The table shows a proportional relationship between x and y.
x
1
2
3
4
y
6
12
18
24
Which equation represents this relationship?
Hint
Find \frac{y}{x} for more than one column. Is it always the same number?
Step-by-step solution
A proportional relationship has the form y = kx, where k = \frac{y}{x}.
Compute the ratio for every column: \frac{6}{1} = 6, \frac{12}{2} = 6, \frac{18}{3} = 6, \frac{24}{4} = 6.
The ratio is always 6, so k = 6.
The equation is y = 6x.
Answer: y = 6x
Check:x = 3: 6(3) = 18 ✓ and x = 4: 6(4) = 24 ✓ — every column works.
Tip: Always test at least two columns. y = x + 5 works for the first column only, which is exactly why it is a trap.
Question 2 of 36
Fill in
The graph shows the cost of buying gasoline at a gas station.
What is the price of gasoline, in dollars per gallon?
$ per gallon
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Pick a point on the line where you can read both numbers exactly. Divide cost by gallons.
Step-by-step solution
The line passes through the origin, so the relationship is proportional: \text{cost} = k \times \text{gallons}.
Use a marked point, (3, 12): 3 gallons cost $12.
Unit rate: k = \frac{12}{3} = 4 dollars per gallon.
Confirm with the other point (5, 20): \frac{20}{5} = 4 ✓
Answer: $4 per gallon
Check: At $4 per gallon, 8 gallons cost 4 \times 8 = \text{\textdollar}32, which is exactly where the line ends at the top right ✓
Tip: Dividing the y-value by the x-value of any point (other than the origin) on a proportional graph gives the unit rate. Divide cost by gallons, not gallons by cost (which would give 0.25 gallon per dollar).
Question 3 of 36
Maria types at a steady rate of 45 words per minute. The number of words w that Jake types in m minutes is given by the equation
w = 52m
Who types faster, and by how many words per minute?
Hint
Find each person's words per minute. For Jake, which number in the equation is the rate?
Step-by-step solution
Maria's unit rate is given in the description: 45 words per minute.
Jake's equation w = 52m has the form y = kx, so his unit rate is k = 52 words per minute.
Compare: 52 > 45, so Jake is faster.
Difference: 52 - 45 = 7 words per minute.
Answer: Jake, by 7 words per minute
Check: In 10 minutes, Jake types 52 \times 10 = 520 words and Maria types 45 \times 10 = 450 words. 520 - 450 = 70 = 7 \times 10 ✓
Tip: In y = kx the number multiplying x is the unit rate. To compare, subtract the two unit rates — do not add them.
Question 4 of 36
No calculator
A garden center sells mulch at a constant price per bag. 4 bags cost $22.
Which equation gives the cost c, in dollars, of b bags of mulch?
Hint
Find the price of one bag first. That number goes in front of b.
Step-by-step solution
A constant price per bag means the relationship is proportional: c = kb, where k is the price of 1 bag.
Find the unit rate: k = \dfrac{\text{cost}}{\text{bags}} = \dfrac{22}{4} = 5.5.
So each bag costs $5.50, and the equation is c = 5.5b.
Answer: c = 5.5b
Check: Put b = 4 into the equation: 5.5 \times 4 = 22 ✓. Also, 0 bags cost $0, as a proportional relationship must.
Tip: In y = kx, k is always "output per 1 input" — here dollars per bag. Divide cost by bags, never bags by cost.
Question 5 of 36
Fill in
The graph shows how many pages an office printer prints over time. The printer works at a constant rate.
How many pages does the printer print per minute?
pages per minute
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Pick a marked point (x, y) and divide pages by minutes.
Step-by-step solution
The line passes through the origin, so the relationship is proportional and the unit rate is \dfrac{y}{x}.
Use the marked point (2, 36): \dfrac{36 \text{ pages}}{2 \text{ min}} = 18 pages per minute.
Use the other marked point to be sure: (5, 90) gives \dfrac{90}{5} = 18.
Answer: 18 pages per minute
Check: At 1 minute the line should be at 18 pages, a little below the 20 gridline — and it is. Also 18 \times 6 = 108, where the line ends ✓.
Tip: On a proportional graph, the unit rate is the y-value when x = 1. Using a labeled point is more accurate than guessing between gridlines.
Question 6 of 36
Jada hikes at a steady 3.2 miles per hour. The distance d (miles) that Omar hikes in t hours is given by
d = 2.8t
Who hikes faster, and by how much?
Hint
Write both speeds as miles per hour. In d = kt, the number k is the speed.
Step-by-step solution
Jada's unit rate is given in words: 3.2 mph.
Omar's equation has the form d = kt, so his unit rate is k =2.8 mph.
Compare: 3.2 > 2.8, so Jada is faster.
Find the difference: 3.2 - 2.8 = 0.4 mile per hour.
Answer: Jada, by 0.4 mile per hour
Check: In 5 hours Jada hikes 3.2 \times 5 = 16 miles and Omar hikes 2.8 \times 5 = 14 miles. The 2-mile gap over 5 hours is 2 \div 5 = 0.4 mph ✓.
Tip: "By how much" asks for a difference — subtract the two unit rates, do not report either rate on its own.
Question 7 of 36
Two pumps are emptying a flooded basement. Each pump works at a constant rate.
Pump A
Time (minutes)
2
5
10
Water pumped (gallons)
18
45
90
Pump B
If both pumps run for 30 minutes, how many more gallons will Pump B pump than Pump A?
Hint
Find each pump's gallons per minute first. Then think about 30 minutes, not 1 minute.
Step-by-step solution
Pump A's unit rate from the table: \frac{18}{2} = 9, \frac{45}{5} = 9, \frac{90}{10} = 9 → 9 gallons per minute.
Pump B's unit rate from the graph: the line passes through (4, 48), so \frac{48}{4} = 12 → 12 gallons per minute.
Difference per minute: 12 - 9 = 3 gallons per minute.
Tip: Estimate first: $148.75 for 8.5 h is a bit over $17 per hour; $99 for 6 h is a bit over $16. So the weekly difference should be about $40 — a quick way to catch a slip.
Question 10 of 36
Drop-down
Car X travels 351 miles on 13 gallons of gas. The distance m (miles) that Car Y travels on g gallons is m = 31g.
Complete the sentence.
gets better gas mileage, by miles per gallon.
Choose an answer from each menu.
Hint
Find each car's miles per gallon. Better mileage means more miles for each gallon.
Step-by-step solution
Car X: \frac{351 \text{ miles}}{13 \text{ gallons}} = 27 miles per gallon.
Car Y: from m = 31g, the unit rate is k = 31 miles per gallon.
More miles per gallon is better, and 31 > 27, so Car Y has better mileage.
Difference: 31 - 27 = 4 miles per gallon.
Answer: Car Y gets better gas mileage, by 4 miles per gallon.
Check: On 13 gallons, Car Y goes 31 \times 13 = 403 miles; Car X goes 351 miles. 403 - 351 = 52 = 4 \times 13 ✓
Tip: With miles per gallon, higher is better. With cost per gallon, lower is better. Decide which kind of rate you have before choosing the winner.
Question 11 of 36
Maya's bike ride is shown on the graph. Leo rides at a constant speed of 15 miles per hour.
If both ride at their constant speeds for 2 hours, how many more miles does Leo ride than Maya?
Hint
Look closely at the units on the horizontal axis. Are they hours?
Step-by-step solution
Read a point on Maya's line: (30, 6) means 6 miles in 30 minutes.
Convert to miles per hour: 30 minutes is half an hour, so in 60 minutes she rides 6 \times 2 = 12 miles. Maya's speed is 12 mph. (Or: \frac{6}{30} = 0.2 mile per minute, and 0.2 \times 60 = 12 mph.)
Check: Leo: 15 \times 2 = 30 miles. Maya: 12 \times 2 = 24 miles (the graph confirms 12 miles at 60 minutes). 30 - 24 = 6 ✓
Tip: Before comparing, make the units identical — here, both in miles per hour. Graph axes in minutes are a classic GED trap.
Question 12 of 36
Drag and drop
Three stores sell bulk almonds at a constant price per pound. A fourth store's pricing is not proportional.
Drag each proportional store into the box that matches its price per pound. One store is not used.
Lowest price per pound
Middle price per pound
Highest price per pound
Tap a tile, then tap a box to place it.
Hint
Find dollars per pound for each store. Which one has an added amount and is therefore not proportional?
Step-by-step solution
Store A:y = 1.25x → $1.25 per pound.
Store B: $6.50 for 4 pounds → 6.50 \div 4 = \text{\textdollar}1.625 per pound.
Store C: table (3, 4.20), (6, 8.40) → 4.20 \div 3 = \text{\textdollar}1.40 and 8.40 \div 6 = \text{\textdollar}1.40 per pound.
Store D:y = x + 1.25 has an added $1.25, so it is not proportional (it does not pass through the origin) — not used.
Order the unit prices: 1.25 < 1.40 < 1.625.
Answer: Lowest — Store A; Middle — Store C; Highest — Store B.
Check: Compare by buying 4 pounds at each: A = 1.25 \times 4 = \text{\textdollar}5.00, C = 1.40 \times 4 = \text{\textdollar}5.60, B = \text{\textdollar}6.50 — same order ✓
Tip: Turning every representation into a single decimal "dollars per pound" makes ordering effortless.
Question 13 of 36
Select all
A store manager wants every price rule to match $0.75 per item, with no extra fees. Which of these relationships are proportional and have a constant of proportionality k = 0.75? Select all that apply.
Select all that apply.
Hint
For each one, check two things: does it go through (0, 0), and is \frac{y}{x} always 0.75?
Step-by-step solution
y = \tfrac{3}{4}x: proportional, and \tfrac{3}{4} = 0.75 ✓.
Line through (0, 0) and (8, 6): passes through the origin; \tfrac{6}{8} = 0.75 ✓.
12 cans for $9 at the same price each:\tfrac{9}{12} = 0.75 dollar per can ✓.
y = \tfrac{4}{3}x: proportional, but \tfrac{4}{3} \approx 1.33, not 0.75 ✗.
y = 0.75x + 2: the +2 means (0, 2) is on the line, not (0, 0) — not proportional ✗.
Table (2, 1.5), (4, 2.5), (6, 3.5):\tfrac{1.5}{2} = 0.75 but \tfrac{2.5}{4} = 0.625 — the ratio changes ✗.
Answer: y = \tfrac{3}{4}x; the line through (0, 0) and (8, 6); 12 cans for $9
Check: Each correct choice gives y = 7.5 when x = 10: \tfrac{3}{4}(10) = 7.5; \tfrac{6}{8}(10) = 7.5; 0.75(10) = 7.5 ✓.
Tip: A fraction and a decimal can be the same rate. Convert everything to decimals before you compare.
Question 14 of 36
Drag and drop
Three survey drones fly at constant speeds. A fourth drone's distance is not proportional to time. Distance d is in meters and time t is in seconds.
Drag each proportional drone into the box that matches its speed. One drone is not used.
Fastest
Middle
Slowest
Tap a tile, then tap a box to place it.
Hint
Turn each proportional drone into meters per second. Then sort the three numbers.
Step-by-step solution
Drone A:d = 14t → 14 meters per second.
Drone B: table (5, 60) and (8, 96): \tfrac{60}{5} = 12 and \tfrac{96}{8} = 12 → 12 m/s.
Drone D:d = t + 13 adds 13 instead of multiplying, so at t = 0 it is already 13 m away — not proportional, leave it out.
Order from fastest: 14 > 13 > 12.
Answer: Fastest = Drone A, Middle = Drone C, Slowest = Drone B
Check: After 10 seconds: A flies 140 m, C flies 130 m, B flies 120 m — same order ✓.
Tip: A sum like t + 13 hides a starting value. Only y = kx (no added number) is proportional.
Question 15 of 36
A grocery store and a farm stand both sell bananas at a constant price per pound.
Grocery store
Pounds
3
5
Cost ($)
1.74
2.90
Farm stand:c = 0.48p, where c is cost in dollars and p is pounds.
Nico has exactly enough money to buy 6 pounds at the grocery store. How many pounds can he buy at the farm stand for the same amount of money?
Hint
First find how much 6 pounds cost at the grocery store. Then ask: how many pounds at the farm stand cost that much?
Step-by-step solution
Grocery unit rate: \tfrac{1.74}{3} = 0.58 and \tfrac{2.90}{5} = 0.58 → $0.58 per pound.
Cost of 6 pounds at the grocery: 6 \times 0.58 = \$3.48.
At the farm stand, set the cost equal to $3.48: 0.48p = 3.48.
Divide: p = \tfrac{3.48}{0.48} = 7.25 pounds.
Answer: 7.25 pounds
Check:0.48 \times 7.25 = 3.48 ✓. It makes sense that he gets more than 6 pounds, because the farm stand is cheaper per pound.
Tip: Work-backwards problems have two steps: use one relationship to find the output, then divide by the other rate to find the input.
Question 16 of 36
Fill in
A nurse compares two IV pumps. Pump 1's output is shown on the graph. Pump 2 is set to deliver fluid at a constant 120 mL per hour.
How many more milliliters per hour does Pump 1 deliver than Pump 2?
mL per hour
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The graph's time axis is in minutes. Change Pump 1's rate to milliliters per hour before you compare.
Step-by-step solution
Read a point on Pump 1's line: (20, 50) means 50 mL in 20 minutes.
Unit rate per minute: \tfrac{50}{20} = 2.5 mL per minute.
Convert to per hour (60 minutes): 2.5 \times 60 = 150 mL per hour.
Compare with Pump 2: 150 - 120 = 30 mL per hour.
Answer: 30 mL per hour
Check: Pump 1 delivers 100 mL in 40 minutes (the point (40, 100)). In 60 minutes that is 100 \times \tfrac{60}{40} = 150 mL ✓, and 150 - 120 = 30.
Tip: Before you subtract two rates, make sure both use the same units — here, mL per hour.
Question 17 of 36
Two lemonade recipes mix lemon juice and water in proportional amounts.
Recipe A:j = \tfrac{3}{8}w, where j is cups of lemon juice and w is cups of water.
Recipe B: shown in the table.
Water (w, cups)
4
10
Lemon juice (j, cups)
1.5
3.75
The recipe with more lemon juice per cup of water tastes stronger. Which statement is true?
Hint
Write both unit rates (cups of juice per 1 cup of water) as decimals.
Step-by-step solution
Recipe A: k = \tfrac{3}{8} = 3 \div 8 = 0.375 cup of juice per cup of water.
Recipe B: \tfrac{1.5}{4} = 0.375 and \tfrac{3.75}{10} = 0.375 cup of juice per cup of water.
The two unit rates are equal, so the recipes are equally strong.
Answer: The recipes are equally strong; both use 0.375 cup of juice per cup of water.
Check: Use Recipe A with 10 cups of water: \tfrac{3}{8} \times 10 = 3.75 cups of juice — exactly what Recipe B's table shows ✓.
Tip: Bigger numbers in a table do not mean a bigger rate. Always reduce to "per 1" before you compare.
Question 18 of 36
No calculator
Two hoses fill a pool at constant rates. The graph shows the water each hose adds over time.
Which statement is true?
Hint
Find each hose's gallons per minute from its marked point. The steeper line has the greater rate.
Step-by-step solution
Hose A passes through (5, 40): \tfrac{40}{5} = 8 gallons per minute.
Hose B passes through (8, 40): \tfrac{40}{8} = 5 gallons per minute.
Hose A's line is steeper, and its rate is greater: 8 > 5.
Difference: 8 - 5 = 3 gallons per minute.
Answer: Hose A fills faster, by 3 gallons per minute.
Check: At 10 minutes, Hose A's line is at 80 gallons and Hose B's at 50. The gap is 30 gallons in 10 minutes, or 3 gallons per minute ✓.
Tip: On one graph with the same axes, the steeper line through the origin is the faster rate. Time to reach a value is not the rate — less time means faster.
Question 19 of 36
Fill in
Store A sells fabric for c = 6.75y, where c is the cost in dollars and y is the number of yards. Store B charges the same price per yard. Part of Store B's price list is shown.
Yards
2
4
?
Cost ($)
13.50
27.00
40.50
How many yards are in the missing column?
yards
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The unit rate is the same at both stores. Which number of yards, times that rate, gives $40.50?
Step-by-step solution
Both stores charge $6.75 per yard (Store A's equation gives k = 6.75).
Check Store B's known columns: 2 \times 6.75 = 13.50 ✓ and 4 \times 6.75 = 27.00 ✓.
For the missing column, solve 6.75y = 40.50.
Divide: y = \tfrac{40.50}{6.75} = 6 yards.
Answer: 6 yards
Check:6 \times 6.75 = 40.50 ✓. Also, $40.50 is 13.50 \times 3, so the yards must be 2 \times 3 = 6 ✓.
Tip: When the output is known and the input is missing, divide the output by the unit rate.
Question 20 of 36
Two bakeries bake cookies at constant rates.
Bakery A
Bakery B
Time (hours)
3
7
Cookies baked
156
364
In an 8-hour shift, how many more cookies does the faster bakery bake?
Hint
Find cookies per hour for each bakery. Then think about the gap over 8 hours.
Step-by-step solution
Bakery A (graph): the point (2, 90) gives \tfrac{90}{2} = 45 cookies per hour. (\tfrac{270}{6} = 45 too.)
Bakery B (table): \tfrac{156}{3} = 52 and \tfrac{364}{7} = 52 cookies per hour.
Bakery B is faster by 52 - 45 = 7 cookies per hour.
Tip: Gap over time = (difference in unit rates) × time. It is the same answer as finding both totals and subtracting.
Question 21 of 36
The graph shows the cost of floor tile, which is sold at a constant price per square foot.
Which equation represents the cost c, in dollars, of f square feet of tile?
Hint
Divide the y-value of a marked point by its x-value to get k in c = kf.
Step-by-step solution
The line goes through (0, 0), so the relationship is proportional: c = kf.
Use the point (10, 35): k = \tfrac{35}{10} = 3.5 dollars per square foot.
Confirm with (20, 70): \tfrac{70}{20} = 3.5 ✓.
The equation is c = 3.5f.
Answer: c = 3.5f
Check: At f = 40 the equation gives 3.5 \times 40 = 140, which is exactly where the line ends on the graph ✓.
Tip: The constant k is \tfrac{y}{x} (cost ÷ square feet). Reading only one coordinate of a point, or dividing in the wrong order, are the two classic slips.
Question 22 of 36
No calculator
The graph shows the cost of ribbon at a craft store.
Which statement correctly describes the unit rate shown by the graph?
Hint
The unit rate is the cost of exactly 1 yard. Where is the line when x = 1?
Step-by-step solution
The line passes through (0, 0), so it is proportional.
Use the marked point (5, 6): 5 yards cost $6.
Unit rate = dollars ÷ yards = \tfrac{6}{5} = 1.20 dollars per yard.
The same rate comes from (10, 12): \tfrac{12}{10} = 1.20.
Answer: Ribbon costs $1.20 per yard.
Check: At x = 1 the line is a little above the $1 level — consistent with $1.20. Also 1.20 \times 10 = 12, matching the point (10, 12) ✓.
Tip: "Unit rate" always means "per 1" of the input on the horizontal axis. Read the axis labels to get the units right.
Question 23 of 36
Drop-down
Two lawn sprinklers use water at constant rates.
Sprinkler A
Time (minutes)
15
40
Water used (gallons)
33
88
Sprinkler B uses 156 gallons in 1 hour.
Complete the sentence.
uses more water per minute, by gallon(s) per minute.
Choose an answer from each menu.
Hint
Write both rates as gallons per minute. Remember 1 hour = 60 minutes.
Step-by-step solution
Sprinkler A: \tfrac{33}{15} = 2.2 and \tfrac{88}{40} = 2.2 gallons per minute.
Sprinkler B: 156 gallons in 60 minutes → \tfrac{156}{60} = 2.6 gallons per minute.
Compare: 2.6 > 2.2, so Sprinkler B uses more water per minute.
Difference: 2.6 - 2.2 = 0.4 gallon per minute.
Answer: Sprinkler B uses more water per minute, by 0.4 gallon per minute.
Check: In 1 hour, Sprinkler A uses 2.2 \times 60 = 132 gallons and B uses 156. Gap: 156 - 132 = 24 gallons per hour, and 24 \div 60 = 0.4 ✓.
Tip: Never compare totals from different time spans (88 gallons in 40 min vs. 156 in 60 min). Convert to the same unit rate first.
Question 24 of 36
Two machines fill water bottles at constant rates.
Machine A fills bottles according to p = 35h, where p is bottles and h is hours.
Machine B's output is shown in the table.
Hours (h)
3
5
Bottles (p)
90
150
Machine B runs for 7 hours. How many hours must Machine A run to fill the same number of bottles?
Hint
First find how many bottles Machine B fills in 7 hours. Then ask: how long does A need for that many?
Step-by-step solution
Machine B's unit rate: \frac{90}{3} = 30 and \frac{150}{5} = 30 → 30 bottles per hour.
Machine B in 7 hours: 30 \times 7 = 210 bottles.
Machine A fills 35 bottles per hour. Solve 35h = 210.
h = \frac{210}{35} = 6 hours.
Answer: 6 hours
Check: Machine A in 6 hours: 35 \times 6 = 210 bottles — the same as Machine B's 210 ✓ It makes sense that A needs less time, because it is faster.
Tip: "Work backwards" questions are solved with the same equation y = kx: when you know y and k, divide: x = \frac{y}{k}.
Question 25 of 36
A painter mixes blue and white paint in proportional amounts.
Mix A:b = \tfrac{2}{3}w, where b is cups of blue and w is cups of white.
Mix B: shown in the table.
White (w, cups)
6
12
Blue (b, cups)
5
10
The mix with more blue paint per cup of white is darker. Which statement is true?
Hint
Find cups of blue per 1 cup of white for each mix. Compare the two fractions as decimals.
Step-by-step solution
Mix A: the constant in b = \tfrac{2}{3}w is \tfrac{2}{3} \approx 0.667 cup of blue per cup of white.
Mix B: \frac{b}{w} = \frac{5}{6} \approx 0.833 (and \frac{10}{12} = \frac{5}{6} ✓) cup of blue per cup of white.
Compare: \frac{5}{6} \approx 0.833 > \frac{2}{3} \approx 0.667. With common denominators: \frac{2}{3} = \frac{4}{6} < \frac{5}{6}.
Mix B has more blue per cup of white, so Mix B is darker.
Answer: Mix B, because it uses \tfrac{5}{6} cup of blue per cup of white, and \tfrac{5}{6} > \tfrac{2}{3}.
Check: With 6 cups of white, Mix A uses \tfrac{2}{3} \times 6 = 4 cups of blue; Mix B uses 5 cups. 5 > 4 ✓
Tip: When comparing the "other way" (white per blue), the larger number means less blue. Keep the quantity the question cares about (blue) on top.
Question 26 of 36
Select all
Six relationships are described below. In which of them is the second quantity proportional to the first — that is, it can be written as y = kx for one constant k? Choose every correct answer.
Select all that apply.
Hint
Proportional means y = kx: the ratio \frac{y}{x} never changes and (0, 0) fits. A constant difference is not enough.
Step-by-step solution
Test each relationship for a constant ratio \tfrac{y}{x} and the point (0, 0).
c = 0.5n — the form y = kx with k = 0.5 → proportional ✓.
3 gallons of paint cover 1,050 sq ft, at the same coverage per gallon — \tfrac{1050}{3} = 350 sq ft per gallon, and 0 gallons cover 0 sq ft → proportional ✓.
Table (0, 0), (2, 9), (5, 22.5) — \tfrac{9}{2} = 4.5, \tfrac{22.5}{5} = 4.5, and it includes (0, 0) → proportional ✓.
Table (1, 5), (2, 8), (3, 11) — y goes up by 3 each time, but \tfrac{5}{1} = 5 while \tfrac{8}{2} = 4 → not proportional ✗ (the line would cross the y-axis at 2).
Gym: $25 to join plus $10 per month — c = 10m + 25; at 0 months you already pay $25 → not proportional ✗.
A line through (0, 4) and (2, 8) — it misses the origin → not proportional ✗, even though \tfrac{8}{2} = 4.
Answer: c = 0.5n; the paint coverage; the table with (0, 0), (2, 9), (5, 22.5)
Check: Double the input in each correct choice and the output doubles: n = 4 \to 2, n = 8 \to 4; 6 gallons → 2,100 sq ft; x = 4 \to 18, x = 10 \to 45 ✓.
Tip: "Same amount added each time" makes a line; "same ratio every time" makes a proportional line. One ratio that matches is not proof — check every point.
Question 27 of 36
Four car-share services charge a constant price per mile.
Service A:c = 0.62m
Service B:
Miles
15
40
Cost ($)
8.40
22.40
Service C: a 30-mile trip costs $19.50.
Service D: a \tfrac{1}{2}-mile trip costs $0.30.
Which list orders the services from lowest to highest cost per mile?
Hint
Find each service's dollars per mile. Be careful with Service D — the trip is only half a mile.
Step-by-step solution
A:k = 0.62 dollar per mile.
B:\tfrac{8.40}{15} = 0.56 and \tfrac{22.40}{40} = 0.56 dollar per mile.
C:\tfrac{19.50}{30} = 0.65 dollar per mile.
D:\tfrac{0.30}{0.5} = 0.60 dollar per mile (half a mile costs $0.30, so a whole mile costs $0.60).
Order from lowest: 0.56 < 0.60 < 0.62 < 0.65 → B, D, A, C.
Answer: B, D, A, C
Check: Price a 10-mile trip: B $5.60, D $6.00, A $6.20, C $6.50 — increasing ✓.
Tip: Dividing by a fraction less than 1 makes the number bigger: 0.30 \div \tfrac{1}{2} = 0.60. A price for half a unit is not the unit price.
Question 28 of 36
Fill in
Two robot vacuums clean floors at constant rates.
Lena's vacuum cleans 48 square feet in 3 minutes.
Raj's vacuum cleans according to a = 20m, where a is the area in square feet and m is minutes.
Raj's vacuum runs for 36 minutes. How many minutes must Lena's vacuum run to clean the same area?
minutes
Type your answer in the box. Fractions like 3/4 are fine.
Hint
First find the area Raj's vacuum cleans. Then divide by Lena's rate.
Step-by-step solution
Area cleaned by Raj's vacuum: a = 20 \times 36 = 720 square feet.
Lena's unit rate: \tfrac{48}{3} = 16 square feet per minute.
Time for Lena to clean 720 sq ft: 16m = 720, so m = \tfrac{720}{16} = 45 minutes.
Answer: 45 minutes
Check:16 \times 45 = 720 ✓. Lena's vacuum is slower (16 < 20 sq ft per minute), so it needs more than 36 minutes — 45 makes sense.
Tip: Sanity-check direction: the slower machine always needs more time for the same job.
Question 29 of 36
A marine biologist tracks two animals swimming at constant speeds.
A sea turtle swims 1.5 kilometers in 20 minutes.
A dolphin's distance is d = 10t, where d is in kilometers and t is in hours.
If both swim for 90 minutes, how many more kilometers does the dolphin swim than the turtle?
Hint
Change the turtle's speed to kilometers per hour, and change 90 minutes to hours.
Step-by-step solution
Turtle: 20 minutes is \tfrac{1}{3} hour, so its speed is 1.5 \times 3 = 4.5 km per hour.
Dolphin: k = 10 km per hour.
Difference in speed: 10 - 4.5 = 5.5 km per hour.
Time: 90 minutes = 1.5 hours.
Extra distance: 5.5 \times 1.5 = 8.25 km.
Answer: 8.25 km
Check: Dolphin: 10 \times 1.5 = 15 km. Turtle: 90 minutes is 4.5 twenty-minute blocks, so 1.5 \times 4.5 = 6.75 km. 15 - 6.75 = 8.25 ✓.
Tip: Mixed units (minutes and hours) are the #1 trap in rate comparisons. Convert everything to one unit before any arithmetic.
Question 30 of 36
Fill in
A gardener can mix liquid fertilizer two ways. Each mix is proportional.
Mix A:\tfrac{2}{3} cup of fertilizer for every 1\tfrac{1}{2} gallons of water.
Mix B:f = \tfrac{1}{2}g, where f is cups of fertilizer and g is gallons of water.
She needs to fill an 18-gallon sprayer. How many more cups of fertilizer does the stronger mix use than the weaker mix?
cup(s)
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find cups of fertilizer per 1 gallon for Mix A by dividing two fractions. Then compare the totals for 18 gallons.
Step-by-step solution
Mix A unit rate: \dfrac{2/3}{3/2} = \dfrac{2}{3} \times \dfrac{2}{3} = \dfrac{4}{9} cup per gallon.
Mix B unit rate: \tfrac{1}{2} cup per gallon. Compare: \tfrac{4}{9} \approx 0.444 < 0.5, so Mix B is stronger.
Mix A for 18 gallons: \tfrac{4}{9} \times 18 = 8 cups.
Mix B for 18 gallons: \tfrac{1}{2} \times 18 = 9 cups.
Difference: 9 - 8 = 1 cup.
Answer: 1 cup
Check: 18 gallons is 18 \div 1.5 = 12 batches of Mix A, and 12 \times \tfrac{2}{3} = 8 cups ✓. Mix B: 9 cups. 9 - 8 = 1 ✓.
Tip: To divide by a fraction, multiply by its reciprocal. On the TI-30XS, use the n/d key to enter \tfrac{2}{3} \div \tfrac{3}{2} and get \tfrac{4}{9} exactly.
Question 31 of 36
No calculatorHot spot
The graph shows the cost of mobile data on two plans. Each plan charges a constant price per gigabyte (GB).
Click the dot that shows the unit rate of the plan with the lower cost per GB.
Click on the figure to mark your answer (or focus it and use the arrow keys).
Hint
First decide which line is less steep. Then find the dot on that line whose x-value is 1.
Step-by-step solution
Plan A's dots: (2, 15) gives \tfrac{15}{2} = 7.5 dollars per GB.
Plan B's dots: (2, 10) gives \tfrac{10}{2} = 5 dollars per GB.
Plan B has the lower cost per GB (its line is less steep).
The unit rate is the cost of 1 GB, so it is the point on Plan B with x = 1: the dot at (1, 5).
The dot (1, 7.5) is a unit-rate point too, but it belongs to the more expensive Plan A.
Answer: the Plan B dot at (1, 5)
Check: Plan B's other dots fit $5 per GB: 5 \times 2 = 10 and 5 \times 4 = 20 ✓.
Tip: On a proportional graph, the point (1, r) shows the unit rate r directly. The less steep line is the cheaper one when cost is on the vertical axis.
Question 32 of 36
Marco's pay is given by p = 17.40h, where p is pay in dollars and h is hours worked. Ellie is paid a constant hourly wage that is $2.50 more per hour than Marco's. Part of Ellie's pay record is shown.
Hours
6
15
Pay ($)
119.40
?
What number belongs in the missing cell?
Hint
Find Ellie's hourly wage first, then multiply by 15. Check it against the 6-hour column.
Step-by-step solution
Marco's unit rate: $17.40 per hour.
Ellie's unit rate: 17.40 + 2.50 = 19.90 dollars per hour.
Check with the table: 6 \times 19.90 = 119.40 ✓.
Missing pay: 15 \times 19.90 = 298.50.
Answer: $298.50
Check: Ellie's ratio must be the same in both columns: \tfrac{298.50}{15} = 19.90 and \tfrac{119.40}{6} = 19.90 ✓.
Tip: A condition like "$2.50 more per hour" changes the rate, not the total. Add it to the unit rate, then multiply.
Question 33 of 36
Fill in
Store A's coffee prices are shown on the graph. Store B sells the same coffee from bulk bins at $7.20 for every 12 ounces.
A café owner needs 5 pounds of coffee. How much money does she save by buying it at the cheaper store? (1 pound = 16 ounces.)
$
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Store B's price is per ounce, Store A's is per pound. Put both in dollars per pound before comparing.
Step-by-step solution
Store A's rate from the graph: the line passes through (3, 27), so \frac{27}{3} = \text{\textdollar}9 per pound.
Store B's rate per ounce: \frac{7.20}{12} = \text{\textdollar}0.60 per ounce.
Convert Store B to dollars per pound: 0.60 \times 16 = \text{\textdollar}9.60 per pound.
Store A is cheaper by 9.60 - 9.00 = \text{\textdollar}0.60 per pound.
For 5 pounds: 0.60 \times 5 = \text{\textdollar}3.00.
Tip: If you compared $7.20 with $9 directly you would wrongly think Store B is cheaper. Units must match: ounces → pounds (×16) before you compare.
Question 34 of 36
Fill in
Two crews pick apples at constant rates, starting at the same time.
Crew A picked 1,365 pounds in 7 hours.
Crew B:
Time (hours)
2.5
4
Apples (pounds)
450
720
After how many hours will the faster crew have picked 300 pounds more than the slower crew?
hours
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The gap between the crews grows by the same amount every hour. How much is that amount?
Step-by-step solution
Crew A: \tfrac{1365}{7} = 195 pounds per hour.
Crew B: \tfrac{450}{2.5} = 180 and \tfrac{720}{4} = 180 pounds per hour.
Crew A is faster. The gap grows by 195 - 180 = 15 pounds every hour.
Set the gap equal to 300: 15t = 300, so t = \tfrac{300}{15} = 20 hours.
Answer: 20 hours
Check: After 20 hours, Crew A has 195 \times 20 = 3{,}900 lb and Crew B has 180 \times 20 = 3{,}600 lb. 3{,}900 - 3{,}600 = 300 ✓.
Tip: The gap between two proportional relationships is itself proportional: gap = (difference of rates) × time. Divide to work backwards.
Question 35 of 36
The graph shows how far Ana and Ben walk at their constant speeds.
Cara walks 8 miles in 2.5 hours at a constant speed. If a line for Cara's walk were drawn on the same graph, where would it be?
Hint
Find all three speeds in miles per hour. The faster the speed, the steeper the line.
Step-by-step solution
Ana: point (2, 7) → \tfrac{7}{2} = 3.5 mph.
Ben: point (3, 9) → \tfrac{9}{3} = 3 mph.
Cara: \tfrac{8}{2.5} = 3.2 mph.
Cara's walk is proportional (0 miles at 0 hours), so her line starts at (0, 0).
Since 3 < 3.2 < 3.5, her line is steeper than Ben's but less steep than Ana's: it lies between the two lines.
Answer: It would start at (0, 0) and lie between Ana's line and Ben's line.
Check: At 4 hours: Ben 12 mi, Cara 3.2 \times 4 = 12.8 mi, Ana 14 mi. 12.8 is between 12 and 14 ✓.
Tip: You can place a new proportional line without drawing it: compare its unit rate with the slopes of the lines already shown.
Question 36 of 36
Fill in
In a science lab, Solution A contains 18 grams of salt in 400 milliliters of solution. A technician must make Solution B with the same concentration (grams of salt per milliliter). She needs 1.5 liters of Solution B.
How many grams of salt does Solution B need? (1 liter = 1,000 milliliters.)
grams
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find grams per milliliter for Solution A. Convert 1.5 liters to milliliters before multiplying.
Step-by-step solution
Unit rate of Solution A: \tfrac{18 \text{ g}}{400 \text{ mL}} = 0.045 gram per mL.
Same concentration means the same unit rate: salt = 0.045 \times 1{,}500 = 67.5 grams.
Answer: 67.5 grams
Check: Ratio check: \tfrac{67.5}{1500} = 0.045 = \tfrac{18}{400} ✓. Or scale up: 1,500 mL is 3.75 times 400 mL, and 18 \times 3.75 = 67.5 ✓.
Tip: "Same concentration" or "same strength" means equal unit rates. Using 1.5 instead of 1,500 mL would give a far-too-small 0.0675 g — always match units.
