Chapter 8 — Student-Produced Responses

Answer: 7/2 or 3.5

Explanation

$$x = \frac{7}{2}$$

Now the part this question is really about. $\frac72$ is $3\frac12\mathclose{,}$ and you must not enter it that way. 3 1/2 is read as $\frac{31}{2}\mathclose{,}$ which is 15.5, and scores zero.

Two acceptable entries:

• 7/2 — the improper fraction, 3 characters
• 3.5 — the decimal, 3 characters

Both fit and both are accepted. The fraction is the safer habit because it is exact.

Takeaway: Never enter a mixed number. Improper fraction or decimal — those are the only two forms the grid understands.

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Answer: 1/3 or .3333 or 0.333

Explanation

$$x = \frac{1}{3} = 0.3333\ldots$$

The decimal never ends, so it cannot be entered in full. Three valid entries:

• 1/3 — exact, 3 characters. This is the best answer.
• .3333 — 5 characters, cut off at the fourth digit
• 0.333 — 5 characters too: with a leading zero the fourth digit comes one place sooner

What would not work is .333 or 0.33, which stop a digit early, or 0.3333333, which is too long for the grid. A decimal that does not end has to fill the grid — the official keys list .8824 and .8823 for $\frac{15}{17}\mathclose{,}$ never .882.

Notice that the exact fraction sidesteps the entire question of how many digits to give. That is why "enter the fraction when it fits" is the rule to build a habit around.

Takeaway: When a decimal repeats, enter the fraction. If you must use a decimal, fill the grid: four digits, truncated or rounded — both are accepted, and a shorter decimal is not.

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Answer: -3

Explanation

$$4x = 7 - 19 = -12 \quad\Longrightarrow\quad x = -3$$

Check: $4(-3) + 19 = -12 + 19 = 7$ ✓

Negative answers are perfectly acceptable. Enter -3 — two characters, and the grid allows six for a negative answer, so there is plenty of room.

Do not be tempted to drop the minus sign because the answer "looks wrong". Of the 112 grid-in answers across the eight official tests examined, four were negative. They are uncommon but they are real.

Takeaway: Enter the minus sign. It counts as one of your six characters and the answer is wrong without it.

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Answer: 9 or -3

Explanation

Factor: two numbers multiplying to $-27$ and adding to $-6$ are $-9$ and $3\mathclose{.}$

$$(x - 9)(x + 3) = 0 \quad\Longrightarrow\quad x = 9 \text{ or } x = -3$$

Both are correct. The question says "one possible value", and the grid takes one answer, so enter either 9 or -3.

This is a real pattern in the official keys — one of them literally lists 15; -5 as the accepted answers, meaning both were valid. Do not waste time deciding which one the test "wants". Enter the first one you found.

Watch the wording, though. If a question instead says "what is the positive solution", or "the least value of $x$", then only one answer scores.

Takeaway: "One possible value" means either answer scores. Enter whichever you found first and move on — but re-read the question in case it restricted you to one of them.

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Section 2 — Checking Without Options

With no options to test, the check has to come from you. Three methods, in order of how often they apply.

Substitute back into the original. The most reliable. Put your answer into the question's own equation and confirm both sides match. This catches sign errors, which are the most common mistake and the hardest to spot by re-reading.

Check the size against the situation. A word problem's answer has to make sense as a real quantity. A negative number of people, 2.6 text messages, a ladder shorter than its distance from the wall — each of these should stop you.

Re-read the final sentence. Grid-ins ask for a specific thing, and the most expensive error on the whole test is solving correctly and reporting the wrong quantity. You found $x\mathclose{;}$ did it want $x\mathclose{,}$ or $2x\mathclose{,}$ or $y\mathclose{?}$

Budget about ten seconds for the check. On a question you have spent ninety seconds on, that is a good trade.

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Answer: 7

Explanation

Multiply both sides by 2 first, to clear the fraction:

$$3x - 5 = 16$$

Then:

$$3x = 21 \quad\Longrightarrow\quad x = 7$$

Now check, which is the whole point of this question:

$$\frac{3(7) - 5}{2} = \frac{21 - 5}{2} = \frac{16}{2} = 8 \; ✓$$

The common wrong answer here is $x = \frac{26}{3}\mathclose{,}$ from adding 5 to 8 before multiplying by 2 — that is, treating the fraction bar as though it only covered the $3x\mathclose{.}$ Substituting catches it instantly: $\frac{26 - 5}{2} = 10.5\mathclose{,}$ not 8.

Takeaway: The fraction bar groups everything above it. Multiply the whole side by the denominator first, then substitute your answer back.

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Answer: 24

Explanation

Substitute $y = 5\mathclose{:}$

$$2x + 5 = 17 \quad\Longrightarrow\quad 2x = 12 \quad\Longrightarrow\quad x = 6$$

But the question asks for $4x\mathclose{,}$ not $x\mathclose{:}$

$$4x = 24$$

There is a faster route. You reached $2x = 12$ along the way, and $4x$ is simply double that: $4x = 24\mathclose{.}$ You never needed $x$ at all.

On a multiple-choice question, 6 and 12 would both be offered. Here there are no options to warn you — which is exactly why re-reading the last line matters more on grid-ins than anywhere else.

Takeaway: Read the final sentence again before entering. Then look for whether the quantity wanted is a simple multiple of something you already have.

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Answer: 81

Explanation

A 35% profit means the selling price is 135% of the cost:

$$60 \times 1.35 = 81$$

Check the size, which is the habit this question is for. The shop makes a profit, so the selling price must be more than $60. It is ✓

Two wrong answers this eliminates immediately:

• 21 — the profit itself, not the selling price. Less than 60, so it fails the check.
• 39 — from $60 \times 0.65\mathclose{,}$ taking 35% off. Also less than 60.

Neither survives a single glance at whether the number should be bigger or smaller than 60.

Enter 81. Not $81 — the dollar sign is not allowed.

Takeaway: Before entering, ask whether the answer should be bigger or smaller than the number you started from. It is the fastest check there is, and it needs no arithmetic.

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Answer: 45/8 or 5.625

Explanation

$$\text{length} = \frac{\text{area}}{\text{width}} = \frac{45}{8} = 5.625$$

Both entries work here, and it is worth seeing why:

• 45/8 — 4 characters, exact, fits comfortably
• 5.625 — 5 characters, exactly at the limit, and exact as well

This decimal happens to terminate, so nothing is lost either way. Compare with the official key that accepted 361/8, 45.12 or 45.13: there the decimal is $45.125\mathclose{,}$ which is 6 characters and does not fit, so it had to be cut to four digits — and both cutting and rounding were accepted.

Takeaway: Divide, then check the character count. A fraction that fits is always safe; a decimal is only safe when it fits and terminates within the grid.

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Section 3 — Mixed Practice

Every question here is a grid-in, as they would appear at positions 6, 7, 13, 14, 20, 21 and 27 of a paper practice module. Work each one on paper, then check it before entering.

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Answer: 5

Explanation

The first equation gives $y = 8 - x\mathclose{.}$ Substitute it into the second:

$$5x + 2(8 - x) = 31 \quad\Longrightarrow\quad 3x + 16 = 31 \quad\Longrightarrow\quad 3x = 15 \quad\Longrightarrow\quad x = 5$$

Then $y = 8 - 5 = 3\mathclose{.}$ Check both: $5 + 3 = 8$ ✓ and $25 + 6 = 31$ ✓

Two numbers come out of a system and the grid takes one. The question asked for $x\mathclose{,}$ so enter 5; entering the 3 would score nothing, and there are no options here to warn you.

Takeaway: Solve the system, check both equations, then read which variable was asked for before entering anything.

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Answer: 36

Explanation

$$0.24 \times 150 = 36$$

Check the size: 24% is about a quarter, and a quarter of 150 is about 37 ✓

Enter 36. Not 36% — the question asked how many, and percent signs are not accepted in the grid regardless.

Takeaway: "What number is $r$% of $b$" means multiply. Estimate first with a friendly percentage to confirm the size.

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Answer: 5

Explanation

$$x = -\frac{b}{2a} = -\frac{-10}{2} = 5$$

Check either side: $f(4) = 16 - 40 + 3 = -21\mathclose{,}$ $f(5) = 25 - 50 + 3 = -22\mathclose{,}$ $f(6) = 36 - 60 + 3 = -21\mathclose{.}$ Lowest at 5 ✓

The question asked where the minimum occurs, so the answer is 5, not $-22\mathclose{.}$

Takeaway: $x = -\frac{b}{2a}\mathclose{,}$ and read whether the question wants where the minimum is or what it is.

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Answer: 15/2 or 7.5

Explanation

$$\frac23 x = 5 \quad\Longrightarrow\quad x = 5 \times \frac32 = \frac{15}{2}$$

Check: $\frac23 \times \frac{15}{2} = 5\mathclose{,}$ and $5 + 4 = 9$ ✓

Enter 15/2 or 7.5. Not 7 1/2.

Takeaway: To undo a fraction, multiply by its reciprocal. Then choose a legal form — improper fraction or decimal.

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Answer: 24

Explanation

The hypotenuse is given, so subtract:

$$a^2 = 26^2 - 10^2 = 676 - 100 = 576 \quad\Longrightarrow\quad a = 24$$

Check: $100 + 576 = 676$ ✓ This is the 5–12–13 triangle doubled.

Check the size too: a leg must be shorter than the hypotenuse, and $24 < 26$ ✓

Takeaway: Given the hypotenuse, subtract. Given two legs, add. Then confirm your leg is shorter than the hypotenuse.

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Answer: 6750

Explanation

Count the periods rather than the hours:

$$\frac{6}{2} = 3 \text{ periods}$$

Each triples, so the factor is $3^3 = 27\mathclose{:}$

$$250 \times 27 = 6750$$

Check by stepping: $250 \to 750$ (2 hrs) $\to 2250$ (4 hrs) $\to 6750$ (6 hrs) ✓

Takeaway: Divide the time by the period length to count the periods. The exponent counts periods, never hours.

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Answer: 8

Explanation

Split into two cases:

$$2x - 5 = 11 \Longrightarrow x = 8 \qquad 2x - 5 = -11 \Longrightarrow x = -3$$

Both are genuine solutions. But the question says positive, so only $x = 8$ scores.

Contrast this with Q4, where "one possible value" made either answer acceptable. The difference is one word, and it is worth reading twice — an absolute-value grid-in almost always has two solutions, and the question decides which of them you may enter.

Takeaway: Solve both cases, then let the wording pick. "One possible value" takes either; "positive", "least" or "greatest" takes exactly one.

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