Unit Rates & Proportions
A car travels 150 miles in 3 hours. What is its unit rate in miles per hour?
Correct answer: 50 miles per hour.
Step 1: Divide distance by time: 150 ÷ 3 = 50. Step 2: Label the rate: 50 miles per hour.
A rule says y = 4x. Find y when x = 3.
Correct answer: 12.
Step 1: Substitute 3 for x: y = 4(3). Step 2: Multiply: y = 12.
Find the mean of these numbers: 2, 4, 6.
Correct answer: 4.
Step 1: Add the numbers: 2 + 4 + 6 = 12. Step 2: Divide by 3 numbers: 12 ÷ 3 = 4.
A school surveys 20 randomly selected seventh graders about their favorite lunch. What is the sample?
Correct answer: The 20 randomly selected seventh graders.
Step 1: Identify who was surveyed. Step 2: The sample is the smaller group that provides the data.
A student buys 3 folders for $6. What is the unit price?
Correct answer: $2 per folder.
Step 1: Divide the total cost by the number of folders: 6 ÷ 3 = 2.
Five notebooks cost $15. What is the cost per notebook?
Correct answer: $3 per notebook.
Step 1: Divide total cost by number of notebooks: 15 ÷ 5 = 3. Step 2: Each notebook costs $3.
A table shows x = 1, 2, 3 and y = 5, 10, 15. Write an equation for the relationship.
Correct answer: y = 5x.
Step 1: Divide y by x for each pair: 5 ÷ 1 = 5, 10 ÷ 2 = 5, 15 ÷ 3 = 5. Step 2: The constant multiplier is 5, so y = 5x.
Find the mean absolute deviation (MAD) of 2, 4, 6.
Correct answer: MAD = 1⅓, or about 1.33.
Step 1: Mean = (2 + 4 + 6) ÷ 3 = 4. Step 2: Find absolute distances from 4: |2 − 4| = 2, |4 − 4| = 0, |6 − 4| = 2. Step 3: Average the distances: (2 + 0 + 2) ÷ 3 = 4/3 ≈ 1.33.
A random sample of 50 students finds that 30 prefer soccer. What fraction of the sample prefers soccer?
Correct answer: 3/5, or 60%.
Step 1: Write the fraction: 30/50.
Step 2: Simplify by dividing by 10: 3/5. Step 3: Convert to a percent: 3 ÷ 5 = 0.6 = 60%.
For y = 7x, find y when x = 8.
Correct answer: 56.
Step 1: Substitute 8 for x: y = 7(8). Step 2: Multiply: y = 56.
A recipe uses 4 cups of flour for 10 cookies. How many cups are needed for 25 cookies?
Correct answer: 10 cups.
Step 1: Find the unit rate: 4 ÷ 10 = 0.4 cup per cookie. Step 2: Multiply by 25: 0.4 × 25 = 10 cups.
A game center charges $6 to enter and $2 per game. Write an equation for the total cost y after playing x games.
Correct answer: y = 2x + 6.
Step 1: The cost per game is $2, so write 2x. Step 2: Add the $6 entry fee. The equation is y = 2x + 6.
Find the MAD of 3, 5, 7, 9.
Correct answer: MAD = 2.
Step 1: Mean = (3 + 5 + 7 + 9) ÷ 4 = 6. Step 2: Absolute deviations are 3, 1, 1, 3. Step 3: MAD = (3 + 1 + 1 + 3) ÷ 4 = 8 ÷ 4 = 2.
In a random sample of 40 students, 10 ride the bus. If the school has 400 students, estimate how many ride the bus.
Correct answer: About 100 students.
Step 1: Find the sample proportion: 10 ÷ 40 = 0.25.
Step 2: Multiply by the population: 0.25 × 400 = 100.
Find the mean of 4, 6, 8, 10.
Correct answer: 7.
Step 1: Add: 4 + 6 + 8 + 10 = 28.
Step 2: Divide by 4: 28 ÷ 4 = 7.
A student walks 12 miles in 4 hours. At the same rate, how far will the student walk in 7 hours?
Correct answer: 21 miles.
Step 1: Find the rate: 12 ÷ 4 = 3 miles per hour. Step 2: Multiply by 7 hours: 3 × 7 = 21 miles.
Use y = 3x + 2 to find y when x = 6.
Correct answer: 20.
Step 1: Substitute 6 for x: y = 3(6) + 2. Step 2: Multiply: 18 + 2. Step 3: Add: y = 20.
Two teams have these scores. Team A: 8, 10, 12. Team B: 9, 10, 11. Both have a mean of 10. Which team has the smaller MAD?
Correct answer: Team B.
Step 1: Team A's deviations from 10 are 2, 0, 2; MAD = 4 ÷ 3 ≈ 1.33.
Step 2: Team B's deviations are 1, 0, 1; MAD = 2 ÷ 3 ≈ 0.67.
Step 3: 0.67 is smaller, so Team B's scores vary less.
A student asks only members of the basketball team whether students like basketball. Is this a good sample of the whole school? Why?
Correct answer: No. It is biased because basketball team members are more likely to like basketball.
Step 1: Ask whether everyone in the school had a fair chance of being selected.
Step 2: Only basketball team members were surveyed.
Step 3: The sample is not representative of the whole school.
A random sample of 25 students finds that 15 bring lunch from home. If there are 200 students, estimate how many bring lunch from home.
Correct answer: About 120 students.
Step 1: Find the proportion: 15 ÷ 25 = 0.6.
Step 2: Multiply by 200: 0.6 × 200 = 120.
Eight pounds of dog food cost $20. At this rate, what is the cost of 14 pounds?
Correct answer: $35.
Step 1: Find the unit price: $20 ÷ 8 = $2.50 per pound. Step 2: Multiply: $2.50 × 14 = $35.
A line passes through (2, 7) and (5, 16). Find the rate of change.
Correct answer: 3.
Step 1: Find the change in y: 16 − 7 = 9. Step 2: Find the change in x: 5 − 2 = 3. Step 3: Divide: 9 ÷ 3 = 3.
Data set A is 10, 10, 10, 10. Data set B is 7, 9, 11, 13. Which data set has a greater MAD? Show why.
Correct answer: Data set B.
Step 1: Both means are 10.
Step 2: Set A's deviations are all 0, so its MAD is 0.
Step 3: Set B's deviations are 3, 1, 1, 3. Step 4: MAD = (3 + 1 + 1 + 3) ÷ 4 = 2. Data set B has the greater MAD
In a random sample of 80 students, 48 prefer a later school start time. The school has 600 students. Estimate how many prefer a later start.
Correct answer: About 360 students.
Step 1: Find the proportion: 48 ÷ 80 = 0.6.
Step 2: Multiply by 600: 0.6 × 600 = 360.
Step 3: This is an estimate, not an exact count.
A store charges $10 plus $4 per notebook order. Another store charges $6 per notebook with no fee. At how many notebooks do the costs match?
Correct answer: 5 notebooks; both cost $30.
Step 1: Write the equations: Store A: y = 4x + 10; Store B: y = 6x.
Step 2: Set them equal: 4x + 10 = 6x.
Step 3: Subtract 4x: 10 = 2x.
Step 4: Divide by 2: x = 5.
Step 5: Check: 4(5) + 10 = 30 and 6(5) = 30.