Proportional Reasoning and Percents
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This is a paper math packet about unit rates, proportional relationships, and percents. Work by yourself with a pencil. A calculator is not required, but a basic calculator is allowed if you have one. Show your work — every answer should show the numbers you used, not just a final number. Everything you need (formulas, examples) is printed right here. If a question is hard, skip it, keep going, and come back. Circle or box your final answers.
Start Warm-Up: Retrieval 5 min
Bring back what you already know about rates and percents.
1Unit rate. A grocery store sells 3 pounds of apples for $6.00. What is the price for 1 pound (the unit rate in dollars per pound)? Show the division you used.
2Simple percent. A class has 25 students, and 20 of them ride the bus. What percent of the class rides the bus? Show your work.
Build Study the Math 5–10 min
What you need to know (with formulas)
Unit rate is a rate written as an amount "per 1." To find it, divide the two quantities. Example: $6 for 3 lb → $6 ÷ 3 lb = $2 per pound.
A proportional relationship connects two quantities x and y that always share the same ratio. The number you multiply x by to get y is the constant of proportionality, written k:
k = y ÷ x → y = k · x
The same relationship can be shown four ways, and they all give the same k:
- Table: every row's y ÷ x equals k.
- Graph: a straight line through the point (0, 0) (the origin). The point (1, k) is on the line.
- Equation: y = k · x.
- Situation: a real story with a constant "per one" rate.
Percent means "out of 100," so a percent is a fraction over 100 (25% = 25/100 = 0.25).
- Percent of a number: multiply. percent (as a decimal) × whole. Example: 15% of 40 = 0.15 × 40 = 6.
- Percent increase: new = whole + (percent × whole). A 20% tip on $10 = $10 + (0.20 × $10) = $12.
- Percent decrease: new = whole − (percent × whole). A 20% discount on $10 = $10 − (0.20 × $10) = $8.
Tip: "percent of" asks for the part; "percent off" (a discount) asks you to subtract the part from the original.
Worked Example — one relationship, four forms
A water dispenser fills cups at a steady rate. In 4 seconds it pours 10 ounces. Because it is steady, this is proportional.
Find k (ounces per second): k = y ÷ x = 10 oz ÷ 4 s = 2.5 oz per second.
Equation: y = 2.5x, where x = seconds and y = ounces.
Table (each y ÷ x = 2.5) and graph below. The line goes through (0, 0), and the point (1, 2.5) shows the unit rate.
| Time x (seconds) | Water y (ounces) | y ÷ x |
|---|---|---|
| 0 | 0 | — |
| 2 | 5 | 2.5 |
| 4 | 10 | 2.5 |
| 6 | 15 | 2.5 |
3Using the worked example, how many ounces would the dispenser pour in 8 seconds? Use y = 2.5x and show your work.
Apply Use What You Know 20–25 min
Show the numbers you use in every item. A basic calculator is allowed.
4Match the four forms. A bakery sells muffins at a steady price. The relationship is described four ways below, but one form is missing a value and one situation must be matched. Study Table A, then answer 4a–4c.
| Muffins x | Cost y (dollars) |
|---|---|
| 2 | 3.00 |
| 4 | 6.00 |
| 6 | 9.00 |
| 10 | ? |
4a. Find the constant of proportionality k (dollars per muffin). Show y ÷ x.
4b. Write the equation for cost y in terms of muffins x, then fill in the missing cost for 10 muffins.
4c. Which graph point would be on this line: (1, 1.50) or (1, 3.00)? Explain in one sentence why.
5Real-world percent (tax and tip). A family's restaurant bill is $40.00. Sales tax is 8%, and they want to leave a 15% tip (the tip is figured on the $40 food amount, before tax).
5a. How much is the 8% tax? Show 0.08 × 40.
5b. How much is the 15% tip on the $40 food? Show your work.
5c. What is the total the family pays (food + tax + tip)?
6Error analysis. A student solved this problem: "A $50 jacket is marked 30% off. What is the sale price?" Their work is shown in the box. Find the mistake, explain it, and give the correct sale price.
Student's work:
30% of $50 = 0.30 × 50 = $15.
So the sale price is $15.
What is wrong, and what is the correct sale price?
7Multi-step application (scaling a recipe). A muffin recipe makes 12 muffins and uses 3 cups of flour and 2 cups of sugar. A baker wants to make 30 muffins, keeping the same proportions.
7a. Find the flour needed per muffin (the unit rate), then the flour for 30 muffins.
7b. How many cups of sugar are needed for 30 muffins? Show your work.
Explain Justify Your Strategy 5–10 min
Question 8. A phone case costs $20. Store A takes 25% off. Store B takes $4 off. Which store gives the lower price, and why? Write a Claim (your answer), Evidence (the numbers you calculated), and Reasoning (why your strategy works).
Sentence stems you may use: "The lower price is at Store ___ because…" · "I found Store A's price by… and Store B's price by…" · "25% of $20 is… so Store A costs…" · "This strategy works because a percent means…"
Close ACE Wrap-Up 5 min
In your own words, explain how to find the constant of proportionality k from a table.
Point to one item in this packet where you used k = y ÷ x or a unit rate. Give its number.
Give a new real-life example of a proportional relationship (a steady "per one" rate) that was not in this packet.
Continue Early Finisher optional · ~15 min
Two discounts in a row. A $60 pair of shoes is first marked 20% off. At the register, a coupon takes an extra 10% off the already-reduced price. On the back of this page: (1) find the final price step by step; (2) explain why the total discount is not 30% off the original $60; and (3) find the single percent off the original $60 that would give the same final price. Show all your work.