Question

Express each of the following mixed fractions as an improper fraction: (a) \(3 \frac{2}{3}\) (b) \(5 \frac{2}{5}\) (c) \(7 \frac{1}{2}\) (d) \(-9 \frac{3}{4}\) (e) \(10 \frac{4}{7}\)

Short answer

Question: Convert the following mixed fractions to improper fractions: a) \(3 \frac{2}{3}\) b) \(5 \frac{2}{5}\) c) \(7 \frac{1}{2}\) d) \(-9 \frac{3}{4}\) e) \(10 \frac{4}{7}\) Answer: a) \(\frac{11}{3}\) b) \(\frac{27}{5}\) c) \(\frac{15}{2}\) d) \(\frac{-33}{4}\) e) \(\frac{74}{7}\)

Step-by-step solution

(a) Convert \(3 \frac{2}{3}\) to an improper fraction

First, multiply the whole number (3) by the denominator (3): \(3 \times 3 = 9\). Then add the numerator (2) to the result: \(9 + 2 = 11\). The improper fraction is \(\frac{11}{3}\).

(b) Convert \(5 \frac{2}{5}\) to an improper fraction

First, multiply the whole number (5) by the denominator (5): \(5 \times 5 = 25\). Then add the numerator (2) to the result: \(25 + 2 = 27\). The improper fraction is \(\frac{27}{5}\).

(c) Convert \(7 \frac{1}{2}\) to an improper fraction

First, multiply the whole number (7) by the denominator (2): \(7 \times 2 = 14\). Then add the numerator (1) to the result: \(14 + 1 = 15\). The improper fraction is \(\frac{15}{2}\).

(d) Convert \(-9 \frac{3}{4}\) to an improper fraction

First, multiply the whole number (-9) by the denominator (4): \(-9 \times 4 = -36\). Then add the numerator (3) to the result: \(-36 + 3 = -33\). The improper fraction is \(\frac{-33}{4}\).

(e) Convert \(10 \frac{4}{7}\) to an improper fraction

First, multiply the whole number (10) by the denominator (7): \(10 \times 7 = 70\). Then add the numerator (4) to the result: \(70+ 4 = 74\). The improper fraction is \(\frac{74}{7}\).

Key concepts

mixed fractions

Mixed fractions are a combination of a whole number and a proper fraction. They are called mixed because they mix together two kinds of numbers. Here's what you need to understand about them:

• The whole number represents the number of full units.
• The proper fraction represents the part of whole unit that remains.

For example, the mixed fraction \(3 \frac{2}{3}\) tells us there are 3 full parts, and then \(\frac{2}{3}\) of another part. Mixed fractions can often be more intuitive or easier to visualize, especially in everyday situations where we deal with whole items and leftovers, like slices of pie. However, for mathematical operations, converting them to improper fractions can be helpful.

fraction conversion

Fraction conversion is the process of changing a number from one form to another, primarily from mixed fractions to improper fractions, or vice versa. When converting mixed fractions to improper fractions, the steps are straightforward:

• Multiply the whole number by the denominator of the fraction.
• Add the numerator to this product.
• The sum from the previous step becomes the new numerator, while the denominator remains unchanged.

For instance, converting \(5 \frac{2}{5}\) involves multiplying 5 by 5, resulting in 25, and then adding 2 to get 27, thus giving us \(\frac{27}{5}\). This conversion is crucial because improper fractions are easier to use in various calculations.

numerators and denominators

Understanding numerators and denominators is essential when working with fractions. In any fraction, the number on top is the numerator, and the number on the bottom is the denominator. Here's what they signify:

• Numerator: Represents how many parts we have.
• Denominator: Represents the total number of equal parts the whole is divided into.

For example, in the fraction \(\frac{3}{4}\), 3 is the numerator, indicating three parts taken, and 4 is the denominator, indicating the whole is split into four equal parts. This basic understanding is vital to manipulate fractions correctly, whether adding, subtracting, or converting them.

mathematical operations

Mathematical operations with fractions include addition, subtraction, multiplication, and division. Each operation requires a good grasp of how fractions work:

• Addition and Subtraction: Convert fractions to have a common denominator before performing the operation.
• Multiplication: Multiply the numerators together to get the new numerator, and the denominators to get the new denominator.
• Division: Invert the second fraction (take its reciprocal) and multiply.

Improper fractions often simplify these operations, as they eliminate the need to handle mixed numbers. For example, adding \(\frac{11}{3}\) and \(\frac{27}{5}\) requires converting these to have a common denominator before combining them. Understanding how these operations function with fractions enables you to solve more complex mathematical problems easily.