Mixed Numbers Calculator: Fraction Ops in One Tap

Mixed Numbers: A mixed number combines a whole number and a proper fraction (where the numerator is smaller than the denominator). It represents a value greater than 1.

Example: 2 ¾ (two and three-fourths)

Improper Fractions: An improper fraction is a fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). It also represents a value greater than or equal to 1.

Example: 11/4 (eleven-fourths)

You write a mixed number as: Whole Number ^Numerator/_Denominator.

To convert a mixed number to an improper fraction:

1. Multiply the whole number by the denominator of the fraction.
2. Add the result to the numerator of the fraction.
3. Write this new sum as the numerator of the improper fraction, keeping the original denominator.

Example: Convert 3 ½ to an improper fraction.

1. Multiply whole number by denominator: 3 × 2 = 6
2. Add the result to the numerator: 6 + 1 = 7
3. The improper fraction is 7/2.

This process effectively counts how many "pieces" (defined by the denominator) are in the whole number part and adds them to the pieces in the fractional part.

To convert an improper fraction to a mixed number:

1. Divide the numerator by the denominator.
2. The quotient (the whole number part of the division result) becomes the whole number part of the mixed number.
3. The remainder of the division becomes the numerator of the fractional part.
4. The original denominator stays the same.

Example: Convert 11/4 to a mixed number.

1. Divide 11 by 4: 11 ÷ 4 = 2 with a remainder of 3.
2. The quotient (whole number) is 2.
3. The remainder (new numerator) is 3.
4. The denominator stays 4.
5. The mixed number is 2 ¾.

The easiest way to multiply mixed numbers is to first convert them into improper fractions.

1. Convert all mixed numbers (and whole numbers) to improper fractions.
   • To convert a mixed number, see the FAQ above.
   • To convert a whole number to an improper fraction, write the whole number over 1 (e.g., 5 becomes 5/1).
   • Regular fractions are already in the correct format.
2. Multiply the numerators together to get the new numerator.
3. Multiply the denominators together to get the new denominator.
4. Simplify the resulting fraction if possible, and convert it back to a mixed number if it's improper and the context requires it.

Example 1: Multiply 1 ½ by 2 ¾

1. Convert to improper fractions: 1 ½ = 3/2 and 2 ¾ = 11/4.
2. Multiply numerators: 3 × 11 = 33.
3. Multiply denominators: 2 × 4 = 8.
4. Result: 33/8.
5. Convert back to mixed number: 33 ÷ 8 = 4 with a remainder of 1. So, 33/8 = 4 ⅛.

Example 2: Multiply 2 ⅓ by ⅖ (mixed number by a fraction)

1. Convert 2 ⅓ to 7/3. The fraction ⅖ stays as is.
2. Multiply numerators: 7 × 2 = 14.
3. Multiply denominators: 3 × 5 = 15.
4. Result: 14/15 (already a proper fraction).

Example 3: Multiply 1 ¼ by 3 (mixed number by a whole number)

1. Convert 1 ¼ to 5/4. Convert 3 to 3/1.
2. Multiply numerators: 5 × 3 = 15.
3. Multiply denominators: 4 × 1 = 4.
4. Result: 15/4. Convert to mixed number: 3 ¾.

Similar to multiplication, first convert all mixed numbers (and whole numbers) to improper fractions.

1. Convert all mixed numbers (and whole numbers) to improper fractions. (See multiplication FAQ for conversion details).
2. Keep the first fraction as it is.
3. Change the division sign to a multiplication sign.
4. Flip (invert) the second fraction (the divisor). This is called finding the reciprocal.
5. Multiply the fractions as described in the multiplication FAQ (multiply numerators, multiply denominators).
6. Simplify the result and convert to a mixed number if needed.

Example: Divide 3 ½ by 1 ¾

1. Convert to improper fractions: 3 ½ = 7/2 and 1 ¾ = 7/4.
2. The problem becomes 7/2 ÷ 7/4.
3. Keep the first, change to multiply, flip the second: 7/2 × 4/7.
4. Multiply numerators: 7 × 4 = 28.
5. Multiply denominators: 2 × 7 = 14.
6. Result: 28/14.
7. Simplify: 28/14 = 2.

This process applies when dividing a mixed number by a fraction, a mixed number by a whole number, or a whole number by a mixed number – always convert to improper fractions first.

There are two common methods for adding mixed numbers:

Method 1: Add Whole Numbers and Fractions Separately

1. Add the whole number parts together.
2. Add the fractional parts together.
   • If the fractions have different denominators, find a common denominator first.
3. If the sum of the fractions is an improper fraction, convert it to a mixed number and add its whole number part to the sum of the whole numbers.
4. Combine the results and simplify the final fractional part if needed.

Example: Add 2 ¼ + 1 ½

1. Add whole numbers: 2 + 1 = 3.
2. Add fractions: ¼ + ½. Find common denominator (4): ¼ + 2/4 = ¾.
3. Combine: 3 ¾.

Method 2: Convert to Improper Fractions

1. Convert all mixed numbers to improper fractions.
2. Find a common denominator if the denominators are different.
3. Add the numerators, keeping the common denominator.
4. Convert the resulting improper fraction back to a mixed number and simplify.

Example: Add 2 ¼ + 1 ½ using improper fractions

1. Convert: 2 ¼ = 9/4, 1 ½ = 3/2.
2. Common denominator for 9/4 and 3/2 (which is 6/4) is 4.
3. Add: 9/4 + 6/4 = 15/4.
4. Convert back: 15/4 = 3 ¾.

This applies to adding mixed numbers with fractions (treat the fraction as having a whole part of 0) or with whole numbers (treat the whole number as having a fractional part of 0).

Similar to addition, there are two main methods:

Method 1: Subtract Whole Numbers and Fractions Separately (with potential borrowing)

1. Subtract the fractional parts.
   • Find a common denominator if necessary.
   • If the first fraction is smaller than the second, you'll need to "borrow" 1 from the whole number part of the first mixed number. Add this borrowed 1 (in the form of fraction, e.g., 4/4, 5/5) to the first fraction before subtracting.
2. Subtract the whole number parts.
3. Combine the results and simplify.

Example: Subtract 3 ¼ – 1 ¾

1. Fractions: ¼ – ¾. Since ¼ is smaller than ¾, borrow 1 from the 3. The 3 becomes 2. The borrowed 1 is added to ¼ as 4/4, so ¼ + 4/4 = 5/4. Now subtract fractions: 5/4 – ¾ = 2/4.
2. Whole numbers: 2 – 1 = 1 (remember we borrowed from the 3).
3. Combine: 1 2/4. Simplify: 1 ½.

Method 2: Convert to Improper Fractions

1. Convert all mixed numbers to improper fractions.
2. Find a common denominator if necessary.
3. Subtract the numerators, keeping the common denominator.
4. Convert the result back to a mixed number and simplify if needed.

Example: Subtract 3 ¼ – 1 ¾ using improper fractions

1. Convert: 3 ¼ = 13/4, 1 ¾ = 7/4.
2. Subtract: 13/4 – 7/4 = 6/4.
3. Convert back and simplify: 6/4 = 1 2/4 = 1 ½.

This applies when subtracting a fraction from a mixed number, or a mixed number from a whole number (convert the whole number to a mixed number with a 0 fraction or to an improper fraction).

To convert a mixed number to a decimal:

1. Keep the whole number part as it is; this will be the part of the decimal before the decimal point.
2. Convert the fractional part to a decimal by dividing the numerator by the denominator.
3. Combine the whole number part and the decimal part.

Example: Convert 3 ¾ to a decimal.

1. The whole number part is 3.
2. Convert the fraction ¾: Divide 3 by 4. 3 ÷ 4 = 0.75.
3. Combine: The whole number 3 plus the decimal 0.75 gives 3.75.

Alternatively, convert the mixed number to an improper fraction first, then divide the numerator by the denominator. For 3 ¾ = 15/4, then 15 ÷ 4 = 3.75.

To convert a decimal to a mixed number (if the decimal is greater than 1):

1. The whole number part of the decimal becomes the whole number part of the mixed number.
2. Take the decimal part (the digits after the decimal point) and write it as the numerator of a fraction.
3. The denominator of the fraction will be a power of 10 (10, 100, 1000, etc.) corresponding to the number of decimal places.
   • 1 decimal place: denominator is 10.
   • 2 decimal places: denominator is 100.
   • 3 decimal places: denominator is 1000, and so on.
4. Simplify the fractional part of the mixed number to its lowest terms.

Example: Convert 5.25 to a mixed number.

1. The whole number part is 5.
2. The decimal part is .25. Write this as 25 (numerator).
3. There are two decimal places, so the denominator is 100. The fraction is 25/100.
4. Combine: 5 25/100.
5. Simplify the fraction 25/100 by dividing both numerator and denominator by their greatest common divisor (25): 25÷25 / 100÷25 = 1/4.
6. The mixed number is 5 ¼.

To simplify a mixed number, you simplify its fractional part. The whole number part remains unchanged unless the fractional part was improper and got converted, changing the whole number.

1. Look at the fractional part of the mixed number.
2. Find the greatest common divisor (GCD) of the numerator and the denominator.
3. Divide both the numerator and the denominator by their GCD.

Example: Simplify 4 6/8.

1. The fractional part is 6/8.
2. The GCD of 6 and 8 is 2.
3. Divide numerator by GCD: 6 ÷ 2 = 3.
4. Divide denominator by GCD: 8 ÷ 2 = 4.
5. The simplified fractional part is ¾.
6. The simplified mixed number is 4 ¾.

If an operation results in a mixed number with an improper fraction (e.g., 2 5/3), first convert the improper fraction part to a mixed number (5/3 = 1 2/3), then add the whole parts (2 + 1 2/3 = 3 2/3).