Question

On admission a client weighed \(150^{3 / 4} \mathrm{lb}\). On discharge the client weighed \(1481 / 2 \mathrm{lb}\). How much weight did the client lose? ______

Short answer

The client lost 2 1/4 lb.

Step-by-step solution

Convert Mixed Fraction to Improper Fraction (Admission Weight)

First, convert the mixed fraction \(150^{3/4}\) to an improper fraction. Multiply the whole number by the denominator and add the numerator:\[150 \times 4 + 3 = 600 + 3 = 603\]Thus, the weight on admission, as an improper fraction, is \(\frac{603}{4}\) lb.

Convert Mixed Fraction to Improper Fraction (Discharge Weight)

Next, convert the mixed fraction \(148\frac{1}{2}\) to an improper fraction. Multiply the whole number by the denominator and add the numerator:\[148 \times 2 + 1 = 296 + 1 = 297\]Thus, the weight on discharge, as an improper fraction, is \(\frac{297}{2}\) lb.

Find Common Denominator

Find a common denominator for the two fractions. The fractions are \(\frac{603}{4}\) and \(\frac{297}{2}\). The common denominator is 4. Convert \(\frac{297}{2}\) to a denominator of 4:Multiply both the numerator and the denominator by 2:\[\frac{297 \times 2}{2 \times 2} = \frac{594}{4}\]

Subtract Fractions

Now that both fractions have a common denominator, subtract the discharge weight from the admission weight:\[\frac{603}{4} - \frac{594}{4} = \frac{603 - 594}{4} = \frac{9}{4}\]The client lost \(\frac{9}{4}\) lb.

Convert Improper Fraction to Mixed Number

Convert the improper fraction \(\frac{9}{4}\) to a mixed number to make it easier to understand. Divide 9 by 4:4 goes into 9 two times, which is \(2\cdot4 = 8\) with a remainder of 1.Thus, \(\frac{9}{4} = 2\frac{1}{4}\).The client lost 2 1/4 lb.

Key concepts

Mixed Fractions

A mixed fraction, also known as a mixed number, consists of two parts: a whole number and a proper fraction. It provides a more intuitive way to express quantities that are greater than one whole unit but less than two or more whole units. For example, in the exercise, the admission weight is given as \(150\frac{3}{4}\) lbs. Here, 150 is the whole number, and \(\frac{3}{4}\) is the fraction part.

When converting mixed fractions to improper fractions, we first multiply the whole number by the denominator of the fraction part. Then, we add the numerator to this product. This conversion is crucial because it allows us to carry out arithmetic operations, such as subtraction, more straightforwardly. In our example, \(150\frac{3}{4}\) becomes \(\frac{603}{4}\).

Mixed fractions are practical in real-life scenarios where measurements often include both whole units and parts of a unit. They simplify communication and help in understanding approximate values.

Improper Fractions

Improper fractions have numerators larger than or equal to their denominators. These fractions represent quantities that are one or more whole units plus an additional fractional part. For example, \(\frac{603}{4}\) is an improper fraction that resulted from converting the mixed number \(150\frac{3}{4}\).

Conversions from mixed to improper fractions are common in calculations because improper fractions make mathematical operations like addition and subtraction easier to handle. By using improper fractions, we avoid dealing with both whole and fractional numbers separately.

To convert back, divide the numerator by the denominator. The quotient gives the whole number, and the remainder gives the new numerator of the fraction part. In our exercise, \(\frac{9}{4}\), once converted, results in \(2\frac{1}{4}\), showing that the client lost a little over two pounds.

Calculating Weight Loss

In mathematical problems involving weight loss or gain, precision is key. Calculating these changes using fractions often gives a more accurate measure than using decimals alone. The exercise outlined shows how to find the weight loss of a client admitted to and discharged from a medical facility.

The process begins with ensuring both weights are in comparable formats, typically improper fractions. Then, by subtracting the discharge weight from the admission weight, you find the difference. In this case, \(\frac{603}{4}\) minus \(\frac{594}{4}\) results in \(\frac{9}{4}\).

This weight difference is then interpreted as a mixed fraction, highlighting the actual amount of weight lost in a way that is easy to understand, which here is \(2\frac{1}{4}\) pounds. This detailed process ensures that calculations are both accurate and easily interpretable.

Subtraction of Fractions

Subtracting fractions involves a few important steps, especially when the fractions have different denominators. First, it is critical to have a common denominator so that the numerators can be directly subtracted. In our exercise, the fractions \(\frac{603}{4}\) and \(\frac{297}{2}\) were first converted to a common denominator of 4, making them \(\frac{603}{4}\) and \(\frac{594}{4}\) respectively.

With a common denominator in place, you simply subtract the numerators: \(603 - 594 = 9\), which results in the fraction \(\frac{9}{4}\).

Getting to a common denominator often involves multiplication, adjusting each fraction so that its denominator becomes the same, thereby making subtraction possible. The conversion to improper fractions facilitates this process, ensuring consistency and accuracy in subtraction.