Home / Math / Ratios & Unit Rates: Equivalent Ratios, Tables, Double Number Lines, Compare
Math · Ratios & Rates
Ratios & Unit Rates: Equivalent Ratios, Tables, Double Number Lines, Compare
Move from fraction division into ratios and rates: read ratio language, build equivalent ratios, use tables and double number lines, find unit rates and compare real-world rates.
The idea
A ratio compares two quantities in a chosen order; reversing the order changes the meaning.
Equivalent ratios scale both quantities by the same factor. Tables and double number lines make that factor visible.
A unit rate rewrites a comparison per 1 unit, which makes different package sizes and time intervals easier to compare.
Worked examples
Example 1 · Ratio order
6 red and 9 blue. Simplify red:blue.
- Write 6:9.
- Divide both by 3.
2:3.
Example 2 · Equivalent ratio
4:7 = 12:__.
- 4 becomes 12 by ×3.
- Multiply 7 by 3.
21.
Example 3 · Unit rate
5 notebooks cost $17.50.
- Divide 17.50 by 5.
$3.50 each.
Example 4 · Compare deals
8 apples/$6.40 vs 12/$9.
- Find $0.80 each and $0.75 each.
12 for $9 is cheaper per apple.
Example 5 · Double number line
24 km in 2 h. Distance in 5 h?
- 1 h = 12 km.
- 5 h = 60 km.
60 km.
Example 6 · Part to total
4 red, 6 yellow, 10 green. red:total?
- Total is 20.
- 4:20 simplifies.
1:5.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Equivalent ratios use one common multiplication or division factor.
Why this fails: The labels and order define the comparison.
Why this fails: Convert both to the same unit first.
Why this fails: Identify the two requested quantities first.
Why this fails: A proportional relationship keeps one constant ratio.
Foundation
Foundation practice
Build equivalent ratios.
1. 2:3 = 8:___.
2. 3:5 = 9:___.
3. 4:7 = 20:___.
4. 5:8 = 15:___.
5. 6:11 = 12:___.
6. 7:9 = 28:___.
7. 3:4 = 18:___.
8. 5:6 = 25:___.
9. 8:3 = 32:___.
10. 9:10 = 27:___.
Core
Core practice
Find unit rates.
11. 12$ for 4 muffins. Find the amount per 1 muffin.
12. 15$ for 6 bars. Find the amount per 1 bar.
13. 18 km for 3 hours. Find the amount per 1 hour.
14. 84 pages for 7 minutes. Find the amount per 1 minute.
15. 17.5$ for 5 notebooks. Find the amount per 1 notebook.
16. 36$ for 8 tickets. Find the amount per 1 ticket.
17. 18 L for 2.5 minutes. Find the amount per 1 minute.
18. 6.6$ for 1.5 kg. Find the amount per 1 kg.
19. 30 items for 4 packs. Find the amount per 1 pack.
20. 24 km for 2 hours. Find the amount per 1 hour.
Think
Think practice
Compare and reason.
21. 3 red and 5 blue. Write red:blue.
22. Simplify 12:18.
23. 6 red, 4 blue, 10 green. red:total?
24. Map scale 1 cm:5 km. What does 7 cm represent?
25. 8 pencils for $4 or 12 for $7: lower unit price?
26. Why is 3:4 not equivalent to 6:7?
27. 5/8 = x/24. Find x.
28. 2.4 kg per box for 9 boxes: estimate total.
29. What does a unit rate tell you?
30. Why are unit rates useful for comparing package sizes?
Challenge
Challenge practice
Solve multi-step rates.
31. 72 km/h for 2.5 h. Distance?
32. 14 boxes/h. Time for 49 boxes?
33. Red:white=3:2, total 25 L. Red?
34. 750 g/$5.25 vs 1.2 kg/$7.92. Cheaper per kg?
35. 4/7 = x/31.5. Find x.
36. 45 km in 3/4 h. Rate?
37. At 60 km/h, distance in 2 h?
38. Sugar:flour=2:5. Flour 1.5 kg. Sugar?
39. A taxi charges $4 plus $2/km. Is cost proportional to distance?
40. Difference between ratio and unit rate?
Answer key
- 1. 12
Both terms are multiplied by 4. - 2. 15
Both terms are multiplied by 3. - 3. 35
Both terms are multiplied by 5. - 4. 24
Both terms are multiplied by 3. - 5. 22
Both terms are multiplied by 2. - 6. 36
Both terms are multiplied by 4. - 7. 24
Both terms are multiplied by 6. - 8. 30
Both terms are multiplied by 5. - 9. 12
Both terms are multiplied by 4. - 10. 30
Both terms are multiplied by 3. - 11. 3$ per muffin
12 ÷ 4 = 3. - 12. 2.50$ per bar
15 ÷ 6 = 2.50. - 13. 6 km per hour
18 ÷ 3 = 6. - 14. 12 pages per minute
84 ÷ 7 = 12. - 15. 3.50$ per notebook
17.5 ÷ 5 = 3.50. - 16. 4.50$ per ticket
36 ÷ 8 = 4.50. - 17. 7.2 L per minute
18 ÷ 2.5 = 7.2. - 18. 4.40$ per kg
6.6 ÷ 1.5 = 4.40. - 19. 7.5 items per pack
30 ÷ 4 = 7.5. - 20. 12 km per hour
24 ÷ 2 = 12. - 21. 3:5
Order follows the labels. - 22. 2:3
Divide both by 6. - 23. 3:10
Total is 20; 6:20 simplifies to 3:10. - 24. 35 km
7 × 5 = 35. - 25. 8 for $4
$0.50 each vs about $0.58. - 26. The terms were not scaled by one common factor.
3 doubled but 4 did not. - 27. 15
24 is 3 times 8, so x is 3 times 5. - 28. about 22 kg
Exact total is 21.6 kg. - 29. The amount for one unit of the comparison quantity.
It rewrites a ratio per 1. - 30. They place both comparisons on the same one-unit basis.
Common units make comparisons direct. - 31. 180 km
72 × 2.5 = 180. - 32. 3.5 h
49 ÷ 14 = 3.5. - 33. 15 L
Five ratio parts means 5 L per part; red is 3 parts. - 34. 1.2 kg box
$7/kg vs $6.60/kg. - 35. 18
31.5 ÷ 7 = 4.5; 4 × 4.5 = 18. - 36. 60 km/h
45 ÷ 0.75 = 60. - 37. 120 km
60 × 2 = 120. - 38. 0.6 kg
1.5 × 2/5 = 0.6. - 39. No.
The fixed $4 start fee prevents a constant cost:distance ratio. - 40. A ratio compares quantities; a unit rate is an equivalent ratio per 1 unit.
A unit rate is a special form of ratio.