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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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Every Peakscor full test follows this order. In the standard tests, the second module of each section adapts to how you did on the first.

  1. Reading & Writing · Module 1

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  4. Math · Module 1

    22 questions35 min

  5. Math · Module 2

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98 questions · 2 hr 14 min of testing, plus the break

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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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Linear inequalities · Lesson 1

At least, at most: inequalities in context

Word problems with a limit, a budget or a goal become inequalities. You build them the way you build equations: a fixed amount that happens once stands alone, and a rate that comes with each unit multiplies the variable. The difference is the sign in the middle.

The words choose the sign. At least and no less than mean the amount can equal the number or go above it: ≥\ge. At most, no more than and cannot exceed mean it can equal the number or stay below it: ≤\le. More than and exceeds leave out the number itself: >>. Less than and fewer than also leave it out: <<.

Test the boundary out loud. If Marcus needs at least 600 dollars, does exactly 600 dollars count? Yes, so the sign includes equality. If a runner needs more than 26 kilometers, does exactly 26 count? No, so the sign is strict.

Many context questions ask for a whole number: the greatest number of rides, the least number of tickets. Solve the inequality, then round in the direction the story allows, not to the nearest integer. If n≤29.3n \le 29.3, the greatest whole number is 29. If n≥11.2n \ge 11.2, the least whole number is 12. When the boundary is itself a whole number, the sign decides: n≤28n \le 28 allows 28, but n>98n > 98 does not allow 98.

Words and the inequality signs they mean
Words in the questionSignIs the number itself included?
at least, no less than, a minimum of≥\geYes
at most, no more than, a maximum of, cannot exceed≤\leYes
more than, greater than, exceeds>>No
less than, fewer than<<No

Worked example Easy

A delivery van can carry at most 1,200 kilograms, counting the driver and the cargo together. The driver weighs 75 kilograms, and each box of tiles weighs 25 kilograms.

Which inequality represents the possible numbers of boxes, bb, that the van can carry on a trip with this driver?

  1. 75+25b<120075 + 25b < 1200
  2. 75+25b≤120075 + 25b \le 1200Answer
  3. 75+25b≥120075 + 25b \ge 1200
  4. 75b+25≤120075b + 25 \le 1200

How to solve it

  1. The driver is a fixed amount, 75 kilograms, counted once. Each box adds 25 kilograms, so bb boxes weigh 25b25b kilograms.
  2. At most 1,200 kilograms means the total can equal 1,200 or be less: ≤\le.
  3. So 75+25b≤120075 + 25b \le 1200. (Solving gives b≤45b \le 45, and 45 boxes make exactly 1,200 kilograms, which is allowed.)

Why each choice is right or wrong

  • A. Incorrect. At most includes the limit itself. With <<, a load of exactly 1,200 kilograms, which is 45 boxes, would be ruled out even though the van can carry it.
  • B. Correct. The driver's 75 kilograms plus 25 kilograms for each box can be at most 1,200 kilograms.
  • C. Incorrect. This reads at most as at least. It says the load must be 1,200 kilograms or more, the opposite of a limit.
  • D. Incorrect. This swaps the roles. It counts the driver's weight once per box and a single box's weight only once.

Worked example Medium

At an amusement park, Ana pays an entry fee of 18 dollars and then 4.50 dollars for each ride. She can spend at most 150 dollars in total.

What is the greatest number of rides Ana can take?

  1. 2929Answer
  2. 3030
  3. 3333
  4. 3737

How to solve it

  1. Let nn be the number of rides. The fee is paid once and each ride costs 4.50 dollars: 18+4.5n≤15018 + 4.5n \le 150.
  2. Subtract 18: 4.5n≤1324.5n \le 132. Divide by 4.5: n≤29.33…n \le 29.33\ldots
  3. Rides come in whole numbers, and nn can't go above 29.33, so round down: 29.
  4. Check: 29 rides cost 18+130.5=148.518 + 130.5 = 148.5 dollars, which is within the budget. 30 rides would cost 153 dollars.

Why each choice is right or wrong

  • A. Correct. 18+4.5n≤15018 + 4.5n \le 150 gives n≤29.33…n \le 29.33\ldots, and the greatest whole number that fits is 29, a total of 148.50 dollars.
  • B. Incorrect. This rounds 29.33 up. 30 rides cost 18+4.5(30)=15318 + 4.5(30) = 153 dollars, which is over the 150 dollar limit.
  • C. Incorrect. This ignores the entry fee: 150÷4.5=33.3…150 \div 4.5 = 33.3\ldots, rounded down to 33. The 18 dollar fee must come out of the budget first.
  • D. Incorrect. This adds the fee to the budget: (150+18)÷4.5=37.3…(150 + 18) \div 4.5 = 37.3\ldots, rounded down to 37. The fee is part of the 150 dollars, so subtract it.

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