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Ratios, Proportions, and Rates
Number, Quantity, and Pre-Algebra
· Topic 1.2
Introduction
ACT loves ratio and rate problems because they look simple but hide traps — a single misread of 'boys to girls' vs 'boys to total' can cost you an easy point.
Ratio and proportion questions appear 4-7 times per test. They cluster in the easy-to-medium range (questions 5-35), making them high-value targets.
By the end of this lesson you will be able to:
You'll work a problem where two workers complete a job at different rates and must find how long they take working together.
The Concept
The Core Rule
A ratio a:b means for every a units of one quantity there are b units of another. A proportion is two equal ratios: a/b = c/d, solved by cross-multiplying: ad = bc. Rate = quantity / time. Distance = Rate × Time (D = RT).
How the ACT tests this
Giving a part-to-part ratio and asking for a part-to-whole fraction
Combining two rates (combined work problems or mixture problems)
Scaling a recipe or map using a unit rate
Ratios: Part-to-Part vs Part-to-Whole
If the ratio of boys to girls is 3:5, then boys:total = 3:8 and girls:total = 5:8. Always determine whether the ratio given is part-to-part or part-to-whole before calculating.
Total parts = sum of ratio terms: 3:5 has 8 total parts
Scale factor: if boys:girls = 3:5 and there are 24 boys, girls = 5 × (24/3) = 40
Equivalent ratios: multiply or divide every term by the same number
Proportions
Set up a proportion by matching units in numerators and denominators: (miles/hour) = (miles/hour). Cross-multiply to solve: if 3/x = 7/21 then 7x = 63, x = 9.
Unit conversion: multiply by a fraction equal to 1 (e.g., 5280 ft/1 mile)
Direct proportion: as one quantity increases, the other increases
Inverse proportion: as one quantity increases, the other decreases; use xy = k
Distance-Rate-Time
D = R × T is the foundation. Rearrange: R = D/T and T = D/R. For round trips, average speed = total distance / total time (NOT the average of the two speeds).
Combined rate for two workers: 1/t1 + 1/t2 = 1/T (T = time together)
Relative speed: objects moving toward each other — add speeds; same direction — subtract
Always check units: if D is in miles and T in minutes, R is miles per minute
Your strategy
1
Label every ratio term with its unit to avoid mixing part and whole
2
For word problems, write the proportion with matching units top-and-bottom
3
For work-rate problems, convert 'can do the job in N hours' to rate = 1/N job per hour
4
Backsolve: plug answer choices into D = RT or the ratio equation to verify
Worked Examples
Easy
Example 1
Setting The Proportion Up Backwards (flour/sugar Vs Sugar/flour)
A recipe uses 3 cups of flour for every 2 cups of sugar. How many cups of flour are needed if 8 cups of sugar are used?
A.
10
B.
11
C.
12 (Correct answer)
D.
13
E.
14
Step 1
Set up the proportion: 3/2 = x/8
Step 2
Cross-multiply: 2x = 24
Step 3
Solve: x = 12
Correct answer: C
Why C is correct
Correct: 3/2 = x/8 → x = 12
Why other options are wrong
A: Adds 2 to both recipe amounts instead of scaling; wrong approach
B: Off-by-one arithmetic error
D: Arithmetic error in cross-multiplication
E: Multiplied flour by 4 without scaling correctly
⚠ Trap: Setting the proportion up backwards (flour/sugar vs sugar/flour)
Medium
Example 2
Forgetting To Convert Hours To Minutes At The Final Step
A car travels 180 miles in 3 hours. At the same rate, how many minutes will it take to travel 90 miles?
A.
60
B.
75
C.
80
D.
90 (Correct answer)
E.
120
Step 1
Find the rate: 180 miles / 3 hours = 60 mph
Step 2
Time for 90 miles: T = D/R = 90/60 = 1.5 hours
Step 3
Convert to minutes: 1.5 × 60 = 90 minutes
Correct answer: D
Why D is correct
Correct: 1.5 hours × 60 = 90 minutes
Why other options are wrong
A: 1 hour = 60 minutes; misses that 90 miles takes 1.5 hours not 1 hour
B: Arithmetic error in conversion
C: 80 minutes ≠ 1.5 hours
E: 2 hours = 120 minutes; would cover 120 miles at 60 mph, not 90
⚠ Trap: Forgetting to convert hours to minutes at the final step
Hard
Example 3
Averaging The Two Times (5 Hours) Or Halving The Average Instead Of Using Combined Rates
Pipe A fills a tank in 4 hours. Pipe B fills the same tank in 6 hours. If both pipes are open, how many hours will it take to fill the tank?
A.
2.0
B.
2.2
C.
2.4 (Correct answer)
D.
2.5
E.
3.0
Step 1
Rate of A: 1/4 tank per hour. Rate of B: 1/6 tank per hour
Step 2
Combined rate: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 tank per hour
Step 3
Time to fill 1 tank: T = 1 ÷ (5/12) = 12/5 = 2.4 hours
Correct answer: C
Why C is correct
Correct: 12/5 = 2.4 hours
Why other options are wrong
A: Incorrect formula attempt
B: Off-by calculation error
D: Average of the two times: (4+6)/2 = 5, then /2 = 2.5; incorrect
E: Half of 6; ignores pipe A entirely
⚠ Trap: Averaging the two times (5 hours) or halving the average instead of using combined rates
Strategy Tips
Always write out the ratio in words first: 'boys : girls = 3 : 5' before using numbers
For combined-work problems, always use rates (jobs per hour), never times directly
Cross-multiplication is faster than finding LCD for most proportion problems
Use picking numbers: if ratio is 3:5 and total is unknown, try multiples of 8
For unit conversions, chain multiply fractions so units cancel
Common pitfalls
Using a part-to-part ratio as if it were part-to-whole
Averaging two speeds instead of computing total distance / total time for average speed
Forgetting unit conversions (hours to minutes, feet to miles, etc.)
Simple proportions should take 30-45 seconds. Work/rate problems take 90 seconds — if you're over 2 minutes, pick a number for 'total work' (e.g., 12 units) and use arithmetic rates.
Summary
Part-to-part and part-to-whole ratios are different; always identify which you have
Combined rate = sum of individual rates (1/t1 + 1/t2)
D = RT underpins all distance problems; average speed = total D / total T
Two painters can paint a room: one in 5 hours, one in 10 hours. Working together, how many hours do they need? Use 1/5 + 1/10 = 1/T and verify your answer is less than 5.