SAT question bank

Sample SAT Math questions

One hard question from each Math domain. Try them before you open the answers.

1. Algebra · Linear equations in one variable · hard

   12x + 2 over 2 - c over 4 = d(x - 2) In the equation above, c and d are constants. If the equation has infinitely many solutions, what is the value of c?

   1. A. 52
   2. B. 13
   3. C. 44
   4. D. 4
   Show answer and explanation

   Answer: A. Infinitely many solutions means both sides are the same expression. The left side is 6x + 1 - c over 4, so d = 6 and the right side is 6x - 12. Matching the constants, 1 - c over 4 = -12, so c over 4 = 13 and c = 52. 4 leaves out the -12 on the right side. 13 is c over 4, not c. 44 uses the wrong sign on 1.

2. Advanced Math · Equivalent expressions · hard

   Which expression is equivalent to 3 over x - 5 - 1 over x - 4, where x ≠ 5 and x ≠ 4?

   1. A. 2x - 11 over (x - 5)(x - 4)
   2. B. 2x - 7 over (x - 5)(x - 4)
   3. C. 2 over 2x - 9
   4. D. 4x - 17 over (x - 5)(x - 4)
   Show answer and explanation

   Answer: B. Use the common denominator (x - 5)(x - 4): 3(x - 4) - 1(x - 5) = 2x - 7, so the difference is 2x - 7 over (x - 5)(x - 4). 2 over 2x - 9 subtracts the numerators and adds the denominators, which is never allowed. 4x - 17 over (x - 5)(x - 4) adds the second fraction instead of subtracting it. 2x - 11 over (x - 5)(x - 4) multiplies each numerator by its own denominator's constant.

3. Problem-Solving and Data Analysis · Ratios, rates, and proportions · hard

   A scale model of a storage tank is built so that 1 centimeter on the model represents 3 meters on the actual tank. The model holds 14 cubic centimeters. How many cubic meters does the actual tank hold?

   1. A. 42
   2. B. 27
   3. C. 378
   4. D. 126
   Show answer and explanation

   Answer: C. Each length is multiplied by 3, so a volume is multiplied by 3³ = 27: 14 × 27 = 378. 42 scales the volume as if it were a length. 126 scales it like an area. 27 forgets the model's 14 cubic centimeters.

4. Geometry and Trigonometry · Circles · hard

   A circle has a radius of 4. A sector of the circle has a central angle of 3 over 2π radians. What is the area of the sector?

   1. A. 24π
   2. B. 6π
   3. C. 3π
   4. D. 12π
   Show answer and explanation

   Answer: D. With the angle in radians, sector area = 1 over 2r²θ = 1 over 2(4)²(3 over 2π) = 12π. 6π is the arc length rθ, not the area. 24π leaves out the factor 1 over 2. 3π is half the arc length.

Sample SAT Reading and Writing questions

1. Information and Ideas · Central ideas and details · hard

   Some reef-building corals can host several types of symbiotic algae, and certain types tolerate heat better than others. After a severe bleaching event, marine biologist Keoni Akana and colleagues found that many surviving colonies on a Hawaiian reef had shifted toward hosting mostly heat-tolerant algae. This 'shuffling' might seem like a straightforward rescue for reefs facing warmer seas. Akana cautions, however, that the heat-tolerant algae typically pass fewer nutrients to their hosts, and colonies that relied on them in his study grew noticeably more slowly than colonies hosting other types. Because slow-growing reefs recover more slowly from storms and erosion, he argues that shuffling may help individual colonies survive a heat wave while leaving the reef as a whole less able to rebuild. Which choice best states the main idea of the text?

   1. A. Shuffling toward heat-tolerant algae will allow coral reefs to withstand future warming without lasting harm.
   2. B. Heat-tolerant algae provide corals with more nutrients than other algae do, which speeds reef growth.
   3. C. Although shifting to heat-tolerant algae may help corals survive heat, it may also slow reef growth and recovery.
   4. D. Most corals on Hawaiian reefs died during the bleaching event because they could not host heat-tolerant algae.
   Show answer and explanation

   Answer: C. The text grants that shuffling helps colonies survive heat, then turns ('however') to Akana's point that these algae slow growth and leave the reef 'less able to rebuild.' Choice A is wrong because it states the 'straightforward rescue' view that Akana argues against. Choice B is wrong because it reverses the finding: heat-tolerant algae pass fewer nutrients, and colonies relying on them grew more slowly. Choice D is wrong because the text discusses colonies that survived and never says most corals died or why.

2. Craft and Structure · Cross-text connections · hard

   Text 1 The return of gray wolves to Yellowstone National Park in 1995 is often presented as a textbook trophic cascade. In this account, wolves reduced elk numbers and, just as important, made elk wary of lingering along streams, so willows that elk had browsed for decades began to recover. Ecologist Mira Castell cites the regrowth of streamside willows as evidence that restoring a single top predator can repair an entire ecosystem. Text 2 Hydrologist Arne Solberg does not dispute that willows have rebounded in places, but he notes that decades without wolves also meant decades without beavers, whose dams once raised water tables along those streams. Where streams had cut deeper channels after the beavers vanished, he found, willows stayed stunted even after elk browsing declined, because their roots could no longer reach groundwater. In Solberg's view, the predator was necessary but not sufficient. Based on the texts, how would Solberg (Text 2) most likely respond to Castell's conclusion in Text 1?

   1. A. By arguing that restoring the predator alone may not repair an ecosystem where other losses persist
   2. B. By rejecting the claim that willows recovered anywhere after wolves returned
   3. C. By contending that beavers, rather than wolves, reduced elk browsing along streams
   4. D. By agreeing that reduced browsing was enough to restore willows along every stream
   Show answer and explanation

   Answer: A. Solberg accepts some willow recovery but found willows stunted where beaver loss had lowered groundwater, so a predator alone did not finish the job. Choice B is wrong because Solberg says he does not dispute that willows rebounded in places. Choice C is wrong because in Text 2 beavers matter because their dams raise water tables, not because they affect elk. Choice D is wrong because Solberg found willows that stayed stunted even after browsing declined.

3. Expression of Ideas · Rhetorical synthesis · hard

   While researching a topic, a student has taken the following notes: - Greenland sharks live in the cold, deep waters of the North Atlantic and Arctic Oceans. - They grow slowly, perhaps about 1 centimeter per year. - Many fish are aged by counting yearly growth rings in hard, bony structures, but Greenland sharks lack hard parts that form such rings. - In 2016, a team of researchers instead measured radiocarbon in the lenses of the sharks' eyes, tissue that forms before birth and does not change afterward. - The largest shark in the study was estimated to be about 390 years old, though the estimate had a wide margin of error. - The results suggest that the species is the longest-lived vertebrate known. The student wants to explain why the researchers could not rely on the standard method used to age many fish. Which choice most effectively uses relevant information from the notes to accomplish this goal?

   1. A. Radiocarbon in the sharks' eye lenses suggested that the largest one was about 390 years old, with a wide margin of error.
   2. B. Eye lenses form before a shark is born and do not change afterward, so they preserve a record of its earliest years.
   3. C. Greenland sharks live in cold, deep Arctic and North Atlantic waters and may be the longest-lived vertebrates known.
   4. D. Many fish are aged by counting growth rings in hard parts, but Greenland sharks lack hard parts that form such rings.
   Show answer and explanation

   Answer: D. The goal is why the usual method failed, and this choice names the usual method (counting rings in hard parts) and the reason it cannot work for these sharks. Choice A is wrong because it reports the new method's result, not why the standard method could not be used. Choice B is wrong because it explains why eye lenses were useful, not why the standard method failed. Choice C is wrong because it describes the sharks' habitat and lifespan but says nothing about aging methods.

4. Standard English Conventions · Boundaries · hard

   Although the cores of ancient ice that glaciologist Pema Dorji and her colleagues drilled from a glacier high on the Tibetan Plateau were only a few centimeters across, the air bubbles trapped inside them, each a tiny sample of the atmosphere at the moment the surrounding snow was sealed, proved remarkably ______ the oldest layers, which date back tens of thousands of years, still held measurable traces of methane. Which choice completes the text so that it conforms to the conventions of Standard English?

   1. A. durable, even
   2. B. durable even
   3. C. durable; even
   4. D. durable, even,
   Show answer and explanation

   Answer: C. The first part ("Although the cores ... proved remarkably durable") is a complete sentence, since its main clause is "the air bubbles ... proved remarkably durable." The second part ("even the oldest layers ... still held measurable traces of methane") is also complete. A semicolon joins them. Choice A is wrong because a comma between two complete sentences creates a comma splice, even though the long interrupting phrases make it hard to see. Choice B is wrong because without punctuation, the two complete sentences run together. Choice D is wrong because the comma after "even" cuts that word off from "the oldest layers," and the comma before it still creates a comma splice.