Problem 42 - Prealgebra

Question

The table shows the weight of Leon's dog during its first five years. $$\begin{array}{|c|c|}\hline \text { Age } & \text { Weigh } \\\\\hline(\text {years}) & (\text { pounds }) \\\\\hline 1 & 17 \frac{2}{8} \\\\\hline 2 & 18 \frac{5}{8} \\\\\hline 3 & 19 \frac{4}{8} \\\\\hline 4 & 18 \frac{3}{8} \\\\\hline 5 & 20 \frac{7}{8} \\\\\hline\end{array}$$ How much weight did Leon's dog lose between ages 3 and 4?

Step-by-Step Solution

Answer

Leon’s dog lost \(1 \frac{1}{8}\) pounds between ages 3 and 4.

Step 1: Identify the weights at age 3 and 4

From the given table, we see that at age 3, Leon's dog weighed \(19 \frac{4}{8}\) pounds. At age 4, the weight was \(18 \frac{3}{8}\) pounds.

Step 2: Convert mixed numbers to improper fractions

The weight at age 3, \(19 \frac{4}{8}\), can be converted to an improper fraction: \(19 \cdot 8 + 4 = 152 + 4 = 156\). Therefore, \(19 \frac{4}{8} = \frac{156}{8}\). Similarly, for age 4, \(18 \frac{3}{8}\) converts to: \(18 \cdot 8 + 3 = 144 + 3 = 147\). Thus, \(18 \frac{3}{8} = \frac{147}{8}\).

Step 3: Subtract the fractions

To find the weight loss, subtract the weight at age 4 from the weight at age 3: \(\frac{156}{8} - \frac{147}{8} = \frac{156 - 147}{8} = \frac{9}{8}\).

Step 4: Convert the fraction to a mixed number

The result, \(\frac{9}{8}\), can be simplified by dividing 9 by 8. The quotient is 1 with a remainder of 1, so \(\frac{9}{8} = 1 \frac{1}{8}\) pounds.

Key Concepts

Fractions

Fractions are a fundamental part of mathematics, and they are used to represent parts of a whole. A fraction consists of a numerator, which represents the number of parts you have, and a denominator, which represents the total number of equal parts in a whole. For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator indicating three parts out of a total of four parts. In the context of mixed numbers, such as \(19 \frac{4}{8}\), you have a whole number accompanied by a fraction. To work effectively with mixed numbers, we often convert them to improper fractions, which are fractions where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. This gives you the new numerator with the original denominator staying the same. When performing arithmetic operations, like addition or subtraction with fractions, it’s crucial to have a common denominator. In most cases, fractions are simplified after performing operations to make them easier to interpret.

Subtraction

Subtraction is a basic arithmetic operation that involves finding the difference between two numbers. When dealing with fractions, subtraction involves a couple of key steps:

• Ensure the fractions have the same denominator. If they don't, you'll need to find a common denominator before proceeding.
• Subtract the numerators while keeping the denominator the same.
• If the numerator becomes negative, the result must be processed correctly if applicable.

In the situation described in the original exercise, we’re subtracting one fraction from another: \(\frac{156}{8} - \frac{147}{8}\). Since both fractions already have the same denominator, the subtraction focuses on the numerators: \(156 - 147 = 9\). The resulting fraction is \(\frac{9}{8}\), which can then be simplified or converted back to a mixed number for ease of interpretation.

Weight Loss Calculation

Understanding weight loss calculations often involves basic arithmetic, including subtraction, to determine differences over time. In this case, we are looking at the change in weight between two specific ages of a dog. The steps generally include:

• Identifying the starting and ending weights.
• Using subtraction to find the weight change, focusing on accurate fraction handling.
• Interpreting the result as a meaningful measurement, often converting it to an easily readable format, such as a mixed number.

This process allows us to effectively determine how much weight Leon's dog lost between ages 3 and 4, calculated to be \(1 \frac{1}{8}\) pounds. Understanding these steps is crucial not only for solving similar problems but also for real-world applications, such as tracking the health and diet progress of pets or even human weight management.