Hard SAT Practice Questions: 628-Question Bank With Explanations

Four levels of hard

47Hard13 Reading and Writing, 34 Math

265Harder125 Reading and Writing, 140 Math

267Hardest112 Reading and Writing, 155 Math

49ExtremeMath only. Past anything on the real test, on purpose.

What makes it different

The bank holds 628 questions written to be harder than the five tests: 250 in Reading and Writing and 378 in Math, spread across all 30 of College Board’s skills. None of them appear in the tests.

Built to beat the calculator

Desmos is allowed on the whole Math section, and it can solve any equation you can type in. So the bank’s Math questions are written so that typing it in isn’t enough: the equation has an unknown constant, the answer is an expression or a count, the graph only exists after a word problem is set up, or the figure has no coordinates at all.

Every trap named

Wrong answers are built from the mistakes strong students actually make: an answer true to one sentence but not another, the right idea in the wrong direction, a percent taken of the wrong base, or stopping one step early. Every explanation says which trap each wrong answer sets.

Tailored practice

Margin keeps a running estimate of how likely you are to get a hard question right in each skill, from every answer you give. Tailored sets weigh each skill by how often it shows up on the real test and how often you miss it, pitch questions just past comfortable, mix skills, and bring missed questions back a day later, then 3, 7 and 16 days after each right answer.

How the bank and Tailored practice work, with sources

Four sample questions

Reading and Writing · Inferences · Hardest

Archaeologists excavating a Bronze Age village in Scotland found bronze tools but no traces of copper or tin ore, the two metals needed to make bronze, within 200 kilometers of the site. They did find molds for casting tools and drips of solidified bronze on the workshop floor. Because bronze can be melted down and recast without losing quality, the archaeologists concluded that the village's metalworkers most likely

Which choice most logically completes the text?

Amined small deposits of copper and tin near the village that the chemical analysis failed to detect.
Bimported all of their tools already cast and never worked bronze themselves.
Cobtained finished bronze objects through trade and recast them into new tools.
Dmade most of their tools from a local metal and used imported bronze only for decoration.

Show the answer

Answer: CThree facts must fit together: no ore nearby, casting evidence on site, and bronze can be recast without losing quality. Only recasting traded bronze satisfies all three. A contradicts the absence of ore, B ignores the molds and drips, and D contradicts the bronze tools that were found.

Bank question r2051

Reading and Writing · Transitions · Hardest

Early in her career, the photographer Imogen Hart was told that portraits of factory workers would never sell. She resolved to prove her critics wrong. although she had little money, she spent three years photographing mill workers in their homes and at their machines, then exhibited the portraits in a gallery known for showing paintings by famous artists.

Which choice completes the text with the most logical transition?

ANevertheless,
BIn contrast,
CEven so,
DTo that end,

Show the answer

Answer: DThe sentence describes the steps Hart took to reach the goal stated just before ("prove her critics wrong"), so "To that end" fits. The contrast ("although she had little money") is already handled inside the sentence. Trap: three choices signal contrast, but the relationship to the previous sentence is purpose.

Bank question r4026

Math · Equivalent expressions · Hardest

The expression

\frac{4 x ^{2} - 25}{2 x ^{2} + k x - 10}

, where

k

is a constant, can be rewritten as a quotient of two linear expressions by canceling a common factor. Which of the following could be the value of

k

A $- 9$
B $1$
C $- 3$
D $3$

Show the answer

Answer: BThe numerator factors as

(2 x - 5) (2 x + 5)

, so the denominator must contain one of these factors. With

2 x + 5

(2 x + 5) (x - 2) = 2 x^{2} + x - 10

, so

k = 1

. (With

2 x - 5

(2 x - 5) (x + 2) = 2 x^{2} - x - 10

, so

k = - 1

, which isn't a choice.) For

- 9

- 3

, and

3

, the denominator has no rational zeros, so nothing cancels.

Bank question m2015

Math · Nonlinear equations in one variable and systems of equations in two variables · Extreme

In the equation

\frac{x}{x - 3} - \frac{k}{x + 3} = \frac{18}{x ^{2} - 9}

k

is a constant. What is the sum of all values of

k

for which the equation has no solution?

Show the answer

Answer: 12Multiply both sides by

(x - 3) (x + 3)

x (x + 3) - k (x - 3) = 18

, or

x^{2} + (3 - k) x + 3 k - 18 = 0

. Substituting

x = 3

gives

9 + 9 - 3 k + 3 k - 18 = 0

, so 3 is always a root of this quadratic, and it is always excluded from the original equation. Factoring,

(x - 3) (x - (k - 6)) = 0

, so the only possible solution is

x = k - 6

. It fails exactly when it is an excluded value too:

k - 6 = 3

k - 6 = - 3

, so

k = 9

k = 3

. The sum is 12.

Bank question m2035

Every skill on the test

Skill	Questions
Reading and Writing
Words in Context · Craft and Structure	30
Text Structure and Purpose · Craft and Structure	20
Cross-Text Connections · Craft and Structure	20
Central Ideas and Details · Information and Ideas	15
Command of Evidence (Textual) · Information and Ideas	18
Command of Evidence (Quantitative) · Information and Ideas	17
Inferences · Information and Ideas	20
Boundaries · Standard English Conventions	30
Form, Structure, and Sense · Standard English Conventions	30
Rhetorical Synthesis · Expression of Ideas	25
Transitions · Expression of Ideas	25
Math
Linear equations in one variable · Algebra	15
Linear functions · Algebra	16
Linear equations in two variables · Algebra	15
Systems of two linear equations in two variables · Algebra	20
Linear inequalities in one or two variables · Algebra	16
Equivalent expressions · Advanced Math	36
Nonlinear equations in one variable and systems of equations in two variables · Advanced Math	34
Nonlinear functions · Advanced Math	40
Ratios, rates, proportional relationships, and units · Problem-Solving and Data Analysis	17
Percentages · Problem-Solving and Data Analysis	15
One-variable data: distributions and measures of center and spread · Problem-Solving and Data Analysis	12
Two-variable data: models and scatterplots · Problem-Solving and Data Analysis	6
Probability and conditional probability · Problem-Solving and Data Analysis	10
Inference from sample statistics and margin of error · Problem-Solving and Data Analysis	5
Evaluating statistical claims: observational studies and experiments · Problem-Solving and Data Analysis	4
Area and volume · Geometry and Trigonometry	30
Lines, angles, and triangles · Geometry and Trigonometry	29
Right triangles and trigonometry · Geometry and Trigonometry	28
Circles · Geometry and Trigonometry	30

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