Desmos for the SAT · Technique 11 of 23

Solve Rate and Ratio Problems

Most of these are one division. For the rest, make the rate a line and click where it meets the number you were given.

SolvesAt this rate, how many?The ratio of a to b is 5 to 3Two machines working togetherHow long until the tank is fullUnit rate and price per item

Watch the lesson

The whole move worked on one real SAT question, narrated.

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Two rates added on one line, graphed against a full tank, with the crossing clicked.

Start with the part most Desmos guides leave out: most SAT rate questions are one division, and opening the calculator for them costs you time. A car covers 180 miles in 3 hours, so the rate is 60 miles an hour and 5 hours is 300 miles. You have that answer before the calculator finishes loading.

Three shapes are worth typing. First, the question gives you the finish and asks for the start, so you are solving rather than multiplying. Second, two things run at the same time, which is where rates add, times do not, and the arithmetic turns into fractions. Third, the numbers are ugly: a rate like $\frac{35}{4}$ , or a ratio written as one fraction over another.

The move is the same in all three. Make the rate a line, y equals the rate times x, so the line reads as the running total. Put the number the question gave you on its own line. Two graphs means the answer is a crossing, and Desmos marks that crossing and prints both coordinates once you click it.

Nova

Before you type anything, check whether this is a one-division question: most rate questions are one division, and opening the calculator for them costs you time you want later in the section. The ones worth typing are the ones where two things run at once, or where the question gives you the finish and asks for the start.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps adding two fractions and then taking the reciprocal.

The question

Pump A fills a tank in 12 hours; pump B fills the same tank in 18 hours

Both run at once. How many hours to fill the tank?

The algebra way

Rates first: A is $\frac{1}{12}$ of a tank per hour, B is $\frac{1}{18}$ .
Common denominator: $\frac{1}{12}+\frac{1}{18}=\frac{3}{36}+\frac{2}{36}=\frac{5}{36}$ .
That is tanks per hour, and the question wants hours per tank, so flip it: $\frac{36}{5}$ .
Convert: $\frac{36}{5}=7.2$ hours. Three places to slip, and the flip is the step that gets dropped, because $\frac{5}{36}$ already looks like an answer.

The Desmos way

Type y=x/12, right arrow, +x/18, right arrow.
Type y=1 on the next line.
Click inside either box, then click the crossing: $(7.2,1)$ .

The payoff: The algebra needs a common denominator and then a reciprocal, and the reciprocal is a separate step with nothing to check it against. The graph needs neither: the line is the share of the tank that is done, and the crossing is the hour it reaches one.

Quick reference

What the question gives you, and whether the calculator is worth opening

The question gives you	What you do	Worth typing?
A rate and a time, and the question wants the total	Multiply, in your head	No
A total and a rate, and the question wants the time	`y=` rate `x`, then `y=` the total, then click the crossing	Only if the numbers are ugly
Two things running at once, one whole job	`y=x/a`, right arrow, `+x/b`, then `y=1`	Yes
A ratio, and one of the two amounts	Let `x` be how many times the ratio is used, graph one part, cross it with the amount you know	Yes
A fraction divided by a fraction	the first fraction, right arrow, `/`, the second fraction, then the right arrow twice	Yes, Desmos does the flip

Step by step

The whole move, all 6 steps

Type the rate on the first line as a fraction with an x after it: the amount on top, the time on the bottom, then the right arrow, then x. That line is the running total.
Put the number the question gave you on its own line, as y= that number. When the target is one whole job rather than a quantity, that line is y=1.
Click inside either equation box so the points of interest appear, then click the crossing. Desmos pins the coordinates and offers an icon for sending the point to the expression list.
Read which coordinate the question asked for. The y is the total you were already told; the x is the time or the amount you were asked to find.
When two things run at once, add the two rates on the same line before you graph. Press the right arrow after each denominator, or the next term lands inside the fraction you just typed.
Check whether you needed the graph at all. A rate and a time, with the total asked for, is one multiplication, and reading it is faster than typing it.

The question

Working alone, pump A can fill an empty tank in 12 hours. Working alone, pump B can fill the same empty tank in 18 hours.

If both pumps start at the same time on an empty tank, how many hours will it take to fill the tank?

A6
B7.2
C15
D30

Step 1 of 5

Both pumps run at the same time, so their shares add. In $x$ hours pump A has filled $\frac{x}{12}$ of the tank and pump B has filled $\frac{x}{18}$ . Type that total: y=x/12, press the right arrow, then +x/18 and the right arrow again.

In the calculator: y=x/12→+x/18

Rates add. Times do not. Twelve hours and eighteen hours is not a thirty hour job, and it is not a fifteen hour job either.

Typed straight through with no right arrow, the live calculator builds $y=\frac{x}{12+\frac{x}{18}}$ instead. A fraction keeps hold of everything you type after it until the right arrow steps you out of the denominator.

Now try it on a real SAT question

Head or scratch paper. 156 divided by 6 is 26 miles per gallon, and 26 goes into 234 nine times. Two clean divisions, and you can do them faster than you can type two lines. Spotting which questions to read instead of type is worth as many points as the typing is.

A car travels 156 miles using 6 gallons of gasoline.

At this rate, how many gallons of gasoline would the car need to travel 234 miles?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

Add the rates, never the times. Two machines running together are faster than either one by itself, so any answer longer than the slower machine takes alone is wrong.
Most rate questions need none of this. A rate and a time, with the total asked for, is one multiplication, and the typing is the slow part.
Check which coordinate the question wanted. The crossing prints both, and one of the two is a number the question already gave you.
For a ratio, let x be how many times the ratio is used rather than one of the amounts. Each part is then a fixed multiple of that x, and the amount you were told becomes the second line.

If the calculator does something you did not expect

Press the right arrow after each denominator. Typed straight through at the keyboard, y=x/12+x/18 builds $y=\frac{x}{12+\frac{x}{18}}$ in the live calculator, because a fraction keeps hold of everything typed after it.
A fraction stacked on a fraction needs the right arrow twice. Typing y=2/3, right arrow, /1/4, right arrow, x puts the x in the denominator; a second right arrow first keeps it outside.

Nova

The clue is which piece the question leaves out. Rate and time, and it wants the total? Multiply and move on. A total, and it wants the time? That is a crossing, and the graph beats the algebra.

All Desmos techniques

Put this move into real practice

"Solve Rate and Ratio Problems" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

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