Mixed Numbers ↔ Improper Fractions

Statement

Mixed numbers and improper fractions are two different ways of expressing the same value. A mixed number combines a whole number and a fraction, such as \(2 \tfrac{1}{3}\). An improper fraction is a fraction where the numerator is equal to or larger than the denominator, such as \(\tfrac{7}{3}\). Converting between the two forms is a fundamental skill that helps in simplifying calculations, especially when adding, subtracting, multiplying, or dividing fractions.

\[ a \tfrac{b}{c} = \tfrac{ac+b}{c}, \quad \tfrac{p}{q} = \left\lfloor \tfrac{p}{q} \right\rfloor \tfrac{p-q\lfloor p/q \rfloor}{q} \]

Why it’s true

• A mixed number like \(a \tfrac{b}{c}\) literally means \(a + \tfrac{b}{c}\). By putting them over a common denominator, we get \(\tfrac{ac+b}{c}\).
• Reversing the process, dividing \(p\) by \(q\) tells us how many whole parts fit inside, and the remainder is written as a fraction of the denominator.

Recipe (how to use it)

1. Mixed to improper: Multiply the whole part by the denominator. Add the numerator. Write the result over the original denominator.
2. Improper to mixed: Divide numerator by denominator. The quotient is the whole number. The remainder becomes the new numerator.

Spotting it

Look for questions that mix whole numbers with fractions (e.g., “3 and two fifths”) or fractions with numerators larger than denominators (e.g., \(22/7\)). These are clear signals you may need to convert.

Common pairings

• Adding and subtracting fractions — improper forms make calculations easier.
• Word problems with measurements (e.g., “2 ½ metres”).
• Algebraic fractions where conversion helps simplify expressions.

Mini examples

1. Given: \(3 \tfrac{2}{5}\). Find: improper fraction. Answer: \(\tfrac{17}{5}\).
2. Given: \(\tfrac{14}{3}\). Find: mixed number. Answer: \(4 \tfrac{2}{3}\).

Pitfalls

• Forgetting to multiply the whole number by the denominator before adding the numerator.
• Leaving remainders as whole numbers instead of writing them as fractions.
• Not simplifying the fraction after conversion (e.g., writing \(6/4\) instead of \(1 1/2\)).
• Confusing mixed numbers with multiplication (e.g., misreading \(3 \tfrac{1}{2}\) as \(3 \times \tfrac{1}{2}\)).

Exam strategy

• If the question asks for addition, subtraction, multiplication, or division, first decide which form (mixed or improper) will be easier to work with.
• Always simplify the final answer if possible.
• Check that your mixed number remainder is smaller than the denominator.

Summary

Mixed numbers and improper fractions are two sides of the same coin. Converting between them is straightforward: multiply then add for mixed-to-improper, divide then remainder for improper-to-mixed. Being fluent in switching between the two makes handling more complex fraction problems much smoother.