This page is for studying, it is not the test and nothing on it is graded. Vocabulary and worked examples come straight from class; the practice questions below are new, written to match, not copies of any quiz.
Scale Factor & Scale Drawings
scale drawing
A drawing of a figure enlarged or reduced from the original: every angle stays exactly the same, every side length changes by the same scale factor.
scale factor
The single number you multiply a drawing length by to get the actual length (or: actual length ÷ drawing length). The same factor works for every pair of corresponding sides.
Every side of the "actual" figure is the matching "drawing" side times the same scale factor.
Worked example
A scale drawing shows a bookshelf 3 cm tall on paper. The actual bookshelf is 90 cm tall.
Scale factor = actual ÷ drawing = 90 ÷ 3 = 30.
Check with a second measurement: a 2 cm handle on the drawing × 30 = 60 cm actual. ✓
Worked example, reverse direction
Scale is 1 in : 5 ft. A driveway is actually 35 ft long. How long on the drawing?
You know the actual length and need the drawing length: divide by 5. 35 ÷ 5 = 7 inches.
Check: 7 in × 5 ft/in = 35 ft. ✓
Try it: A scale drawing uses 1 cm = 5 m. A fence is 4 cm long in the drawing. Show answer
Actual length = 4 cm × 5 m/cm = 20 m. Going from drawing to actual is upscaling, so you multiply by the scale factor (5), the same direction as the bookshelf example above.
Try it: A rectangle is 6 in by 3 in on a drawing with scale 1 in : 4 ft. What is the actual area?
Actual sides: 6×4 = 24 ft and 3×4 = 12 ft. Area = 24 × 12 = 288 sq ft. (Area scales by the factor squared, 4² = 16 times the drawing's area of 18 sq in if you were comparing units, but multiplying both actual sides directly gets you the same answer without that extra step.)
Try it: A scale is 1 in : 8 ft. The actual length of a driveway is 64 ft. How long is the driveway on the drawing?
You know the actual length and need the drawing length, that's downscaling, so divide by the factor: 64 ÷ 8 = 8 inches. Check by going back the other way: 8 in × 8 ft/in = 64 ft. ✓
Try it: On a drawing, a bedroom is 5 cm by 4 cm. The actual bedroom is 15 m by 12 m. What is the scale factor, and does it work for both sides?
Scale factor = actual ÷ drawing, checked on both sides: 15 ÷ 5 = 3, and 12 ÷ 4 = 3. Same factor both times, so the scale factor is 3 (in this case, 1 cm = 3 m). Checking a second pair of sides is exactly how you catch a mistake, if the two calculations don't match, you made an error somewhere.
Unit Rates (including fractions)
unit rate
The amount of one quantity for exactly 1 unit of a second quantity: first quantity ÷ second quantity. Works the same with fractions.
complex fraction
A fraction whose numerator, denominator, or both are themselves fractions. Multiply by the reciprocal of the divisor to simplify one, that's how a unit rate is computed from a ratio of fractions.
Worked example
A runner jogs ½ mile in ¼ hour at a steady pace. Unit rate = ½ ÷ ¼ = ½ × &frac41; = 2 miles per hour.
Try it: A cyclist rides ¾ mile in ⅙ hour. What's the speed in miles per hour?
¾ ÷ ⅙ = ¾ × &frac61; = 4.5 miles per hour. Dividing by a fraction means multiplying by its reciprocal (flip ⅙ to &frac61;), that's what turns a messy fraction-divided-by-fraction into a normal multiplication.
Try it: A hose fills ⅔ of a gallon in ⅕ of a minute. What's the fill rate in gallons per minute?
⅔ ÷ ⅕ = ⅔ × &frac51; = 3⅓ gallons per minute. Same move every time: keep the first fraction, flip the second, multiply.
Try it: A painter covers ⅜ of a wall in ¼ of an hour. What's the rate in walls per hour, and what would you get if you divided the wrong way?
Correct: ⅜ ÷ ¼ = ⅜ × &frac41; = 1½ walls per hour. If you divided the wrong way (¼ ÷ ⅜) you'd get ⅔, which isn't nonsense, it's actually "hours per wall" instead of "walls per hour," the exact mix-up this question is built to catch. Always check: your units should match what the question actually asked for.
Proportional Relationships & Constant of Proportionality
proportional relationship
Two quantities where the ratio between them stays the same, constant, value for every pair. Check more than one row, one matching row is not enough.
constant of proportionality
The constant ratio, same number as the unit rate. On a graph it's the y-coordinate where x = 1, and the graph is always a straight line through the origin (0, 0).
A straight line through (0, 0) that also passes through (4, 10): k = 10 ÷ 4 = 2.5.
Worked example: matching four descriptions
These four all describe the SAME proportional relationship: a table (x: 2,4,6,8 → y: 5,10,15,20), the equation y = 2.5x, "a parking garage charges $2.50 per hour, no flat fee," and a line through the origin passing through (4, 10). All four give k = 2.5.
A fifth description, "a rideshare costs a $3 pickup fee plus $2.50 per mile," is not proportional, it doesn't pass through (0, 0) because of the flat fee.
Try it: y = 42 when x = 6. What is the constant of proportionality k (where y = kx)?
k = 42 ÷ 6 = 7. That means the equation for this relationship is y = 7x, and every other (x, y) pair in it will divide out to exactly 7 as well.
Try it: A proportional relationship has y = 18 when x = 3. What is y when x = 10?
First find k: 18 ÷ 3 = 6. Then apply it to the new x: y = 6 × 10 = 60. Once you know k, you can find y for any x without needing a full table.
Try it: A table shows (3, 8) and (6, 15). Is this relationship proportional?
Check the ratio at both points: 8÷3 ≈ 2.67, but 15÷6 = 2.5. The ratios don't match, so no, it is not proportional. A single matching pair is never enough to prove proportionality, you have to check that y/x stays constant across every row.
Try it: A gym membership costs a $20 sign-up fee plus $15 per month. Is cost proportional to number of months?
No. At 1 month, cost is $35 ($20 + $15), giving a ratio of 35. At 2 months, cost is $50, giving a ratio of 25. The ratios don't match, and more directly: the relationship doesn't pass through (0, 0), since even 0 months still costs $20. A flat fee added on top is the classic signal a relationship is not proportional.
Start here: labs
Scale & Magnitude lab · scale factors, powers of ten, order-of-magnitude estimation, direct practice for the section above
The Going Rate lab · 30 real rates, rate builder, live ticker, direct practice for unit rates