Integer Operations - ISEE Upper Level Mathematics Achievement

1. What is the remainder when $(-47)$ is divided by $8$?
   Answer: $1$ 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.
2. If a bank account changes by $-24 \div 6$ and then by $+15$, what is the net change?
   Answer: $11$ 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 bank account changes involving division, requiring precise calculation to determine the net change after two transactions. The correct answer, choice B ($11$), accurately applies integer operations: first calculating $-24 ÷ 6 = -4$, then adding $15$ to get $-4 + 15 = 11$, demonstrating understanding of order of operations and signed division. A common error, as seen in choice D ($-11$), might involve incorrectly handling the signs in division or addition, which results from confusion about negative divided by positive yielding negative. To improve, students should practice integer operations involving division, ensuring they apply order of operations correctly. Teachers can use step-by-step breakdowns to help students see how division is performed before addition in multi-step problems.
3. 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?
   Answer: $42$ feet below sea level 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.
4. Calculate: $\frac{(-8) \times 15}{(-6) + (-4)}$
   Answer: $12$ 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.
5. A hiker climbs $+120$ ft, descends $-75$ ft, then climbs $+40$ ft; what is the net change?
   Answer: $85$ 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.
6. A warehouse ships $-8$ boxes each hour for $5$ hours, then receives $+12$ boxes; what is the net change?
   Answer: $-28$ 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.
7. A store has a loss of $-12$ dollars per day for $4$ days, then a gain of $+30$; what is total profit/loss?
   Answer: $-18$ 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.
8. What is the value of $\frac{(-4)^3 \times (-2)^2}{(-8) \div 2}$?
   Answer: $64$ 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.
9. 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?
   Answer: $20$ 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.
10. If a bank account starts at $-20$ and changes by $+45$, $-30$, then $+10$, what is the final balance?
    Answer: $5$ 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 bank account transactions, requiring precise calculation to determine the final balance after multiple deposits and withdrawals. The correct answer, choice B ($5$), accurately applies integer operations: starting at $-20$, then adding $45$ gives $25$, subtracting $30$ gives $-5$, and adding $10$ results in $5$, demonstrating understanding of how to handle positive and negative values. A common error, as seen in choice C ($-5$), involves stopping the calculation one step early or miscalculating the final addition, which results from not carefully tracking all transactions. To improve, students should practice integer operations by applying them in real-world contexts like banking, ensuring they check each step of their calculations. Teachers can use number lines or visual aids to help students conceptualize these operations, especially when dealing with negative starting values.
11. What is the value of $(-3)^4 - (-2)^5 + (-1)^{10}$?
    Answer: $114$ 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 $(-2)^5$ as $-30$ instead of $-32$, or making an arithmetic error in the final addition. Choice C (116) could result from incorrectly evaluating $(-1)^{10}$ as $-1$ 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.
12. Simplify: $\frac{(-12) + (-18)}{(-6) - (-3)}$
    Answer: $10$ 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.
13. A bank account has a balance of $-\$127$. After three transactions of $+\$45$, $-\$38$, and $+\$92$, what is the new balance?
    Answer: $-\$28$ 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 $45 - 38$ as $5$ 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.
14. A hiker drops $-18$ ft each minute for $3$ minutes, then climbs $+20$ ft; what is the net change?
    Answer: $-34$ 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.
15. An inventory has $35$ notebooks; $-12$ are sold and $+18$ arrive; how many are there now?
    Answer: $41$ 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.
16. If $x = -4$ and $y = -6$, what is the value of $|x - y| - |x + y|$?
    Answer: $-8$ 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.
17. What is the value of $(-7) - (-12) + (-5) - 8$?
    Answer: $-8$ 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 $(-7) - (-12)$ as $-19$ instead of $5$, then getting $-19 + (-5) - 8 = -32$, but making an arithmetic error. Choice C ($-4$) might result from treating $- (-12)$ as $-12$ 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.
18. The temperature dropped from $7°C$ to $-14°C$ over a 6-hour period. If the temperature dropped at a constant rate, what was the temperature after the first 2 hours?
    Answer: $0°C$ For a constant rate of change, find the total change, divide by the time to get the rate, and then apply the rate for the time asked. The temperature fell from $7°C$ to $-14°C$, a drop of $7 - (-14) = 21$ degrees. Over 6 hours, that is $21 \div 6 = 3.5$ degrees per hour. After 2 hours, the temperature has dropped $3.5 \times 2 = 7$ degrees, so it is $7 - 7 = 0°C$. An answer of $-1°C$ comes from rounding the rate up to 4 degrees per hour. An answer of $1°C$ comes from rounding it down to 3 degrees per hour. An answer of $3°C$ comes from using a drop of only 2 degrees per hour. The temperature after 2 hours was $0°C$.
19. Calculate the total change in temperature: it rises $+7 \times 3$ degrees, then falls $-25$ degrees.
    Answer: $-4$ 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.
20. The expression $(-3)^n$ equals $81$ when $n$ is a positive even integer. What is the value of $(-3)^{n+1}$?
    Answer: $-243$ 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 $n$ to $n+1$. 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.

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