Problem 14 - Prealgebra

Question

Solve each of the following problems. What percent of 70 is \(14 ?\)

Step-by-Step Solution

Answer

14 is 20% of 70.

Step 1: Understanding the Problem

We need to find out what percent 14 is of 70. This means determining the ratio of 14 to 70 and then converting that ratio into a percentage.

Step 2: Setting Up the Equation

To find what percent one number is of another, set up the equation as \( x\% = \frac{14}{70} \times 100 \). Here, \( x \) represents the percentage of 70 that is 14.

Step 3: Calculating the Ratio

First, calculate the fraction \( \frac{14}{70} \). Divide 14 by 70, which gives \( \frac{14}{70} = 0.2 \).

Step 4: Converting to Percentage

Now, convert the decimal value, 0.2, into a percentage by multiplying it by 100. Hence, \( 0.2 \times 100 = 20\% \).

Key Concepts

Understanding Ratios

In mathematics, a ratio represents a relationship between two numbers, indicating how many times the first number contains the second. It's a comparison between numbers, often expressed as "a to b" or using a colon, i.e., "a:b". Ratios are used in a variety of real-life situations, like cooking recipes or comparing the scores in sports. In the context of percentages, understanding ratios is essential because percentages are a way to express ratios where the second number is 100. When you say "20% of a number," you are saying "20 out of 100," which parallels the concept of a ratio. For example, in the problem "What percent of 70 is 14?", the ratio aspect defines the relationship of 14 to 70. Dividing 14 by 70 gives the ratio, which is later converted into a decimal to facilitate further calculations.

Decimal Conversion Basics

Decimal conversion involves transforming a fraction into a decimal number. This is an important step in percentage calculations. When you have a fraction like \( \frac{14}{70} \), you can convert it into a decimal by performing the division.

• In this example, dividing 14 by 70 yields the decimal 0.2.
• This simple operation of division converts the fraction to a decimal.

Once in decimal form, the number becomes much easier to work with, especially in further calculations where percentages are involved. Decimal numbers are essentially fractions with a denominator of powers of ten, simplifying the transition to percentages, which are out of a hundred. Always remember, before converting into a percentage, ensure your ratio is calculated and turned into a decimal form.

Using the Percentage Formula

The percentage formula is a method used to find out what percentage one number is of another. The basic formula used is:\[ x\% = \frac{Part\,of\,Whole}{Whole} \times 100 \]In our example, the "part of whole" is 14, and the "whole" is 70. By rearranging the basic percentage formula, we substitute the values:

• Substitute the values into the formula: \( x\% = \frac{14}{70} \times 100 \).
• Calculate the fraction: \( \frac{14}{70} = 0.2 \).
• Multiply by 100 to convert the decimal into a percentage: \( 0.2 \times 100 = 20\% \).

The final result tells us that 14 is 20% of 70. This method streamlines the process of finding percentages by combining fraction and decimal conversions directly into one formula. Once you get the hang of the formula, you'll find percentage calculations straightforward and useful in various contexts.