Probability: Background You’ll Need 1

  • How to write fractions

Ratios and Integers

Ratios are an integral part of our daily lives, from baking recipes to financial planning. They allow us to compare two or more quantities in order to determine the relative size of one compared to the other. Ratios are expressed as a fraction, with the number on the left representing the first quantity and the number on the right representing the second quantity. Ratios can be used to compare anything from the price of two different items to the number of cars in a parking lot. Ratios are a great tool for understanding relationships between different values and can be used in many aspects of life.

Ratio

A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of [latex]a[/latex] to [latex]b[/latex] is written [latex]a\text{ to }b,{\Large\frac{a}{b}},\text{or} \ \mathit{\text{a}}\text{:}\mathit{\text{b}}\text{.}[/latex]

In this section, we will use fraction notation. When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number. Because a ratio compares two quantities, we would leave a ratio as [latex]{\Large\frac{4}{1}}[/latex] instead of simplifying it to [latex]4[/latex] so that we can see the two parts of the ratio.

Ratios Involving Decimals

We will often work with ratios of decimals, especially when we have ratios involving money. In these cases, we can eliminate the decimals by using the Equivalent Fractions Property to convert the ratio to a fraction with whole numbers in the numerator and denominator.

For example, consider the ratio [latex]0.8\text{ to }0.05[/latex]. We can write it as a fraction with decimals and then multiply the numerator and denominator by [latex]100[/latex] to eliminate the decimals.

A fraction is shown with 0.8 in the numerator and 0.05 in the denominator. Below it is the same fraction with both the numerator and denominator multiplied by 100. Below that is a fraction with 80 in the numerator and 5 in the denominator.

 

Do you see a shortcut to find the equivalent fraction? Notice that [latex]0.8={\Large\frac{8}{10}}[/latex] and [latex]0.05={\Large\frac{5}{100}}[/latex]. The least common denominator of [latex]{\Large\frac{8}{10}}[/latex] and [latex]{\Large\frac{5}{100}}[/latex] is [latex]100[/latex]. By multiplying the numerator and denominator of [latex]{\Large\frac{0.8}{0.05}}[/latex] by [latex]100[/latex], we ‘moved’ the decimal two places to the right to get the equivalent fraction with no decimals. Now that we understand the math behind the process, we can find the fraction with no decimals like this:

  The top line says 0.80 over 0.05. There are blue arrows moving the decimal points over 2 places to the right.
“Move” the decimal 2 places. [latex]{\Large\frac{80}{5}}[/latex]
Simplify. [latex]{\Large\frac{16}{1}}[/latex]

 

You do not have to write out every step when you multiply the numerator and denominator by powers of ten. As long as you move both decimal places the same number of places, the ratio will remain the same.

Ratios and Mixed Numbers

Some ratios compare two mixed numbers. They are used to compare two fractions with different numerators and denominators. Ratios of mixed numbers are used in a variety of applications, from solving algebraic equations to calculating the cost of a product. Understanding how to work with ratios of mixed numbers is essential for a strong foundation in mathematics.

How to: Divide Mixed Numbers

  1. Transform Mixed Numbers into Improper Fractions: Begin by converting each mixed number into an improper fraction. Multiply the whole number by the denominator of the fraction and add the numerator to this product. The sum becomes the new numerator, with the original denominator remaining the same.
  2. Reciprocate the Divisor: For division, invert the second improper fraction (the divisor), which means you’ll swap its numerator and denominator. This is also known as finding the reciprocal.
  3. Multiply the Fractions: Now, multiply the first improper fraction by the reciprocal of the second. Multiply the numerators together for the new numerator, and multiply the denominators together for the new denominator.
  4. Simplify the Result: If possible, reduce the resulting fraction to its simplest form. This may involve dividing both the numerator and the denominator by their greatest common divisor.

Divide the mixed numbers [latex]2\frac{3}{4}[/latex] by [latex]3\frac{1}{2}[/latex].