Бесплатный пробный SHSAT, который ведёт себя как экзамен

Пять вопросов из банка заданий — с полным разбором

Без регистрации. Это те же вопросы, объяснения и разборы неверных ответов, которые ваш ребёнок получает на практике: каждый объясняет, почему соблазнительный ответ неверен, а не только какой верный. Попробуйте решить, прежде чем открыть ответ.

Вопросы на английском, как на настоящем SHSAT. Объяснений на русском пока нет, поэтому они показаны на английском.

1. 1 · Revising & Editing

   Which revision best combines these sentences so that the reason for the practice is clear? The team practiced every day. They wanted to win the championship.

   • A.The team practiced every day, and they wanted to win the championship.
   • B.The team practiced every day because they wanted to win the championship.
   • C.The team practiced every day, they wanted to win the championship.
   • D.The team practiced every day although they wanted to win the championship.
   Показать ответ и разбор

   Ответ: B. The team practiced every day because they wanted to win the championship.

   Read what the question asks for before comparing the choices: the revision has to make the reason for the practice clear. That rules out any version that merely places the two facts side by side, however correctly it is punctuated.

   "Because" is the word that states a reason. Putting it in front of the second sentence turns it into a subordinate clause and attaches it to the first: "The team practiced every day because they wanted to win the championship." Now the sentence says both what the team did and why, and no comma is needed, since a because-clause at the end of a sentence is usually not set off.

   Compare the near miss. Joining with "and" produces a grammatically clean sentence that reports two facts and never connects them — practicing and wanting to win simply sit next to each other. Another version drops in a comma between two complete sentences, a comma splice, and still does not state the reason, so it fails twice over. The last uses "although," which reverses the logic: it would mean the team practiced in spite of wanting to win.

   Tip: Match the connecting word to the relationship the question names — "because" for a reason, "but" or "although" for a contrast, "so" for a result. A correctly punctuated sentence that states the wrong relationship is still the wrong revision.

   Почему другие варианты неверны

   • A. This is a correctly punctuated sentence, but and only sets the two facts side by side. It never says that one caused the other, which is what the question asks for.
   • C. Two complete sentences are joined with only a comma. The relationship is not stated either, so the revision fails twice over.
   • D. Although reverses the relationship. This version says the team practiced in spite of wanting to win, which makes no sense.

2. 2 · Number Sense

   A price is increased by 20%, then decreased by 20%. What is the net percent change?

   • A.0%
   • B.4% decrease
   • C.4% increase
   • D.10% decrease
   Показать ответ и разбор

   Ответ: B. 4% decrease

   1. A 20% increase multiplies the price by $1.2$; the following 20% decrease multiplies by $0.8$. 2. Chaining them gives $1.2\times 0.8=0.96$. 3. Ending at 96% of the start means a 4% decrease.

   A faster route uses a concrete price. Starting from 100, the rise takes it to 120, and 20% of 120 is 24, so the fall takes it to 96 — a loss of 4 from 100. Choosing 100 as the starting value turns every percent question into simple arithmetic.

   Check the size before committing. The decrease is applied to a LARGER amount than the increase was, so the fall must outweigh the rise and the net change must be downward. That rules out 0% and any increase before the arithmetic is even done.

   The mistake to avoid is assuming the two changes cancel because the percents match. They do not, because they act on different bases — this asymmetry is the entire point of the question, and it is why a stock that falls 50% must rise 100% to recover.

   Tip: Percent changes multiply. Test with a starting value of 100, and remember that a rise then an equal-percent fall always lands below where it began.

   Почему другие варианты неверны

   • A. No change assumes the two moves cancel because both read 20%, but the drop acts on the raised price: $100\times 1.2=120$ and $120\times 0.8=96$.
   • C. The size is right but the direction is not: the fall acts on the larger amount, so $1.2\times 0.8=0.96$ lands below the starting price, never above it.
   • D. A 10% drop would end at $100\times 0.9=90$, but chaining the two moves gives 96, so the loss is 4 for every 100, not 10.

3. 3 · Word Problems

   A circle's radius is increased by 50%. By what percent does the area increase?

   • A.100%
   • B.50%
   • C.125%
   • D.150%
   Показать ответ и разбор

   Ответ: C. 125%

   1. Let the original radius be $r$. Increasing it by $50\mathrm{\%}$ makes it $1.5r$. 2. Area scales with the square of the radius: the new area is $\pi (1.5r{)}^2=2.25\pi r^2$. 3. That is $2.25$ times the original area — an increase of $2.25-1=1.25$ of the original. 4. So the area increases by $125\mathrm{\%}$.

   Concrete numbers make the same point if the algebra feels abstract. Take $r=10$: the area starts at $100\pi$ and becomes $\pi \times {15}^2=225\pi$. The rise is $125\pi$, and $\frac{125\pi }{100\pi }=125\mathrm{\%}$.

   The step that decides the question is the difference between the new area and the INCREASE. The new area is $225\mathrm{\%}$ of the old one; the increase is what was added, which is $125\mathrm{\%}$. Read the last few words of the question to see which is wanted.

   The trap is answering $50\mathrm{\%}$, carrying the radius's percentage straight over to the area, or $100\mathrm{\%}$ from assuming the area simply doubles.

   Tip: Percentage changes never carry across a squaring. Convert to a multiplier $(1.5)$, square it $(2.25)$, then convert back.

   Почему другие варианты неверны

   • A. A rise equal to $100\mathrm{\%}$ assumes the area simply doubles, but squaring the radius multiplier gives ${1.5}^2=2.25$ — the area grows to well over twice its original size.
   • B. Carrying the radius's percentage straight over to the area skips the squaring; with $r=10$ the area jumps from $100\pi$ to $225\pi$, far more than a half-again rise.
   • D. As a percent, the new radius is $150\mathrm{\%}$ — handing that in mistakes the radius's new size for the area's increase, which is $2.25-1=1.25$, meaning $125\mathrm{\%}$.

4. 4 · Algebra

   Which values of x satisfy −3x + 7 > 19 ?

   • A.x > −4
   • B.x < −4
   • C.x < 4
   • D.x > 4
   Показать ответ и разбор

   Ответ: B. x < −4

   1. Subtract $7$ from both sides of $-3x+7>19$: that leaves $-3x>12$. 2. 🔑 Divide by $-3$ — a NEGATIVE — so the inequality reverses: x < −4.

   The flip is the whole question. Dividing without reversing gives $x>-4$, which is offered and which is exactly the trap.

   Test values on each side. At $x=-5$, which satisfies the answer: $-3(-5)+7=15+7=22$, and $22>19$ ✓. At $x=0$, which does not: $0+7=7$, and $7>19$ is false ✓.

   Why the reversal happens is visible on a number line: $2<5$, but multiplying both by $-1$ gives $-2$ and $-5$, and now the first is the LARGER. Negating reverses every comparison, so the symbol must turn round.

   There is a route that avoids the rule. Add $3x$ to both sides for $7>3x+19$, subtract $19$ for $-12>3x$, then divide by the positive $3$ to get $-4>x$ — the same statement written the other way round.

   Tip: Flip the sign the moment you divide by a negative, then test one value from each side of the boundary to confirm.

   Почему другие варианты неверны

   • A. This forgets to flip the inequality.
   • C. Correct direction but wrong sign on the value.
   • D. Sign error on both the value and the direction.

5. 5 · Number Sense

   A backpack is priced at $80. It is discounted by 25%, and then 8% sales tax is added to the discounted price. What is the final price, in dollars? Enter your answer as a number.

   Grid-in — без вариантов ответа; ученик вводит число.

   Показать ответ и разбор

   Ответ: 64.8

   1. A 25% discount leaves 75% of the price: $80\times 0.75=60$ dollars. 2. Sales tax of 8% is then charged on that discounted 60, not on the original 80. 3. So the final price is $60\times 1.08=64.8$ dollars.

   A faster route chains both multipliers at once: $80\times 0.75\times 1.08$. Because multiplication can be done in any order, you can also read it as $80\times 0.81=64.8$. Chaining is quicker and, more importantly, it makes the ORDER of the two steps visible so you cannot accidentally tax the pre-discount price.

   Check the size before committing. The discount is much larger than the tax, so the final figure has to land well below 80 but above the 60 you paused at. 64.8 sits exactly in that window.

   The mistake to avoid is adding and subtracting the percents first. Treating this as "25% off then 8% on" as if it were 17% off gives $80\times 0.83=66.4$ — close enough to look right and still wrong, because the 8% is charged on the smaller amount.

   Tip: Successive percent changes MULTIPLY, they never add. Apply them one at a time in the order the question states them.