Learning Goal: I’m working on a writing discussion question and need a sample draft to help me learn.
Refer to the Learning Activity titled Introduction to Ratio and Rates: Comparing Values. Describe the technique that was used to solve Example 3. What other approaches might you use to solve the same problem? Suppose a customer prefers to work with the criterion How many items can be purchased for $1? Explain why that might be a meaningful comparison for the customer to use, and explain how the comparison would then be able to compare two different items having different selling prices.
Introduction to Ratio and Rates: Comparing Values
A discount warehouse offers a box of 55 individual instant oatmeal servings for $11.10. The supermarket offers smaller boxes of the same product containing 12 individual servings for $3.60. Which store offers the better value?
© Magone/iStock/ThinkstockHow would you compare the value of the two packages?
| Discount Warehouse | Supermarket |
|---|---|
| per serving | per serving |
Answer: The discount warehouse offers a better value. Each serving only costs 20 cents, compared to the supermarkets unit price of 30 cents per serving. We can also express this as a ratio of the cost at the warehouse to the supermarket cost which is 2:3. In other words, for every 2 cents you spend on oatmeal from the warehouse, you would be spending an equivalent 3 cents if you were to buy your oatmeal from the supermarket.
WHAT ARE RATIOS AND UNIT RATES?
A ratio is a relationship between two numbers or quantities usually expressed as a quotient. Ratios are typically expressed using the following equivalent forms:
However, the most familiar way to express a ratio is in the form of a fraction. When writing ratios, it is important to pay attention to the units. If the units are the same, then the ratio can be written without them.
Example 1: Express the ratio 12 feet to 48 feet in reduced form.
Solution:
Answer: 1 to 4
If the units are different, the ratio represents a rate. In this case, be sure to include the units.
Example 2: Express the rate 220 miles to 4 hours in reduced form.
Solution:
Answer: 55 miles to 1 hour (or 55 miles per hour)
When describing rates, the word per is used often. As in the previous example, 55 miles per hour indicates the change in position with respect to time. Another example is found in monthly payments, such as cellular service. If a company offers cellular service at a rate of $34.95 per month, you can find the equivalent yearly rate by multiplying it by 12 months.
That is, the yearly rate of the service is $419.40 per year. Furthermore, rates are useful when determining unit cost, or the price of each unit. Unit cost is used to compare values when the quantities are not the same. To determine the unit cost, divide the cost by the number of units.
Example 3: A local supermarket offers a pack of 12 sodas for $3.48 on sale, and the local discount warehouse offers the soda in a 36-can case for $11.52. Which is the better value?
Solution: Divide the cost by the number of cans to obtain the unit price.
| Supermarket | Discount Warehouse |
|---|---|
Answer: The supermarket sale price of $3.48 for a 12-pack is a better value at $0.29 per can.
TRY THIS!
PROBLEM 1
Jerry can assemble computers in hours and Mark can assemble computers in hours. Who is faster?
© zentilia/iStock/ThinkstockStep 1. How many computers can Jerry assemble in 1 hour?
Step 2. How many computers can Mark assemble in 1 hour?
Step 3. Compare Jerrys and Marks rates of computer assembly per hour and write a complete solution statement.
Solution: Mark is a bit faster at about 1.68 computers per hour.
PROBLEM 2
A webmaster runs two websites and keeps track of earnings and unique visitors. The healthcare-related site earns $15 per 1,000 unique visitors and the automotive-related website earns $27.50 per 2,500 unique visitors. Which is the more valuable site?
© Ellagrin /ThinkstockStep 1. How much does the healthcare-related site earn per unique visitor?
Step 2. How much does the automotive-related website earn per unique visitor?
Step 3. Compare the earnings of the healthcare- and automotive-related sites to answer the original question in a complete sentence.
Solution: The healthcare-related site is more valuable, returning $0.015 per unique visitor.
Note. Adapted from Elementary Algebra, by John Redden, 2011, Ch 2, Section 6. Copyright 2011 Flat World Knowledge, Inc.
