Three of the four choices are true. Only one of them answers the question that was asked.
A pool is being drained. The equation V = 12,000 − 450t gives the volume V, in gallons, of water in the pool t minutes after draining begins. Which of the following is the best interpretation of 450 in this context?
450 is multiplied by t, the number of minutes, so it is a rate: 450 gallons leave the pool every minute. Now look at what you turned down — every one of the other three is a true statement. A is true, and it is the meaning of 12,000, the other constant. B is true, because 12,000 ÷ 450 = 26.67, but that is the answer to "when is the pool empty", a different question. D is true, and it is the meaning of V itself. Being true is not the test; answering the question asked is.
1
Every linear model is a starting amount plus a rate times something.
A storage company charges a one-time setup fee plus a weekly rate. The total cost C, in dollars, of renting a unit for w weeks is C = 15w + 200.
C = 15w + 200
Piece
Its units
What it means here
200
dollars
the one-time setup fee
15
dollars per week
the weekly rental rate
15w
dollars
the total rent for the w weeks, setup fee not included
C
dollars
everything the customer pays in total
Four different quantities, four different meanings, and the SAT can ask about any
one of them. Before you look at the choices, decide in your own words which one
the stem named — 15, or 200, or 15w, or C.
2
The constant on its own is the value when the variable is zero.
Substitute 0 and read what is left. That is always the fastest way to say what the constant means.
C = 15w + 200
the model
w = 0 → C = 15(0) + 200 = 200
the cost before a single week has passed
so 200 is the setup fee
a one-time charge, in dollars
a fixed feea starting valuea depositthe amount already there
The same constant wears different clothes depending on the story: an initial
population, the height a ball is thrown from, the value of a machine when it was
new, the flat delivery charge before any items are counted.
"When the variable is zero" is not always "at the start"
If a model is written P = 480 − 25(t − 3) and the stem says it applies for t ≥ 3,
then 480 is the value at t = 3, not at t = 0 — substitute and see: 480 − 25(0) = 480.
Read the domain the stem gives you before you call a constant "the initial amount".
3
The coefficient is a rate, and it always has a "per" in it.
Say it out loud with the units: 15 is dollars per week. If your sentence has no "per" or "each", you have described the wrong thing.
w goes from 4 to 5
C goes from 260 to 275
275 − 260 = 15
one more week costs 15 dollars more
w goes from 9 to 10
335 to 350 — the same 15 again
Model
The coefficient means
C = 15w + 200
the cost rises by 15 dollars for each additional week
V = 12,000 − 450t
the volume falls by 450 gallons for each additional minute
d = 58h + 12
the car travels 58 more miles for each additional hour
A minus sign in front of the coefficient is part of the meaning, not a detail.
−450 does not mean "450 gallons"; it means the pool loses 450
gallons a minute. A choice that says the volume increases is wrong no matter how
right the number is.
4
The whole term 15w is an amount, not a rate.
Multiply the units and the answer falls out: dollars per week × weeks = dollars.
15dollars / weeka rate — how fast the bill grows
wweeksthe count
15wdollarsthe rent for those w weeks
15w + 200dollarsthe rent plus the setup fee — the whole bill
At w = 6: 15(6) = 90 dollars of rent, plus the 200-dollar setup fee, for a total of
290 dollars. "90" is what 15w means; "290" is what C means.
The favourite wrong answer is the coefficient's meaning
Asked about 15w, most students pick "the cost per week", which is the meaning of
15 alone. The extra w changes it from a rate into a total. The same trap runs
through every version of this item: 0.06s is not "6 percent", it is the amount of
salt in s liters of brine; 5b is not "the price of a bus ticket", it is what was
spent on b bus tickets.
5
The same question, asked about a graph.
A marked point, an intercept and a slope carry exactly the meanings you have just named.
On the graph
In the equation F = 24 − 3h
In the story
(0, 24)
the constant 24
the tank held 24 gallons when the generator was switched on
(5, 9)
F = 24 − 3(5) = 9
9 gallons are left after 5 hours
(8, 0)
24 − 3h = 0
the generator runs out of fuel after 8 hours
the steepness
the coefficient −3
3 gallons are burned each hour
One item type Desmos cannot answer for you
Graphing F = 24 − 3h does make the picture concrete — the intercepts appear at
(0, 24) and (8, 0) and you can click either one. But no calculator will tell you
which sentence about a generator is the right one. This is a reading question
wearing an algebra costume, and the time you save elsewhere with the calculator is
the time to spend here.
6
"Best interpretation" means three of the choices are true and irrelevant.
Do not ask "is this statement true?" — all but one usually are. Ask "does this statement describe the exact quantity the stem named?"
Choice, for 450 in V = 12,000 − 450t
True?
What it actually describes
The pool holds 12,000 gallons when draining begins.
yes
the constant 12,000
The pool is empty about 26.7 minutes after draining begins.
yes
the solution of 12,000 − 450t = 0
The pool loses 450 gallons each minute.
yes
the coefficient 450 — the answer
The pool holds V gallons t minutes after draining begins.
yes
the variable V
underline the quantity in the stemsay its meaning before reading the choicesthen find the sentence that matches
Covering the choices and finishing the sentence "450 is the number of gallons that
…" takes about five seconds and makes the right answer obvious. Reading the four
choices first invites you to agree with all of them.
Watch the units inside the sentence
Two choices often differ only in a unit or a direction: "gallons each minute"
versus "minutes for each gallon", or "loses" versus "gains". Check the unit of
the quantity you were asked about — 450 is gallons per minute — and the near-miss
disappears.
✓
Check: name every piece before you answer.
If you can label all four parts of a model, no version of this question can surprise you.
200 in C = 15w + 200dollarsthe one-time setup fee
15 in C = 15w + 200dollars per weekthe weekly rate
15w in C = 15w + 200dollarsthe rent for w weeks alone
−3 in F = 24 − 3hgallons per hourfuel burned each hour ✓
450 in V = 12,000 − 450tgallons per minutethe opening problem ✓
Remember the flow
Find the quantity · Give it units · Say the meaning · Then read the choices
Three true sentences and one answer — tell them apart by the question, not the truth.
Now you try
Practice problem
A machine shop bills a customer using the equation B = 95 + 62h, where B is the total bill, in dollars, for a repair that takes h hours. Which of the following is the best interpretation of 62h in this context?
62 is dollars per hour and h is hours, so 62h is dollars — the labor charge for those h hours. All four sentences are true of this model, which is exactly the difficulty. A interprets 62 alone, the rate, and is the trap the extra h is there to set. B interprets 95, the fixed part. D interprets B, the whole bill, which is 62h plus the 95. Test it with numbers: a 4-hour repair bills 62(4) = 248 dollars of labor and 343 dollars in total, so 62h is the 248.
Hint: the stem named 62h, not 62. Multiply the units —
dollars per hour times hours — and see what kind of quantity you are being asked to
describe before you look at a single choice.
