TEAS Math: Understanding Ratios, Rates, and Unit Rates
Ratios and rates are useful mathematical tools for comparing quantities. They can appear in questions involving prices, distance, time, measurements, or quantities. The most important skill is understanding what two numbers are being compared and keeping track of their units.
What Is a Ratio?
A ratio compares two quantities. For example, if a classroom contains 12 nursing students and 8 medical assistant students, the ratio of nursing students to medical assistant students is: $$ 12:8 $$ This ratio can be simplified by dividing both numbers by 4: $$ 3:2 $$ Always pay attention to the order. A ratio of nursing students to medical assistant students is different from a ratio of medical assistant students to nursing students.
What Is a Rate?
A rate compares two quantities with different units. For example, suppose a car travels 120 miles in 3 hours. The rate is: $$ \frac{120\text{ miles}}{3\text{ hours}} $$ $$ =40\text{ miles per hour} $$ The car's rate is 40 miles per hour.
Understanding Unit Rates
A unit rate expresses a comparison for one unit of something. Suppose six notebooks cost $18. The cost per notebook is: $$ 18\div6=3 $$ Therefore, the unit price is $3 per notebook. Unit rates make it easier to compare different options. For example: Store A: 6 notebooks for $18 Store B: 10 notebooks for $25 Store A costs $3 per notebook. Store B costs: 25÷10=2.50 Therefore, Store B has the lower price per notebook.
Watch Your Units
Units are an important part of rate problems. Suppose a student walks 4 miles in 80 minutes. If the question asks for miles per minute: $$ 4\div80=0.05 $$ The answer is 0.05 miles per minute. If the question asks for miles per hour, however, you need to convert the time appropriately. Do not assume that a numerical answer is complete without checking the required unit.
Common Mistakes
One common mistake is reversing the quantities. Another is forgetting to simplify the ratio. Students also sometimes calculate correctly but give the answer using the wrong unit. Before submitting an answer, ask:
- What two quantities am I comparing?
- What unit should my answer use?
How to Practice
Practice with everyday situations such as:
- Price per item
- Miles per hour
- Pages per minute
- Calories per serving
- Students per classroom
- Distance per unit of time
For each problem, write the units alongside your numbers.
Final Thoughts
Ratios, rates, and unit rates become easier when you focus on what is being compared. Write the quantities clearly, simplify when necessary, and always check your units before finalizing your answer.
