Problem 65 - Prealgebra

Question

According to a survey, \(85 \%\) of people eat a salad at least once a week. Which ratio represents this portion? A 17 to 20 B 13 to 20 C 9 to 10 D 4 to 5

Step-by-Step Solution

Answer

The correct ratio is 17 to 20 (option A).

Step 1: Understanding Percentage to Fraction Conversion

The given percentage, 85%, represents a part of the whole group. A percentage is simply a fraction out of 100. Thus, we can express 85% as the fraction \( \frac{85}{100} \).

Step 2: Simplifying the Fraction

We need to simplify the fraction \( \frac{85}{100} \) to find an equivalent, simpler fraction. To do this, we find the greatest common divisor (GCD) of 85 and 100. Since 85 can be divided by 5 to get 17 and 100 can be divided by 5 to get 20, the GCD is 5. So, simplifying \( \frac{85}{100} \) by dividing both the numerator and denominator by 5 gives us \( \frac{17}{20} \).

Step 3: Matching with Provided Options

Now that we have the simplified fraction \( \frac{17}{20} \), we look at the answer choices: A 17 to 20, B 13 to 20, C 9 to 10, D 4 to 5. We can see that option A, 17 to 20, matches our simplified fraction.

Key Concepts

Percentage to Fraction Conversion

When we talk about percentages, we're actually dealing with fractions of 100. This makes it easy to convert any percentage into a fraction. If you have a percentage like 85%, it means you have 85 parts out of 100 total parts. To turn this into a fraction, you simply write it as \( \frac{85}{100} \).

This fraction represents the portion of the whole you're interested in—in this case, the part of people who eat a salad once a week. Remember:

• Percentage is always out of 100.
• To convert a percentage to a fraction, use the number as the numerator and 100 as the denominator.

By understanding this simple conversion, you can move back and forth from percentages to fractions smoothly, making data easier to interpret.

Simplifying Fractions

Fractions often become simpler when we reduce them to their lowest terms. The process of simplifying means dividing the numerator and the denominator by the same factor until you can't anymore without going below 1. It's about finding the most concise way to say exactly the same thing.

For your fraction \( \frac{85}{100} \), simplifying involves finding the largest number that evenly divides both 85 and 100, then dividing them by that number. This largest number is known as the Greatest Common Divisor (GCD).

Simplifying a fraction makes it easier to work with, especially in equations or comparisons. Simplified fractions are more intuitive and often preferred for their clarity and ease of understanding.

• Simplified fractions retain the same value.
• The goal is to make calculations or comparisons easier.

By simplifying \( \frac{85}{100} \) to \( \frac{17}{20} \), you make the fraction neater and just as accurate.

Greatest Common Divisor

Finding the Greatest Common Divisor (GCD) is key to simplifying fractions. The GCD is the largest number that can completely divide both the numerator and the denominator of a fraction without leaving a remainder.

In the case of the fraction \( \frac{85}{100} \), we look for the GCD of 85 and 100. To find it, you can list the factors of each number and identify the largest one they share. However, a quicker way is by using the division method, where you repeatedly divide until a common factor is found.

For 85 and 100, both can be divided evenly by 5, and no larger number divides both, so 5 is their GCD. From here, dividing 85 and 100 by their GCD gives the simplified fraction \( \frac{17}{20} \).

Understanding GCD helps to:

• Simplify fractions effectively.
• Maintain the original value without altering it.

Grasping how the GCD works makes simplifying fractions a straightforward task.