Comparison Shopping: Unit Price and Related Measurements

Comparison Shopping: Unit Price and Related Measurements exercise

• 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 to 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 figurines. 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.