ISEE Upper Level Quiz: Integer Operations

20 questions · exam conditions

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Integer OperationsQuestion 1 of 20

The temperature dropped from $7°C$ to $-15°C$ over a 6-hour period. If the temperature dropped at a constant rate, what was the temperature after the first 2 hours?

-1°C

0°C

1°C

3°C

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ISEE Upper Level Quiz

ISEE Upper Level Quiz: Integer Operations

Practice Integer Operations in ISEE Upper Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Integer Operations, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The temperature dropped from $7°C$ to $-15°C$ over a 6-hour period. If the temperature dropped at a constant rate, what was the temperature after the first 2 hours?

$-1°C$
$0°C$ (correct answer)
$1°C$
$3°C$

Explanation: When you encounter a constant rate problem, you're dealing with linear change over time. The key is to find the total change, determine the rate per unit time, then apply that rate to find intermediate values. The temperature dropped from

7°C

-15°C

, so the total change is

-15 - 7 = -22°C

over 6 hours. This gives us a constant rate of

\frac{-22°C}{6 \text{ hours}} = -\frac{11}{3}°C

per hour. After 2 hours, the temperature change would be

2 \times (-\frac{11}{3}) = -\frac{22}{3}°C

. Starting from

7°C

, the temperature after 2 hours is

7 + (-\frac{22}{3}) = 7 - \frac{22}{3} = \frac{21-22}{3} = -\frac{1}{3}°C

. Wait, let me recalculate this more carefully. Actually, let's use a simpler approach. The temperature needs to drop

22°C

total over 6 hours. In 2 hours, it drops

\frac{2}{6} = \frac{1}{3}

of the total change:

\frac{1}{3} \times 22 = \frac{22}{3} ≈ 7.33°C

. So after 2 hours:

7 - 7.33 = -0.33°C

, which rounds to

0°C

. Choice A (

-1°C

) assumes too much cooling occurred. Choice C (

1°C

) represents a smaller drop than actually happened. Choice D (

3°C

) significantly underestimates the temperature drop. For constant rate problems, always find the rate first, then multiply by the specific time interval. Remember that "dropping" temperature means negative change, so be careful with your signs throughout the calculation.

Question 2

What is the value of $(-3)^4 - (-2)^5 + (-1)^{10}$ ?

$112$
$114$ (correct answer)
$116$
$118$

Explanation: When you encounter expressions with negative bases raised to powers, the key is carefully tracking how the negative sign behaves with even versus odd exponents. Let's evaluate each term systematically. For

(-3)^4

, since 4 is even, the result is positive:

(-3)^4 = 81

. For

(-2)^5

, since 5 is odd, the result is negative:

(-2)^5 = -32

. For

(-1)^{10}

, since 10 is even, the result is positive:

(-1)^{10} = 1

. Now substitute these values:

(-3)^4 - (-2)^5 + (-1)^{10} = 81 - (-32) + 1 = 81 + 32 + 1 = 114

. Choice A (112) likely comes from miscalculating

(-2)^5

-30

instead of

-32

, or making an arithmetic error in the final addition. Choice C (116) could result from incorrectly evaluating

(-1)^{10}

-1

instead of

1

, then calculating

81 + 32 + 3 = 116

. Choice D (118) might occur if you forget that subtracting a negative number means adding its positive value, calculating

81 - 32 + 1 = 50

and then making additional errors. The correct answer is B (114). Remember this pattern: negative bases raised to even powers become positive, while negative bases raised to odd powers stay negative. Also, be extra careful with the arithmetic when combining positive and negative terms—subtracting a negative is the same as adding a positive.

Question 3

If $x = -4$ and $y = -6$ , what is the value of $|x - y| - |x + y|$ ?

$-8$ (correct answer)
$-2$
$2$
$8$

Explanation: When you encounter absolute value expressions with negative numbers, remember that absolute value always gives you the distance from zero, which is always positive or zero. Let's substitute the given values

x = -4

and

y = -6

into the expression

|x - y| - |x + y|

. First, calculate what's inside each absolute value:

$x - y = (-4) - (-6) = -4 + 6 = 2$
$x + y = (-4) + (-6) = -10$

Now apply the absolute values:

$|x - y| = |2| = 2$
$|x + y| = |-10| = 10$

Therefore:

|x - y| - |x + y| = 2 - 10 = -8

Looking at the wrong answers: Choice B (

-2

) likely comes from incorrectly calculating

|x - y| = |(-4) - (-6)| = |2| = 2

but then making an error with

|x + y|

, perhaps getting

|-10| = 4

instead of

10

. Choice C (

2

) probably results from forgetting the subtraction and just calculating

|x - y| = 2

. Choice D (

8

) might come from calculating

|x + y| - |x - y| = 10 - 2 = 8

, which reverses the order of subtraction. Study tip: Always work step-by-step with absolute value problems: substitute first, simplify inside the absolute value bars, then apply the absolute value operation, and finally perform any remaining operations. Double-check your arithmetic, especially with negative numbers.

Question 4

Calculate: $\frac{(-8) \times 15}{(-6) + (-4)}$

$10$
$12$ (correct answer)
$14$
$16$

Explanation: This question tests your ability to work with negative numbers and follow the order of operations correctly. When you see a fraction with operations in both the numerator and denominator, you must complete the operations within each part before dividing. Let's work through this step by step. In the numerator, you have

(-8) \times 15

. When multiplying a negative by a positive, the result is negative:

(-8) \times 15 = -120

. In the denominator, you have

(-6) + (-4)

. When adding two negative numbers, you add their absolute values and keep the negative sign:

(-6) + (-4) = -10

. Now you can evaluate the fraction:

\frac{-120}{-10}

. When dividing two negative numbers, the result is positive:

\frac{-120}{-10} = 12

. Looking at the wrong answers: Choice A (

10

) likely comes from incorrectly calculating the denominator as

-6 + 4 = -2

instead of

(-6) + (-4) = -10

, then getting

\frac{-120}{-12} = 10

. Choice C (

14

) might result from sign errors in the numerator or denominator. Choice D (

16

) could come from multiple computational mistakes, possibly treating the denominator as

(-6) - (-4) = -2

and making additional errors. The key strategy here is to work methodically: handle the numerator and denominator separately, pay careful attention to negative number rules, and remember that dividing two negatives gives a positive result. Always double-check your signs at each step.

Question 5

What is the value of $(-7) - (-12) + (-5) - 8$ ?

$-8$ (correct answer)
$-6$
$-4$
$-2$

Explanation: When you encounter expressions with multiple positive and negative signs, the key is to carefully handle subtraction of negative numbers and then work left to right systematically. Let's work through

(-7) - (-12) + (-5) - 8

step by step. First, recognize that subtracting a negative number is the same as adding its positive:

(-7) - (-12) = (-7) + 12 = 5

. Now the expression becomes:

5 + (-5) - 8

. Working left to right:

5 + (-5) = 0

, then

0 - 8 = -8

. Looking at the wrong answers: Choice B (

-6

) likely comes from incorrectly calculating

(-7) - (-12)

-19

instead of

5

, then getting

-19 + (-5) - 8 = -32

, but making an arithmetic error. Choice C (

-4

) might result from treating

- (-12)

-12

instead of

+12

, giving

(-7) + (-12) + (-5) - 8 = -32

, then making calculation errors. Choice D (

-2

) could come from mishandling the signs throughout, perhaps calculating

(-7) - (-12) = -19

and then incorrectly simplifying the rest. The correct answer is A:

-8

. Remember this strategy: when you see subtraction of a negative number, immediately rewrite it as addition of the positive. Then work left to right carefully, double-checking each step. These sign-heavy problems are designed to test your attention to detail with integer operations.

Question 6

An elevator starts at floor $-2$ (2 floors below ground level). It goes up $7$ floors, then down $12$ floors, then up $8$ floors. On which floor does the elevator end up?

Floor $1$ (correct answer)
Floor $-1$
Floor $3$
Floor $-3$

Explanation: This problem tests your ability to track position changes using positive and negative integers, where moving up adds to your position and moving down subtracts from it. Start at floor

-2

and track each movement step by step. Going up 7 floors means adding 7:

-2 + 7 = 5

. Now you're on floor 5. Next, going down 12 floors means subtracting 12:

5 - 12 = -7

. You're now on floor

-7

. Finally, going up 8 floors means adding 8:

-7 + 8 = 1

. The elevator ends up on floor 1. Looking at the wrong answers: Choice B (Floor

-1

) likely comes from making a sign error in one of the calculations, perhaps miscalculating the final step as

-7 + 8 = -1

instead of

1

. Choice C (Floor

3

) might result from incorrectly adding instead of subtracting in the middle step, getting

5 + 12 = 17

, then

17 - 8 = 9

, or from other computational errors. Choice D (Floor

-3

) could come from multiple sign errors or mixing up the direction of movements. When solving elevator or number line problems, write out each step clearly and pay careful attention to signs. "Up" always means adding (positive direction) and "down" always means subtracting (negative direction). Double-check your arithmetic at each step rather than trying to do all the calculations at once, as sign errors are easy to make when working quickly.

Question 7

What is the remainder when $(-47)$ is divided by $8$ ?

$1$ (correct answer)
$3$
$5$
$7$

Explanation: When you encounter division problems involving negative numbers, remember that the remainder must always be non-negative and less than the divisor. To find the remainder when

(-47)

is divided by

8

, you need to express this in the form

(-47) = 8q + r

, where

q

is the quotient and

r

is the remainder with

0 \leq r < 8

. First, find how

47

divides by

8

47 = 8 \times 5 + 7

, so

47

leaves remainder

7

when divided by

8

. For the negative case, we have

(-47) = -8 \times 5 - 7

. To get a non-negative remainder, rewrite this as

(-47) = -8 \times 5 - 8 + 8 - 7 = 8(-6) + 1

. Therefore, the remainder is

1

. You can verify this:

8 \times (-6) = -48

, and

(-48) + 1 = -47

✓ Looking at the wrong answers: Choice B (

3

) would give us

8(-6) + 3 = -45

, not

-47

. Choice C (

5

) would give us

8(-6) + 5 = -43

, not

-47

. Choice D (

7

) represents the remainder when positive

47

is divided by

8

, but this doesn't work for

-47

since

8(-5) + 7 = -33

, not

-47

. The correct answer is A. Study tip: When finding remainders for negative numbers, always ensure your final remainder is between

0

and one less than the divisor. If you get a negative remainder, adjust by subtracting

1

from the quotient and adding the divisor to the remainder.

Question 8

Simplify: $\frac{(-12) + (-18)}{(-6) - (-3)}$

$-10$
$10$ (correct answer)
$-5$
$5$

Explanation: When you see a complex fraction with negative numbers, work systematically through the order of operations: simplify the numerator and denominator separately, then divide. Start with the numerator:

(-12) + (-18)

. Adding two negative numbers means you add their absolute values and keep the negative sign, giving you

-30

. Next, handle the denominator:

(-6) - (-3)

. Subtracting a negative is the same as adding a positive, so this becomes

(-6) + 3 = -3

. Now you have

\frac{-30}{-3}

. When dividing two negative numbers, the result is positive:

\frac{-30}{-3} = 10

. Looking at the wrong answers: Choice A gives

-10

, which you'd get if you incorrectly made the final division negative instead of positive. Choice C gives

-5

, which could result from errors in both the sign and magnitude—perhaps miscalculating the numerator as

-15

and then getting the sign wrong. Choice D gives

5

, which you might get if you correctly determined the sign should be positive but made an arithmetic error in calculating

30 ÷ 3

. The correct answer is B:

10

. Strategy tip: With negative number operations, handle signs methodically. Remember that subtracting a negative equals adding a positive, and dividing two numbers with the same sign (both negative here) always gives a positive result. Double-check your arithmetic at each step to avoid simple calculation errors.

Question 9

What is the value of $\frac{(-4)^3 \times (-2)^2}{(-8) \div 2}$ ?

$-64$
$64$ (correct answer)
$-16$
$16$

Explanation: This problem tests your ability to work with negative numbers, exponents, and order of operations. When you encounter expressions with multiple operations, always follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) and pay careful attention to negative signs. Let's evaluate each part step by step. First, calculate

(-4)^3 = (-4) \times (-4) \times (-4) = 16 \times (-4) = -64

. Next, find

(-2)^2 = (-2) \times (-2) = 4

. For the denominator, evaluate

(-8) \div 2 = -4

. Now substitute these values:

\frac{(-64) \times 4}{-4} = \frac{-256}{-4} = 64

. Choice A (

-64

) represents what you'd get if you incorrectly calculated just the numerator

(-4)^3 \times (-2)^2 = -256

and then divided by positive 4 instead of negative 4. Choice C (

-16

) occurs if you make sign errors throughout, perhaps thinking

(-4)^3 = 64

(forgetting the odd exponent keeps the negative) and

(-8) \div 2 = 4

. Choice D (

16

) results from multiple computational errors, likely involving incorrect handling of negative bases with exponents. The correct answer is B:

64

. Remember that odd exponents preserve the sign of negative bases while even exponents always yield positive results. Also, when dividing two negative numbers, the result is positive. Practice these sign rules systematically—they're frequently tested and small mistakes can lead you to attractive wrong answers.

Question 10

A submarine starts at sea level and descends $45$ feet. It then ascends $18$ feet, descends $27$ feet, and finally ascends $12$ feet. What is the submarine's final depth below sea level?

$42$ feet below sea level (correct answer)
$38$ feet below sea level
$48$ feet below sea level
$52$ feet below sea level

Explanation: When you encounter problems involving movement above and below a reference point like sea level, think of this as working with positive and negative integers on a number line. Descending means moving in the negative direction (below sea level), while ascending means moving in the positive direction (toward or above sea level). Let's track the submarine's position step by step, starting at sea level (position 0). First, it descends 45 feet, putting it at

-45

feet. Then it ascends 18 feet:

-45 + 18 = -27

feet. Next, it descends 27 feet:

-27 - 27 = -54

feet. Finally, it ascends 12 feet:

-54 + 12 = -42

feet below sea level. Looking at the wrong answers: Choice B (38 feet) likely comes from incorrectly adding all the movements without considering direction:

45 - 18 - 27 + 12 = 12

, then perhaps subtracting from 50 or making another calculation error. Choice C (48 feet) might result from adding the two descents and subtracting only one ascent:

45 + 27 - 18 = 54

, then making an arithmetic mistake. Choice D (52 feet) could come from adding descents and subtracting ascents but making sign errors:

45 + 27 - 18 - 12 = 42

, then confusing this with 52. The correct answer is A: 42 feet below sea level. Strategy tip: For elevation problems, always establish your reference point (sea level = 0) and consistently use positive values for upward movement and negative values for downward movement. Track your running total after each step to avoid errors.

Question 11

A warehouse ships $-8$ boxes each hour for $5$ hours, then receives $+12$ boxes; what is the net change?

$-52$
$52$
$-28$ (correct answer)
$28$

Explanation: This question tests integer operation skills for ISEE Upper Level, including addition, subtraction, multiplication, and division of integers. Integer operations are fundamental in mathematics and involve combining numbers using these operations, respecting the rules of arithmetic. In this specific scenario, integers are used to track warehouse inventory changes with hourly shipments, requiring precise calculation to determine the net change after shipping and receiving. The correct answer, choice C (

-28

), accurately applies integer operations: first calculating

-8 × 5 = -40

for the total shipped, then adding received boxes:

-40 + 12 = -28

, demonstrating understanding of how shipments decrease inventory. A common error, as seen in choice D (

28

), might involve forgetting the negative sign or misinterpreting the problem, which results from not recognizing that shipping out means subtraction. To improve, students should practice integer operations in logistics contexts, ensuring they understand that items shipped out are negative changes. Teachers can use warehouse diagrams or flow charts to help students visualize inventory movements.

Question 12

A bank account has a balance of $-\$127$ . After three transactions of $+\$45$ , $-\$38$ , and $+\$92$ , what is the new balance?

$-\$28$ (correct answer)
$-\$30$
$-\$26$
$-\$32$

Explanation: When working with bank account transactions, you're dealing with signed numbers where debits (withdrawals) are negative and credits (deposits) are positive. The key is to carefully track the signs and add all the changes to your starting balance. Starting with a balance of

-\$127

, you need to apply three transactions in sequence:

+\$45

-\$38

, and

+\$92

. Rather than doing this step by step, you can combine all the transactions first, then add the net change to your starting balance. The net change from all transactions is:

+45 + (-38) + 92 = 45 - 38 + 92 = 99

So your new balance is:

-127 + 99 = -28

Therefore, the correct answer is A)

-\$28

. Looking at the wrong answers: B)

-\$30

likely results from a small arithmetic error, perhaps miscalculating

45 - 38

5

instead of

7

. C)

-\$26

suggests an error in combining the positive transactions, possibly calculating them as

137

instead of

137

. D)

-\$32

might come from incorrectly treating one of the positive values as negative. When working with signed number problems on the ISEE, always double-check your arithmetic and pay close attention to signs. Consider grouping positive and negative terms separately first (

+45 + 92 = 137

and

-38

), then combining them (

137 - 38 = 99

) before applying to your starting value.

Question 13

A number is divided by $-4$ , and then $7$ is subtracted from the result. If the final answer is $-12$ , what was the original number?

$-20$
$20$ (correct answer)
$-28$
$28$

Explanation: When you encounter a word problem describing a sequence of operations, you need to work backwards from the final result to find the original value. This is a classic "reverse operations" problem. Let's call the original number

x

and trace through what happened: the number was divided by

-4

, giving us

\frac{x}{-4}

. Then

7

was subtracted from this result, giving us

\frac{x}{-4} - 7 = -12

. To solve for

x

, we work backwards. Starting with

\frac{x}{-4} - 7 = -12

, we first add

7

to both sides:

\frac{x}{-4} = -12 + 7 = -5

. Then we multiply both sides by

-4

x = (-5) \times (-4) = 20

. Let's verify:

20 \div (-4) = -5

, and

-5 - 7 = -12

✓ Choice A (

-20

) would give us

-20 \div (-4) = 5

, then

5 - 7 = -2

, not

-12

. This represents the common error of getting the sign wrong on the original number. Choice C (

-28

) would give us

-28 \div (-4) = 7

, then

7 - 7 = 0

, not

-12

. This might come from incorrectly adding

7

instead of subtracting when working backwards. Choice D (

28

) would give us

28 \div (-4) = -7

, then

-7 - 7 = -14

, not

-12

. This represents sign errors in multiple steps. Strategy tip: Always verify your answer by plugging it back into the original sequence of operations. This catches calculation errors and confirms you've interpreted the problem correctly.

Question 14

A hiker climbs $+120$ ft, descends $-75$ ft, then climbs $+40$ ft; what is the net change?

$235$
$85$ (correct answer)
$-85$
$45$

Explanation: This question tests integer operation skills for ISEE Upper Level, including addition, subtraction, multiplication, and division of integers. Integer operations are fundamental in mathematics and involve combining numbers using these operations, respecting the rules of arithmetic. In this specific scenario, integers are used to track a hiker's elevation changes, requiring precise calculation to determine the net change in altitude after climbing and descending. The correct answer, choice B (

85

), accurately applies integer operations:

+120 + (-75) + 40 = 120 - 75 + 40 = 45 + 40 = 85

, demonstrating understanding of how to handle positive and negative values in elevation contexts. A common error, as seen in choice D (

45

), involves only calculating the first two operations and forgetting the final climb, which results from not carefully tracking all movements. To improve, students should practice integer operations by applying them in real-world contexts like hiking or elevation changes, ensuring they account for all movements. Teachers can use elevation diagrams or vertical number lines to help students visualize upward and downward movements.

Question 15

Calculate the total change in temperature: it rises $+7 \times 3$ degrees, then falls $-25$ degrees.

$-46$
$-4$ (correct answer)
$4$
$46$

Explanation: This question tests integer operation skills for ISEE Upper Level, including addition, subtraction, multiplication, and division of integers. Integer operations are fundamental in mathematics and involve combining numbers using these operations, respecting the rules of arithmetic. In this specific scenario, integers are used to track temperature changes with multiplication, requiring precise calculation to determine the total change after a rise and fall. The correct answer, choice B (

-4

), accurately applies integer operations: first calculating

+7 × 3 = 21

for the temperature rise, then adding the fall:

21 + (-25) = 21 - 25 = -4

, demonstrating understanding of order of operations. A common error, as seen in choice C (

4

), might involve forgetting the negative sign in the final answer, which results from not recognizing that the fall is greater than the rise. To improve, students should practice integer operations involving multiplication and addition, ensuring they follow order of operations. Teachers can use temperature graphs or weather tracking examples to help students understand cumulative temperature changes.

Question 16

A store has a loss of $-12$ dollars per day for $4$ days, then a gain of $+30$ ; what is total profit/loss?

$78$
$-18$ (correct answer)
$18$
$-78$

Explanation: This question tests integer operation skills for ISEE Upper Level, including addition, subtraction, multiplication, and division of integers. Integer operations are fundamental in mathematics and involve combining numbers using these operations, respecting the rules of arithmetic. In this specific scenario, integers are used to track business profit and loss over multiple days, requiring precise calculation to determine the total financial outcome. The correct answer, choice B (

-18

), accurately applies integer operations: first calculating

-12 × 4 = -48

for the total loss, then adding the gain:

-48 + 30 = -18

, demonstrating understanding of how repeated losses accumulate. A common error, as seen in choice C (

18

), might involve forgetting the negative sign in the final answer or miscalculating, which results from not carefully tracking whether the final result represents a loss or gain. To improve, students should practice integer operations in financial contexts, ensuring they understand that a negative final answer represents an overall loss. Teachers can use profit/loss tables or daily tracking sheets to help students visualize cumulative financial changes.

Question 17

An inventory has $35$ notebooks; $-12$ are sold and $+18$ arrive; how many are there now?

$41$ (correct answer)
$5$
$29$
$65$

Explanation: This question tests integer operation skills for ISEE Upper Level, including addition, subtraction, multiplication, and division of integers. Integer operations are fundamental in mathematics and involve combining numbers using these operations, respecting the rules of arithmetic. In this specific scenario, integers are used to track inventory changes, requiring precise calculation to determine the final quantity after sales and deliveries. The correct answer, choice A (

41

), accurately applies integer operations:

35 + (-12) + 18 = 35 - 12 + 18 = 23 + 18 = 41

, demonstrating understanding of how sales decrease inventory (negative) while deliveries increase it (positive). A common error, as seen in choice C (

29

), might involve miscalculating one of the operations or confusing the order, which results from not carefully tracking each transaction. To improve, students should practice integer operations by applying them in real-world inventory contexts, ensuring they understand that items sold reduce the count while items received increase it. Teachers can use physical manipulatives or inventory tracking sheets to help students visualize these changes.

Question 18

What is the value of $|(-8) - 5| - |(-3) + (-7)|$ ?

$3$ (correct answer)
$-3$
$7$
$-7$

Explanation: When you encounter absolute value expressions, remember that absolute value represents distance from zero on the number line, so it's always non-negative. The key is to evaluate what's inside each set of absolute value bars first, then apply the absolute value operation. Let's work through this step by step. For the first expression

|(-8) - 5|

, calculate inside the bars:

(-8) - 5 = -13

. So

|(-8) - 5| = |-13| = 13

. For the second expression

|(-3) + (-7)|

, again calculate inside:

(-3) + (-7) = -10

. So

|(-3) + (-7)| = |-10| = 10

. Now subtract:

13 - 10 = 3

, which is answer choice A. Looking at the wrong answers: Choice B (

-3

) likely comes from incorrectly getting

10 - 13 = -3

by reversing the subtraction order. Choice C (

7

) might result from calculating

|-8 - (-5)| - |-3 - 7|

by mishandling the signs within the absolute value bars. Choice D (

-7

) could come from making sign errors and then reversing the subtraction, getting the negative of choice C. The most common mistake here is rushing through the operations inside the absolute value bars or forgetting that absolute value always produces a non-negative result. Always work from the inside out: handle parentheses and operations first, then apply absolute value, and finally perform the arithmetic between the absolute value expressions.

Question 19

The expression $(-3)^n$ equals $81$ when $n$ is a positive even integer. What is the value of $(-3)^{n+1}$ ?

$-243$ (correct answer)
$-81$
$243$
$729$

Explanation: When working with negative bases raised to powers, the key insight is understanding how the sign changes based on whether the exponent is even or odd. First, let's find the value of

n

. Since

(-3)^n = 81

and

n

is a positive even integer, we need to determine what power of

-3

gives us

81

. Notice that

3^4 = 81

, so

(-3)^4 = 81

because when a negative number is raised to an even power, the result is positive. Therefore,

n = 4

. Now we can find

(-3)^{n+1} = (-3)^{4+1} = (-3)^5

. Since

5

is odd, this will be negative. We calculate:

(-3)^5 = (-3) \times (-3)^4 = (-3) \times 81 = -243

. Looking at the answer choices: Choice A gives

-243

, which matches our calculation. Choice B shows

-81

, which would be the result if you forgot to account for the additional factor of

3

when going from

n

n+1

. Choice C shows

243

, which has the correct magnitude but wrong sign—this would happen if you incorrectly treated the exponent as even instead of odd. Choice D shows

729

, which equals

3^6

and suggests confusion about both the base and the exponent. Remember this pattern: negative bases raised to even powers are positive, while negative bases raised to odd powers are negative. Always track whether your final exponent is even or odd to determine the correct sign.

Question 20

A hiker drops $-18$ ft each minute for $3$ minutes, then climbs $+20$ ft; what is the net change?

$-34$ (correct answer)
$34$
$-74$
$-16$

Explanation: This question tests integer operation skills for ISEE Upper Level, including addition, subtraction, multiplication, and division of integers. Integer operations are fundamental in mathematics and involve combining numbers using these operations, respecting the rules of arithmetic. In this specific scenario, integers are used to track a hiker's elevation changes with repeated descents, requiring precise calculation to determine the net change after multiple drops and a climb. The correct answer, choice A (

-34

), accurately applies integer operations: first calculating

-18 × 3 = -54

for the total drop, then adding the climb:

-54 + 20 = -34

, demonstrating understanding of multiplication with negative numbers. A common error, as seen in choice D (

-16

), might involve only multiplying

-18 × 1

or making arithmetic errors, which results from not recognizing that the drop happens for 3 minutes. To improve, students should practice integer operations involving multiplication of negative numbers, ensuring they understand that negative times positive yields negative. Teachers can use repeated addition models or time-based scenarios to help students understand multiplication in context.