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MathProblem-Solving and Data AnalysisHard

A store raises the price of a jacket by 20%. Later, it lowers the new price by 20%. The final price is what percent of the original price?

A80%
B96%Correct
C100%The trap
D104%

Check with $100: $100then$120then$96

  1. Turn each change into a multiplier.

    Up 20% means × 1.20. Down 20% means × 0.80.

  2. Apply them in order.

    1.20 × 0.80 = 0.96

  3. Read the answer.

    The final price is 96% of the original. That’s B.

  4. Why not 100%?

    The 20% cut is taken from the higher price, so it removes more than the 20% increase added.

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The Daily, plus 36 practice questions every day: 12 easy, 12 medium and 12 hard, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too. No card needed.

Pro is $20 a month or $200 a year: unlimited practice with no daily limit, plus every full-length test and Challenge test. Peak is $29 a month or $280 a year: everything in Pro, plus the full Learn course and Peak Plan. Cancel anytime from your account.

98 questions across four modules, in the digital SAT’s order: Reading & Writing Module 1 and Module 2 (27 questions each), a 10-minute break, then Math Module 1 and Module 2 (22 questions each). On official timing that’s 2 hr 14 min of testing. You get estimated section scores and a total at the end.

Yes. In the standard full-length tests, Module 2 of each section is the harder version if you get 60% or more of Module 1 right, and the easier version if you don’t. Challenge tests use the hardest material throughout.

Official timing, extended 1.5× time, or untimed, on every test, including the free ones. Extended time allows 48 minutes for each Reading & Writing module and about 52 for each Math module. Untimed removes the clock so you can focus on accuracy.

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Lines, angles and triangles · Lesson 2

Parallel lines cut by a transversal

A transversal is a line that crosses two other lines. When it crosses two parallel lines, it makes eight angles, four at each crossing, and the two crossings look exactly alike.

Corresponding angles sit in the same position at the two crossings, such as above the line and to the right of the transversal at both. They are equal. Alternate interior angles are between the parallel lines, on opposite sides of the transversal. They are equal too, and so are alternate exterior angles, outside the parallel lines on opposite sides. Same-side interior angles are between the parallel lines on the same side of the transversal. They add to 180∘180^\circ.

A shortcut: unless the transversal is perpendicular to the lines (then all eight angles are 90∘90^\circ), the eight angles come in just two sizes, an acute one and an obtuse one, and the two sizes add to 180∘180^\circ. In the figure below, every acute angle is 55∘55^\circ and every obtuse angle is 125∘125^\circ. So once you know one angle, you know all eight: an angle of the same kind (both acute or both obtuse) is equal to it, and an angle of the other kind is its supplement.

When a bent path between two parallel lines makes an angle at a point between them, draw a third line through that point, parallel to the other two. It splits the angle into two pieces, and each piece is an alternate interior angle with an angle at one of the parallel lines.

These facts need the lines to be parallel. The question will say so, or write ℓ∥m\ell \parallel m. They also work backward: if corresponding angles or alternate interior angles are equal, the lines are parallel. When the angles are written with xx, set equal angles equal to each other, or set the sum of same-side interior angles equal to 180.

Parallel lines ℓ and m cut by transversal t55°125°55°125°55°125°55°125°ℓmt
Parallel lines ℓ and m cut by transversal t Two horizontal parallel lines, ℓ on top and m below, are cut by a transversal t that slants up to the right. All four angles at each crossing are labelled. Going counterclockwise from the angle above the horizontal line and to the right of t, they are 55°, 125°, 55° and 125°, the same at both crossings.

Worked example Easy

Lines ℓ and m cut by transversal t72°x°ℓmt
Lines ℓ and m cut by transversal t Two horizontal lines, ℓ on top and m below, are cut by a transversal t that slants up to the right. Where t crosses ℓ, the angle above ℓ and to the right of t is labelled 72°. Where t crosses m, the angle above m and to the left of t is labelled x°.

In the figure, line ℓ\ell is parallel to line mm. What is the value of xx?

  1. 1818
  2. 7272
  3. 108108Answer
  4. 288288

How to solve it

  1. At line mm, the angle above mm and to the right of tt is in the same position as the 72∘72^\circ angle at line ℓ\ell. They are corresponding angles, so it is 72∘72^\circ.
  2. The x∘x^\circ angle sits next to that angle along line mm, and the two make a straight line: x=180−72=108x = 180 - 72 = 108.
  3. Check with the shortcut: x∘x^\circ is obtuse and 72∘72^\circ is acute, so they are supplementary, not equal.

Why each choice is right or wrong

  • A. Incorrect. This subtracts 72 from 90, as if the angles were complementary. Nothing in the figure is a right angle. Two angles that make a straight line add to 180.
  • B. Incorrect. This treats xx as equal to the 72∘72^\circ angle. The angle equal to 72∘72^\circ at line mm is above mm and to the right of tt. The x∘x^\circ angle is beside it, so it is the supplement.
  • C. Correct. The corresponding angle at mm, above mm and right of tt, is 72∘72^\circ, and x∘x^\circ forms a straight line with it: x=180−72=108x = 180 - 72 = 108.
  • D. Incorrect. 288 comes from subtracting 72 from 360. Two angles that make a straight line add to 180, not 360.

Worked example Medium

Lines ℓ and m cut by transversal t(3x − 32)°x°ℓmt
Lines ℓ and m cut by transversal t Two parallel lines, ℓ and m, run from upper left to lower right, with ℓ above m. A transversal t crosses both, rising to the right. Both labelled angles are between ℓ and m and on the same side of t, the side below and to the right of it. At ℓ, the angle between ℓ and t is labelled (3x − 32)°. At m, the angle between m and t is labelled x°.

In the figure, line ℓ\ell is parallel to line mm. What is the value of xx?

  1. 1616
  2. 5353Answer
  3. 9898
  4. 127127

How to solve it

  1. Both angles are between the parallel lines and on the same side of tt, so they are same-side interior angles. They add to 180∘180^\circ: (3x−32)+x=180(3x - 32) + x = 180.
  2. 4x−32=1804x - 32 = 180, so 4x=2124x = 212 and x=53x = 53.
  3. Check: the other angle is 3(53)−32=1273(53) - 32 = 127, and 127+53=180127 + 53 = 180. One angle is obtuse and the other acute, as the figure shows.

Why each choice is right or wrong

  • A. Incorrect. This sets the two angles equal: 3x−32=x3x - 32 = x gives 2x=322x = 32 and x=16x = 16. Same-side interior angles add to 180, so they are equal only when both are 90∘90^\circ. Here x=16x = 16 makes both angles 16∘16^\circ, which add to 32, not 180.
  • B. Correct. Same-side interior angles add to 180: 4x−32=1804x - 32 = 180, so x=53x = 53. The angles are 127∘127^\circ and 53∘53^\circ.
  • C. Incorrect. This uses 360 as the total: 4x−32=3604x - 32 = 360 gives 4x=3924x = 392 and x=98x = 98. Same-side interior angles add to 180.
  • D. Incorrect. 127 is the measure of the (3x−32)∘(3x - 32)^\circ angle, 3(53)−323(53) - 32. The question asks for xx.

Trap Equal or supplementary

The SAT can list the given angle when the answer is its supplement, and the supplement when the answer is the given angle. Among the eight angles a transversal makes with two parallel lines, an acute angle and an obtuse angle are never equal, so if one is acute and the other obtuse, they add to 180∘180^\circ. Same-side interior angles are the usual case: they add to 180, so they are equal only when both are 90∘90^\circ.

Trap Using the facts without parallel lines

Corresponding and alternate interior angles are equal only when the two lines are parallel. Use these facts when the question says the lines are parallel, not because two lines look parallel.

Desmos When Desmos isn’t faster

The real work is deciding whether two angles are equal or add to 180, and Desmos can't read the figure for you.

Once you have an equation, you can solve it the Lesson 1 way. For the second example, type y=(3x−32)+xy = (3x - 32) + x on one line and y=180y = 180 on the next, zoom out with the minus button at the top right of the graph until the lines cross on screen, then click the intersection, (53,180)(53, 180), so x=53x = 53. For a two-step equation like 4x−32=1804x - 32 = 180, solving by hand is just as fast.

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