Section 2.2: Ratios, Proportions, and Rates

Section 2.2: Ratios, Proportions, and Rates

The EDPT requires mastery of relationships between quantities. Ratios, proportions, and rates are mathematical ways of comparing amounts. While they might seem distinct, they are deeply interconnected concepts governed by the same fundamental algebraic rules.

1. Ratios: Comparing Quantities

A ratio is a relationship between two numbers indicating how many times the first number contains the second. It can be written with a colon (a:b), using the word "to" (a to b), or as a fraction (a/b).

Part-to-Part vs. Part-to-Whole

This is the most critical distinction in ratio problems.

• Part-to-Part: Compares two distinct subgroups. Example: The ratio of boys to girls is 3:4.
• Part-to-Whole: Compares one subgroup to the total population. Example: The ratio of boys to total students is 3:7 (since 3 + 4 = 7).

Strategic Tip: Always convert part-to-part ratios to part-to-whole ratios when a total quantity is given. Introduce an "unknown multiplier" (let's call it x). If the ratio of red to blue marbles is 5:2, then there are 5x red marbles and 2x blue marbles. The total is 7x.

Worked Example 1: The ratio of cats to dogs in an animal shelter is 3:5. If there are 64 total animals, how many dogs are there?

Step 1: Represent the quantities using the multiplier x. Cats = 3x, Dogs = 5x. Step 2: Set up the total equation. Total animals = 3x + 5x = 8x 8x = 64 Step 3: Solve for x. x = 8 Step 4: Find the number of dogs. Dogs = 5(8) = 40. There are 40 dogs.

2. Proportions and Cross-Multiplication

A proportion is simply a statement that two ratios are equal (a/b = c/d). The universally preferred method for solving proportions is cross-multiplication: if a/b = c/d, then ad = bc.

Direct vs. Inverse Variation

Understanding whether two variables move in the same direction or opposite directions is essential.

• Direct Variation (Direct Proportion): As one variable increases, the other increases at a constant rate. Equation: y = kx or y1/x1 = y2/x2. Example: If 5 apples cost $2, then 10 apples cost $4.
• Inverse Variation (Inverse Proportion): As one variable increases, the other decreases. Equation: xy = k or x1*y1 = x2*y2. Example: If 4 workers take 6 days to build a wall, 8 workers will take 3 days.

Worked Example 2: If 6 machines can produce 150 widgets in an hour, how many widgets can 14 machines produce in an hour?

This is a direct proportion. More machines = more widgets. 6/150 = 14/w Cross-multiply: 6w = 150 × 14 6w = 2100 w = 350 widgets.

3. Distance, Rate, and Time

The fundamental formula governing motion problems is Distance = Rate × Time (D = RT).

Strategic Tip: The D-R-T Table

For complex motion problems involving multiple vehicles or segments, draw a table with columns for Distance, Rate, and Time, and rows for each entity. Fill in what you know and use the D = RT formula to express the unknowns.

Worked Example 3: Train A leaves a station traveling at 60 mph. Two hours later, Train B leaves the same station on a parallel track traveling at 90 mph. How long will it take for Train B to catch up to Train A?

Step 1: Define the time variable. Let t = the time Train B travels. Since Train A left two hours earlier, its time is t + 2.

Step 2: Express distances. Distance for A = 60(t + 2) Distance for B = 90(t)

Step 3: Set up the equation. When B catches A, their distances are equal. 60(t + 2) = 90t 60t + 120 = 90t 120 = 30t t = 4

It will take Train B 4 hours to catch up.

4. Work and Rate Problems

Work problems ask how long it takes multiple people or machines working together to complete a single task. The secret is to convert everything to a rate of work per unit of time.

The Combined Work Formula

If Person A can do a job in A hours, their rate is 1/A of the job per hour. If Person B can do the job in B hours, their rate is 1/B. Working together, their combined rate is 1/T, where T is the total time. Formula: 1/A + 1/B = 1/T

Worked Example 4: Pipe 1 can fill a tank in 4 hours. Pipe 2 can fill the same tank in 6 hours. How long will it take them to fill the tank working together?

Step 1: Set up the rate equation. 1/4 + 1/6 = 1/T Step 2: Find a common denominator to add the fractions (12). 3/12 + 2/12 = 1/T 5/12 = 1/T Step 3: Cross-multiply or invert both sides to solve for T. T = 12/5 = 2.4 hours.

Common Traps in Work Problems

• Adding times instead of rates: A common mistake is assuming that if Person A takes 4 hours and Person B takes 6 hours, together they take 10 hours. This is completely illogical—two people working together will always be faster than the fastest individual. You MUST add their rates (the reciprocals of their times).
• Working against each other: Sometimes a problem involves a pipe filling a tank (positive work) and a drain emptying it (negative work). If the drain empties the tank in 8 hours, the equation becomes 1/A - 1/B = 1/T.

Final Review of Strategic Tips

1. Always establish whether a relationship is a direct or inverse variation before setting up a proportion.
2. In D-R-T problems, ensure all units match (e.g., if rate is mph, time must be in hours, not minutes).
3. Check your work rate answers for logic. The combined time must be smaller than the smallest individual time.