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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.

Sagot :

GinnyAnsweridn
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.
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