Proportional Reasoning
Determine whether the two quantities are proportional: x: 3, 2x: 6.
Yes. 2x:6 = 2*(x):6; for x=3 gives 6:6 -> equal ratios.
Find the unit rate: 45 miles in 3 hours.
15 miles/hour.
Simplify the expression: 3(x + 4
3x + 12.
Write an expression for total cost if each ticket costs $7 and a student buys t tickets.
7t.
A recipe uses 2 cups of flour for every 3 cups of sugar. If you use 8 cups of flour, how much sugar is needed? Show ratio reasoning
12 cups sugar. (Scale factor: 8/2 =4;
A table shows hours and pages: (1, 20), (2, 40), (3, 60). Is this proportional? What is the unit rate and the graph’s slope?
Yes proportional; unit rate 20 pages/hour; slope = 20.
ewrite 0.75m as an equivalent expression showing subtraction from m (context hint: discount).
0.75m = m − 0.25m
A subway card costs $10 plus $1.25 per ride. Write an expression for total cost C after r rides.
C = 10 + 1.25r.
Identify the unit rate and constant of proportionality: A car travels 150 miles in 3 hours.
Unit rate 50 miles/hour
A bike rental costs $18 for 3 hours. Write the proportional equation relating cost C to hours h,
A bike rental costs $18 for 3 hours. Write the proportional equation relating cost C to hours h,
Expand and simplify: 2(3x + 5). Then write another equivalent form by factoring.
6x + 10; factored: 2(3x + 5).
You earn $6 per hour and have a $20 bonus. Write an expression for your pay P for h hours. Then evaluate for h = 5.
P = 6h + 20; for h=5, P = 6(5) + 20 = 50.
Two quantities are proportional. Their table shows (2, 6) and (5, y). Find y and explain using equivalent ratios
y = 15. (2→6 unit rate 3; 5*3=15)
On a coordinate plane, a proportional relationship is represented by the line through (0,0) and (1, 2.5). What does the point (4, 10) mean in context? If this is a speed scenario, state the unit rate.
(4,10) means in 4 units of time/distance the quantity is 10; unit rate is 2.5 per unit time.
Given the expression for total cost T: T = 5n + 10. Rewrite T as T = 5(n + 2) and explain how both forms help interpret the situation.
Both are equivalent; T = 5n + 10 shows base cost + extra; T = 5(n + 2)
Write an expression for the area of a rectangular poster with length (x + 2) and width 3. Then expand and interpret each term.
Area = 3(x + 2) = 3x + 6. 3x is variable area; 6 is constant part from width*2.
A student says: “If y is proportional to x and y = 12 when x = 4, then y = 3x for all x.” Explain why this is correct,
constant = 3 so y = 3x. Graph is line through origin with slope 3. (Points: (0,0), (1,3)
Given a graph of a line through the origin with points (0,0), (2, 7), determine if it represents a proportional relationship. If not, explain and find the unit rate if possible.
Not proportional because slope would be 7/2 = 3.5, but (0,0) is present — actually line through origin with (2,7) would give slope 3.5, so it is proportional; unit rate 3.5 per unit.
The discounted price P is expressed as P = C − 0.25C. Rewrite P in a simplified equivalent form and explain the meaning of each form in a shopping context.
P = 0.75C. Interpretation: C − 0.25C shows original minus discount; 0.75C shows final price is 75% of original.
Multi-step: A cell phone plan charges $12 per month plus $0.05 per text message. You have a budget of $30 for one month. Write an inequality for the number of texts t you can send and solve it. Show the solution set and interpret.
12 + 0.05t ≤ 30. Solve: 0.05t ≤ 18 → t ≤ 360. Interpretation: up to 360 texts. Graph solution on number line