Problem 86 - Beginning and Intermediate Algebra

Question

The Preakness, one of the horse races in the Triple Crown, is a \(1 \frac{3}{16}\) -mile race. The Belmont, another of the three races, is \(\frac{5}{16}\) of a mile longer than the Preakness. How long is the Belmont?

Step-by-Step Solution

Answer

The Belmont is \(1\frac{1}{2}\) miles long.

Step 1: Understanding the Problem

We are given the length of the Preakness race as \(1\frac{3}{16}\) miles and need to find the length of the Belmont race, which is \(\frac{5}{16}\) of a mile longer than the Preakness.

Step 2: Convert Mixed Number to Improper Fraction

Convert the mixed number \(1\frac{3}{16}\) into an improper fraction. To do this, multiply the whole number (1) by the denominator (16) and add the numerator (3): \(1 \times 16 + 3 = 19\). So, \(1\frac{3}{16} = \frac{19}{16}\).

Step 3: Calculate the Length of the Belmont

Since the Belmont is \(\frac{5}{16}\) of a mile longer than the Preakness, add \(\frac{5}{16}\) to \(\frac{19}{16}\). Add the fractions: \(\frac{19}{16} + \frac{5}{16} = \frac{19+5}{16} = \frac{24}{16}\).

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{24}{16}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 8. Therefore, \(\frac{24}{16} = \frac{24 \div 8}{16 \div 8} = \frac{3}{2}\).

Step 5: Convert the Improper Fraction to a Mixed Number

Convert \(\frac{3}{2}\) into a mixed number by dividing the numerator (3) by the denominator (2). The quotient is 1 with a remainder of 1, so \(\frac{3}{2} = 1\frac{1}{2}\).

Key Concepts

Converting Mixed Numbers

Mixed numbers are fractions that include both a whole number and a fraction. When solving problems involving mixed numbers, it is often easier to work with improper fractions.

To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator to the result from the first step.
• Write this sum as the new numerator, keeping the original denominator.

For example, the mixed number \( 1\frac{3}{16} \) becomes an improper fraction by multiplying 1 (the whole number) by 16 (the denominator), resulting in 16. Then, add the 3 from the numerator to get 19. Thus, \( 1\frac{3}{16} = \frac{19}{16} \).

With improper fractions, addition and subtraction become straightforward, making them very useful in calculations.

Improper Fractions

Improper fractions have numerators that are equal to or larger than their denominators. These fractions can be used perfectly for arithmetic operations like addition, subtraction, multiplication, and division.

Using improper fractions in calculations can ease the process. For instance, when adding \( \frac{5}{16} \) to the improper fraction \( \frac{19}{16} \), it's easy to align the numerators since the denominators are the same.

The result \( \frac{24}{16} \) indicates an improper fraction where the numerator is larger than the denominator, showing that you've effectively added your fractions.

Improper fractions can later be simplified or converted back to mixed numbers to better understand the size of the fraction in terms of whole units.

Fraction Simplification

Simplifying fractions means reducing them to their simplest form. This makes them easier to understand and compare.

In our exercise, we had the fraction \( \frac{24}{16} \). To simplify, find the greatest common divisor (GCD) of the numerator and the denominator. In this case, both 24 and 16 are divisible by 8.

By dividing both numerator and denominator by their GCD, we get \( \frac{24 \div 8}{16 \div 8} = \frac{3}{2} \). This fraction represents the same value as \( \frac{24}{16} \), but is simpler.

Simplification helps in various math problems, as it provides a clear, concise form without altering the value of the fraction. Always look to simplify any fraction where possible, to keep math calculations as easy and accurate as possible.