What is a Ratio vs. What is a Proportion?
Understanding the distinction makes solving these problems effortless:
- •A Ratio is a comparison of two numbers or quantities.
- •Example: If there are 3 dogs and 2 cats, the ratio of dogs to cats is 3$ to $2.
- •Ratios can be written three ways:
- •A Proportion is a mathematical statement that two ratios are equal.
- •Example: 32 = 64
When one of the four numbers in a proportion is missing, we use algebra or mental math to solve for it!
Method 1: The "Scale Factor" Shortcut (Mental Math)
If the numbers are friendly, you don't even need to cross-multiply. Just look for the multiplier:
- Look at the bottom denominators: What do you multiply 5 by to get 20?
- Multiply the top numerator by the exact same number:
- Therefore:
When to use this: Use the scale factor method when the relationship between numbers is an obvious whole-number multiple (e.g. × 2, × 5, × 10).
Method 2: The Universal "Cross-Multiplication" Method
What if the numbers aren't friendly (e.g., 47 = x18)?
The Cross-Product Property states:
Worked Example:
Solve for x:
- Cross-multiply diagonally:
- Divide by the number next to x:
Answer: x = 22.5
Method 3: Visual Bar Models (Tape Diagrams) for Word Problems
In modern classrooms and Singapore Math, word problems involving ratios are often solved with Bar Models (blocks representing equal units).
Problem:
The ratio of boys to girls in a school club is 3:4. If there are 35 students in total, how many girls are in the club?
Step-by-Step with Bar Models:
- Draw the units:
- •Boys get 3 blocks: [ ][ ][ ]
- •Girls get 4 blocks: [ ][ ][ ][ ]
- Find the total number of blocks (parts):
- Find the value of 1 block (unit rate):
- Multiply by the requested quantity:
- •We need the number of girls (4 blocks):
- •(Check: Boys = 3 × 5 = 15. And 15 + 20 = 35 total students!)
Final Answer: There are 20 girls in the club.
Real-World Recipe & Scaling Word Problem
Problem:
A lemonade recipe calls for 3 cups of lemon juice for every 8 cups of water. If you are making a giant batch using 24 cups of water, how many cups of lemon juice do you need?
Setting up the Proportion:
Cross-Multiplying:
Final Answer: You need 9 cups of lemon juice.
The 3 Most Common Ratio Mistakes
- Flipping the order: If a problem states the ratio of cats to dogs is 2:5, writing 52 is backwards. Order matters!
- Confusing part-to-part with part-to-whole: In a ratio of 2 boys to 3 girls, the whole is 2 + 3 = 5, not 3.
- Mismatched units: Always convert units before setting up a proportion (e.g., convert 2 hours to 120 minutes if the other ratio is in minutes).
Try It Yourself! (Practice Problem)
Practice:
A car travels 150 miles in 3 hours at a constant speed. At this rate, how many miles will it travel in 7 hours?
🔍 Click to View the Step-by-Step SolutionSolution
- Set up the proportion:
- Simplify the left ratio first (Unit Rate shortcut):
- Calculate for 7 hours:
(Alternatively, cross-multiply: 150 × 7 = 3x ⟹ 1050 = 3x ⟹ x = 350)
Answer: The car will travel 350 miles.
Learning proportions with Sparky: When you ask Sparky a ratio problem, Sparky can draw visual bar models and check each step with you so you never get lost in the numbers!