Ratio Tables - Additive and Multiplicative Structure

Ratio Tables - Additive and Multiplicative Structure exercise

• Identify a pattern and apply it to a table of ratios.

  Hints

  Look for patterns in the table to find the missing numbers. Looking in two directions is helpful, vertically and horizontally.

  Looking at the ratio table, can you determine the difference between $7$ and $14$? What is the pattern?

  Solution

  The table here shows the missing values. The pattern in the left column is added by 7. The pattern in the right column is added by 1. The ratio of Coffee to Chocolate is 7:1.

• Determine the ratio for a given table of values.

  Hints

  One way to find the ratio is to see what the table's additive structure is. Look at the first column and see what each row is increasing by. Then look at the second column to see what each row is increasing by.

  Notice in this table that the "cream" column is increasing by 1 and the "coffee" column is increasing by 3. The ratio of cream to coffee is $\frac{1}{3}$.

  Remember to simplify all ratios.

  The ratio $2:8$, or $\frac{2}{8}$ can be simplified by dividing the top and bottom by $2$.

  $\frac{2}{8}=\frac{1}{4}$

  Solution

  To find the ratio of $x$ to $y$, set them each up as a fraction comparing the two columns.

• Apply the additive structure of ratio tables to extend the table.

  Hints

  You can use the additive property to add the same amount vertically to extend the table.

  You can use the multiplicative property to multiply horizontally to extend the table.

  The amount added each time for caramel is $1.5$ and for strawberries is $6$.

  Solution

  $\begin{array}{|c|c|} \text{Caramel} & \text{Strawberries}\\ \hline 2.5 &10\\ 4&16\\ 5.5&22\\ 7&28\\ 8.5&34\\ 10&40\\ 11.5&46\\ 13&52\\ 14.5&58\\ \end{array}$

  In the left column, add $1.5$ to $8.5$ to get $10$. Similarly, add $6$ to $34$ to get $40$ in the right column.

  Continue this pattern by adding $1.5$ to $10$ in the left column to get $11.5$. Similarly, add $6$ to $40$ to get $46$ in the right column.

  Continue this pattern by adding $1.5$ to $11.5$ in the left column to get $13$. Similarly, add $6$ to $46$ to get $52$ in the right column.

  Continue this pattern by adding $1.5$ to $13$ in the left column to get $14.5$. Similarly, add $6$ to $52$ to get $58$ in the right column.

  The multiplicative strategy can also be used to see the pattern between the Caramel and Strawberries. The caramel can be multiplied by $4$ to find out the quantity of strawberries.

• Identify key features of a given ratio table.

  Hints

  What it the pattern you notice in each column? What are the numbers increasing or decreasing by?

  The multiplier is the reciprocal of the simplified ratio.

  If the multiplier is $7$, the reciprocal is $\frac{1}{7}$

  Solution

  The following statements are true regarding the famous chocolate milk recipe:

  The ratio of the recipe is 1 part chocolate syrup for every 6 parts of milk.

  The additive structure of this table is adding 1 in the chocolate syrup column, and adding 6 in the milk column.

  The multiplicative structure of this table is $6$.

• Identify a ratio in a table.

  Hints

  There are a few different ways to look at the table to find a pattern to help you determine the ratio. One way is to look horizontally and figure out the relationship between the syrup and coffee.

  You will notice that the syrup values can all be multiplied by $8$ to find the amount of coffee.

  You can also look at the table vertically to notice that the syrup column is increasing by 1, and the coffee column is increasing by 8.

  The multiplicative structure is $8$. The additive structure is $\frac{1}{8}$.

  If you notice these numbers are reciprocals.

  Solution

  The ratio for the given table is $\frac{1}{8}$.

  1 part peppermint syrup for every 8 parts of coffee.

• Describe patterns in ratio tables.

  Hints

  The common ratio of cream to coffee is the simplest fraction with the amount of cream in the numerator and the amount of coffee in the denominator.

  Make sure to put the amount of cream in the numerator and the amount of coffee in the denominator of the fraction.

  Solution

  $\begin{array}{|c|c|} cream & coffee\\ 2.5 &1\\ 5&2\\ 7.5&3\\ 10&4\\ 12.5&5\\ \end{array}$

  This ratio table has a pattern of adding $2.5$ in the left column and $1$ in the right column. This results in a common ratio of $5/2$ in simplest form. The multiplicative property shows that the value in the left column is multiplied by $0.4$ to get the value in the right column.

  $\begin{array}{|c|c|} cream & coffee\\ 1 & 4\\ 2 & 8\\ 3&12\\ 4&16\\ 5&20\\ \end{array}$

  This ratio table has a pattern of adding $1$ in the left column and $4$ in the right column. This results in a common ratio of $1/4$ in simplest form. The multiplicative property shows that the value in the left column is multiplied by $4$ to get the value in the right column.

  $\begin{array}{|c|c|} cream & coffee\\ 1 & 1.5\\ 2 & 3\\ 3&4.5\\ 4&6\\ 5&7.5\\ \end{array}$

  This ratio table has a pattern of adding $1$ in the left column and $1.5$ in the right column. This results in a common ratio of $2/3$ in simplest form. The multiplicative property shows that the value in the left column is multiplied by $1.5$ to get the value in the right column.

  $\begin{array}{|c|c|} cream & coffee\\ 1 & 1\\ 2 & 2\\ 3&3\\ 4&4\\ 5&5\\ \end{array}$

  This ratio table has a pattern of adding $1$ in the left column and $1$ in the right column. This results in a common ratio of $1$ in simplest form. The multiplicative property shows that the value in the left column is multiplied by $1$ to get the value in the right column.