Question

A client consumed the following: \(2^{1} /_{4}\) ounces of tea, \(1 / 3\) ounce of juice, \(11 / 2\) ounces of Jello. What is the total number of ounces consumed by the client? ______

Short answer

The client consumed a total of \( 8\frac{1}{12} \) ounces.

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, convert the mixed number from the tea into an improper fraction. The tea consumed is \(2^{1} /_{4}\), so we use the formula for conversion: \( \text{Whole number} \times \text{Denominator} + \text{Numerator} \). Thus, \(2 \frac{1}{4} = \frac{2\times4 + 1}{4} = \frac{9}{4}\).

List the Fractional Consumptions

Note that the client consumed \(\frac{9}{4}\) ounces of tea, \(\frac{1}{3}\) ounce of juice, and \(\frac{11}{2}\) ounces of Jello. Now all consumptions are in fractional form.

Find a Common Denominator

To add these, find a common denominator for the fractions. The denominators we have are 4, 3, and 2. The least common multiple of these is 12.

Convert all Fractions to Common Denominator

Convert \(\frac{9}{4}\) by multiplying the numerator and denominator with 3: \(\frac{9\times3}{4\times3} = \frac{27}{12}\). Convert \(\frac{1}{3}\) by multiplying the numerator and denominator with 4: \(\frac{1\times4}{3\times4} = \frac{4}{12}\). Convert \(\frac{11}{2}\) by multiplying the numerator and denominator with 6: \(\frac{11\times6}{2\times6} = \frac{66}{12}\).

Add the Fractions

Add all these fractions now that they have a common denominator: \(\frac{27}{12} + \frac{4}{12} + \frac{66}{12} = \frac{97}{12}\).

Convert Improper Fraction to Mixed Number

Convert \(\frac{97}{12}\) back to a mixed number by dividing 97 by 12. This gives 8 with a remainder of 1, so \(\frac{97}{12} = 8\frac{1}{12}\).

Key concepts

Fractions

Fractions are a way to represent parts of a whole. They are composed of two main parts: a numerator and a denominator. The numerator is the top part of the fraction and shows how many parts we have. The denominator is the bottom part and indicates the total number of equal parts the whole is divided into. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator.

Understanding fractions is crucial for performing operations like addition, subtraction, multiplication, and division on them. Fractions can be less than one, equal to one, or greater than one, depending on the size of the numerator relative to the denominator.

Mixed Numbers

Mixed numbers are numbers that combine a whole number and a fraction. They are often used when the fraction is greater than one and it is convenient to express the number in terms of both whole and fractional parts. For example, \(2\frac{1}{4}\) is a mixed number combining the whole number 2 and the fraction \(\frac{1}{4}\).

Mixed numbers provide an intuitive way of understanding values greater than one. However, when performing operations like addition, subtraction, or multiplication, it is often easier to first convert mixed numbers into improper fractions. This simplifies calculations and makes it easier to work with them.

Addition of Fractions

Adding fractions involves a few key steps, especially when they have different denominators.

- **Common Denominator:** First, find a common denominator for all the fractions involved. A common denominator is a shared multiple of the denominators of all fractions. - **Equivalent Fractions:** Once a common denominator is found, convert each fraction to an equivalent fraction with this denominator. Multiply the numerator and the denominator of each fraction by necessary numbers so that the denominators match. - **Add the Numerators:** With equivalent fractions, you can simply add the numerators while keeping the common denominator the same. - **Simplify the Result:** Finally, simplify the resulting fraction if possible. If the numerator is larger than the denominator, you might need to convert it back into a mixed number.

Improper Fractions

Improper fractions have numerators that are equal to or greater than their denominators. For example, \(\frac{9}{4}\) and \(\frac{11}{2}\) are improper fractions. These represent values that are greater than or equal to one.

To handle improper fractions, they can be converted to mixed numbers by dividing the numerator by the denominator. The quotient becomes the whole number part, and the remainder over the denominator becomes the fractional part. This conversion depends on what operation you're performing or what format you require the answer in. In many calculations, working with improper fractions directly can make addition or multiplication easier, before converting them back to mixed numbers for an answer.