SAT Word Problems on Desmos: Graph It First

To solve an SAT word problem on Desmos, write each quantity in terms of the same variable, graph both, and click the point where the graphs cross. That point holds your answer. Then turn its coordinates back into the words of the question and check the result by hand.

Watch: SAT Math Tricks: Consecutive Even Integers (Don't Write x + 1) (Short)

SAT Math Tricks: Consecutive Even Integers (Don't Write x + 1)

You can type every line on this page into the College Board version of the calculator on our Desmos on the SAT page. Bluebook, the app you take the SAT in, has Desmos built in, and College Board allows a calculator on the whole Math section. Below, you watch two SAT-style word problems solved step by step, finish two with help, and then solve one on your own.

The method in four steps

1. Read the last sentence first. Know what the question wants, and in what unit, before you read the story.
2. Pick one variable and write down what it means. Desmos graphs in x and y, so use x. Write "x = hours after Train B leaves" on your scratch paper, not just "x".
3. Write each quantity as y = ... in terms of x. Two amounts that end up equal become two lines. Data readings go in a table. A formula gets its values on separate lines.
4. Read the graph, then translate. Click a line, select the gray point where the graphs cross, and ask what its x and y mean in the story.

Which setup fits the problem

Example 1, worked for you: distance and rate

Train A leaves a station traveling at 60 miles per hour. Train B leaves the same station 1 hour later on the same track, traveling at 90 miles per hour in the same direction. How many hours after Train B leaves does it catch up to Train A?

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SAT-style practice problem.

The question counts hours after Train B leaves, so I let x be exactly that. Train A has been moving 1 hour longer, so after x hours it has traveled 60(x + 1) miles. Train B has traveled 90x miles.

Line 1: y = 60(x + 1)
Line 2: y = 90x

I write both in y = form. Desmos's help pages say coordinates show most reliably for equations entered that way. I click either line, and gray dots appear at the intercepts and at the crossing. I select the crossing. It reads (2, 180).

Now I translate. x = 2 means 2 hours after Train B leaves. y = 180 means both trains are 180 miles from the station at that moment. The answer is 2 hours.

By hand: 60(x + 1) = 90x gives 60x + 60 = 90x, so 60 = 30x and x = 2. If the question had asked how long Train A had been traveling, the answer would be 3 hours. That is why step 2 says to write down what x means.

Example 2, worked for you: a rate from two readings

A candle burns down at a constant rate. It is 24 centimeters tall 2 hours after it is lit and 18 centimeters tall 5 hours after it is lit. How many hours after it is lit does the candle burn down completely?

SAT-style practice problem.

A constant rate means a line. The problem gives two readings, so I put them in a table. Typing table in an expression line creates one.

Table: x₁ column 2, 5 and y₁ column 24, 18
Line 2: y_1 ~ mx_1 + b

The tilde tells Desmos to fit the model to the table. It reports m = -2 and b = 28. With only two points, the fitted line passes through both of them, so these values are exact. m = -2 means the candle loses 2 centimeters per hour. b = 28 means it was 28 centimeters tall when it was lit.

The candle is gone when its height is 0. From 0 = mx + b, that happens at x = -b/m. Desmos stores m and b from the regression, so I can use them on a new line.

Line 3: -b/m

Desmos shows 14. The answer is 14 hours.

By hand: the slope is (18 - 24)/(5 - 2) = -6/3 = -2. Then 24 = -2(2) + b, so b = 28. Setting -2x + 28 = 0 gives x = 14.

Example 3, you finish it: volume with named values

A steel pipe is a right circular cylinder with an outside diameter of 4 3/4 inches, a wall thickness of 5/8 inch, and a length of 144 inches. To the nearest cubic inch, what is the volume of the steel in the pipe's wall?

SAT-style practice problem.

The steel is the outer cylinder minus the empty space inside it, so V = πr_o²h - πr_i²h. There is nothing to graph here. Desmos works as a calculator that keeps every value on its own line, so you can fix one typo without retyping the formula.

Line 1: d = 4.75
Line 2: w = 5/8
Line 3: h = 144
Line 4: r_o = d/2
Line 5: r_i = r_o - w

Line 4 turns the diameter into a radius, 2.375. Line 5 subtracts the wall once, because the wall sits on each side of the hole but the radius only crosses one side. That gives 1.75.

Line 6 is yours. Type the volume formula with these names, read the value, and round it the way the question asks.

Check your answer: pi(r_o^2 - r_i^2)h gives 1,166.32..., so the answer is 1,166 cubic inches. The exact volume is 371.25π. If the answer choices are written with π, type (r_o^2 - r_i^2)h without the pi to get 371.25. If you got about 1,979, you subtracted the wall thickness twice from the radius.

Example 4, you finish it: a mixture

A chemist mixes a 20% acid solution with a 50% acid solution to make 90 liters of a 30% acid solution. How many liters of each solution does she use?

SAT-style practice problem.

Let x be the liters of the 20% solution. The rest of the 90 liters, 90 - x, is the 50% solution. Track the acid, since the acid from both parts adds up to the acid in the mix. The mix holds 30% of 90 liters, which is 27 liters of acid.

Line 2: y = 27

Line 1 is yours. Write the liters of acid in the mix using x, with each percent typed as a decimal. Then find the crossing and give both amounts.

Check your answer: Line 1 is y = 0.2x + 0.5(90 - x). The lines cross at (60, 27), so she uses 60 liters of the 20% solution and 30 liters of the 50% solution. By hand: 0.2(60) + 0.5(30) = 12 + 15 = 27. If you typed 20 and 50 in place of 0.2 and 0.5, the line has the wrong slope and crosses in the wrong place.

Your turn

Tank A holds 500 liters of water and drains at 12 liters per minute. Tank B holds 80 liters and fills at 9 liters per minute. After how many minutes do the two tanks hold the same amount of water?

SAT-style practice problem written for this guide.

Solve it before you read the answer below.

Answer: 20 minutes. Graph y = 500 - 12x and y = 80 + 9x. They cross at (20, 260), so after 20 minutes each tank holds 260 liters. By hand: 500 - 12x = 80 + 9x gives 420 = 21x, so x = 20.

When algebra is faster

College Board's calculator tips say some Math questions are better done without a calculator. Example 1 is one of them: 60 = 30x takes a single line by hand. Graph first when the setup has several parts, like the mixture, or when you want a second check on an answer you found by hand.

Mistakes to avoid

• Typing a list with curly braces. Desmos lists use square brackets, as in x_1 = [0, 8].
• Typing percents as whole numbers. Type 20% as 0.2, not 20.
• Leaving commas inside numbers. Desmos uses commas to separate values, as in the point (2, 180). Type 1200, not 1,200.
• Answering with the wrong coordinate. Reread the last sentence. In Example 1, x counts hours after Train B leaves, not after Train A leaves.

Practice next

Work through Word Problems: Translating Situations into Equations, free lesson notes with practice questions. For mixed Math practice, open the TTA SAT question bank. Related Desmos guides: how to solve systems of equations on Desmos, linear functions and rate of change on Desmos, SAT regression practice problems on Desmos, cylinder and cone volume on Desmos, and find x- and y-intercepts on Desmos.

Checked on September 26, 2026 against College Board's SAT Suite calculator policy, College Board's digital SAT launch announcement (March 12, 2024), and the Desmos Help Center pages FAQs, Regressions and Lists. We reworked every answer on this page by hand. The Test Advantage is independent of College Board and Desmos.