Question

Add the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{2}{5}+\frac{1}{3}+\frac{7}{10}=\) ______

Short answer

1 \(\frac{13}{30}\)

Step-by-step solution

Find the Least Common Denominator (LCD)

The fractions have different denominators: 5, 3, and 10. To find the least common denominator, look for the smallest number that all these denominators can divide into without a remainder. The least common multiple of 5, 3, and 10 is 30.

Convert Each Fraction to Have the LCD

Convert each fraction to an equivalent fraction with the denominator of 30. - For \(\frac{2}{5}\), multiply the numerator and denominator by 6 to get \(\frac{12}{30}\).- For \(\frac{1}{3}\), multiply the numerator and denominator by 10 to get \(\frac{10}{30}\).- For \(\frac{7}{10}\), multiply the numerator and denominator by 3 to get \(\frac{21}{30}\).

Add the Fractions

Now that the fractions have a common denominator, add the numerators: \(\frac{12}{30} + \frac{10}{30} + \frac{21}{30} = \frac{43}{30}\).Keep the common denominator, 30.

Simplify the Fraction

The fraction \(\frac{43}{30}\) is an improper fraction and cannot be reduced further. To express as a mixed number, divide 43 by 30, which gives 1 remainder 13. Hence, \(\frac{43}{30} = 1 \frac{13}{30}\).

Key concepts

Understanding the Least Common Denominator

When adding fractions with different denominators, such as \(\frac{2}{5}\), \(\frac{1}{3}\), and \(\frac{7}{10}\), it's crucial to find the least common denominator (LCD). This LCD is the smallest number that all the denominators can divide into evenly. Why is this important? Because it allows us to convert each fraction into an equivalent fraction that shares the same denominator, making the addition possible. For example:

• The denominators in our example are 5, 3, and 10.
• First, consider the multiples of each denominator: multiples of 5 (5, 10, 15,...), multiples of 3 (3, 6, 9, 12, 15,...), and multiples of 10 (10, 20, 30,...).
• The smallest common multiple that appears in all lists is 30.

This is why the least common denominator of these three fractions is 30. With this common denominator, you can then adjust each fraction so they align well for easy addition.

Improper Fractions Made Simple

An improper fraction is simply a fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). For example, in the step-by-step solution when you get \(\frac{43}{30}\) after adding \(\frac{12}{30}\), \(\frac{10}{30}\), and \(\frac{21}{30}\), it results in an improper fraction.To understand better:

• Improper fractions are important in mathematics as they allow representations of quantities greater than one.
• They can be easily converted into mixed numbers, which might be more intuitive depending on the context.

Knowing how to handle improper fractions is essential, as they frequently arise in operations like addition, subtraction, and between mixed numbers.

Exploring Mixed Numbers

A mixed number combines a whole number with a fraction, like 1 \(\frac{13}{30}\). Mixed numbers are a common way to express results of addition or subtraction when dealing with improper fractions. They make the fraction easier to understand by showing the whole number part separately from the remainder.Here’s how to convert \(\frac{43}{30}\) into a mixed number:

• Divide 43 by 30, which gives a quotient of 1.
• The remainder is 13, which becomes the new numerator over the original denominator of 30.
• Thus, \(\frac{43}{30}\) is equivalent to 1 \(\frac{13}{30}\).

This method provides clarity because it distinctly shows how much more than one whole the improper fraction represents. Mixed numbers are often preferred in everyday scenarios, as they are easier to visualize and use.