Problem 44 - Prealgebra

Question

The table shows the weight of Leon's dog during its first five years. $$\begin{array}{|c|c|}\hline \text { Age } & \text { Weigh } \\\\\hline(\text {years}) & (\text { pounds }) \\\\\hline 1 & 17 \frac{2}{8} \\\\\hline 2 & 18 \frac{5}{8} \\\\\hline 3 & 19 \frac{4}{8} \\\\\hline 4 & 18 \frac{3}{8} \\\\\hline 5 & 20 \frac{7}{8} \\\\\hline\end{array}$$ Suppose Leon's dog gained \(2 \frac{7}{8}\) pounds between years 5 and 6. How much does it weigh now?

Step-by-Step Solution

Answer

The dog weighs \(23 \frac{3}{4}\) pounds at year 6.

Step 1: Convert Mixed Fractions to Improper Fractions

First, convert the dog's weight at year 5 into an improper fraction. The dog's weight at year 5 is \(20 \frac{7}{8}\). To convert this, multiply 20 by 8 to get 160, then add 7 to get 167, making it \(\frac{167}{8}\).

Step 2: Convert Weight Gain to Improper Fraction

Next, convert the weight gain of \(2 \frac{7}{8}\) into an improper fraction. Multiply 2 by 8 to get 16, then add 7 to get 23, making it \(\frac{23}{8}\).

Step 3: Add the Fractions

To find the dog's weight at year 6, add the two improper fractions: \(\frac{167}{8} + \frac{23}{8}\). Since the denominators are the same, add the numerators: 167 + 23 = 190. This gives us \(\frac{190}{8}\).

Step 4: Convert the Fraction to a Mixed Number

Convert \(\frac{190}{8}\) back to a mixed number. Divide 190 by 8, which results in 23 with a remainder of 6. Thus, the dog's weight at year 6 is \(23 \frac{6}{8}\).

Step 5: Simplify the Mixed Number

Lastly, simplify \(23 \frac{6}{8}\). The fraction \(\frac{6}{8}\) can be simplified to \(\frac{3}{4}\). Therefore, the dog's weight at year 6 is \(23 \frac{3}{4}\) pounds.

Key Concepts

Mixed Fractions

A mixed fraction combines a whole number with a fraction. It is a convenient way to represent numbers larger than one but not whole numbers. For example, the weight of Leon's dog when it was one year old is represented as \( 17 \frac{2}{8} \). This means the dog weighs 17 whole pounds plus an additional \( \frac{2}{8} \) of a pound.
To understand mixed fractions:

• The number before the fraction is the whole part.
• The fraction part shows what portion of a whole follows.

Mixed fractions make it easier to read and understand quantities that are more than one but not complete whole numbers. They are frequently used in measurements, such as weights or distances.

Improper Fractions

An improper fraction is a type of fraction where the numerator (top number) is larger than the denominator (bottom number). This means the fraction is greater than one. For example, when you convert the mixed fraction \( 20 \frac{7}{8} \) to an improper fraction, it becomes \( \frac{167}{8} \). This shows how many eighths are in the total weight.
To convert a mixed fraction to an improper fraction, follow these simple steps:

• Multiply the whole number by the fraction's denominator (bottom part).
• Add this result to the numerator (top part) of the fraction.
• Place the total above the original denominator.

Converting to improper fractions is particularly useful for computations like addition and subtraction of fractions.

Addition of Fractions

Adding fractions involves combining their values. When the denominators are the same, as with the dog's weight change from year five to six \( \frac{167}{8} + \frac{23}{8} \), the process is straightforward. You simply add the numerators.
Here's a simple way to add fractions with the same denominator:

• Keep the denominator the same.
• Add the numerators together.

The result, in this case, is \( \frac{190}{8} \), still in an improper fraction form. This step prepares you for further computation, like converting it back to a mixed number if needed.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than one. When we have the fraction \( 23 \frac{6}{8} \), the fraction part \( \frac{6}{8} \) can be simplified.
Here's how you simplify:

• Break down both the numerator and the denominator into their prime factors.
• Divide both by their greatest common factor.

For \( \frac{6}{8} \), the greatest common factor is 2. Dividing both 6 and 8 by 2 gives us \( \frac{3}{4} \). Therefore, the simplified weight of Leon's dog at year six is \( 23 \frac{3}{4} \) pounds. Simplifying fractions makes them easier to interpret and use in further calculations.