A cookie recipe uses 3 cups of flour to make 24 cookies. How many cups of flour are needed to make 60 cookies?
Hint
Write a proportion with cups on top and cookies on the bottom on both sides.
Step-by-step solution
Set up the proportion (cups over cookies on both sides): \frac{3}{24} = \frac{x}{60}
Cross-multiply: 24x = 3 \times 60 = 180
Divide both sides by 24: x = \frac{180}{24} = 7.5
You need 7.5 cups of flour.
Check:\tfrac{3}{24} = 0.125 and \tfrac{7.5}{60} = 0.125 — equal ratios ✓ (Also: 60 is 2.5 times 24, and 3 \times 2.5 = 7.5.)
Tip: Keep the same units in the same positions. Mixing them (\tfrac{3}{24} = \tfrac{60}{x}) gives 480 cups — always ask whether your answer makes sense for a batch of cookies.
Question 2 of 36
No calculatorFill in
A shelf is 4.5 feet long. How long is the shelf in inches? (1 foot = 12 inches)
inches
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Inches are smaller than feet, so you will need more of them. Multiply or divide?
Step-by-step solution
Inches are a smaller unit than feet, so the number of inches will be larger — multiply.
Use the conversion factor \dfrac{12\text{ in}}{1\text{ ft}}: 4.5\text{ ft} \times \frac{12\text{ in}}{1\text{ ft}} = 54\text{ in}
By hand: 4 \times 12 = 48 and 0.5 \times 12 = 6; 48 + 6 = 54.
The shelf is 54 inches long.
Check:54 \div 12 = 4.5 ft ✓
Tip: Smaller unit → bigger number → multiply. Larger unit → smaller number → divide. If you divided (4.5 \div 12 = 0.375), you'd claim a 4.5-ft shelf is less than half an inch long — a quick sense check catches that.
Question 3 of 36
No calculator
In a class of 32 students, the ratio of boys to girls is 3 : 5. How many girls are in the class?
Hint
3 : 5 compares boys with girls. How many equal parts make up the whole class?
Tip: A part-to-part ratio (boys : girls) is not a fraction of the whole. Add the ratio numbers to get the whole: girls are \tfrac{5}{8} of the class, not \tfrac{3}{5} or \tfrac{5}{3}.
Question 4 of 36
One 2-gallon can of deck stain covers 700 square feet. Rosa's deck and fence together measure 1,925 square feet. How many gallons of stain does she need to cover the whole area once?
Hint
Write a proportion with gallons on top and square feet on the bottom on both sides.
Step-by-step solution
Set up the proportion with the same units in the same places (gallons over square feet): \frac{2}{700} = \frac{g}{1{,}925}
Cross-multiply: 700g = 2 \times 1{,}925 = 3{,}850
Divide both sides by 700: g = \frac{3{,}850}{700} = 5.5
Rosa needs 5.5 gallons of stain.
Check: One gallon covers 700 \div 2 = 350 square feet, and 5.5 \times 350 = 1{,}925 square feet ✓ Also, 1,925 is a bit less than 3 times 700, so a bit less than 3 \times 2 = 6 gallons makes sense.
Tip: Before you compute, estimate. A deck about 3 times bigger needs about 3 times as much stain, so any answer in the hundreds or thousands is impossible.
Question 5 of 36
No calculatorFill in
A bag of dog treats weighs 3.5 pounds. How much does the bag weigh in ounces? (1 pound = 16 ounces)
ounces
Type your answer in the box. Fractions like 3/4 are fine.
Hint
An ounce is smaller than a pound. Will the number of ounces be bigger or smaller than 3.5?
Step-by-step solution
Ounces are a smaller unit than pounds, so the number of ounces will be larger. That means multiply.
Multiply by the conversion factor \dfrac{16\text{ oz}}{1\text{ lb}}: 3.5\text{ lb} \times \frac{16\text{ oz}}{1\text{ lb}} = 56\text{ oz}
By hand: 3 \times 16 = 48 and 0.5 \times 16 = 8, so 48 + 8 = 56.
The bag weighs 56 ounces.
Check:56 \div 16 = 3.5 pounds ✓
Tip: Going from a big unit to a small unit, multiply; going from a small unit to a big unit, divide. Dividing here (3.5 \div 16 \approx 0.22) would say a bag of treats weighs less than a quarter of an ounce, which makes no sense.
Question 6 of 36
A farmers market sells local honey for $0.75 per ounce. What is the price of a 2-pound jar of honey? (1 pound = 16 ounces)
Hint
The price is per ounce, so first find how many ounces are in the jar.
Step-by-step solution
The price is given per ounce, but the jar is measured in pounds. Convert pounds to ounces first: 2\text{ lb} \times \frac{16\text{ oz}}{1\text{ lb}} = 32\text{ oz}
Multiply the number of ounces by the price per ounce: 32 \times \$0.75 = \$24.00
Tip: Before you multiply by a price, make sure the amount is in the same unit as the price ("per ounce" means you need ounces). If you skip the conversion, you get 2 \times 0.75 = \$1.50, far too cheap for two pounds of honey.
Question 7 of 36
Mike's truck uses 9 gallons of gas to travel 252 miles. He is planning a 420-mile trip, and gas costs $3.60 per gallon. How much will the gas for the trip cost?
Hint
First find how many gallons the 420-mile trip needs. Then find the cost of that many gallons.
Step-by-step solution
Gallons for 420 miles — set up a proportion (gallons over miles): \frac{9}{252} = \frac{g}{420}
Cross-multiply and divide: g = \frac{9 \times 420}{252} = \frac{3{,}780}{252} = 15\text{ gallons}
Cost: 15 \times \$3.60 = \$54.00
The gas will cost $54.00.
Check: The truck gets 252 \div 9 = 28 miles per gallon, and 15 \times 28 = 420 miles ✓
Tip: In a two-step problem the middle result (15 gallons) is almost always a trap answer. Before choosing, reread the question to confirm the unit it asks for — here, dollars.
Question 8 of 36
Fill in
A car is traveling at 60 miles per hour. What is its speed in feet per second? (1 mile = 5,280 feet; 1 hour = 3,600 seconds)
feet per second
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Change miles to feet and hours to seconds. Arrange each conversion factor so the old unit cancels.
Step-by-step solution
Write the rate with units: \dfrac{60\text{ mi}}{1\text{ h}}.
Multiply by conversion factors so miles and hours cancel: \frac{60\text{ mi}}{1\text{ h}} \times \frac{5{,}280\text{ ft}}{1\text{ mi}} \times \frac{1\text{ h}}{3{,}600\text{ s}}
Check:88\ \text{ft/s} \times 3{,}600\ \text{s} = 316{,}800 ft in one hour, and 316{,}800 \div 5{,}280 = 60 miles ✓
Tip: In a chain conversion, every old unit must appear once on top and once on the bottom so it cancels. If you only convert miles (316,800 "feet per hour") or only hours (0.0167 "miles per second"), the units left over tell you a step is missing.
Question 9 of 36
Curtain fabric costs $6.75 per yard. Elena needs 14 feet of fabric. How much will the fabric cost? (1 yard = 3 feet)
Hint
The price is per yard, but the length is in feet. Convert first.
Step-by-step solution
Convert feet to yards (yards are larger, so divide): 14\text{ ft} \times \frac{1\text{ yd}}{3\text{ ft}} = \frac{14}{3}\text{ yd} \approx 4.67\text{ yd}
Multiply by the price per yard — keep the fraction to avoid rounding error: \frac{14}{3} \times 6.75 = \frac{94.5}{3} = 31.50
The fabric costs $31.50.
Check: $6.75 per yard is 6.75 \div 3 = \$2.25 per foot, and 14 \times 2.25 = \$31.50 ✓
Tip: Price and amount must use the same unit. You can convert the length to yards or convert the price to dollars per foot. Rounding 4.67 yd first gives $31.52 — so keep fractions or extra decimals until the end.
Question 10 of 36
Drop-down
A medicine is given at 5 mg for every 10 pounds of body weight. The patient weighs 154 pounds. The liquid medicine contains 25 mg in every 5 mL.
Choose the correct answer for each blank.
The patient's dose is . The nurse should give of the liquid.
Choose an answer from each menu.
Hint
Do it in two proportions: first weight → milligrams, then milligrams → milliliters.
Check: The liquid has 25 \div 5 = 5 mg per mL, and 15.4 \times 5 = 77 mg ✓; also 77 \div 154 = 0.5 mg per lb = 5 mg per 10 lb ✓
Tip: Dosage problems are chains of proportions. Label each ratio with units (mg/lb, mg/mL) — the units show you which numbers go on top, and they catch upside-down setups.
Question 11 of 36
Select all
8 pounds of potatoes cost $14. Let c be the cost, in dollars, of 20 pounds at the same price per pound.
Select all the equations that can be used to find c.
Select all that apply.
Hint
In a correct proportion, both ratios compare the same things in the same order (pounds to dollars, or dollars to pounds).
Step-by-step solution
The true answer is c = 20 \times \tfrac{14}{8} = \$35. Test each equation.
\dfrac{8}{14} = \dfrac{20}{c}: pounds over dollars on both sides ✓ → 8c = 280, c = 35.
\dfrac{14}{8} = \dfrac{c}{20}: dollars over pounds on both sides ✓ → 8c = 280, c = 35.
c = 20 \times \dfrac{14}{8}: pounds times price per pound ($1.75) ✓ → c = 35.
c = 20 \times \dfrac{8}{14}: uses pounds per dollar as if it were a price ✗ → c \approx 11.43.
Correct equations: \tfrac{8}{14} = \tfrac{20}{c}, \tfrac{14}{8} = \tfrac{c}{20}, and c = 20 \times \tfrac{14}{8}.
Check: $35 for 20 lb is 35 \div 20 = \$1.75 per lb, the same as 14 \div 8 = \$1.75 ✓
Tip: A proportion is correct when the units line up: the top of each fraction has the same unit, and so does the bottom. Equations that mix the order give a cost that is smaller than the cost of 8 pounds — an easy red flag.
Question 12 of 36
A rectangular bedroom is 15 feet by 12 feet. Carpet costs $28 per square yard. How much will it cost to carpet the whole floor? (1 yard = 3 feet)
Hint
Find the area in square feet. How many square feet are in one square yard? (Picture a square 1 yard on each side.)
Step-by-step solution
Area in square feet: 15 \times 12 = 180\text{ ft}^2
One square yard is 3\text{ ft} \times 3\text{ ft} = 9\text{ ft}^2. Convert: 180 \div 9 = 20\text{ yd}^2
Cost: 20 \times \$28 = \$560
The carpet costs $560.
Check: Convert the sides first: 15 \text{ ft} = 5 yd and 12\text{ ft} = 4 yd, so the area is 5 \times 4 = 20 sq yd ✓
Tip: For area, the conversion factor is squared: 1\text{ yd}^2 = 9\text{ ft}^2, not 3. The safest method is to convert each side to yards before multiplying.
Question 13 of 36
Select all
An office printer prints 45 pages in 3 minutes. Let p be the number of pages it prints in 14 minutes at the same rate.
Select all the equations that can be used to find p.
Select all that apply.
Hint
In a correct proportion, both fractions compare the same things in the same order (pages to minutes, or minutes to pages).
Step-by-step solution
The true answer is p = 14 \times \tfrac{45}{3} = 14 \times 15 = 210 pages. Test each equation.
\dfrac{45}{3} = \dfrac{p}{14}: pages over minutes on both sides ✓ → 3p = 630, p = 210.
\dfrac{3}{45} = \dfrac{14}{p}: minutes over pages on both sides ✓ → 3p = 630, p = 210.
p = 15 \times 14: the unit rate (15 pages per minute) times 14 minutes ✓ → p = 210.
p = 14 \times \dfrac{3}{45}: uses minutes per page as if it were pages per minute ✗ → p \approx 0.93.
\dfrac{45}{14} = \dfrac{p}{3}: puts 45 pages over 14 minutes, which are not a matching pair ✗ → p \approx 9.64.
Correct equations: \tfrac{45}{3} = \tfrac{p}{14}, \tfrac{3}{45} = \tfrac{14}{p}, and p = 15 \times 14.
Check: 210 pages in 14 minutes is 210 \div 14 = 15 pages per minute, the same as 45 \div 3 = 15 ✓
Tip: Each fraction in a proportion must be built from a matching pair (45 pages go with 3 minutes; p pages go with 14 minutes). A printer that runs longer must print more than 45 pages, so any equation giving less than 1 page is wrong.
Question 14 of 36
The table shows how much Tasha earns at her part-time job. Her pay is proportional to the number of hours she works.
Hours worked
Pay
3
$46.50
5
$77.50
8
$124.00
How much will Tasha earn for 13 hours of work?
Hint
Find the constant of proportionality first: pay divided by hours.
Step-by-step solution
In a proportional table, pay \div hours is the same in every row. Find this unit rate: \frac{\$124.00}{8} = \$15.50 \text{ per hour}
Confirm with another row: \$46.50 \div 3 = \$15.50 ✓ and \$77.50 \div 5 = \$15.50 ✓
Multiply the unit rate by 13 hours: 13 \times \$15.50 = \$201.50
Tasha will earn $201.50.
Check: 13 hours = 5 + 8 hours, so the pay should be \$77.50 + \$124.00 = \$201.50 ✓ (This shortcut works only because the relationship is proportional.)
Tip: In any proportional table, divide y by x to get the constant of proportionality k, then use y = kx for any new value. Always test two rows; if the quotients differ, the table is not proportional.
Question 15 of 36
No calculator
In a parking lot, the ratio of cars to trucks is 7 : 2. There are 35 more cars than trucks. How many vehicles (cars and trucks) are in the lot?
Hint
In the ratio 7 : 2, how many more "parts" do cars have than trucks? Those parts are worth 35 vehicles.
Step-by-step solution
Think of the ratio in equal parts: cars have 7 parts and trucks have 2 parts.
Cars have 7 - 2 = 5 more parts than trucks, and that difference is 35 vehicles: 5 \text{ parts} = 35 \;\Rightarrow\; 1 \text{ part} = 35 \div 5 = 7
Find each group: cars = 7 \times 7 = 49 and trucks = 2 \times 7 = 14.
Check:49 - 14 = 35 more cars ✓ and 49 : 14 simplifies (divide by 7) to 7 : 2 ✓
Tip: When a problem gives the difference between two groups, match it to the difference in ratio parts. Then you know what one part is worth and can find anything.
Question 16 of 36
No calculator
Dana wants to know how many cups are in 2.5 gallons of lemonade. She uses 1 gallon = 4 quarts, 1 quart = 2 pints, and 1 pint = 2 cups. Her work:
Step 1:2.5 \text{ gal} \times 4 = 10 quarts
Step 2:10 \text{ qt} \times 2 = 20 pints
Step 3:20 \text{ pt} \div 2 = 10 cups
Which statement about Dana's work is true?
Hint
At every step she goes from a larger unit to a smaller unit. What should happen to the number each time?
Step-by-step solution
Step 1: Gallons to quarts is big unit → small unit, so multiply: 2.5 \times 4 = 10 quarts ✓
Step 2: Quarts to pints is big unit → small unit, so multiply: 10 \times 2 = 20 pints ✓
Step 3: Pints to cups is also big unit → small unit, so she should multiply, not divide: 20 \text{ pt} \times \frac{2 \text{ c}}{1 \text{ pt}} = 40 \text{ cups}
Step 3 is wrong; it should be 20 \times 2 = 40 cups.
Check: 1 gallon = 4 \times 2 \times 2 = 16 cups, so 2.5 \times 16 = 40 cups ✓ Her answer of 10 cups is less than the 20 pints she just found, but cups are smaller than pints, so there must be more cups.
Tip: In a conversion chain, check the direction at each link: smaller unit → bigger number. If a number shrinks while the units get smaller, that step has an error.
Question 17 of 36
Fill in
A contractor is pouring a concrete pad that is 12 feet long, 9 feet wide, and 4 inches thick. Concrete is ordered in cubic yards. How many cubic yards of concrete does the pad need? Round to the nearest hundredth. (1 foot = 12 inches; 1 cubic yard = 27 cubic feet)
cubic yards
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Put all three measurements in feet first. Then remember that a cubic yard is 3 \times 3 \times 3 cubic feet.
Step-by-step solution
Convert the thickness to feet so all three measurements match: 4 \text{ in} \div 12 = \tfrac{1}{3} \text{ ft}
Find the volume in cubic feet: V = 12 \times 9 \times \tfrac{1}{3} = 108 \times \tfrac{1}{3} = 36 \text{ cubic feet}
Convert cubic feet to cubic yards. One cubic yard is 3 \times 3 \times 3 = 27 cubic feet, so divide by 27: 36 \div 27 = 1.333\ldots \approx 1.33 \text{ cubic yards}
The pad needs about 1.33 cubic yards of concrete (exactly 1\tfrac{1}{3}).
Check: In yards the pad is 4 \text{ yd} \times 3 \text{ yd} \times \tfrac{1}{9} \text{ yd} (4 inches is \tfrac{4}{36} = \tfrac{1}{9} yard), and 4 \times 3 \times \tfrac{1}{9} = \tfrac{12}{9} = 1.33 ✓
Tip: For cubic units, the conversion factor is cubed: 1 \text{ yd}^3 = 27 \text{ ft}^3, not 3. And never multiply feet by inches. Put every length in the same unit first.
Question 18 of 36
A resting adult's heart beats about 72 times per minute. At this rate, about how many times does the heart beat in one day?
Hint
First change one minute into a full day: how many minutes are in an hour, and how many hours are in a day?
Step-by-step solution
Find the number of minutes in one day: 60 \text{ min/h} \times 24 \text{ h} = 1{,}440 \text{ minutes}
Multiply the rate by the number of minutes: 72 \text{ beats/min} \times 1{,}440 \text{ min} = 103{,}680 \text{ beats}
As a single chain: 72 \frac{\text{beats}}{\text{min}} \times \frac{60 \text{ min}}{1 \text{ h}} \times \frac{24 \text{ h}}{1 \text{ day}} = 103{,}680 \frac{\text{beats}}{\text{day}}
The heart beats about 103,680 times in one day.
Check: Estimate: about 70 beats \times about 1,500 minutes \approx 105{,}000, close to 103,680 ✓
Tip: Write the units in every factor. Minutes cancel minutes and hours cancel hours; if a unit is left over that you didn't want, a factor is missing or flipped.
Question 19 of 36
The graph shows the relationship between the gas a car uses and the distance it travels.
At this rate, how many gallons of gas will the car use on a 434-mile trip?
Hint
Use a marked point to find the miles per gallon. The line goes through the origin, so the relationship is proportional.
Step-by-step solution
The line passes through the origin, so miles are proportional to gallons. Use a marked point to find the unit rate: \frac{280 \text{ mi}}{10 \text{ gal}} = 28 \text{ miles per gallon}
Confirm with the other point: 112 \div 4 = 28 ✓
Find the gallons for 434 miles by dividing miles by miles per gallon: 434 \div 28 = 15.5 \text{ gallons}
(Or with a proportion: \tfrac{10}{280} = \tfrac{g}{434} \Rightarrow 280g = 4{,}340 \Rightarrow g = 15.5.)
The car will use 15.5 gallons.
Check:15.5 \times 28 = 434 miles ✓ The trip is a bit longer than 280 miles \times 1.5 = 420 miles, so a bit more than 10 \times 1.5 = 15 gallons makes sense.
Tip: On a graph through the origin, the constant of proportionality is y \div x at any marked point. Use the labeled points; they are exact, while points you read between grid lines may not be.
Question 20 of 36
Fill in
A community center is sewing a rectangular banner. The ratio of its length to its width is 3 : 2. The width is 26 inches. How many inches of trim are needed to go all the way around the edge (the perimeter)?
inches
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Width matches the 2 in the ratio. How many inches is one part? Then find the length before you find the perimeter.
Step-by-step solution
The width matches 2 parts of the ratio: 2 \text{ parts} = 26 \text{ in} \;\Rightarrow\; 1 \text{ part} = 13 \text{ in}
The length is 3 parts: 3 \times 13 = 39 \text{ inches}
(Or with a proportion: \tfrac{3}{2} = \tfrac{L}{26} \Rightarrow 2L = 78 \Rightarrow L = 39.)
Perimeter of a rectangle: P = 2L + 2W = 2(39) + 2(26) = 78 + 52 = 130 \text{ inches}
The banner needs 130 inches of trim.
Check:39 : 26 divides by 13 to give 3 : 2 ✓ and 39 + 26 + 39 + 26 = 130 ✓
Tip: Match the given measurement to its own number in the ratio (width ↔ 2), not to the first number. Then finish what the question actually asks; here the length is only a middle step.
Question 21 of 36
Drag and drop
Three siblings, Ana, Ben, and Cruz, share an inheritance of $54,000 in the ratio 2 : 3 : 4 (Ana : Ben : Cruz).
Drag the correct amount to each sibling. Not every amount is used.
Ana
Ben
Cruz
Tap a tile, then tap a box to place it.
Hint
Add the ratio numbers to find how many equal parts the money is split into.
Step-by-step solution
Add the parts of the ratio: 2 + 3 + 4 = 9 \text{ parts}
Find the value of one part: \$54{,}000 \div 9 = \$6{,}000
Multiply each sibling's number of parts by $6,000:
Ana: 2 \times \$6{,}000 = \$12{,}000
Ben: 3 \times \$6{,}000 = \$18{,}000
Cruz: 4 \times \$6{,}000 = \$24{,}000
Ana $12,000; Ben $18,000; Cruz $24,000.
Check:\$12{,}000 + \$18{,}000 + \$24{,}000 = \$54{,}000 ✓ and 12 : 18 : 24 divided by 6 is 2 : 3 : 4 ✓
Tip: With a total and a ratio, always divide the total by the sum of the parts. Then the shares must add back up to the total; that's a built-in check.
Question 22 of 36
A grocery store surveyed shoppers and found that 18 out of every 50 prefer paper bags to plastic bags. Last Saturday, 1,375 shoppers visited the store. Based on the survey, about how many of them would prefer paper bags?
Hint
Write a proportion: paper-bag shoppers over all shoppers, on both sides.
Step-by-step solution
Set up a proportion with "paper-bag shoppers" on top and "all shoppers" on the bottom: \frac{18}{50} = \frac{x}{1{,}375}
Check:\tfrac{18}{50} = 36\%, and 36% of 1,375 is 0.36 \times 1{,}375 = 495 ✓
Tip: Read carefully which group the question asks for. Here "18 out of 50" is a part-to-whole ratio; the other 50 - 18 = 32 out of 50 prefer plastic, which leads to a tempting wrong answer.
Question 23 of 36
No calculator
A movie's running time is listed as 2.4 hours. How long is the movie in hours and minutes?
Hint
The ".4" is a part of one hour, not a number of minutes. How many minutes is 0.4 of 60 minutes?
Step-by-step solution
Keep the whole number of hours: 2 hours.
Convert the decimal part of an hour to minutes. One hour is 60 minutes, so multiply: 0.4 \text{ h} \times \frac{60 \text{ min}}{1 \text{ h}} = 24 \text{ minutes}
Tip: Time is not a decimal system: an hour has 60 minutes, not 100. Never read the digits after the decimal point as minutes; multiply the decimal part by 60.
Question 24 of 36
A concrete mix uses cement, sand, and gravel in the ratio 1 : 2 : 4 by weight. A contractor needs 420 pounds of the mix. How many pounds of sand are needed?
Hint
Add all three numbers in the ratio to find how many equal parts make up the 420 pounds.
Tip: With a three-part ratio, the whole is the sum of all the parts. Sand's share is \tfrac{2}{7} of the mix — not \tfrac{1}{2} (that ignores the third ingredient) and not \tfrac{1}{3} (that counts ingredients instead of parts).
Question 25 of 36
Fill in
A leaky faucet drips 2 cups of water every 15 minutes. At this rate, how many gallons of water does it waste in one day? (16 cups = 1 gallon)
gallons
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find cups per hour first. Then find cups per day, and finally change cups to gallons.
Step-by-step solution
Cups per hour: there are 60 \div 15 = 4 fifteen-minute periods in an hour, so 2 \times 4 = 8\text{ cups per hour}
Cups per day: 8 \times 24 = 192\text{ cups}
Convert to gallons (gallons are larger, so divide): 192 \div 16 = 12\text{ gallons}
The faucet wastes 12 gallons per day.
Check: One chain with units that cancel: \dfrac{2\text{ cups}}{15\text{ min}} \times \dfrac{1{,}440\text{ min}}{1\text{ day}} \times \dfrac{1\text{ gal}}{16\text{ cups}} = \dfrac{2 \times 1{,}440}{15 \times 16} = \dfrac{2{,}880}{240} = 12 ✓
Tip: Break long conversions into small steps you trust (per hour → per day → gallons), and write the unit after every result so you know where you are.
Question 26 of 36
Drag and drop
Kenji wants to convert a speed of 45 miles per hour to feet per second. He writes:
Check:66\text{ ft/s} \times 3{,}600\text{ s} = 237{,}600 ft = 237{,}600 \div 5{,}280 = 45 miles in one hour ✓
Tip: Every conversion factor equals 1, so you can flip it freely — choose the version that puts the unwanted unit opposite where it currently sits. Using \tfrac{1\text{ mi}}{5{,}280\text{ ft}} would leave \text{mi}^2 in the answer, a sure sign of an error.
Question 27 of 36
A factory conveyor belt moves at 100 yards per minute. What is this speed in inches per second? (1 yard = 36 inches; 1 minute = 60 seconds)
Hint
Both units change: yards become inches (on top) and minutes become seconds (on the bottom). Use one factor for each.
Step-by-step solution
Write the rate as a fraction with units: 100 \dfrac{\text{yd}}{\text{min}}.
Cancel yards (on top) with a factor that has yards on the bottom: \dfrac{36 \text{ in}}{1 \text{ yd}}.
Cancel minutes (on the bottom) with a factor that has minutes on top: \dfrac{1 \text{ min}}{60 \text{ s}}.
Check: In 60 seconds the belt moves 60 \times 60 = 3{,}600 inches, and 3{,}600 \div 36 = 100 yards, which matches 100 yards per minute ✓
Tip: When a rate changes both units, use two conversion factors. A smaller distance unit makes the number bigger; a smaller time unit (per second instead of per minute) makes it smaller.
Question 28 of 36
Fill in
A baker uses 1.75 pounds of flour for every 2 loaves of bread. She needs to bake 50 loaves for a fundraiser. Flour is sold only in 5-pound bags that cost $3.80 each. How much will she spend on flour?
$
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the pounds of flour first. Then think: can she buy part of a bag?
Step-by-step solution
Find the flour needed with a proportion (pounds over loaves): \frac{1.75}{2} = \frac{f}{50} \;\Rightarrow\; 2f = 87.5 \;\Rightarrow\; f = 43.75 \text{ pounds}
Find the number of 5-pound bags: 43.75 \div 5 = 8.75 \text{ bags}
She can't buy 0.75 of a bag, and 8 bags (40 pounds) is not enough, so she must round up to 9 bags.
Find the cost: 9 \times \$3.80 = \$34.20
She will spend $34.20 on flour.
Check: 9 bags hold 9 \times 5 = 45 pounds, which covers the 43.75 pounds she needs, while 8 bags (40 pounds) would fall short ✓
Tip: When things are sold in whole packages, round up even if the decimal is small. A real-life answer must be something you can actually buy. Paying for 8.75 bags (\$33.25) is not possible.
Question 29 of 36
During a storm, 3 inches of rain fell in 4 hours. Jamal wants to know how long it would take, at the same rate, for 7.5 inches of rain to fall. His work:
Step 1:\dfrac{3}{4} = \dfrac{x}{7.5}
Step 2:4x = 22.5
Step 3:x = 5.625 hours
Which statement is true?
Hint
In Step 1, what does x stand for, and is it on top with the inches or on top with the hours?
Step-by-step solution
Look at the units in Jamal's Step 1. The left fraction is \dfrac{3 \text{ in}}{4 \text{ h}} (inches over hours). On the right, x is a number of hours, but he put it on top, over 7.5 inches. The units don't match, so Step 1 is wrong.
Correct setup (inches over hours on both sides): \frac{3}{4} = \frac{7.5}{x}
Cross-multiply: 3x = 4 \times 7.5 = 30
Divide: x = 30 \div 3 = 10 \text{ hours}
(Steps 2 and 3 of Jamal's work are done correctly for his wrong equation, but they can't fix a wrong setup.)
Step 1 is wrong; the proportion should be \tfrac{3}{4} = \tfrac{7.5}{x}, so it takes 10 hours.
Check: The rate is 3 \div 4 = 0.75 inch per hour, and 0.75 \times 10 = 7.5 inches ✓ Also, 7.5 inches is more than double 3 inches, so it must take more than double 4 hours; 5.625 hours is too short.
Tip: Label the units on every number in a proportion before cross-multiplying. Most proportion mistakes happen in the setup, not the arithmetic.
Question 30 of 36
Drop-down
A city charges for water at $4.20 per 100 cubic feet. Last month, the Garcia household used 5,984 gallons of water. (1 cubic foot = 7.48 gallons)
Choose the correct answer for each blank.
The household used of water, so the water charge was .
Choose an answer from each menu.
Hint
Gallons are smaller than cubic feet, so there will be fewer cubic feet than gallons. Then count how many "hundreds" of cubic feet were used.
Step-by-step solution
Step 1: gallons to cubic feet
One cubic foot holds 7.48 gallons, so a cubic foot is the bigger unit. Divide: 5{,}984 \text{ gal} \times \frac{1 \text{ ft}^3}{7.48 \text{ gal}} = 800 \text{ cubic feet}
Step 2: the charge
The price is per 100 cubic feet. Count the hundreds: 800 \div 100 = 8
Multiply by the price: 8 \times \$4.20 = \$33.60
The household used 800 cubic feet, and the charge was $33.60.
Check:800 \times 7.48 = 5{,}984 gallons ✓ and the price per cubic foot is \$4.20 \div 100 = \$0.042, so 800 \times 0.042 = \$33.60 ✓
Tip: Watch for prices "per 100" or "per 1,000" units (water, electricity, shipping). Divide the amount by 100 first, or change the price to a price per one unit.
Question 31 of 36
Select all
A bottling machine fills 150 bottles every 4 minutes. Let b be the number of bottles it fills in 2.5 hours at the same rate.
Select all the equations that give the correct value of b.
Select all that apply.
Hint
The rate uses minutes, but the time is in hours. An equation can only be right if the time units match.
Step-by-step solution
First find the true value. 2.5 hours = 2.5 \times 60 = 150 minutes, and the rate is 150 \div 4 = 37.5 bottles per minute, so b = 37.5 \times 150 = 5{,}625 bottles. Now test each equation.
\dfrac{150}{4} = \dfrac{b}{150}: bottles over minutes on both sides (150 minutes) ✓ → 4b = 22{,}500, b = 5{,}625.
b = 37.5 \times 2.5 \times 60: bottles per minute \times hours \times minutes per hour ✓ → b = 5{,}625.
\dfrac{4}{150} = \dfrac{150}{b}: minutes over bottles on both sides ✓ → 4b = 22{,}500, b = 5{,}625.
\dfrac{150}{4} = \dfrac{b}{2.5}: the left side is per minute but 2.5 is in hours ✗ → b = 93.75.
\dfrac{4}{150} = \dfrac{b}{150}: minutes/bottles on the left but bottles/minutes on the right ✗ → b = 4.
Correct equations: \tfrac{150}{4} = \tfrac{b}{150}, b = 37.5 \times 2.5 \times 60, and \tfrac{4}{150} = \tfrac{150}{b}.
Check: 5,625 bottles in 150 minutes is 5{,}625 \div 150 = 37.5 bottles per minute, matching 150 \div 4 = 37.5 ✓
Tip: A proportion needs matching units, not just matching positions. Convert hours to minutes (or the rate to bottles per hour) before you set up the equation.
Question 32 of 36
A window pane is 30 inches by 42 inches. Replacement glass costs $8 per square foot. How much will the glass for this pane cost? (1 foot = 12 inches)
Hint
Find the area in square inches first. How many square inches are in one square foot? (Hint: it is not 12.)
Step-by-step solution
Find the area in square inches: 30 \times 42 = 1{,}260 \text{ square inches}
Check: Convert the sides first instead: 30 \text{ in} = 2.5 ft and 42 \text{ in} = 3.5 ft, so the area is 2.5 \times 3.5 = 8.75 sq ft, and 8.75 \times 8 = \$70 ✓
Tip: A square unit converts by the square of the length factor: 1 \text{ ft}^2 = 12^2 = 144 \text{ in}^2 and 1 \text{ yd}^2 = 3^2 = 9 \text{ ft}^2. The safest method is to convert each side to feet before you multiply.
Question 33 of 36
Fill in
A hospital IV pump delivers 1.5 milliliters every 10 seconds. At this rate, how many liters of fluid does it deliver in 8 hours? (1 liter = 1,000 milliliters)
liters
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find how many 10-second intervals are in 8 hours, or find the rate per second first. Change milliliters to liters at the end.
Step-by-step solution
Find the rate per second: 1.5 \text{ mL} \div 10 \text{ s} = 0.15 \text{ mL per second}
Find the number of seconds in 8 hours: 8 \text{ h} \times 60 \frac{\text{min}}{\text{h}} \times 60 \frac{\text{s}}{\text{min}} = 28{,}800 \text{ seconds}
Convert to liters (a liter is the bigger unit, so divide): 4{,}320 \div 1{,}000 = 4.32 \text{ L}
The pump delivers 4.32 liters in 8 hours.
Check: Per hour: 0.15 \times 3{,}600 = 540 mL. Then 540 \times 8 = 4{,}320 mL = 4.32 L ✓
Tip: Break long rate problems into small, labeled steps: rate per one unit, total time in that unit, then multiply and convert the answer. Metric conversions only move the decimal point (1,000 mL = 1 L: move 3 places).
Question 34 of 36
No calculator
A jar holds red and blue beads in the ratio 3 : 5. After 12 red beads are added (and no blue beads), the numbers of red and blue beads are equal. How many beads were in the jar before the red beads were added?
Hint
The blue beads don't change. How many more "parts" of red are needed to catch up with blue?
Step-by-step solution
Write the starting amounts in parts: red = 3k and blue = 5k, where k is the size of one part.
Adding 12 red beads makes red equal to blue: 3k + 12 = 5k
Solve: 12 = 2k \;\Rightarrow\; k = 6
Starting amounts: red = 3 \times 6 = 18, blue = 5 \times 6 = 30. Total before = 18 + 30 = 48.
There were 48 beads in the jar before.
Check:18 : 30 divides by 6 to give 3 : 5 ✓ After adding 12 red: 18 + 12 = 30 red and 30 blue, which are equal ✓
Tip: When a ratio changes, find what stays the same (here the blue beads), then write both groups with the same part size k. The 12 added beads fill the 2-part gap between 3 and 5, so each part is 6, not 12.
Question 35 of 36
A lawn-feed label says: mix 2 tablespoons of concentrate into each gallon of water, and spray 3.5 gallons of mixed water on every 100 square feet of lawn. Omar's lawn is 1,800 square feet. He works out how much concentrate he needs, in cups (16 tablespoons = 1 cup):
Step 3: A cup is bigger than a tablespoon (1 cup = 16 tbsp), so there should be fewer cups than tablespoons. Divide, don't multiply: 126 \text{ tbsp} \times \frac{1 \text{ cup}}{16 \text{ tbsp}} = 7.875 \text{ cups}
Check:7.875 \times 16 = 126 tablespoons ✓ And 2,016 cups would be more than 100 gallons of pure concentrate, far more than the 63 gallons of water it is mixed into, so it can't be right.
Tip: In a multi-step chain, sense-check the size of each result. When you convert to a bigger unit, the number must get smaller.
Question 36 of 36
Drop-down
A landscaping company lists the cost of delivered mulch.
Mulch (cubic yards)
Total cost
2
$95
5
$200
8
$305
Choose the correct answer for each blank.
The relationship between cubic yards and total cost . At these prices, 12 cubic yards would cost .
Choose an answer from each menu.
Hint
Divide cost by cubic yards in each row. Are the results equal? Then look at how much the cost goes up for each extra cubic yard.
Step-by-step solution
Is it proportional?
Divide cost by cubic yards in each row: \frac{95}{2} = 47.5, \quad \frac{200}{5} = 40, \quad \frac{305}{8} = 38.125
The ratios are not equal, so the relationship is not proportional. (Also, 0 cubic yards would not cost $0.)
Find the pattern
Each extra 3 cubic yards adds $105: 200 - 95 = 105 and 305 - 200 = 105. So each cubic yard costs 105 \div 3 = \$35
Find the fixed part (delivery fee): 95 - 2 \times 35 = 95 - 70 = \$25
So cost = 35y + 25.
Not proportional, because cost ÷ cubic yards is not constant; 12 cubic yards cost $445.
Check: The rule works for every row: 35(5) + 25 = 200 ✓ and 35(8) + 25 = 305 ✓
Tip: A table is proportional only if y \div x is the same in every row (the graph would pass through the origin). Equal jumps alone don't make it proportional; a fixed fee breaks proportionality, so don't scale one row up.
