TSI Math Study Guide
Ratios and Proportions
Ratios and proportions compare quantities and describe relationships between them. On the TSI they usually appear in real-world problems involving rates, unit rates, scale drawings, similar figures, and proportional relationships.
Ratios
If a class has 12 boys and 18 girls, the ratio of boys to girls is:
Order matters
A bag has 8 red marbles and 5 blue marbles.
Red to blue
85
Blue to red
58
Always identify exactly which quantity comes first.
Part-to-part vs. part-to-whole
A basket has 6 apples and 4 oranges — 10 pieces of fruit altogether.
Part to part — apples to oranges
64 = 32
Part to whole — apples to all fruit
610 = 35
Before setting up a ratio, ask
Does the question want part to part, or part to whole?
Rates and unit rates
Unit rate — a rate where the second quantity equals 1.
180 miles3 hours = 60 miles per hour
Example
Store A
$10.506 = $1.75
Store B — lower
$13.208 = $1.65
When comparing prices, speeds, or wages, converting each to the same unit rate makes the comparison easy.
Proportions
23 = 812
Example
Keep corresponding quantities together
Either arrangement works, as long as both sides use the same order.
Cross multiplication
If ab = cd, then ad = bc
Cross multiplication is only as good as the setup. First make sure the proportion actually represents the situation.
Proportional relationships
A worker earns $18 per hour. If x is hours and y is earnings:
y = 18x, so k = 18
In a table
| Hours | Earnings | Earnings ÷ hours |
|---|---|---|
| 2 | $30 | 15 |
| 4 | $60 | 15 |
| 6 | $90 | 15 |
| 8 | $120 | 15 |
Constant ratio → proportional
Divide earnings by hours in every row. It’s always 15, so:y = 15x
On a graph
A proportional relationship graphs as a straight line through the origin (0, 0), because y = k(0) = 0.
Proportional
y = 3x
Linear, not proportional
y = 3x + 5
This distinction matters again in Linear Equations and Functions.
Scale factors
A 4-inch-wide photo is enlarged to 10 inches wide:
104 = 2.5
Example
Scale drawings and maps
A map uses the scale 1 inch = 25 miles. Two cities are 3.5 inches apart:
3.5(25) = 87.5 miles
Ratios in word problems
Problems rarely say “use a proportion.” Recognize the relationship from phrases like:
Unit rate method
Proportion method
Choosing between them
Unit rate
Convenient when the amount for one unit is easy to find. $24 for 6 tickets → $4 per ticket, and any number of tickets is easy from there.
Proportion
More convenient when the numbers don’t reduce cleanly.
The goal isn’t one method every time — choose whichever makes the relationship easiest to see.
TSI strategy: label your ratios
Before solving a proportion, write what the numerator and denominator represent.
mileshours = mileshours costitems = costitems
This one habit prevents the easiest proportion error: reversing one of the ratios. Also check the units — if one rate is in minutes and another quantity is in hours, convert before calculating.
When a problem says “at the same rate,” immediately think unit rate or proportion. Ask:
What two quantities are being compared, and does their ratio stay constant?
Once you see that relationship, most ratio and proportion problems organize themselves.