Question: Tiara is trying to start an Intramural Club at her school. The principal tells she she must get signatures from students to show support. Each filled sheet contains 25 signatures.
a. If the number of signatures, $y$, Tiara gets varies directly with the number of filled sheets, $x$, write an equation to represent this relationship.
Enter the answer in the space provided. Use numbers instead of words.
b. The principal tells Tiara she must have 7 sheets filled with signatures.
If she fills all of these, how many signatures will she have in all?
Enter the answer in the space provided. Use numbers instead of words.
c. On Tuesday, Tiara has 225 signatures.
How many sheets has she filled?
Enter the answer in the space provided. Use numbers instead of words.
d. What is the unit rate in this situation?
Enter the answer in the space provided. Use numbers instead of words.
What does it represent?
○ For each increase in the number of sheets filled, the number of signatures increases by 25.
○ For each increase in the number of signatures, the number of sheets filled increases by 25.
○ For each increase in the number of sheets filled, the number of signatures decreases by 25.
○ For each increase in the number of signatures, the number of sheets filled decreases by 25.
e. Graph your equation from part (a) on a coordinate plane.
Use the graphing tool to graph the equation.
1. **State the problem:** Tiara needs to relate the number of signatures $y$ to the number of filled sheets $x$, knowing each sheet has 25 signatures.
2. **Write the equation for direct variation:** Since $y$ varies directly with $x$, the formula is $$y = kx$$ where $k$ is the constant of proportionality.
3. **Find the constant $k$:** Each sheet has 25 signatures, so $$k = 25$$
4. **Write the equation:** $$y = 25x$$
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5. **Calculate signatures for 7 sheets:** Substitute $x=7$ into the equation:
$$y = 25 \times 7$$
$$y = 175$$
So, Tiara will have 175 signatures.
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6. **Find number of sheets for 225 signatures:** Given $y=225$, solve for $x$:
$$225 = 25x$$
Divide both sides by 25:
$$\cancel{25}x = \frac{225}{\cancel{25}}$$
$$x = 9$$
Tiara has filled 9 sheets.
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7. **Unit rate:** The unit rate is the number of signatures per sheet, which is 25.
8. **Interpretation:** For each increase in the number of sheets filled, the number of signatures increases by 25.
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9. **Graph:** The graph of $y=25x$ is a straight line through the origin with slope 25, showing the direct variation between sheets and signatures.