Question

Express the following mixed fractions as improper fractions: (a) \(1 \frac{1}{2}\) (b) \(2 \frac{1}{3}\) (c) \(3 \frac{1}{4}\) (d) \(3 \frac{2}{5}\) (e) \(-10 \frac{2}{5}\)

Short answer

Question: Convert the following mixed fractions to improper fractions: 1) 1 1/2 2) 2 1/3 3) 3 1/4 4) 3 2/5 5) -10 2/5 Answer: 1) 3/2 2) 7/3 3) 13/4 4) 17/5 5) -48/5

Step-by-step solution

Converting 1 1/2 to an improper fraction

Step 1: Multiply the whole number (1) by the denominator of the fraction (2). This results in 2. Step 2: Add the result (2) to the numerator of the fraction (1). This results in 3. Step 3: Write the result (3) over the original denominator (2). The improper fraction is 3/2.

Converting 2 1/3 to an improper fraction

Step 1: Multiply the whole number (2) by the denominator of the fraction (3). This results in 6. Step 2: Add the result (6) to the numerator of the fraction (1). This results in 7. Step 3: Write the result (7) over the original denominator (3). The improper fraction is 7/3.

Converting 3 1/4 to an improper fraction

Step 1: Multiply the whole number (3) by the denominator of the fraction (4). This results in 12. Step 2: Add the result (12) to the numerator of the fraction (1). This results in 13. Step 3: Write the result (13) over the original denominator (4). The improper fraction is 13/4.

Converting 3 2/5 to an improper fraction

Step 1: Multiply the whole number (3) by the denominator of the fraction (5). This results in 15. Step 2: Add the result (15) to the numerator of the fraction (2). This results in 17. Step 3: Write the result (17) over the original denominator (5). The improper fraction is 17/5.

Converting -10 2/5 to an improper fraction

Step 1: Multiply the whole number (-10) by the denominator of the fraction (5). This results in -50. Step 2: Add the result (-50) to the numerator of the fraction (2). This results in -48. Step 3: Write the result (-48) over the original denominator (5). The improper fraction is -48/5.

Key concepts

Improper Fractions

Improper fractions may sound a bit like they're breaking the rules, but they're a perfectly valid form of mathematical expression. They're called 'improper' because the numerator (the top part) is larger than or equal to the denominator (the bottom part), making them greater than or equal to one.

For example, in the fraction \(\frac{7}{3}\), the numerator 7 is larger than the denominator 3. It's helpful to use improper fractions in calculations and algebraic expressions because they can make addition, subtraction, multiplication, and division of fractions simpler. Since they represent whole numbers and parts combined into a single fraction, they're indispensable in fraction operations and solving equations.

Mixed Fractions

A mixed fraction is essentially a whole number mixed with a proper fraction, one where the numerator is smaller than the denominator. For instance, \(1\frac{3}{4}\) combines the whole number 1 with the proper fraction \(\frac{3}{4}\).

Mixed fractions are great when you want to describe a quantity that's more than one but not a whole number. They're often used in everyday situations, like cooking or measuring. However, for mathematical calculations, it's usually easier to convert mixed fractions to improper fractions. This way, you don't have to deal with separate whole numbers and fractions in your calculations.

Fraction Operations

Fraction operations involve adding, subtracting, multiplying, and dividing fractions and mixed numbers. To perform these operations, especially adding and subtracting, it's often necessary to have a common denominator. Multiplying and dividing fractions can be more straightforward because you multiply the numerators together and the denominators together, or simply flip the divisor fraction for division.

Converting mixed fractions to improper fractions simplifies these operations. It's like turning a jigsaw puzzle into a simpler shape before putting it together with other pieces. By using improper fractions, you eliminate the need to juggle between whole numbers and fractions, allowing for smoother calculations and fewer mistakes.

Mathematical Notation

Mathematical notation is a system of symbols and signs used to communicate math concepts clearly and concisely. For fractions, this includes the use of a horizontal bar (or slash) to separate the numerator and the denominator, like in \(\frac{numerator}{denominator}\).

Proper notation is key in math. It ensures that others can understand your calculations and also helps to avoid errors. When you write \(2\frac{1}{3}\), for example, the notation signifies the whole number 2 plus the fraction \(\frac{1}{3}\) and not the number 21 divided by 3. Learning to convert between mixed fractions and improper fractions is one aspect of mastering mathematical notation, a critical skill for succeeding in math.