Problem 22 - Elementary and Intermediate Algebra

Question

Translate each ratio into a fraction in simplest form. 45 people to 30 people

Step-by-Step Solution

Answer

The ratio 45 people to 30 people in simplest fraction form is \( \frac{3}{2} \).

Step 1: Understand the Ratio

The ratio given is 45 people to 30 people. Ratios compare two quantities in the same units, so this is a comparison of the number of people.

Step 2: Express the Ratio as a Fraction

To convert a ratio into a fraction, write the ratio as a fraction such that the first term of the ratio becomes the numerator and the second term becomes the denominator. Therefore, the fraction is \( \frac{45}{30} \).

Step 3: Simplify the Fraction

To simplify \( \frac{45}{30} \), divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 45 and 30 is 15. So, divide both 45 and 30 by 15: \( \frac{45 \div 15}{30 \div 15} = \frac{3}{2} \).

Step 4: Final Simplified Fraction

The simplified form of the ratio 45 to 30, when expressed as a fraction, is \( \frac{3}{2} \). This fraction is in its simplest form as the numerator and denominator have no common factors other than 1.

Key Concepts

Ratios

Understanding ratios is a fundamental concept in mathematics. A ratio is a way to compare quantities of the same kind by showing how much of one thing there is compared to another. Think of it as a way to express relationships between two numbers. For example, in the ratio of 45 people to 30 people, we are comparing the number of people in two groups.

Ratios can be written in three different ways:

• Using the colon symbol, like 45:30
• As a fraction, such as \( \frac{45}{30} \)
• In words, saying "45 to 30"

When you see a problem involving ratios, it’s important to remember that ratios must maintain the order of the values. The order tells us which quantity is being compared to which. Recognizing and understanding the order will help to correctly express the ratio as a fraction and make sure any simplifications are accurate.

Greatest Common Divisor

The greatest common divisor, or GCD, plays a crucial role in simplifying fractions. The GCD is the largest number that can evenly divide two or more numbers. Finding the GCD is necessary when reducing fractions to their simplest form.

There are several methods to find the GCD:

• List out all factors of both numbers and choose the largest one that they have in common.
• Use the Euclidean algorithm, which involves repeated division and finding remainders.

In the exercise given, the GCD of 45 and 30 is 15. This is the largest number that divides both 45 and 30 without leaving a remainder. By dividing both the numerator and denominator of the fraction by their GCD, we simplify the fraction effectively, ensuring it is reduced to the simplest terms possible while retaining the same value.

Fraction Simplification

Simplifying fractions is making them easier to work with by reducing them to their most basic form. This involves dividing the numerator and the denominator by their greatest common divisor (GCD). Doing so will give you the simplest form of the fraction.

To illustrate, with a fraction like \( \frac{45}{30} \):- Find the GCD of 45 and 30, which is 15.- Divide the numerator and denominator by 15.- You obtain \( \frac{3}{2} \), which is the simplified fraction.

It's important to check that the new numerator and denominator have no common factor other than 1, which confirms it's the simplest form. Simplifying fractions doesn’t change the proportion or the value, it just reduces it to its simplest terms for easier calculations and understanding. This is useful in both everyday math tasks and complex mathematical problems.