1
A ratio compares two quantities in equal parts.
3 to 8 does not mean 3 pages and 8 pages. It means 3 parts to 8 parts, and every part is the same size.
color
color
color
b&w
b&w
b&w
b&w
b&w
b&w
b&w
b&w
3 parts color, 8 parts black-and-white, 11 parts in all
A ratio never fixes a size on its own. 3 to 8 fits 3 and 8 pages, 300 and 800 pages, or 600 and 1,600 pages. The stem has to supply one more number — a total, a difference, or one of the two amounts — before any of them becomes the answer.
2
Part to part is not part to whole.
Both fractions are built from the same two numbers, and the SAT always offers the other one as a choice.
color to black-and-white
38
part to part — one share against the other
color to all pages
311
part to whole — one share against the total
“the ratio of A to B” → part to part
“what fraction of the total” → part to whole
“what percent of the pages” → part to whole
Where 825 comes from
Multiplying the total by 38 gives 0.375 × 2,200 = 825, and 825 is sitting in the choices waiting for it. The word to in “3 to 8” compares the two parts with each other; only after adding them do you have anything to compare with the total.
3
To split a total, add the parts and divide once.
This is the opening problem, in three lines.
3 + 8 = 11 parts
count the parts before anything else
2,200 ÷ 11 = 200
one part, in pages
color = 3 × 200 = 600
the share the question asked for
b&w = 8 × 200 = 1,600
the other share, for the check
600 color pages — and 600 + 1,600 = 2,200 ✓
Add the shares back together every time. It costs one line and catches a division slip, a mis-copied total, and the part-to-part trap all at once.
4
A proportion is two equal ratios — line the labels up.
On a map, 2 inches represents 35 miles. Two towns are 5.6 inches apart on the map. How many miles apart are they?
milesinches
=
352
=
d5.6
2d = 35(5.6)
cross-multiply
2d = 196
tidy the right side
d = 98 miles
÷ 2 both sides
Sanity check: 5.6 inches is 2.8 times 2 inches, and 2.8 × 35 = 98. Scaling the ratio directly and cross-multiplying are the same move written two ways.
Miles over inches on both sides
If one side reads miles over inches and the other reads inches over miles, you get 352 = 5.6d, and cross-multiplying gives 35d = 11.2, so d ≈ 0.32 — two towns a third of a mile apart. Write the two words into the fraction before you write any numbers, and the set-up cannot flip.
5
A unit rate is a ratio with 1 underneath.
A machine stamps parts at a constant rate. The table records one day of production.
| Time (hours) | Parts produced | Parts ÷ hours |
|---|
| 4 | 120 | 120 ÷ 4 = 30 |
|---|
| 8 | 240 | 240 ÷ 8 = 30 |
|---|
| 12 | 360 | 360 ÷ 12 = 30 |
|---|
| 20 | 600 | 600 ÷ 20 = 30 |
|---|
Every row gives the same quotient, 30. “At a constant rate” is the stem's promise that it will, so one row is enough to find it and the others only confirm it.
rate = 240 parts8 hours = 30 parts per hour
parts on top, so “per hour” reads off the bottom
in 14 hours: 30 × 14 = 420 parts
multiply by the unit rate
rate = 8 hours240 parts = 2 minutes per part
the same fact upside down: 130 hour is 2 minutes
for 900 parts: 900 ÷ 30 = 30 hours
divide by the unit rate to get back to time
Which quantity goes on top? The one the phrase names first. “Parts per hour” puts parts on top; “minutes per part” puts minutes on top. A scaled recipe is the same decision: 12 servings need 9 cups of broth, so 912 = 0.75 cup per serving, and 20 servings need 0.75 × 20 = 15 cups.
Where the calculator earns its keep
Desmos will hold a rate exactly: type 2408 and you get 30, type 912 and you get 0.75. For a proportion, put the unknown on the left as x — entering x5.6 = 352 draws a vertical line, and clicking it reads out x = 98. Faster still, define r = 2408 on one line and then type 14r on the next, so every later question in the same stem is one keystroke.
6
Ask whether more means more, or more means less.
Direct and inverse proportion look alike in words and behave in opposite directions.
| Situation | More of the first gives | What stays fixed |
|---|
| more hours, more parts | more | the quotient — direct, so cross-multiply |
|---|
| more machines, fewer hours | less | the product — inverse, so multiply across |
|---|
Four machines finish a job in 15 hours. How long would six identical machines take?
4 machines × 15 hours = 60
the job is 60 machine-hours of work
6 machines × t = 60
same job, more machines
t = 10 hours
fewer hours, as common sense demands
Cross-multiplying an inverse relationship
Setting up 415 = 6t gives 4t = 90 and t = 22.5 hours — adding machines and taking longer. Before solving, decide which way the answer must move; an answer that moves the wrong way is wrong no matter how clean the algebra was.
✓
Check: does the answer keep the ratio and the units?
the print shop
600 : 1,600 = 3 : 8
divide both by 200 ✓
the map
98 ÷ 5.6 = 17.5 = 35 ÷ 2
miles per inch matches ✓
the machine
30 × 14 = 420 parts
parts per hour × hours = parts ✓
the machines
6 × 10 = 60 machine-hours
product held, time fell ✓
Remember the flow
Count the parts · Find one part or one unit · Multiply back up
Then read the question again and answer the quantity it actually named.
