A mixed number has a whole number and a fraction together, such as 234, which means 2 + 34. It represents a value greater than one whole when the whole-number part is positive. Every mixed number can be rewritten as an improper fraction, and back: 234 = 114.
Two whole circles and three fourths of another make 2¾. Counting every fourth gives 11 fourths, so 2¾ and 11/4 are the same amount.
Formal definition and notation
A mixed number qrb is a whole numberq written next to a proper fractionrb (with 0 < r < b); it means the sum q + rb, not a product. It equals the improper fractionqb + rb. A negative mixed number such as −213 means −(2 + 13) = −73.
Symbol
Read it as
Meaning
234
two and three fourths
2 wholes plus 3 fourths; the word and marks the plus
2 3/4
two and three fourths
the same mixed number typed on one line, with a space
The fraction part of a mixed number is always less than 1. If it is not, regroup: 354 = 414.
On a number line, 234 is between 2 and 3, three fourths of the way from 2 to 3.
To multiply or divide mixed numbers, change them to improper fractions first. To add or subtract, you can work with the wholes and the fractions separately.
Mixed numbers are common in measurement: 112 cups, 538inches, 214 hours.
Mixed number to improper fraction
Mixed number to improper fraction
qrb = q × b + rb
q
the whole-number part
rb
the fraction part, less than 1
q × b
the wholes rewritten as parts: each whole is b parts
Back again: divide the numerator by the denominator. 235: 23 ÷ 5 = 4 R 3, so 235 = 435.
Try it: locate a mixed number
Interactive
Drag the point along the number line, which is marked in thirds, or select it and use the arrow keys. The readout gives the point as an improper fraction, a decimal and a mixed number. Find 323, then 113.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Mixed Number examples, step by step
Example 1: Mixed number to improper fraction
Warm-up
Write 325 as an improper fraction.
Each whole is 55, so 3 wholes are 3 × 5 = 15 fifths.
An improper fraction writes the whole amount as one fraction with a numerator at least as big as the denominator: 114. The mixed number 234 is the same value split into wholes and a part.
A decimal shows the fraction part with place value: 234 = 2.75. Both name the same number.
Quick check: 3 questions
Question 1 of 3
Write 423 as an improper fraction.
Show answer and explanation
Answer: C.4 × 3 + 2 = 14, so 423 = 143. 63 adds 4 and 2, 83 multiplies them, and 103 subtracts the 2 instead of adding it.
Question 2 of 3
Write 175 as a mixed number.
Show answer and explanation
Answer: A.17 ÷ 5 = 3 R 2, so 175 = 325. Choice B flips the remainder fraction, choice C takes too few wholes, and choice D uses the wrong remainder.
Question 3 of 3
Add 134 + 224.
Show answer and explanation
Answer: D. Wholes: 1 + 2 = 3. Fractions: 34 + 24 = 54 = 114. Total 414. Choice A adds the denominators, choice B drops the carried whole, and choice C misreads 54.
Mixed Number: frequently asked questions
How do you change a mixed number to an improper fraction?
Multiply the whole number by the denominator, add the numerator, and put the result over the same denominator. For 527: 5 × 7 + 2 = 37, so 527 = 377. This works because each whole is 7 sevenths, so 5 wholes are 35 sevenths, plus 2 more sevenths.
How do you add mixed numbers?
Add the whole numbers, then add the fractions using a common denominator. If the fraction sum is 1 or more, carry the extra whole. For 323 + 112: wholes give 4, fractions give 46 + 36 = 76 = 116, so the total is 516. You can also convert both to improper fractions first.
Can a mixed number be negative?
Yes. −213 means the opposite of 213, so the negative sign applies to both parts: −213 = −(2 + 13) = −73. It is not −2 + 13, which would be −123. On a number line it sits a third of the way from −2 to −3.
Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5Proper FractionA proper fraction is a fraction whose numerator is smaller than its denominator, such as 34 or 29. Because fewer parts are taken than make one whole, a positive proper fraction is always less than 1. If the numerator is equal to or greater than the denominator, the fraction is improper instead.Number System · Grade 3, Grade 4, Grade 5Improper FractionAn improper fraction has a numerator that is greater than or equal to its denominator, so its value is 1 or more. Examples are 74, 92 and 55. Every improper fraction can be rewritten as a whole number or a mixed number: 74 = 134. Improper does not mean wrong.Number System · Grade 4, Grade 5, Grade 6AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6InchAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5Common DenominatorA common denominator is a number that can be used as the denominator of two or more fractions at once. It is any common multiple of the denominators. For 13 and 14, 12 works: 412 and 312. You need one to add, subtract or compare fractions with different denominators.Number System · Grade 4, Grade 5, Grade 6MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5DecimalA decimal is a number written with a decimal point to show parts of a whole. Decimals use place value to represent tenths, hundredths, thousandths and more: in 3.14, the 1 means 1 tenth and the 4 means 4 hundredths. So 0.75 = 75100, and 0.5 is one half.Number System · Grade 4, Grade 5, Grade 6Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2FractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5Least Common DenominatorThe least common denominator (LCD) of two or more fractions is the smallest number that every denominator divides into evenly. It is the least common multiple of the denominators. For 14 and 56, the LCD is 12, so you rewrite them as 312 and 1012 before you add, subtract or compare.Number System · Grade 5, Grade 6, Grade 7Equivalent FractionsEquivalent fractions are fractions that name the same value, like 12, 24 and 36. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.Number System · Grade 3, Grade 4, Grade 5