Comparison Shopping: Unit Price and Related Measurements

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Erin K.

Description Comparison Shopping: Unit Price and Related Measurements

After this lesson, you will be able to use unit price to make comparisons in tables, equations, graphs, and real-world scenarios.

The lesson begins with how to determine unit rate, using tables, graphs, and equations. This leads to a strategy for comparing unit rates, to determine the best value. The video concludes with a scenario that uses unit price to make comparisons.

Learn about unit price by helping Gabe get the best value when upgrading his Galactic Gorillas action figures!

This video includes key concepts, notation, and vocabulary such as: unit rate (the ratio of two measurements in which the second term is 1); unit price (a unit rate which measures cost per item); the graph of a direct proportion (a line that goes through the origin); slope (the rate of change in a linear equation).

Before watching this video, you should already be familiar with unit rates and setting up and solving proportions.

After watching this video, you will be prepared to use rates to convert from one measurement unit to another.

Common Core Standard(s) in focus: 6.RP.2, 6.RP.3.b A video intended for math students in the 6th grade Recommended for students who are 11-12 years old

Comparison Shopping: Unit Price and Related Measurements Exercise

Would you like to practice what you’ve just learned? Practice problems for this video Comparison Shopping: Unit Price and Related Measurements help you practice and recap your knowledge.
  • Find the rate and unit rate from multiple representations.

    Hints

    To find a unit rate from a table or graph, set up the ratio for the unit rate using information given in either the table or graph.

    To find a unit rate from an equation, use opposite operations to bring both variables to one side. For example in $y=\frac{2}{5}x$, bring the $x$ to the left side by dividing both sides by $x$ to get $\frac{y}{x}=\frac{2}{5}$.

    The unit rate is the ratio of two measurements in which the second term is $1$.

    Solution
    • To find the unit rate from a table, pick any row except for the one that contains $(0,0)$. ($(0,0)$ is the origin, which is a part of every proportional relationship, so it will not help us find the unit rate for any specific relationship.) Put the values in the ratio $\frac{\text{Distance}}{\text{Time}}$. Simplify the fraction to get a unit rate.
    • To find the unit rate from a graph, first check to see if the graph goes through the origin. This graph does, so the graph represents a direct proportion. To start, pick any point on the line, other than $(0,0)$. Then put the values into the ratio $\frac{\text{Distance}}{\text{Time}}$. Simplify the fraction to get the unit rate.
    • To find the unit rate from an equation, isolate the ratio by dividing both sides by $t$. This gives us the proportion $\frac{d}{t}=$unit rate. Simplify the fraction to get a unit rate.
  • Determine the best price by identifying the unit price.

    Hints

    Determine the unit price for each deal.

    Set up the ratio of $\frac{\text{cost}}{\text{unit}}$, then divide to simplify.

    For example, let's say a five pack of figurines sold for $\$63.75$. The $\frac{\text{cost}}{\text{unit}}$ ratio would be $\frac{63.75}{5}$. Simplified $\frac{63.75}{5}=12.75$. So the unit price for this example is $\$12.75$.

    Solution

    The order from least to most expensive is:

    $1)$ A four pack of Cat Hero figurines for $\$41.50$.

    • First, set up the $\frac{\text{cost}}{\text{unit}}$ ratio: $\frac{41.50}{4}$.
    • Then, divide $\frac{41.50}{4}=10.35$.
    • So, the unit price for this deal is $\$10.35$.
    $2)$ A single Cat Hero figurine for $\$11.50$.
    • First, set up the $\frac{\text{cost}}{\text{unit}}$ ratio: $\frac{11.50}{1}$.
    • Then, divide $\frac{11.50}{1}=11.50$.
    • So, the unit price for this deal is $\$11.50$.
    $3)$ A two pack of Cat Hero figurines for $\$24.00$.
    • First, set up the $\frac{\text{cost}}{\text{unit}}$ ratio: $\frac{24.00}{2}$.
    • Then, divide $\frac{24.00}{2}=12.00$.
    • So, the unit price for this deal is $\$12.00$.
    $4)$ A three pack of Cat Hero figurines for $\$36.75$.
    • First, set up the $\frac{\text{cost}}{\text{unit}}$ ratio: $\frac{36.75}{3}$.
    • Then, divide $\frac{36.75}{3}=12.25$.
    • So, the unit price for this deal is $\$12.25$.

  • Find the unit rate.

    Hints

    Convert each representation to the same rate: cost to quantity.

    $\frac{\text{cost}}{\text{quantity}}=\frac{y}{x}$.

    Once you have the ratio of $\frac{\text{cost}}{\text{quantity}}$, we want to have a denominator of one. Simplify by dividing both the numerator and denominator by the value in the denominator.

    Solution

    To be successful in this problem you need to set up a ratio of cost to quantity for each representation. It is important to note that the directions labeled $x$ as quantity and $y$ as cost. Therefore $\frac{\text{cost}}{\text{quantity}}=\frac{y}{x}$.

    $~$

    To determine the unit rate from a table, choose any point, set up the ratio, then divide to simplify.

    • Grapefruit: Let's choose the first point, $(4,6)$, substitute into the ratio $\frac{6}{4}$, then simplify by dividing $\frac{6}{4}=1.5$. Therefore the unit rate for grapefruit is $1.5$.
    • Apples: Let's choose the first point, $(10,4)$, substitute into the ratio $\frac{4}{10}$, then simplify by dividing $\frac{4}{10}=0.4$. Therefore the unit rate for grapefruit is $0.4$.
    $~$

    To determine the unit rate from a graph, choose any point on the line, other than $(0,0)$, set up the ratio, then divide to simplify. $~$

    • Watermelon: Let's choose the point $(1,4)$. Substitute this point into the ratio $\frac{4}{1}$, then simplify by dividing $\frac{4}{1}=4$. Therefore the unit rate for watermelon is $4$.
    • Mangos: Let's choose the point $(2,5)$. Substitute this point into the ratio $\frac{5}{2}$, then simplify by dividing $\frac{5}{2}=2.5$. Therefore the unit rate for mangos is $2.5$.
    $~$

    To find the unit rate from an equation isolate the ratio by multiplying both sides by $x$.

    • Strawberries: First, isolate the ratio to get $\frac{y}{x}=\frac{1}{4}$. Now, simplify $\frac{1}{4}$ by dividing to get $\frac{1}{4}=0.25$. Therefore the unit rate for strawberries is $0.25$.

  • Identify the steps used to determine a unit rate.

    Hints

    Identify the ratio.

    Use the numbers given in the example to substitute into the ratio.

    Recall that a unit rate is a ratio of two measurements in which the second term is $1$.

    Solution

    There are three basic steps to follow when comparison shopping:

    1. Set up the ratio.
    2. Substitute in values.
    3. Reduce to the unit rate.
    In this problem, Gabe has created an example for his sister, so the example steps need to be added within the basic steps.
    1. Set up the ratio.
    2. $\frac{\text{cost}}{\text{unit}}$
    3. Substitute in values.
    4. Deal A: $\frac{28.50}{3}$ and Deal B: $\frac{46.25}{5}$
    5. Reduce to the unit rate by dividing.
    6. Deal A: $\frac{28.50}{3}=9.50$ and Deal B: $\frac{46.25}{5}=9.25$
    7. The unit rate for Deal A is $\$9.50$ and the unit rate for Deal B is $\$9.25$. Therefore, Deal B has a smaller price per unit.

  • Define rate and unit rate.

    Hints

    Some examples of rates are: $7$ apples to $6$ oranges, $4$ cups of flour per $2$ cups of milk, and $70$ miles per $2$ hours.

    Some examples of unit rates are: defense per point, cost per unit, and miles per hour. Point, unit, and hour are singular.

    When simplifying a unit rate, the goal is to make the denominator $1$.

    Solution

    The definition of unit rate is the ratio of two measurements in which the second term is $1$.

    The definition of rate is a ratio that compares two different quantities.

    Unit rate and rate are similar because they both compare two measurements, but in a unit rate the second measurement is always reduced to $1$.

  • Determine the best deal using unit rate.

    Hints

    Set up the $\frac{\text{cost}}{\text{unit}}$ ratio and simplify.

    Make a list of the possible combinations of packs Gabe's sister can buy that will give her a total of five figures. For example $2+3=5$, what are some other combinations? Think outside of the box, she doesn't only have to buy two packs.

    Substitute in the cost for each for each pack to determine which combination is the cheapest. For example a two-pack and a three-pack will give her five figurines. The cost of a two-pack is $\$9.50$ and the cost of a three pack is $\$13.50$, so the total cost would be $\$23.00$.

    Solution

    To answer this problem, first find the cost per unit for each pack of figurines.

    • One figurine for $\$5.00$, so $\frac{5.00}{1}=5.00$.
    • Two figurines for $\$9.50$, so $\frac{9.50}{2}=4.75$
    • Three figurines for $\$13.50$, so $\frac{13.50}{3}=4.50$
    • Four figurines for $\$17.00$, so $\frac{17.00}{4}=4.25$
    $~$

    The cheapest cost per unit is $\$4.25$ from the four-pack.

    $~$

    Second, think of all possible combinations of packs that will result in five figurines.

    • $2+3=5$
    • $1+4=5$
    • $2+2+1=5$
    • $3+1+1=5$
    • $2+1+1+1=5$
    • $1+1+1+1+1=5$
    $~$

    Now, substitute in each cost per pack to determine the cheapest option for five figurines.

    • $2+3=5\\ 9.50+13.50=23.00$
    • $1+4=5\\5.00+17.00=22.00$
    • $2+2+1=5\\ 9.50+9.50+5=24.00$
    • $3+1+1=5\\ 13.50+5+5=23.50$
    • $2+1+1+1=5\\ 9.50+5.00+5.00+5.00=24.50$
    • $1+1+1+1+1=5\\ 5.00+5.00+5.00+5.00+5.00=25.00$
    $~$

    The cheapest total is $\$22.00$ from the combination of $1+4$, so Gabe's sister should buy $1$ one-pack and $1$ four-pack.