Problem 43 - Prealgebra

Question

Express each ratio or rate as a fraction in simplest form. 155 apples to 75 oranges

Step-by-Step Solution

Answer

The ratio 155 apples to 75 oranges simplifies to \( \frac{31}{15} \).

Step 1: Write the Ratio as a Fraction

Express the ratio of 155 apples to 75 oranges as a fraction: \( \frac{155}{75} \). This fraction represents the comparison of apples to oranges.

Step 2: Find the Greatest Common Divisor (GCD)

To simplify the fraction, find the GCD of 155 and 75. The factors of 155 are 1, 5, 31, and 155; the factors of 75 are 1, 3, 5, 15, 25, and 75. The largest common factor is 5.

Step 3: Divide the Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their GCD (5): \( \frac{155 \div 5}{75 \div 5} = \frac{31}{15} \).

Step 4: Confirm the Fraction is in Simplest Form

Check to ensure that the fraction \( \frac{31}{15} \) is in simplest form by verifying that there are no common factors between 31 and 15 other than 1. This is true, so \( \frac{31}{15} \) is the simplest form.

Key Concepts

Simplifying Fractions Made Easy

Simplifying fractions means expressing a fraction in its simplest form. This involves reducing a fraction to its lowest terms, where the numerator and the denominator have no common factors except 1. Let’s break it down further:

• Identify the Fraction: To simplify a fraction like \( \frac{155}{75} \), first understand that this represents the ratio of apples to oranges in the exercise. It means comparing 155 apples to 75 oranges.
• Find Common Factors: Find numbers that divide both the numerator and the denominator evenly. For \( \frac{155}{75} \), these are numbers like 1 and 5.
• Simplify: Once the greatest common factor (which we'll discuss next) is identified, divide both the numerator and the denominator by it to reduce the fraction. This will convert your original fraction into its simplest form, such as \( \frac{31}{15} \).

Understanding how to simplify fractions is crucial, as it ensures you work with the simplest and clearest ratios.

Understanding the Greatest Common Divisor (GCD)

Finding the greatest common divisor (GCD) is a vital step when simplifying fractions or ratios. The GCD of two numbers is the largest number that divides both exactly without leaving a remainder.

Suppose you have to find the GCD of 155 and 75. Here’s how you'd do it:

• List the Factors: Write down all possible factors of each number. For 155, that’s 1, 5, 31, and 155. For 75, it’s 1, 3, 5, 15, 25, and 75.
• Identify the Common Factors: Look for numbers that appear in both lists of factors. In this case, 1 and 5 are common.
• Choose the Largest: The largest number is your GCD. Here, it’s 5.

The GCD helps you determine how to shrink down your numbers in a fraction. Once you divide by the GCD, the result is a simpler fraction where the numerator and denominator have no common factors other than one.

Numerator and Denominator Demystified

Every fraction has two parts: the numerator and the denominator. These are fancy words but are quite simple once you understand them!

• Numerator: This is at the top of the fraction. It tells you how many parts of a whole you have. For example, in the ratio \( \frac{155}{75} \), 155 shows how many apples you are comparing.
• Denominator: This is at the bottom. It tells you into how many parts the whole is divided. In \( \frac{155}{75} \), 75 represents how many oranges you are comparing the apples to.

When simplifying a fraction, you will divide both the numerator and the denominator by the GCD to keep the fraction proportionate. This will result in a simpler form which is easier to understand at a glance, like \( \frac{31}{15} \) for the apples to oranges ratio.