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SAT Math for high scorers

Become the person who can solve it.

308 original SAT math problems, harder than the real test and never out of scope.

See a complete Build UpExplore 22 free problems
  • 308 original problems
  • 4 to 5 layered nudges each
  • Built-in Desmos
  • Progress across devices

Harder problems. Clearer thinking.

One problem, start to finish
The reader, on a phone

What harder means here

Same topic. Same tools. One of them makes you connect them.

Harder does not mean longer, and it does not mean off-syllabus. Both of these are circle questions you already have every tool for.

Standard difficulty

A circle has equation x2+y2−14x+22y+154=0x^2 + y^2 - 14x + 22y + 154 = 0. What is its radius?

One method, applied carefully. Complete the square twice and read the answer off. The only difficulty is not slipping, so getting it right means you practiced being careful.

Layered difficulty

In the xyxy-plane, the line y=x+ky = x + k is tangent to the circle x2+y2=18x^2 + y^2 = 18. What is a possible value of kk?

Every tool here is standard and none of them is named. Tangent has to become “one solution”, which has to become “discriminant is zero”. Three ideas you learned separately, and the question mentions none of them. Once they connect, it is two lines.

How Build Ups work

A solution shows you the reasoning. A Build Up helps you produce it.

Hard questions squeeze several ideas into one step. A Build Up unpacks that step into a few short questions, checks each answer, then asks you to put it back together.

Miss the original problem

You answer, and it is wrong. Instead of handing over the solution, the app offers to go back a step.

Two kinds of help

Take the smallest thing that gets you moving.

A nudge keeps you on the problem. A Build Up repairs what sits underneath it. Different jobs, and you should not have to guess which one you need.

Nudge

When you almost see it

One question, pointed at the next idea. They come one at a time, so you take only what you need and the problem stays yours.

See a nudge

Build Up

When the pieces underneath are missing

Easier questions rebuild the ideas, check they transfer to a new problem, then send you back to the original.

See a Build Up

Featured problem

Try it first. Then pick how much help you want.

A real problem in the real workspace, with its Build Up open and no account needed. Nothing you do here is recorded.

Problem

An electronics store is offering two different discounts on its headphones: a volume discount for large orders and a promo code discount. For orders with nn headphones, where 10<n<5010 < n < 50, a volume discount of n%n\% off the original price of $60 is applied to the price of each headphone. If a promo code is applied to the order, a discount of an additional $3 is applied to the price of each headphone, after the volume discount is applied. For a certain order with both the volume discount and the promo code applied, the total price after the discounts for the headphones purchased was $1,050. How many headphones were purchased in this order?
Choose an answer

Full solution7 steps

Problem 1 of 5

Free

Try 22 complete problems free.

Whole problems, not extracts: answer checking, the same nudges, the full worked solution. Spread across the set, so you see the range and not the easy end.

Explore free problems

Questions

Before you start.

  • How were these problems made?

    One at a time, by hand, each built from a specific move the test asks you to perform rather than from a released question with the numbers swapped. Every one is reviewed in three tracked passes, the stem, the nudges and the worked solution, then verified by computer algebra.

  • How do these match the real test?

    Released official forms set the style, phrasing and length, so these read like the last few questions of a real section. The difficulty is in what has to be connected, never in off-syllabus content. Coverage is checked against the domain mix of official pools.

  • How is this different from official College Board practice?

    Official practice is the best picture of the whole test and you should work through it. It is also built for the whole range, so a strong scorer spends most of their time on questions they were never going to miss. This is 308 problems pitched at the end of that range.

  • Is this math only?

    For now. Volume I is SAT math, and nothing in it covers Reading and Writing. ACT math and SAT/ACT grammar are the next planned products, and they arrive when they are written. For grammar, the plan is an interactive platform that teaches the rules, not merely another problem set.

  • Does this work for the ACT?

    Partly, and not by design. The math overlaps and the habit of connecting ideas under time carries over, but the ACT moves faster and reaches content the SAT does not test. Treat it as a side effect rather than a reason to buy.

  • I am around a 650. Is this too hard for me?

    It will be challenging, which is the point. What decides it is whether your content is solid: if the gap is between ideas rather than in them, the nudges and Build Ups will get you through these. If there are whole topics you have not met yet, cover those first.

  • Do I need to work all 308?

    No. It is not a course and there is no order to fall behind in. Domains interleave on purpose, so any stretch of them practices working out which technique applies. Ten problems done properly, with the retry, beat fifty skimmed.

  • How much time do I need before my test date?

    A few problems a week for a couple of months. The retry matters more than the volume and it wants a gap after the Build Up. In the last couple of weeks, timed official practice is the better use of the hours.

Armaan Singh, founder of Pencils Down Labs

Built by Armaan Singh

Built by Armaan Singh, a University of Chicago math graduate who has tutored the SAT and ACT for seven years. It started with a gap: strong students had nothing hard enough to practice on, and nothing better than a worked solution when they got stuck.

Why I Built Pencils Down