Melissa made three batches of oatmeal cookies. Her recipe calls for the amounts of sugar (in cups) and flour (in cups) in the following ratios.

\begin{tabular}{lrrr}
\hline
& Batch 1 & Batch 2 & Batch 3 \\
Cups of sugar & 2 & 3 & 4 \\
Cups of flour & [tex]$4 \frac{1}{2}$[/tex] & [tex]$6 \frac{3}{4}$[/tex] & 9 \\
\hline
\end{tabular}

What is the unit rate of change of flour [tex]$(y)$[/tex] with respect to sugar [tex]$(x)$[/tex]? That is, how much flour corresponds to one cup of sugar?

The unit rate of change is [tex]$\square$[/tex].

Graph the proportional relationship described above, with the [tex]$x$[/tex]-coordinate representing cups of sugar, and the [tex]$y$[/tex]-coordinate representing cups of flour.



Answer :

To determine the unit rate of change of flour with respect to sugar from the given data, we can start by calculating the ratios of flour to sugar for each batch.

### Step 1: Convert Mixed Numbers to Improper Fractions or Decimals
First, let's convert the mixed numbers for flour into improper fractions or decimals.

- Batch 1: [tex]\( 4 \frac{1}{2} = 4 + \frac{1}{2} = 4.5 \)[/tex]
- Batch 2: [tex]\( 6 \frac{3}{4} = 6 + \frac{3}{4} = 6.75 \)[/tex]
- Batch 3: [tex]\( 9 \)[/tex]

### Step 2: Calculate the Ratios
Next, we compute the ratio of flour to sugar for each batch:

- Batch 1: [tex]\( \frac{4.5}{2} = 2.25 \)[/tex]
- Batch 2: [tex]\( \frac{6.75}{3} = 2.25 \)[/tex]
- Batch 3: [tex]\( \frac{9}{4} = 2.25 \)[/tex]

### Step 3: Determine the Unit Rate
Since the ratio is consistent across all batches, the unit rate of change of flour with respect to sugar is:

[tex]\[ \text{Unit rate} = 2.25 \][/tex]

This means that for every 1 cup of sugar, there are 2.25 cups of flour.

### Step 4: Graph the Relationship
To graph the proportional relationship, we plot the data points (cups of sugar, cups of flour), which are:

- [tex]\( (2, 4.5) \)[/tex]
- [tex]\( (3, 6.75) \)[/tex]
- [tex]\( (4, 9) \)[/tex]

Since the relationship is proportional, all points should lie on a line that passes through the origin (0, 0). The line's slope corresponds to the unit rate we calculated.

Steps to plot the graph:
1. Draw the x-axis (cups of sugar) and y-axis (cups of flour).
2. Mark points (2, 4.5), (3, 6.75), and (4, 9) on the graph.
3. Draw a straight line passing through these points and the origin (0, 0).

### Step 5: Draw and Label the Graph
- X-axis: Labeled as "Cups of Sugar".
- Y-axis: Labeled as "Cups of Flour".
- Plot: Points [tex]\((2, 4.5)\)[/tex], [tex]\((3, 6.75)\)[/tex], and [tex]\((4, 9)\)[/tex].
- Line: A straight line passing through the origin and these points, illustrating the proportional relationship.

To summarize, the unit rate of change of flour with respect to sugar is [tex]\(2.25\)[/tex]. The graph should portray a straight line through (0,0), (2, 4.5), (3, 6.75), and (4, 9), illustrating the consistent ratio.