Topic Quiz Q.3: Ratios, rates, proportions and percents
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Question 1 of 10 · Scale factors and scale drawings
On a landscape plan, a square patio is drawn 4 inches on each side. The actual patio covers 576 square feet. What is the scale of the plan?
Hint
A scale compares lengths, not areas. What is the length of one side of the real patio?
Step-by-step solution
Find the real side length. A square's area is side², so s = \sqrt{576} = 24\text{ ft}
4 inches on the plan stands for 24 feet: \frac{24\text{ ft}}{4\text{ in}} = 6\text{ ft per inch}
The scale is 1 inch = 6 feet.
Check: The real patio is 4 \times 6 = 24 ft on a side, and 24 \times 24 = 576 ft² ✓
Tip: A scale compares lengths. If you are given an area, take the square root first to get back to a length, then divide by the drawing length.
Question 2 of 10 · Proportions and unit conversion
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 3 of 10 · Unit rates
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Tamika is paid $22.40 per hour for her first 40 hours each week. For every hour over 40, she is paid 1.5 times her regular rate. Last week she earned $1,097.60.
How many total hours did she work last week?
hours
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find her pay for the first 40 hours and her overtime rate. The leftover pay must come from overtime hours.
Step-by-step solution
Regular pay for 40 hours: 40 \times \$22.40 = \$896.00
Pay left over that must be overtime: \$1{,}097.60 - \$896.00 = \$201.60
Overtime rate: 1.5 \times \$22.40 = \$33.60\text{ per hour}
Overtime hours (work backwards — divide by the rate): \frac{\$201.60}{\$33.60\text{ per h}} = 6\text{ h}
Tip: When pay has two different rates, split the total into pieces that each have one rate. To work backwards from a total, divide the right piece by its rate. Don't forget to add the 40 regular hours back at the end — "6" is the most common wrong answer.
Question 4 of 10 · Unit rates
A copy machine prints at a constant rate. The table shows some of its print times.
Minutes
4
10
?
Pages printed
26
65
117
How many minutes does it take to print 117 pages?
Hint
Find the pages per minute from a complete column, then work backwards.
Step-by-step solution
Unit rate from a complete column: \frac{26\text{ pages}}{4\text{ min}} = 6.5\text{ pages per minute} Check: 65 \div 10 = 6.5 ✓
Check:6.5 \times 18 = 117 ✓, so the ratio \tfrac{117}{18} = 6.5 matches the other columns.
Tip: Don't continue a pattern in one row of the table (4, 10, 16, …). In a proportional table, the ratio pages ÷ minutes stays the same, so use the unit rate to fill in a missing value.
Question 5 of 10 · Unit rates
A water truck fills a 1,250-gallon tank in 2 hours 30 minutes. What is the filling rate in gallons per hour?
Hint
Write 2 hours 30 minutes as a decimal number of hours. How many minutes are in an hour?
Step-by-step solution
Write the time in hours. 30 minutes is \tfrac{30}{60} = 0.5 hour, so 2\text{ h }30\text{ min} = 2.5\text{ h}
Divide gallons by hours: \frac{1{,}250\text{ gal}}{2.5\text{ h}} = 500\text{ gal per hour}
The tank fills at 500 gallons per hour.
Check:500 \times 2.5 = 1{,}250 gallons ✓ (500 in the first hour, 500 in the second, and 250 in the last half hour)
Tip: "2 hours 30 minutes" is 2.5 hours, not 2.3 hours. Minutes are sixtieths of an hour, so divide the minutes by 60 before writing a decimal.
Question 6 of 10 · Scale factors and scale drawings
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A 4-inch by 6-inch photo is enlarged. The enlarged photo has the same shape, and its area is 216 square inches.
What is the width of the enlarged photo (the side that was 4 inches)?
inches
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Compare the new area with the original area. Area grows by the square of the scale factor.
Step-by-step solution
Original area: 4 \times 6 = 24\text{ in}^2
How many times larger is the new area? 216 \div 24 = 9
Area scales by k^2, so k^2 = 9 \;\Rightarrow\; k = \sqrt{9} = 3
New width: 4 \times 3 = 12\text{ in}
The enlarged photo is 12 inches wide.
Check: New length = 6 \times 3 = 18 in, and 12 \times 18 = 216 sq in ✓
Tip: Working backwards from area, take the square root of the area ratio to get the length scale factor. Using k = 9 would give a 36-in width — an area of 1,944 sq in, far too big.
Question 7 of 10 · Scale factors and scale drawings
On a blueprint, a 30-foot wall is drawn 7.5 inches long. Written as a ratio with no units (drawing : actual), what is the scale of the blueprint? (1 foot = 12 inches)
Hint
Both parts of a unitless ratio must be in the same unit. How many inches are in 30 feet?
Step-by-step solution
A unitless ratio needs both lengths in the same unit. Change 30 feet to inches: 30 \times 12 = 360\text{ in}
Write drawing : actual and divide both parts by 7.5: 7.5 : 360 = 1 : 48
The scale is 1 : 48.
Check:7.5 \times 48 = 360 in, and 360 \div 12 = 30 ft ✓. (This is the common "\tfrac{1}{4} inch = 1 foot" architect's scale.)
Tip: Before writing a ratio scale, convert both lengths to one unit. Then divide both parts by the drawing length so the ratio starts with 1.
Question 8 of 10 · Proportions and unit conversion
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).
As part of a doctor's health plan, Marcus went from 240 pounds to 207 pounds. What was the percent decrease in his weight?
%
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the number of pounds he lost, then divide by his starting weight.
Step-by-step solution
Find the change: 240 - 207 = 33 \text{ pounds}
Divide by the original weight (240 pounds): \frac{33}{240} = 0.1375
Change to a percent: 0.1375 \times 100\% = 13.75\%
His weight decreased by 13.75%.
Check: 13.75% of 240 is 0.1375 \times 240 = 33 pounds, and 240 - 33 = 207 ✓ (Dividing by the new weight gives 33 \div 207 \approx 15.9\%, the classic wrong-base mistake.)
Tip: A percent decrease always uses the starting value as the base, even when the end value is "nicer" to divide by. Estimate first: 10% of 240 is 24 and 15% is 36, so a 33-pound loss must be between 10% and 15%.
The table shows the prices of three grocery items in two different years.
Item
2021 price
2024 price
Eggs (1 dozen)
$2.50
$3.40
Milk (1 gallon)
$4.00
$3.70
Bread (1 loaf)
$3.20
$3.60
Drag the correct percent change (from 2021 to 2024) to each item. A positive value is an increase and a negative value is a decrease. Not every value is used.
Eggs
Milk
Bread
Tap a tile, then tap a box to place it.
Hint
For each item: (2024 price − 2021 price) ÷ 2021 price. Keep the sign.
Step-by-step solution
Use percent change = \dfrac{\text{new} - \text{original}}{\text{original}} \times 100\%, with 2021 as the original.
Tip: The same dollar change can be very different percents: eggs rose 90 cents (36%), but bread rose 40 cents (12.5%). Always divide by each item's own starting price, and keep the minus sign for a decrease.