SAT Math Review · Problem-Solving and Data Analysis
Unit conversion and dimensional analysis
Multiply by a fraction that equals 1, arranged so the unit you do not want cancels. The SAT hands you the factor; the work is the arrangement.
What is the value of the quantity described?
A car travels at a constant speed of 45 miles per hour. What is the car's speed, in feet per second? (1 mile = 5,280 feet)
Both halves of the rate change. Miles become feet, so multiply by 5,280; hours become seconds, so divide by 3,600, because one hour holds 60 × 60 = 3,600 seconds. That gives 45 × 5,280 ÷ 3,600 = 66 feet per second. A converts only the time, leaving 0.75 miles per minute — the right arithmetic on the wrong half. C stops at feet per minute: 45 × 5,280 ÷ 60 = 3,960, one time conversion short. D converts only the distance and reports 237,600 feet per hour, which is the same speed with the bottom untouched.
1
A conversion factor is a fraction that equals 1.
1 foot = 12 inches, so 12 in1 ft = 1 and 1 ft12 in = 1. Multiplying by either one changes the label without changing the size.
Anything unusual — fathoms, bushels, furlongs, cubic centimeters per gram — is written into the stem in parentheses. The SAT is not testing whether you remember it. It is testing whether you can put it the right way up.
Two miles is 10,560 feet and 10,560 feet is two miles. The quantity never moved; only the unit it is counted in did. That is why you are allowed to multiply by a conversion factor at any point in a problem without disturbing anything else.
2
Put the unit you want to lose on the bottom.
Units cancel exactly like variables do: a unit on top and the same unit underneath divide out and disappear.
A camera sits at a depth of 34 fathoms. What is that depth in feet? (1 fathom = 6 feet)
34 fathoms × 6 ft1 fathom
fathoms on the bottom, because fathoms must go
= 34 × 6 ft = 204 feet
only feet is left standing
34 fathoms × 1 fathom6 ft = 5.67 fathoms²ft
the upside-down version, and the units say so
You do not have to decide whether to multiply or divide. Write the factor so the unwanted unit cancels, and the arithmetic that survives is automatically right.
The one-second direction check
A fathom is bigger than a foot, so a depth measured in feet must come out as a bigger number: 204, not 5.67. Ask “should this number grow or shrink?” before you press anything. Every wrong arrangement fails that question, which is why it catches so much.
3
Chain the factors and let the middle units cancel.
There is no rule that says one factor at a time. Line them up and cross off matching units down the chain.
2.5 × 5,280 = 13,200 ft
first factor done
13,200 × 12 = 158,400 inches
second factor done
The same chain runs backwards just as well: 204 feet × 1 yd3 ft = 68 yards, because feet is now the unit being cancelled and so it goes underneath.
4
In a rate, both the top and the bottom change.
Miles per hour to feet per second is two conversions in one expression — this is the opening problem.
45 mi1 hr
×
5,280 ft1 mi
×
1 hr3,600 s
mi cancels top-and-bottom; hr cancels top-and-bottom
what is left is ft over s
45 × 5,280 = 237,600
feet per hour
237,600 ÷ 3,600 = 66
feet per second
Note where the hours went. To clear an hour off the bottom, the factor must carry an hour on its top — 1 hr3,600 s, not 3,600 s1 hr. A unit in a denominator cancels against a unit in a numerator, never against another denominator.
Two conversions, two chances to stop early
237,600 is feet per hour and 3,960 is feet per minute. Both are honest numbers for this car and both are the wrong answer, and both sit in the choices. Read the units the question asked for, then read the units your expression ended with, and make them match before you look at the list.
Worth carrying: 60 miles per hour is exactly 88 feet per second, since 60 × 5,280 ÷ 3,600 = 88. Any speed in miles per hour scales from there — 45 mph is three quarters of 60 mph, and three quarters of 88 is 66.
5
For square and cube units, the factor is squared or cubed.
1 yard = 3 feet, but 1 square yard = 9 square feet and 1 cubic yard = 27 cubic feet. This is the conversion most students get wrong.
The picture is the proof. A square yard is 3 ft across and 3 ft down, so it holds 3 × 3 = 9 square feet. Stack that a yard deep and it holds 3 × 3 × 3 = 27 cubic feet.
2.5 ft³ × 12 in1 ft × 12 in1 ft × 12 in1 ft
three feet on top, so the factor is used three times
12³ = 1,728
1 cubic foot is 1,728 cubic inches
2.5 × 1,728 = 4,320 in³
and 4,320 is far bigger than 2.5, as it must be
The trap, stated plainly
Converting 2.5 ft³ by multiplying by 12 once gives 30, and converting an area of 4,500 ft² by dividing by 3 once gives 1,500. Both are the linear factor applied to a quantity that is not linear. Write the exponent on the unit — ft², ft³ — and it tells you how many copies of the factor the expression needs.
6
Write the whole chain, then evaluate once.
Converting one factor at a time invites a rounding slip at every stop and loses track of which unit you are standing in.
start withthe given quantity, units attached45 mi/hr
multiply byone factor per unit that must change×5,280 ft/mi, ×1 hr3,600 s
arrange soevery unwanted unit appears once up, once downmi and hr both cancel
thenread the surviving units and evaluate66 ft/s
Where the calculator earns its keep
Type the chain into Desmos as one expression — 45 · 52803600 — and it returns 66 with nothing rounded in the middle. For the awkward factors the SAT supplies, such as 3.785 liters to a gallon, that matters: 9 · 3.785 gives 34.065 in one line, so the rounding happens once, at the end, where the question asks for it.
✓
Check: did every unit you did not want disappear?
the camera34 fathoms → 204 ftbigger number, smaller unit ✓
the chain2.5 mi → 158,400 inmi and ft both cancelled ✓
the car45 mi/hr → 66 ft/stop and bottom both converted ✓
the tank2.5 ft³ → 4,320 in³factor cubed, not used once ✓
Remember the flow
Unwanted unit on the bottom · One factor per unit · Square it for ², cube it for ³
Then ask whether the number should have grown or shrunk.
Now you try
Practice problem
What is the value of the quantity described?
A room has a floor area of 4,500 square feet. Carpet for the room costs $23 per square yard. What is the total cost of the carpet for this room? (1 yard = 3 feet)
Square feet must become square yards first, and the factor for an area is 3² = 9: 4,500 ÷ 9 = 500 square yards. Then 500 × 23 = $11,500. B divides by 3 instead of 9 — the linear factor used on an area — giving 1,500 square yards and $34,500. C never converts at all and prices 4,500 square feet at the square-yard rate. D multiplies by 9 instead of dividing; a square yard is the bigger unit, so the count of them has to come out smaller than 4,500, which rules that choice out before any arithmetic.
Hint: write the conversion as a fraction with ft² underneath — 1 yd²9 ft² — and the squared factor is impossible to forget, because it is written into the unit itself.
