Problem 25 - Prealgebra

Question

Express each ratio as a fraction in simplest form. 19 yards in 2.5 minutes

Step-by-Step Solution

Answer

The simplest form of the ratio is \( \frac{38}{5} \).

Step 1: Write the Ratio

First, express the given quantities (19 yards and 2.5 minutes) as a ratio. This can be written as \( \frac{19}{2.5} \).

Step 2: Eliminate the Decimal

To simplify the fraction, eliminate the decimal by multiplying both the numerator and the denominator by 10. This changes the fraction from \( \frac{19}{2.5} \) to \( \frac{190}{25} \).

Step 3: Simplify the Fraction

Find the greatest common divisor (GCD) of 190 and 25. The GCD is 5. Divide both the numerator and the denominator by 5: \( \frac{190 \div 5}{25 \div 5} = \frac{38}{5} \). This is the simplest form.

Key Concepts

Simplifying Fractions

Simplifying fractions is a fundamental skill in mathematics that involves reducing fractions to their smallest form. This is achieved by dividing both the numerator (the top number) and the denominator (the bottom number) by the same number until it is no longer possible.
A simplified fraction makes it easier to understand and compare the values. For instance, simplifying the fraction \( \frac{190}{25} \) involves identifying a common factor for both numbers. By recognizing that both 190 and 25 can be divided by 5, the fraction simplifies to \( \frac{38}{5} \).
Every fraction has a unique simplest form, which is important when adding, subtracting, or comparing fractions with different numerators and denominators. Hence, simplifying makes arithmetic with fractions much more manageable.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that divides two or more numbers without leaving a remainder.
For simplifying fractions, identifying the GCD is a crucial step because it helps simplify the fraction to its lowest terms. Finding the GCD involves listing out the factors of each number and identifying the largest one they share.
In our example, to simplify \( \frac{190}{25} \), the step begins by finding the GCD of 190 and 25, which is 5.

• Factors of 190: 1, 2, 5, 10, 19, 38, 95, 190
• Factors of 25: 1, 5, 25

By identifying the GCD (5), you can reduce the fraction by dividing the numerator and the denominator by this number, resulting in \( \frac{38}{5} \). This makes the GCD a powerful tool in fraction simplification.

Decimal to Fraction Conversion

Converting decimals to fractions is an essential skill in mathematics, especially when dealing with parts of a whole that are not easily expressed as fractions.
To convert a decimal to a fraction, place the decimal number as the numerator with 1 as the denominator, and then multiply both by 10 until there are no more decimal points.
In our case, converting 2.5 to a fraction starts with noting that 2.5 equals \( \frac{25}{10} \) simply because 2.5 is the same as 2.5/1, and multiplying by 10 moves the decimal point. However, it directly becomes \( \frac{25}{10} \) once you have removed the decimal.
This fraction can then be simplified further by finding the GCD of 25 and 10, which is 5: dividing both by this GCD results in \( \frac{5}{2} \). This method of conversion is invaluable for making calculations simpler and more precise when dealing with mixed numbers or ratios that involve decimals.