Math: Ratios and Proportions

• Math, proportions, ratios, and rates are interconnected concepts used to compare quantities and express relationships between them.

Ratios

• Comparison of two quantities
• Indicates how many times one quantity contains or is contained within another
• Ratios can be expressed in several ways:
  • Using a colon: a:b
  • As a fraction: a/b
  • Using the word "to": a to b
• The order of the quantities in a ratio is important; changing the order changes the ratio.
• Ratios are simplified by dividing both terms by their greatest common factor; the ratio 6:8 simplifies to 3:4
• Ratios can compare quantities with the same units or different units.
• When comparing quantities with the same units, the ratio is dimensionless.
• When comparing quantities with different units, the ratio has units and is called a rate.

Proportions

• Statement that two ratios are equal
• Expresses the equivalence of two ratios
• Proportions are written as a/b = c/d or a:b = c:d
• In the proportion a/b = c/d, a and d are called the extremes, and b and c are called the means.
• The fundamental property of proportions is that the product of the means equals the product of the extremes (cross-multiplication).
  • if a/b = c/d, then ad = bc
• Proportions are used to solve problems involving scaling, similarity, and direct variation.
• If two ratios forms a proportion, the two ratios are proportional.
• Direct proportion: when one quantity increases, the other increases proportionally.
• Inverse proportion: when one quantity increases, the other decreases proportionally.

Rates

• Ratio that compares two quantities with different units.
• Expresses how much of one quantity there is for each unit of another quantity.
• Examples of rates include speed (distance per time), price per unit weight (e.g., dollars per kilogram), and flow rate (volume per time).
• A unit rate is a rate where the denominator is 1.
• If a car travels 120 miles in 2 hours, the rate is 120 miles / 2 hours = 60 miles/hour (unit rate).
• Rates are used to convert between different units.
• Rates are essential in various fields, including physics, engineering, economics, and everyday life.
• Average rate is the total change in one quantity divided by the total change in another.

Problem Solving with Ratios, Proportions, and Rates

• Identify the quantities being compared and their units.
• Set up the ratio or rate with the quantities in the correct order.
• Ensure that the units are consistent or convert them to the same units.
• Write the proportion, ensuring that corresponding quantities are in the correct positions.
• Use cross-multiplication to solve for the unknown quantity.
• Check the answer to ensure it is reasonable and has the correct units.
• When working with rates, pay attention to whether you need to find a unit rate or use the rate to convert between units.
• Estimation often is used to determine validity of answer
• Ratios, proportions, and rates are fundamental mathematical tools with wide-ranging applications.