Problem 8 - Calculate With Confidence

Question

Complete the table below. Express each of the following measures in their equivalents where indicated. Reduce fractions and ratios to lowest terms; round decimals to hundreths. $$ \begin{array}{|l|l|l|l|l|} \hline & \text { Percent } & \text { Ratio } & \text { Fraction } & \text { Decimal } \\ \hline 8 . & & 7: 1,000 & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Ratio: 7:1,000, Fraction: \(\frac{7}{1000}\), Decimal: 0.01, Percent: 0.7%.

Step 1: Convert Ratio to Fraction

Given the ratio is 7:1000. To convert this ratio into a fraction, write it as \( \frac{7}{1000} \).

Step 2: Simplify the Fraction

Since 7 is a prime number and does not share any common factors with 1000, \( \frac{7}{1000} \) is already in its simplest form.

Step 3: Convert Fraction to Decimal

To convert \( \frac{7}{1000} \) to a decimal, divide 7 by 1000. This yields 0.007.

Step 4: Round Decimal to Hundredths

The decimal 0.007 is rounded to 0.01 when approximated to the nearest hundredth.

Step 5: Convert Decimal to Percent

To convert the decimal 0.007 to a percent, multiply by 100. This results in 0.7%.

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities by showing how many times one value contains or is contained within the other. They are often expressed in the form 'a:b'. For example, in the exercise provided, the ratio is 7:1000. This tells us that for every 1000 instances of one measure, there are 7 of another measure.

Ratios can be used in various fields such as cooking, where recipes need adjustments, or in scientific experiments for measuring ingredients.

• They help in understanding proportions.
• Ratios are dimensionless, meaning they do not have units.
• They are often simplified by dividing both terms by their greatest common divisor.

In the given task, the greatest common divisor of 7 and 1000 is 1, so the ratio is already simplified.

Exploring Fractions

Fractions represent a part of a whole. They consist of a numerator and a denominator, which divide the whole into parts. In the exercise, we converted the ratio 7:1000 into the fraction \( \frac{7}{1000} \). This means 7 parts out of 1000, and this is the equivalent of the fraction form of the ratio.

When working with fractions, it's essential to simplify them, which involves dividing both the numerator and the denominator by their greatest common factor. For \( \frac{7}{1000} \), 7 is a prime number, meaning it has no divisors other than 1 and itself. Therefore, this fraction is already simplified.

• Always remember to reduce fractions to their simplest form.
• Fractions are useful in showing parts of a whole number.
• They are distinct from decimals and percentages but can easily be converted to and from these forms.

Delving Into Decimals

Decimals are a way to express fractions in a base-10 system. They are often easier to use in calculations, compared to fractions. Converting fractions to decimals provides a useful form that can be used in most mathematics fields.

To find the decimal form of \( \frac{7}{1000} \), you divide 7 by 1000, resulting in 0.007. Decimals can be rounded to various place values. In our case, rounding 0.007 to the nearest hundredth gives us 0.01.

• Decimals are essential in financial calculations, enabling precise monetary representation.
• Always round decimals appropriately based on the context.
• Decimals and percentages are closely related, which makes conversions between them straightforward.

Converting to Percentages

Percentages represent a fraction of 100 and are widely used in various aspects of everyday life. Understanding how to convert other forms, like decimals, into percentages is crucial for interpreting data effectively.

In the given task, we converted the decimal 0.007 to a percentage by multiplying by 100, resulting in 0.7%. This means that the fraction \( \frac{7}{1000} \) is equivalent to 0.7%, showing how small the fraction is in terms of a whole.

• Percentages make comparisons easier across different data sets.
• They are frequently used in statistics, business, and economics to represent data points clearly.
• Understanding percentages helps in grasping discounts, interest rates, and other financial metrics.

By thinking of percentages as fractions over 100, the conversion process becomes clearer and more intuitive.