Comparing Ratios Using Ratio Tables

Comparing Ratios Using Ratio Tables exercise

• Understand how to compare ratios using a ratio table.

  Hints

  A ratio table is composed of equivalent ratios.

  To compare ratios, there must be the same number in the same unit.

  To determine whether something is fruitier, there must be a greater fruit to yogurt ratio than the other combination.

  Solution

  • A ratio table is a structured list of equivalent ratios that helps us to understand the relationship between the ratios and the numbers.

  • Some of the equivalent ratios in Mrs. C's table are $\frac{24}{42}$, $\frac{16}{28}$, and $\frac{8}{14}$. Each of these ratios simplifies to $\frac{4}{7}$. Simplify each ratio by dividing. For example, divide the top and bottom of $\frac{24}{42}$ by $6$ to get $\frac{4}{7}$. These are equivalent ratios. Every ratio in this table will simplify to $\frac{4}{7}$.

  • Some of the ratios in Mr. S's table are $\frac{35}{77}$, $\frac{25}{55}$, and $\frac{10}{22}$. Each of these ratios simplifies to $\frac{5}{11}$. Simplify each ratio by dividing. For example, divide the top and bottom of $\frac{35}{77}$ by $7$ to get $\frac{5}{11}$. These are equivalent ratios. Every ratio in this table will simplify to $\frac{5}{11}$.

  • To compare Mr. C and Mrs. S, look through the ratio tables to see where they have the same amount of blueberries or yogurt. You want to compare ratios that have the same number in the same unit.

  • We can compare $\frac{20}{35}$ from Mrs. C and $\frac{20}{44}$ from Mr. S because they have the same amount of blueberries.

  • These two ratios do not have the same number of cups of yogurt. Mrs. C uses $35$ cups of yogurt for $20$ blueberries, and Mr. S uses $44$ cups of yogurt for $20$ cups of blueberries.

  • Mrs. C's $35$ cups of yogurt is less than Mr. S's $44$ cups of yogurt.

  • So when Mrs. C and Mr. S use the same amount of blueberries, Mrs. C has the fruitier smoothie because she uses less yogurt than Mr. S.

• Extend the ratio table in order to compare ratios.

  Hints

  You could use repeated addition to extend Mr. G's table.

  Find ratios that have the same number in the same unit to compare.

  When comparing a ratio (A) to two other ratios (B and C), first compare the ratio (A) with one of the others (B), then compare the ratio (A) with the other ratio (C).

  Solution

  First, extend Mr. G's table, one method is to use repeated addition, adding $7$ to the top row. $\begin{array}{cccccc} \text{blueberries} & 7 & 14 & 21 & 28 & 35 \\ \hline \text{yogurt} & 10 & 20 & 30 & 40 & 50\\ \end{array}$

  Second, let's compare Mr. G's ratio to Mr. S's ratio.

  • Find a ratio that has the same number in the same unit.
  • Both Mr. G and Mr. S have $35$ cups of blueberries in one of their ratios. $\frac{35}{50}$ for Mr. G and $\frac{35}{77}$ for Mr. C.
  • When they have the same number of cups of blueberries, Mr. G uses $50$ cups of yogurt and Mr. S uses $77$ cups of yogurt. $50$ is less than $77$, so Mr. G has the fruitier smoothie.

  Third, let's compare Mr. G and Mrs. C.

  • Both Mr. G and Mrs. C have $28$ cups of blueberries in one of their ratios. $\frac{28}{40}$ for Mr. G and $\frac{28}{49}$ for Mrs. C.
  • When they have the same number of cups of blueberries, Mr. G uses $40$ cups of yogurt and Mrs. C uses $49$ cups of yogurt. $40$ is less than $49$, so Mr./Mrs. G has the fruitier smoothie.

  Finally, since we found the Mr. G has a fruiter smoothie than Mrs. C and Mr. S, we can say that Mr. G has the fruitiest smoothie out of the three parents.

• Identify which values can be quickly compared and pick the solution.

  Hints

  To compare ratios, you need the same number in the same unit.

  The faster swimmer would complete more laps than the other swimmer in the same amount of time, or would complete the same amount of laps in less time.

  Solution

  • First, decide which lap per minute ratios you can compare from John and Andrew's tables. Both tables have a $6$ in the minutes row, so we can compare $\frac{4}{6}$ from John's table and $\frac{5}{6}$ from Andrew's table.
  • When comparing $\frac{4}{6}$ to $\frac{5}{6}$, both ratios have the same amount of time: minutes. The faster swimmer would complete more laps in the same amount of time. Since $5>4$, Andrew completed more laps ($5$) than John ($4$) in the same amount of time ($6$ minutes).
  • Therefore, Andrew has the faster laps per minute ratio.

• Create a ratio table and compare ratios.

  Hints

  A ratio table is a table of equivalent ratios.

  Use the ratio given to complete the table using the constant increase.

  Find a ratio from each table that has the same number in the same unit, to compare the two ratios.

  Solution

  • To complete the table for China, use the given ratio, $500$ to $1$. This ratio means that $500$ thousand metric tons of garbage is produced every day, so $1000$ thousand metric tons is produced in $2$ days, $1500$ thousand metric tons is produced in $3$ days, and so on. The complete table is below:

  $\begin{array}{r|l} \hline \text{tons of garbage (in thousands)} & \text{number of days} \\ \hline 500 & 1 \\ 1000 & 2 \\ 1500 & 3 \\ 2000 & 4 \\ 2500 & 5 \\ \end{array}$

  • Find a ratio from each table with the same value in the same unit to compare. Both tables have $2500$. The United States table has the ratio $\frac{2500}{4}$, and the China table has the ratio $\frac{2500}{5}$.

  • To compare the two ratios, look at what each ratio represents. $\frac{2500}{4}$ means the United States produces $2500$ thousand metric tons of garbage in four days. $\frac{2500}{5}$ means China produces $2500$ thousand metric tons of garbage in five days.

  • The question is asking you to determine which country has the least garbage production to day ratio, so you are looking for who produces the same amount of garbage in a longer period of time. It takes China five days to produce the same amount of garbage the United States produces in four days. This means that China has the least garbage production to day ratio.

• Find the simplified ratio that describes each ratio table.

  Hints

  All ratios in a ratio table are equivalent.

  Pick a ratio from anywhere in the table and simplify by dividing both values by their greatest common factor.

  Write the ratio with the unit it represents. For example, $3$ cups of blueberries to $5$ cups of yogurt.

  Solution

  • For the Banana Mango Smoothie, the simplified banana to mango ratio from the table is $2$ to $1$. This simplified ratio can be determined in many ways. One way is to pick a ratio from the table, for example $8$ to $4$, and simplify by dividing both values by their greatest common factor. The greatest common factor of $8$ and $4$ is $4$, so $8\div4=2$ and $4\div4=1$. This $2:1$ ratio means for every $2$ bananas you need $1$ mango.

  • For the Spinach Smoothie, the ratio from the table is $3$ to $2$. This means that for every $3$ cups of spinach you need to use $2$ cups of yogurt.

  • For the Strawberry Acai Smoothie, the ratio is $5$ to $4$. This means that for every $5$ packets of acai you need $4$ cups of strawberries.

• Sort ratio tables based on speed.

  Hints

  To compare ratio tables, find a ratio in each table that has the same number in the same unit.

  When comparing multiple ratio tables, compare two tables first, determine who is the fastest, and then compare that table with another table.

  A faster cyclist will travel more miles in the same amount of time, or travel the same amount of miles in less time.

  Solution

  The order of the cyclists from fastest to slowest is: Roger, Lance, Erik, Bradley.

  There are many ways to find the order of the cyclists. One method is to pick two cyclists at a time to compare.

  • Roger vs Bradley: Both tables have $102$ in the miles column, so you can compare Roger's $\frac{102}{3}$ to Bradley's $\frac{102}{6}$. Both cyclists traveled $102$ miles, but Roger traveled $102$ miles in less time than Bradley, so Roger is faster than Bradley.

  At this point the order is: Roger, Bradley.

  • Roger vs. Bradley vs. Erik: All three tables have $9$ in the hours column, so you can compare Roger's $\frac{306}{9}$ to Bradley's $\frac{153}{9}$ to Erik's $\frac{180}{9}$. All three cyclists rode for the same amount of time. In that time, Roger traveled more miles than Erik, who traveled more miles than Bradley, so Roger is faster than Erik, who is faster than Bradley.

  Now the order is: Roger, Erik, Bradley.

  • Bradley vs. Lance: Both tables have $6$ in the hours column, so you can compare Bradley's $\frac{102}{6}$ to Lance's $\frac{180}{6}$. Both cyclists rode for the same amount of time. In that time, Lance traveled more miles than Bradley, so Lance is faster than Bradley.

  We do not know where Lance falls before Bradley on the list, so let's compare Lance and Erik.

  • Lance vs. Erik: Both cyclists have $180$ in the miles column. Compare Lance's $\frac{180}{6}$ to Erik's $\frac{180}{9}. Both cyclists rode the same amount of miles, but Lance rode for less time than Erik, so Lance is faster than Erik.

  Finally, we need to compare Lance to Roger.

  • Lance vs. Roger: Both cyclists have $3$ in the hours column. Compare Lance's $\frac{90}{3}$ to Roger's $\frac{102}{3}$. Both cyclists traveled for the same amount of time, in that time Roger rode more miles than Lance, so Roger is faster than Lance.

  The final list is Roger, Lance, Erik, Bradley.