parts and wholes

Refer to the Learning Activity titled “Introduction to Ratio and Rates: Comparing Values.” Describe the technique that was used to solve Example 3: Soda Purchase. 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
Introduction
Ratios and unit rates are commonly used to compare, calculate, or
convert unit prices, exchange and currency rates, distance or
measurement scales, recipe amounts, and more. As you read this
section, consider how the concept of ratios and unit rates can be
applied to problems you encounter regularly when you are deciding
on which size or amount of product to purchase.
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/Thinkstock
How 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 supermarket’s 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/Thinkstock
Step 1. How many computers can Jerry assemble in 1 hour?
Step 2. How many computers can Mark assemble in 1 hour?
Step 3. Compare Jerry’s and Mark’s 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 automotiverelated website earns $27.50 per 2,500 unique visitors. Which is the more valuable
site?
© Ellagrin /Thinkstock
Step 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.
Introduction to Ratio and
Rates: Comparing Values
Introduction
Ratios and unit rates are commonly used to compare, calculate, or
convert unit prices, exchange and currency rates, distance or
measurement scales, recipe amounts, and more. As you read this
section, consider how the concept of ratios and unit rates can be
applied to problems you encounter regularly when you are deciding
on which size or amount of product to purchase.
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/Thinkstock
How 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 supermarket’s 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/Thinkstock
Step 1. How many computers can Jerry assemble in 1 hour?
Step 2. How many computers can Mark assemble in 1 hour?
Step 3. Compare Jerry’s and Mark’s 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 automotiverelated website earns $27.50 per 2,500 unique visitors. Which is the more valuable
site?
© Ellagrin /Thinkstock
Step 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.