Grade: Grade 6 Subject: Mathematics Unit: Ratios & Rates SAT: ProblemSolving+DataAnalysis ACT: Math

Word Problems: Ratios & Rates

Apply your ratio and rate skills to solve real-world problems involving shopping, travel, cooking, and more.

Problem-Solving Strategy

Read, Plan, Solve, Check

Every word problem can be solved using a simple four-step approach.

The 4-Step Strategy

1
READ - Read the problem carefully. Identify what you know and what you need to find.
2
PLAN - Decide what operation to use. Look for keywords like "per," "each," "for every," or "how many."
3
SOLVE - Set up and solve the problem. Show your work step by step.
4
CHECK - Does your answer make sense? Is it reasonable? Did you include units?

Keywords to Watch For

  • "per," "for each," "for every" - These signal a rate (divide to find unit rate)
  • "how much for," "total cost" - Multiply unit rate by quantity
  • "better deal," "cheaper" - Compare unit rates
  • "how long," "how far" - Use rate to find time or distance

Worked Example

Step-by-Step Solution

Maria is planning a road trip. Her car gets 32 miles per gallon, and gas costs $3.80 per gallon. If her trip is 240 miles each way (480 miles round trip), how much will she spend on gas?
1
READ - What do we know?
- Car gets 32 miles per gallon
- Gas costs $3.80 per gallon
- Total trip is 480 miles
- Need to find: total gas cost
2
PLAN - What's the approach?
Step 1: Find gallons needed (miles / miles per gallon)
Step 2: Find total cost (gallons x price per gallon)
3
SOLVE - Do the math
Gallons needed:
480 miles / 32 mpg = 15 gallons
Total cost:
15 gallons x $3.80/gallon = $57.00
4
CHECK - Is it reasonable?
15 gallons at about $4 each = about $60. Our answer of $57 makes sense!
Answer: Maria will spend $57.00 on gas for her round trip.

🛒 Shopping Problems

Problem 1: Grocery Shopping

At the grocery store, you can buy a 3-pound bag of apples for $5.25 or a 5-pound bag for $8.00.

Which bag is the better deal (lower price per pound)?

Solution:

3-pound bag: $5.25 / 3 = $1.75 per pound

5-pound bag: $8.00 / 5 = $1.60 per pound

The 5-pound bag is cheaper per pound, so B is incorrect - the 5-pound bag IS the better deal! Wait, let me recalculate... $1.60 < $1.75, so the 5-pound bag is actually the better deal.

Problem 2: School Supplies

Pencils are sold in packs. A pack of 12 costs $3.00. Your class needs 180 pencils for a project.

How much will it cost to buy enough pencils for the class?

Solution:

Packs needed: 180 / 12 = 15 packs

Total cost: 15 packs x $3.00 = $45.00

🚗 Travel Problems

Problem 3: Bike Ride

Jake bikes at a constant speed. He covers 15 miles in 1.5 hours.

At this rate, how long will it take him to bike 25 miles?

Solution:

Speed: 15 miles / 1.5 hours = 10 mph

Time for 25 miles: 25 miles / 10 mph = 2.5 hours

Problem 4: Train Journey

A train travels at 80 miles per hour. The next station is 340 miles away. The train left at 9:00 AM.

What time will the train arrive at the next station?

Solution:

Time = Distance / Speed = 340 / 80 = 4.25 hours = 4 hours 15 minutes

Arrival: 9:00 AM + 4 hours 15 minutes = 1:15 PM

🍳 Cooking & Recipe Problems

Problem 5: Scaling a Recipe

A cookie recipe makes 24 cookies and uses 2 cups of flour and 1.5 cups of sugar. You want to make 60 cookies.

How many cups of flour do you need?

Solution:

Scale factor: 60 / 24 = 2.5

Flour needed: 2 cups x 2.5 = 5 cups

Problem 6: Party Planning

For a party, each pizza serves 4 people and costs $14. Each 2-liter soda bottle serves 8 people and costs $3.

How much will food and drinks cost for a party of 32 people?

Solution:

Pizzas: 32 / 4 = 8 pizzas x $14 = $112

Sodas: 32 / 8 = 4 bottles x $3 = $12

Total: $112 + $12 = $124

⚙️ Work & Productivity Problems

Problem 7: Typing Speed

Emma can type 65 words per minute. She needs to type a 2,600-word essay.

How many minutes will it take Emma to type the essay?

Solution:

Time = Total words / Words per minute

Time = 2,600 / 65 = 40 minutes

Problem 8: Assembly Line

A factory machine produces 180 widgets every 3 hours. The factory runs for 10 hours per day.

How many widgets does the machine produce in one day?

Solution:

Rate: 180 widgets / 3 hours = 60 widgets per hour

Daily production: 60 widgets/hour x 10 hours = 600 widgets

🏆 Challenge Problems

Problem 9: Multi-Step Challenge

A painter can paint 400 square feet per hour. A room has 4 walls, each 12 feet wide and 9 feet tall. The room has one door (3 feet by 7 feet) and two windows (each 4 feet by 3 feet) that don't need painting.

How long will it take to paint all the walls (excluding the door and windows)?

Solution:

Total wall area: 4 x (12 x 9) = 4 x 108 = 432 sq ft

Door area: 3 x 7 = 21 sq ft

Window area: 2 x (4 x 3) = 2 x 12 = 24 sq ft

Paintable area: 432 - 21 - 24 = 387 sq ft

Time: 387 / 400 = 0.9675 hours (about 58 minutes)

Problem 10: Rate Comparison

Two delivery services offer different pricing:
Service A: $5 base fee + $2 per mile
Service B: $15 base fee + $1.50 per mile

At what distance do both services cost the same?

Solution:

Set the costs equal: 5 + 2x = 15 + 1.5x

2x - 1.5x = 15 - 5

0.5x = 10

x = 20 miles

Check: A costs $5 + $40 = $45. B costs $15 + $30 = $45. Correct!

Key Takeaways

Remember These Steps

  • Read carefully - Identify what you know and what you need to find
  • Look for keywords - "per," "each," "total," "how many"
  • Set up the problem - Use unit rates to connect quantities
  • Check your answer - Does it make sense in the context?
Next Steps: Ready to test yourself? Move on to Common Mistakes to learn what errors to avoid, then take the Unit Quiz to see how well you've mastered ratios and rates!