Ratios, Rates, and Percents
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This is a paper math packet about ratios, rates, 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, an example) 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 ratios and rates.
1Ratio. A fruit bowl has 6 apples and 9 oranges. Write the ratio of apples to oranges, then write it again in simplest form. Show what you divided by.
2Unit rate. A car travels 120 miles on 4 gallons of gas. What is the unit rate in miles per gallon? Show the division you used.
Build Study the Math 5–10 min
What you need to know (with formulas)
A ratio compares two amounts. The ratio "6 apples to 9 oranges" can be written 6 to 9, 6 : 9, or 6/9. To write a ratio in simplest form, divide both numbers by the same number (their greatest common factor): 6/9 → divide both by 3 → 2 : 3.
Equivalent ratios name the same comparison. You make one by multiplying (or dividing) both numbers by the same number. A ratio table lines up equivalent ratios in columns:
2 : 3 = 4 : 6 = 6 : 9 = 8 : 12 (each ×2, ×3, ×4 …)
A rate compares two amounts with different units (miles and gallons, dollars and pounds). A unit rate is a rate "per 1." To find it, divide:
unit rate = (first amount) ÷ (second amount)
Example: 120 miles ÷ 4 gallons = 30 miles per gallon. A unit price is a unit rate in dollars: dollars ÷ items = price for one. The lower unit price is the "better buy."
Percent means "per 100," so a percent is a ratio out of 100 (25% = 25 out of 100 = 25/100 = 0.25). To turn a ratio into a percent, make an equivalent ratio with 100 on the bottom — this is a ratio-table method:
18/20 → (×5 top and bottom) → 90/100 = 90%
- Percent of a number: multiply. percent (as a decimal) × whole. Example: 15% of 40 = 0.15 × 40 = 6.
- You can also use a ratio table: 15 out of 100 is the same as ? out of 40.
Tip: keep the order the same in every ratio. If you write "apples : oranges," keep apples first every time.
Worked Example — a ratio table and a unit rate
A smoothie recipe mixes 2 cups of yogurt for every 3 cups of fruit. That is the ratio 2 : 3 (yogurt : fruit). To find other amounts that taste the same, multiply both numbers by the same factor. This makes a ratio table:
| Yogurt (cups) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Fruit (cups) | 3 | 6 | 9 | 12 |
Every column is 2 : 3 (4:6 divides to 2:3; 6:9 divides to 2:3; and so on). The unit rate of fruit per 1 cup of yogurt is 3 ÷ 2 = 1.5 cups of fruit per cup of yogurt.
3Using the worked example (yogurt : fruit = 2 : 3), how many cups of fruit go with 10 cups of yogurt? Extend the ratio table or multiply both numbers by the same factor. 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 forms. A paint mix uses 3 cups of blue for every 5 cups of white. The same relationship is shown three ways below: a ratio, a ratio table, and words. Study them, then answer 4a–4c.
| Blue (cups) | 3 | 6 | 9 | ? |
|---|---|---|---|---|
| White (cups) | 5 | 10 | 15 | 20 |
4a. Write the paint ratio blue : white in simplest form, and as a fraction.
4b. Fill in the missing "?" cup of blue that goes with 20 cups of white. Show how you scaled the ratio.
4c. In words: which is the "part-to-part" ratio, blue : white (3 : 5) or blue : total (3 : 8)? Explain in one sentence.
5Best buy (unit price). The same brand of juice comes two ways: a 3-quart bottle for $4.50 and a 5-quart bottle for $7.00. Find each unit price (dollars per quart) and decide which is the better buy.
5a. Unit price of the 3-quart bottle. Show $4.50 ÷ 3.
5b. Unit price of the 5-quart bottle. Show $7.00 ÷ 5.
5c. Which bottle is the better buy? Circle it and write one sentence telling how you know.
6Error analysis. A student solved this problem: "A shirt costs $25. The sales tax is 6%. How much is the tax?" Their work is shown in the box. Find the mistake, explain it, and give the correct tax.
Student's work:
6% means 6, so the tax is 6 out of 25.
So the tax is $6.
What is wrong, and what is the correct tax?
7Multi-step percent of a quantity. A middle school has 300 students. On a survey, 40% chose pizza as their favorite lunch.
7a. How many students chose pizza? Show 0.40 × 300 (or use a ratio table: 40 out of 100 = ? out of 300).
7b. How many students did not choose pizza? Show your work.
Explain Justify Your Decision 5–10 min
Question 8. Two snack packs of the same crackers are on the shelf. Pack A holds 8 crackers for $2.00. Pack B holds 12 crackers for $2.40. Which pack is the better buy, and why? Write a Claim (your decision), Evidence (the unit prices you calculated), and Reasoning (why the unit-price method makes this a fair comparison).
Sentence stems you may use: "The better buy is Pack ___ because…" · "I found Pack A's price per cracker by… and Pack B's by…" · "Pack A costs ___ per cracker and Pack B costs ___ per cracker." · "Comparing unit prices is fair because…"
Close ACE Wrap-Up 5 min
In your own words, explain how to make an equivalent ratio from a ratio you already have.
Point to one item in this packet where you found a unit rate or unit price. Give its number.
Give a new real-life example of a ratio or rate (a "per one" amount) that was not in this packet.
Continue Early Finisher optional · ~15 min
Write and solve your own best-buy problem. On the back of this page: (1) invent two package sizes of the same item with different prices (use whole numbers that divide evenly); (2) find the unit price of each and name the better buy; and (3) explain why a shopper might still choose the pricier package anyway (think about how much they actually need). Show all your work.