Question

Combine and simplify. Don't use your calculator for these numerical problems. The practice you get working with common fractions will help you when doing algebraic fractions. $$2+\frac{3}{5}$$

Short answer

\(2 \frac{3}{5}\) or simplified as \(2 + \frac{3}{5} = 2 \frac{3}{5}\)

Step-by-step solution

Identify the Whole Number and the Fraction

In this expression, 2 is the whole number and \(\frac{3}{5}\) is the fraction. You need to add them together.

Convert the Whole Number to a Fraction

To combine the whole number with the fraction, convert the whole number to a fraction with the same denominator as the given fraction. This can be done by multiplying the whole number by \(\frac{5}{5}\) to get \(\frac{10}{5}\).

Add the Fractions

Now, add the fractions \(\frac{10}{5}\) and \(\frac{3}{5}\) together. Since they have the same denominator, you simply add the numerators together to get \(\frac{13}{5}\).

Simplify the Fraction (If Necessary)

The fraction \(\frac{13}{5}\) is already in simplest form. However, it is an improper fraction and can be expressed as a mixed number. To convert it to a mixed number, divide the numerator by the denominator. The quotient is 2 with the remainder of 3. The mixed number is \(2 \frac{3}{5}\).

Key concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This indicates that the fraction represents a value that is equal to or greater than one whole. For example, in the fraction \(\frac{13}{5}\), 13 is greater than 5, making it improper.

To understand improper fractions, picture a pie split into 5 equal pieces. If you have 13 pieces, that's more than two whole pies, which is why the value is greater than one. In mathematics, we often convert improper fractions to mixed numbers to make them easier to interpret in real life situations. This is because mixed numbers can represent wholes and parts simultaneously, which can be more intuitive.

Mixed Numbers

A mixed number combines a whole number with a proper fraction, which is a fraction where the numerator is less than the denominator. Mixed numbers are used to provide a clearer representation of amounts that are more than a whole but not a whole number on their own.

For instance, \(\frac{13}{5}\) as a mixed number is \(2\frac{3}{5}\). This is because 13 divided by 5 gives us 2 wholes with a remainder of 3, which is the part of the whole we still have left over. Mixed numbers are particularly useful in everyday usage, such as in cooking or measuring where you might see ingredients listed as '2 and a half cups of sugar'.

Adding Fractions

When adding fractions, it's essential that they have a common denominator, which is the bottom part of the fraction. Once the denominators are the same, you can add the numerators - the top numbers - directly.

To illustrate, consider adding \(\frac{10}{5}\) and \(\frac{3}{5}\). Since the denominators are already the same, you sum up the numerators to get \(\frac{10 + 3}{5}\) which equals \(\frac{13}{5}\). It's like saying you have 10 parts of a thing and you're adding 3 more parts - so you end up with 13 parts of that thing.

Fraction Simplification

Simplifying fractions involves rewriting them in the most reduced form, where the numerator and denominator have no common factors other than 1. This makes the fraction as simple as possible without changing its value.

For example, \(\frac{8}{12}\) can be simplified. Both 8 and 12 are divisible by 4, so when you divide them both by 4, you get \(\frac{2}{3}\), which is the simplified form of the fraction. Simplifying fractions makes calculations easier and results clearer to understand. However, in some situations like\(\frac{13}{5}\), the fraction is already in its simplest form as 13 and 5 share no common factors.