📊 Ratios, Rates, Proportions — Problem Types

Master the common problem patterns and systematic solution approaches for ratio, rate, and proportion problems.

Basic Ratio Problems

Simplifying Ratios

Recognition: Questions asking to simplify ratios to lowest terms Template:

1. Find GCF of both numbers
2. Divide both terms by GCF
3. Check if further simplification is possible

Example: Simplify $24:36$

1. GCF: GCF(24, 36) = 12
2. Divide: $24 \div 12 = 2$, $36 \div 12 = 3$
3. Result: $24:36 = 2:3$

Common variations:

• Simplify $a:b$ to lowest terms
• Find equivalent ratios
• Compare ratios

Finding Missing Terms

Recognition: Questions with one missing term in a ratio Template:

1. Set up proportion
2. Use cross multiplication
3. Solve for unknown

Example: If $3:4 = x:12$, find $x$

1. Proportion: $\frac{3}{4} = \frac{x}{12}$
2. Cross multiply: $3 \times 12 = 4x$
3. Solve: $36 = 4x$, so $x = 9$

Common variations:

• Find missing numerator
• Find missing denominator
• Find missing term in equivalent ratios

Rate Problems

Unit Rate Problems

Recognition: Questions asking for rate per unit Template:

1. Identify the two quantities
2. Divide first quantity by second
3. Simplify to unit rate

Example: A car travels 150 miles in 3 hours. What is the speed in mph?

1. Quantities: 150 miles, 3 hours
2. Divide: $150 \div 3 = 50$
3. Result: 50 mph

Common variations:

• Speed problems
• Price per unit problems
• Work rate problems
• Flow rate problems

Rate Conversion Problems

Recognition: Questions asking to convert between different rate units Template:

1. Identify current and target units
2. Use conversion factors
3. Calculate new rate

Example: Convert 60 mph to feet per second

1. Current: 60 mph
2. Target: ft/s
3. Convert: $60 \times \frac{5280 \text{ ft}}{3600 \text{ s}} = 88 \text{ ft/s}$

Common variations:

• Speed unit conversions
• Price unit conversions
• Work rate conversions
• Flow rate conversions

Proportion Problems

Direct Proportion Problems

Recognition: Questions where one quantity varies directly with another Template:

1. Set up proportion
2. Use cross multiplication
3. Solve for unknown

Example: If 3 apples cost $1.50, how much do 8 apples cost?

1. Proportion: $\frac{3}{1.50} = \frac{8}{x}$
2. Cross multiply: $3x = 1.50 \times 8 = 12$
3. Solve: $x = 4$

Common variations:

• Price-quantity problems
• Distance-time problems
• Work-time problems
• Scale problems

Inverse Proportion Problems

Recognition: Questions where one quantity varies inversely with another Template:

1. Set up inverse proportion
2. Use product rule
3. Solve for unknown

Example: If 4 workers can build a wall in 6 days, how long for 8 workers?

1. Inverse proportion: $4 \times 6 = 8 \times x$
2. Product rule: $24 = 8x$
3. Solve: $x = 3$ days

Common variations:

• Work-time problems
• Speed-time problems
• Pressure-volume problems
• Resistance-current problems

Word Problems

Part-Whole Problems

Recognition: Questions about parts of a whole Template:

1. Identify part and whole
2. Set up ratio
3. Use proportion to find unknown

Example: In a class of 30 students, the ratio of boys to girls is 2:3. How many boys are there?

1. Part: boys, Whole: total students
2. Ratio: boys:total = 2:5
3. Proportion: $\frac{2}{5} = \frac{x}{30}$, so $x = 12$ boys

Common variations:

• Gender ratio problems
• Age ratio problems
• Grade ratio problems
• Color ratio problems

Mixture Problems

Recognition: Questions about mixing different concentrations Template:

1. Set up weighted average equation
2. Solve for unknown
3. Check answer

Example: Mix 2 liters of 20% acid with 3 liters of 80% acid. What is the concentration?

1. Weighted average: $\frac{2 \times 0.20 + 3 \times 0.80}{2 + 3}$
2. Calculate: $\frac{0.4 + 2.4}{5} = 0.56$
3. Result: 56% acid

Common variations:

• Acid concentration problems
• Alcohol concentration problems
• Salt concentration problems
• Sugar concentration problems

Work Problems

Recognition: Questions about work rates and time Template:

1. Find individual work rates
2. Use combined rate formula
3. Solve for time

Example: Pipe A fills tank in 3 hours, Pipe B in 6 hours. How long together?

1. Rates: A = $\frac{1}{3}$ tank/hour, B = $\frac{1}{6}$ tank/hour
2. Combined: $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$ tank/hour
3. Time: 2 hours

Common variations:

• Pipe filling problems
• Worker problems
• Machine problems
• Pump problems

Scale and Similarity Problems

Scale Factor Problems

Recognition: Questions about scaling up or down Template:

1. Find scale factor
2. Apply to all dimensions
3. Calculate new measurements

Example: A model car is $\frac{1}{24}$ scale. If the real car is 12 feet long, how long is the model?

1. Scale factor: $\frac{1}{24}$
2. Apply: $12 \times \frac{1}{24} = 0.5$ feet
3. Result: 6 inches

Common variations:

• Model building problems
• Map problems
• Blueprint problems
• Photography problems

Similar Figure Problems

Recognition: Questions about similar shapes Template:

1. Identify corresponding sides
2. Set up proportion
3. Solve for unknown

Example: Two similar triangles have corresponding sides 3 and 6. If the area of the smaller triangle is 9, what is the area of the larger triangle?

1. Scale factor: $\frac{6}{3} = 2$
2. Area ratio: $2^2 = 4$
3. Larger area: $9 \times 4 = 36$

Common variations:

• Similar triangle problems
• Similar rectangle problems
• Similar circle problems
• Similar polygon problems

Complex Problems

Multi-Step Problems

Recognition: Problems requiring multiple ratio/proportion steps Template:

1. Break into steps
2. Solve each step
3. Combine results

Example: A recipe calls for 2 cups flour and 1 cup sugar for 4 servings. How much flour for 10 servings?

1. Step 1: Find ratio of flour to servings: $\frac{2}{4} = \frac{1}{2}$
2. Step 2: Apply to 10 servings: $\frac{1}{2} = \frac{x}{10}$
3. Step 3: Solve: $x = 5$ cups

Rate-Time-Distance Problems

Recognition: Problems involving motion Template:

1. Use $d = rt$ formula
2. Set up equations
3. Solve system

Example: Two cars start 100 miles apart and drive toward each other. One goes 60 mph, the other 40 mph. When do they meet?

1. Distance: $d_1 + d_2 = 100$
2. Time: $t_1 = t_2 = t$
3. Equations: $60t + 40t = 100$, so $t = 1$ hour

Common variations:

• Meeting problems
• Overtaking problems
• Round trip problems
• Upstream/downstream problems

Common Mistakes and Fixes

Ratio Mistakes

Mistake: Adding ratios incorrectly: $3:4 + 1:2 = 4:6$ Fix: Use proper formula: $(3 \times 2 + 4 \times 1):(4 \times 2) = 10:8 = 5:4$

Mistake: Forgetting to simplify Fix: Always divide by GCF

Proportion Mistakes

Mistake: Wrong cross multiplication Fix: Multiply diagonally: $a \times d = b \times c$

Mistake: Forgetting units Fix: Always check unit consistency

Rate Mistakes

Mistake: Wrong unit conversion Fix: Use conversion factors carefully

Mistake: Confusing direct and inverse Fix: Direct means both increase together

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