Exactness, rounding, and output verification

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Learn how to turn Desmos decimals, graph points, and fitted values into exact or correctly rounded SAT answers without losing precision.

Difficulty: GOAT Mode
Estimated time: 50 minutes
Practice: Math question bank

What you’ll learn

Tell an exact result from a decimal approximation.
Decide whether a question wants an exact value, a rounded value, or any answer that fits the entry box.
Keep every digit until the very last step.
Turn a decimal from a graph or regression back into an exact fraction or radical.
Pick between short exact algebra, one Desmos expression, and a table.
Check that your answer is the right quantity, with the right units, sign, conditions, and form.

Prerequisites

Why this matters on the SAT

Turn what Desmos shows into an answer the SAT will score

Desmos might hand you a decimal, a point, or a number from a regression. The SAT might want an exact fraction, one of four exact answer choices, a measurement rounded to a certain place, a percent typed without the $\%$ sign, or a total built from several of those numbers. So the number on the screen is a clue, not automatically the answer. Your job is to turn it into the answer the question asks for.

SAT example

In a laboratory, $x$ and $y$ are positive amounts that satisfy

\begin{aligned} x+y&=12,\\[1.4em] x^2y+xy^2&=135. \end{aligned}

If $x>y$ , which choice gives the value of $x$ ?

Fast Desmos solution

Graph both equations. They meet twice, and the point where $x>y$ is about

(10.975,1.025).

Choice A also works out to about $10.975$ , so it's the choice to check.

Try it yourself:

Click both intersections and find the one where $x>y$ . Compare its $x$ -coordinate with choices A and B, then reset the example.

A matching decimal makes A very likely, but a quick substitution makes it certain. Try

x=6+\frac{3\sqrt{11}}{2} \quad\text{and}\quad y=6-\frac{3\sqrt{11}}{2}.

They add to $12$ , and

\begin{aligned} xy &=36-\left(\frac{3\sqrt{11}}{2}\right)^2\\[1.4em] &=36-\frac{99}{4}\\[1.4em] &=\frac{45}{4}. \end{aligned}

x^2y+xy^2=xy(x+y)=\frac{45}{4}(12)=135.

The answer is A. Desmos found the point, the answer choices gave its exact form, and substituting proved it.

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The graph finds the point where x is greater than y. The choices give its exact form.

When should you use algebra, Desmos, or a table?

Use the least setup that still keeps the answer exact.

Use short exact algebra

Go by hand when one or two steps give the exact answer, like $5x=7$ , which gives $x=\frac75$ right away.
Go by hand when the setup already shows the exact form, like a slope of $\frac{7-2}{4-1}=\frac53$ read from two rows of a table.

Use Desmos

Use one expression or graph to find a hard-to-reach value, like an intersection, compare the answer choices as decimals, or run one long calculation such as a drain time.
Use a table to keep paired $x$ and $y$ values lined up when you fit a regression or check a model.
Use a table to run the same rule on several inputs at once, like the cost of $5$ , $10$ , $15$ , and $20$ tickets.

A table keeps the numbers tidy, but it can’t prove that a decimal is exact.

Check your understanding:

Which setup fits each task? (1) Evaluate $f(6)$ . (2) Test four possible inputs in the same function. (3) Solve $3x=2$ exactly.

Know what Desmos actually gave you

Desmos can show you two kinds of numbers, and they aren't the same thing.

An exact value is the number itself, with nothing cut off, like

\frac{13}{6}.

An approximation is a decimal that's close to it, like

2.167.

That decimal is often close enough to pick the right answer choice. But it isn't equal to $\frac{13}{6}$ . It's only the rounded version Desmos shows.

Points on a graph and values from a regression are decimals, so treat them as approximations. For example, graph

y=2x+1

and

y=-x+3.

Click where they meet, and the label says about

(0.667,2.333).

Now set the two equations equal:

2x+1=-x+3,

3x=2 \quad\text{and}\quad x=\frac23.

Then

y=2\left(\frac23\right)+1=\frac73.

The graph told you where to look. The equations gave you the exact point.

Check your understanding:

A regression shows $m=0.333333$ , and one answer choice says $m=\frac13$ . Does the decimal prove that $m$ is exactly $\frac13$ ? What would you do next?

Common mistake:

Copying every digit Desmos shows and treating the result as exact. If the number came from a graph point or a regression, more digits make it closer, but it’s still an approximation.

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The label shows decimals. The exact point is (2/3, 7/3).

Decide which kind of answer the question wants

Before you touch the number Desmos gave you, reread the last sentence of the question. Every answer falls into one of three kinds.

An exact answer

Keep the value exact, or get it back, when the answer choices have fractions, radicals, or $\pi$ , or when the question asks for an exact coefficient or expression.

On a multiple-choice question, use the Desmos decimal to spot the exact choice that matches. Then check that choice in the original conditions.

A rounded answer

Sometimes the question names a place: nearest whole number, nearest tenth, nearest hundredth, and so on. Then keep every digit while you work, and round once, at the very end.

Nearest whole number means round to the ones place.
Nearest tenth means keep one digit after the decimal point.
Nearest hundredth means keep two digits after the decimal point.

For example, if the full result is

0.656\ldots,

then to the nearest hundredth it's

0.66.

Enter that rounded decimal.

A typed answer with no rounding instruction

When you type your own answer, current College Board directions allow up to $5$ characters for a positive answer and up to $6$ for a negative one, counting the negative sign. The decimal point and the fraction bar each count as a character too.

If the exact fraction fits, it's often the safest thing to enter. For example,

\frac{12}{73}

goes in as 12/73, which is exactly $5$ characters.

If the fraction doesn't fit, use its decimal. When the decimal is too long, the official directions let you round it or truncate it to fit. To truncate means to cut the decimal off without rounding up the last digit you keep.

The character limit decides how many decimal places fit. A positive answer less than $1$ gets four digits after the decimal point if you leave off the leading zero, as in .6666, or three if you include it, as in 0.666.

So for $\frac23$ , all of these are accepted:

2/3;
.6666, truncated;
.6667, rounded;
0.666, truncated; and
0.667, rounded.

Don't enter a mixed number, a percent sign, a comma, or a dollar sign. You'll practice the full entry routine later, in From Desmos to Bluebook: answer entry and final checks.

Check your understanding:

Desmos gives $\frac{12}{73}=0.16438\ldots$ , and the question doesn’t say to round. What’s the simplest, safest thing to enter?

Common mistake:

Fitting the answer to the character limit when the question already says how to round. The question’s instruction wins. If it says nearest hundredth, give the nearest hundredth.

Recover an exact value without guessing

There are three ways to get an exact value back. Start with the first one that fits.

Keep exact inputs exact

Type fractions as fractions and radicals as radicals. Don't swap $\frac57$ for $0.714$ and then use it in the next calculation.

When you calculate directly, Desmos can often turn the decimal it shows into a fraction. Type

36/219

and click the fraction conversion icon. Desmos shows

\frac{12}{73}.

Try it yourself:

Switch the first result between decimal and fraction form. Then change 36/219 to 45/252 and check that Desmos simplifies it to $\frac{5}{28}$ . Reset the example when you’re done.

You can trust that fraction because the numbers you typed were exact. Converting a rounded decimal is different: you can get a fraction for the rounded number, not the original one. For example, $0.667$ is exactly $\frac{667}{1000}$ , which isn't $\frac23$ .

Match the decimal to an answer choice

Say a graph gives

x\approx7.571.

If one answer choice is

\frac{53}{7}\approx7.571,

that's the choice to check. If the match isn't obvious, type each choice into Desmos to see its decimal. Then check the one that matches in the original equation.

Rebuild the exact value from the math

With no exact choices to match, go back to the equations, the geometry, or the model.

If two lines meet at a decimal point, solve the exact equations by substitution or elimination. If a circle calculation shows a decimal radius, square your exact answer or plug it into the exact circle equation. If a regression shows decimal values, first check whether the question promises an exact fit. With an exact fit, check your answer against every given point. A best fit doesn't have to pass through the points, so compare your exact value with the fitted value instead.

Check your understanding:

A graph shows $x\approx12.702$ , and one choice is $\frac{19+\sqrt{41}}{2}$ . What’s the fastest reliable next step?

Common mistake:

Converting a rounded graph label to a fraction and trusting the result. The label $2.167$ becomes $\frac{2167}{1000}$ , not $\frac{13}{6}$ . Get the exact value from exact inputs, the answer choices, or the original equations.

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Both lines have the same value. Click the fraction conversion icon to see the simplified fraction.

Round once, after the complete calculation

If you round partway through, every later step inherits the error. So keep all the digits through the formulas and any unit conversions, and round only the final answer.

Example: Drain a pool without rounding early

Worked example

A rectangular swimming pool measures $25$ meters by $10$ meters with a uniform depth of $2$ meters. The pool is emptied through a straight circular drain pipe with an inner diameter of $0.5$ meter. Water flows through the pipe at a constant speed of $2$ meters per second.

To the nearest whole minute, how long will it take to completely drain the full pool?

Step 1

Find how much water the pool holds

The pool's volume is

V=25(10)(2)=500

cubic meters.

Step 2

Find how fast water leaves the pipe

The pipe's radius is half its diameter:

r=\frac{0.5}{2}=0.25

meter. So the opening has area

A=\pi(0.25)^2=\frac{\pi}{16}

square meter.

Water moves through that opening at $2$ meters per second, so the pipe drains

2A=\frac{\pi}{8}

cubic meter every second.

Step 3

Keep every digit through the change to minutes

Divide the volume by the rate to get the time in seconds:

\frac{500}{\pi/8}=\frac{4000}{\pi}.

Change to minutes before you round:

\frac{4000}{60\pi} =\frac{200}{3\pi} \approx21.220659\ldots.

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The last line is the drain time in minutes, before rounding.

Step 4

Round what the question asks for

The question wants the nearest whole minute. The tenths digit of $21.220659\ldots$ is $2$ , so round down:

\boxed{21}.

Enter 21.

Common mistake:

Rounding the pipe area or the flow rate before you divide. Round the flow rate to $0.4$ , and the time comes out to $20.83$ minutes instead of $21.22$ . That still rounds to $21$ here, but a drift that size can flip the answer on another question.

Verify the output, not just the calculation

A correct calculation can still end in a wrong answer if you submit the wrong number. Before you submit, ask yourself five quick questions.

Is this what it asks for? The question might want $x$ , $y$ , $x+y$ , a coefficient, a radius, an area, or a percent.
Did I use every condition? Watch for words like positive, first, greatest, least, or $x>y$ .
How exact should it be? It might need an exact value, a certain rounding place, or anything that fits the entry box.
Are the units right? You might need to change seconds to minutes, feet to inches, or a decimal rate to a percent.
Is it in the right form? It should match an answer choice or fit the entry box with no extra symbols.

A few habits help. When you click an intersection, write down both coordinates before you decide which one you need. With a regression, know what each fitted value means before you combine them. In a word problem, carry the units while you work, and drop only the symbol the entry box won't take.

Check your understanding:

Desmos gives the intersection $\left(\frac{26}{19},\frac{27}{19}\right)$ , but the question asks for $x+y$ . What do you enter if it doesn’t say to round?

Common mistake:

Doing all the math right and then submitting the wrong number from the screen. When the calculator work is done, reread the last sentence of the question.

Finish the solution

The two line equations are already in the calculator. Use the decimal where they meet to pick the exact choice, then check it.

Turn a decimal intersection into an exact choice

Finish the solution

Lines m and n pass through the four labeled points.

The graph shows two lines, $m$ and $n$ .

Line $m$ passes through $A=(-1,4)$ and $B=(2,2)$ .
Line $n$ passes through $C=(0,-1)$ and $D=(3,3)$ .

Which choice is the $x$ -coordinate of the point where lines $m$ and $n$ intersect?

First steps

Line $m$ is $y=-\frac23x+\frac{10}{3}$ .
Line $n$ is $y=\frac43x-1$ .
Click where they meet. Desmos shows an $x$ -coordinate of about $2.167$ .

Finish it

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Match the decimal x-coordinate to an exact choice.

Practice problems

Now it's your turn. On each one, decide what kind of answer you need before you start calculating, and give each problem a real try before you open its hint.

Round only the final percent

Practice problem

At the start of a year, a company's workforce consists of $40\%$ engineers, $35\%$ designers, and the rest marketing staff.

Over the year:

the number of engineers increases by $25\%$ ;
the number of designers decreases by $20\%$ ; and
the number of marketing staff increases by $10\%$ .

At the end of the year, the total number of employees is $p\%$ of the number at the start of the year. What is the value of $p$ , to the nearest whole number?

Your answer

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Pick an easy starting total, change each group, and round p last.

Get an exact prediction back

Practice problem

A line of best fit models the relationship between the altitude of a weather balloon, $x$ , in thousands of feet, and the outside temperature, $y$ , in degrees Fahrenheit. The line passes through $(2,61)$ and $(11,40)$ .

According to the line of best fit, what is the predicted altitude, in thousands of feet, when the temperature is $48^\circ\text{F}$ ?

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Graph the line and y = 48, then match the decimal to a choice.

Choose the shortest exact method

Practice problem

A data set is shown in the table.

$x$	$y$
$1$	$2$
$4$	$7$
$7$	$12$

The data fit a linear model $y=mx+b$ exactly. What is the exact value of $m$ ?

Your answer

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Try the slope by hand first. A regression is only a check.

Check an exact radical choice

Practice problem

A rectangular mural has a perimeter of $38$ feet and an area of $80$ square feet.

Which choice is the length, in feet, of the longer side of the mural?

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Graph both conditions, find the longer side, and match it to a choice.

Finish the lesson

5 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Points on a graph and values from a regression are approximations, so treat them as candidates to check.
Type fractions, radicals, and other exact numbers exactly whenever the answer has to be exact.
Match a decimal to an exact answer choice, then check that choice in the original conditions.
Use a few lines of algebra for a short exact problem, one Desmos expression for one calculation, and a table for repeated rows.
Follow the question's rounding instruction, and round only after the whole calculation and any unit changes.
With no rounding instruction on a typed answer, enter the exact fraction if it fits.
Reread the last sentence and check the quantity, conditions, units, precision, and form. Only then is a calculator result your SAT answer.

Next lesson

From geometry to coordinates to circle regression

Combine exact setup, regression, rebuilding the exact answer, and a final check into one complete method.

Start next lesson

Practice

Practice SAT Math

Put what you learned to work on real SAT Math questions.

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