}{12} = \frac{49}{12} \)
Converting back to mixed number:
\( \frac{49}{12} = 4\frac{1}{12} \)
Subtraction of Mixed Numbers
Subtraction follows similar principles to addition, but requires careful attention when the fractional part being subtracted is larger. Formula:
\( a\frac{b}{c} - d\frac{e}{f} = \frac{ac + b}{c} - \frac{df + e}{f} \)
Example: Subtract 3¼ - 1⅔
Convert to improper fractions:
\( 3\frac{1}{4} - 1\frac{2}{3} = \frac{13}{4} - \frac{5}{3} \)
LCD (3, 4) = 12
\( \frac{13 \times 3}{12} - \frac{5 \times 4}{12} = \frac{39}{12} - \frac{20}{12} = \frac{19}{12} \)
Converting to a mixed number, the result will be as follows:
\( \frac{19}{12} = 1\frac{7}{12} \)
Multiplication of Mixed Numbers
For mixed numbers multiplication, it's essential to convert them to improper fractions first, then multiply numerators and denominators.
Example: Multiply 2⅖ × 1½
Convert to improper fractions:
\( 2\frac{2}{5} \times 1\frac{1}{2} = \frac{12}{5} \times \frac{3}{2} \)
Multiply:
\( \frac{12 \times 3}{5 \times 2} = \frac{36}{10} \)
Simplify and convert:
\( \frac{36}{10} = \frac{18}{5} = 3\frac{3}{5} \)
Division of Mixed Numbers
Division by a mixed number is equivalent to multiplication by its reciprocal. Formula:
\( a\frac{b}{c} \div d\frac{e}{f} = \frac{ac + b}{c} \times \frac{f}{df + e} \)
Example: Divide 3⅓ ÷ 2¼
Convert to improper fractions:
\( 3\frac{1}{3} \div 2\frac{1}{4} = \frac{10}{3} \div \frac{9}{4} \)
Multiply by reciprocal: \( \frac{10}{3} \times \frac{4}{9} = \frac{40}{27} \)
Convert to mixed number: \( \frac{40}{27} = 1\frac{13}{27} \)
Simplifying Mixed Numbers
After performing operations, always reduce the fractional part to its lowest terms by dividing both numerator and denominator by their greatest common divisor (GCD).
Example: Simplify \( 4\frac{8}{12} \)
Find GCD(8, 12) = 4
\( 4\frac{8}{12} = 4\frac{8 \div 4}{12 \div 4} = 4\frac{2}{3} \)