Fractions, Ratios, and Proportions
Key Takeaways
- Fractions, ratios, and proportions appear in nearly every OAR math section — mastering them eliminates common errors.
- To add or subtract fractions, you must find a common denominator; to multiply, multiply straight across; to divide, multiply by the reciprocal.
- Cross-multiplication is the fastest way to solve proportion equations on a timed test.
- Ratio problems can be solved by finding the total parts and calculating each share accordingly.
- Converting between fractions, decimals, and percentages should be automatic — memorize the common conversions.
Fraction and ratio questions are among the most common on the OAR Math Skills Test. The key is not just knowing the rules but executing them quickly without a calculator.
Fraction Operations
Adding and Subtracting Fractions
You must have a common denominator before adding or subtracting:
Same denominator: 3/7 + 2/7 = 5/7
Different denominators — find the LCD (Least Common Denominator):
2/3 + 1/4
- LCD of 3 and 4 = 12
- 2/3 = 8/12
- 1/4 = 3/12
- 8/12 + 3/12 = 11/12
Finding the LCD efficiently:
| Method | When to Use |
|---|---|
| Multiply denominators | When they share no common factors (e.g., 3 and 7 → 21) |
| Use the larger denominator | When one denominator is a multiple of the other (e.g., 4 and 12 → 12) |
| Find LCM | When denominators share factors (e.g., 6 and 8 → LCM = 24) |
Multiplying Fractions
Multiply numerators together and denominators together, then simplify:
3/4 × 2/5 = (3 × 2) / (4 × 5) = 6/20 = 3/10
Speed tip: Cross-cancel before multiplying to keep numbers small.
3/4 × 2/5: The 2 and 4 share a factor of 2, so reduce to 3/2 × 1/5 = 3/10
Dividing Fractions
Multiply by the reciprocal (flip the second fraction):
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
Mixed Numbers
Convert to improper fractions before performing operations:
2 1/3 × 1 1/2 = 7/3 × 3/2 = 21/6 = 7/2 = 3 1/2
Common Fraction-Decimal-Percent Conversions
Memorize these — they save enormous time on the OAR:
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333... | 33.3% |
| 2/3 | 0.666... | 66.7% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 7/8 | 0.875 | 87.5% |
| 1/10 | 0.1 | 10% |
| 1/6 | 0.1666... | 16.7% |
| 5/6 | 0.8333... | 83.3% |
Ratios
A ratio compares two quantities. Written as a:b, a/b, or "a to b."
Solving Ratio Problems
Example: The ratio of officers to enlisted personnel at a training facility is 2:7. If there are 270 total people, how many are officers?
- Total parts = 2 + 7 = 9
- Value per part = 270 ÷ 9 = 30
- Officers = 2 × 30 = 60
Scaling Ratios
To maintain a ratio when scaling:
- If a recipe calls for 3 cups flour : 2 cups sugar for 12 cookies
- For 36 cookies (3× the batch): 9 cups flour : 6 cups sugar
Proportions
A proportion states that two ratios are equal: a/b = c/d
Cross-Multiplication
The fastest way to solve proportions:
If 3/4 = x/20, then:
- 3 × 20 = 4 × x
- 60 = 4x
- x = 15
Proportion Word Problems
Example: If a ship travels 180 nautical miles in 3 hours, how far will it travel in 5 hours at the same speed?
Set up the proportion: 180/3 = x/5
Cross-multiply: 180 × 5 = 3 × x → 900 = 3x → x = 300 nautical miles
Example: A map scale shows 1 inch = 25 miles. If two bases are 3.5 inches apart on the map, what is the actual distance?
1/25 = 3.5/x → x = 25 × 3.5 = 87.5 miles
Rate Problems as Proportions
Many OAR word problems are proportion problems in disguise:
| Problem Type | Setup |
|---|---|
| Speed/Distance/Time | distance/time = distance/time |
| Unit pricing | cost/quantity = cost/quantity |
| Scale/Maps | map distance/real distance = map distance/real distance |
| Work rate | work/time = work/time |
What is 2/3 + 3/4?
If the ratio of fuel to oil in a mixture is 40:1 and you need 10 gallons of mixture, how much oil do you need?
If 5 machines produce 200 parts in 8 hours, how many parts will 8 machines produce in 8 hours at the same rate?
What is 3/5 ÷ 2/3?
A training class has a student-to-instructor ratio of 12:1. If there are 5 instructors, how many students are there?