Ordering Integers - SSAT Middle Level Quantitative

A sequence of integers follows the pattern: $8, 3, -2, -7, -12, ...$ What is the next integer in this sequence?
Answer: $-17$ When you encounter a sequence problem, your first step is to identify the pattern by examining how each term relates to the previous one. Let's look at the differences between consecutive terms: - From 8 to 3: $8 - 5 = 3$ - From 3 to -2: $3 - 5 = -2$ - From -2 to -7: $-2 - 5 = -7$ - From -7 to -12: $-7 - 5 = -12$ The pattern is clear: each term decreases by 5. This is an arithmetic sequence with a common difference of $-5$ . To find the next term, subtract 5 from the last given term: $-12 - 5 = -17$ . This confirms that choice A is correct. Now let's examine why the other answers are wrong: Choice B ( $-16$ ) would result from subtracting only 4 from -12, which breaks the established pattern of subtracting 5. Choice C ( $-15$ ) would result from subtracting only 3 from -12, also inconsistent with the pattern. Choice D ( $-5$ ) would require adding 7 to -12, which completely ignores the decreasing pattern and suggests a misunderstanding of the sequence direction. Strategy tip: For sequence questions, always calculate the difference between at least three consecutive pairs of terms to confirm the pattern. Don't assume it's arithmetic after checking just one pair—verify consistency throughout the given terms. This habit will help you avoid careless errors and build confidence in your answer.
Integers are whole numbers that can be negative, zero, or positive. A number line helps compare them: left is smaller, right is larger. Morning temperatures were: -8, -4, -2, 5, 10. Which of the following lists integers from least to greatest based on the number line above?
Answer: -8, -4, -2, 5, 10 This question tests the ability to compare and order integers on a number line, a key skill in understanding number properties at the middle school level. Integers include both positive and negative whole numbers, and ordering them involves arranging them from least to greatest or vice versa. In this question, the number line shown includes integers from -10 to 10, providing a visual representation to guide ordering. The correct answer is choice B because it accurately lists the integers in the order from least to greatest based on the number line. Choice A is incorrect because it reverses the order, a common mistake when not paying attention to the direction of the number line. To help students avoid such errors, encourage them to visualize the number line mentally and practice placing real-world contexts, like temperatures or bank balances, on it. This strategy can solidify their understanding of integer order.
An elevator starts at floor $0$ . It goes up $7$ floors, then down $12$ floors, then up $3$ floors. On which floor is the elevator now?
Answer: $-2$ This problem tests your ability to track position changes using positive and negative integers, where movement up is positive and movement down is negative. Starting at floor $0$ , you need to track each movement step by step. Going up $7$ floors means adding $7$ : $0 + 7 = 7$ . Then going down $12$ floors means subtracting $12$ : $7 - 12 = -5$ . Finally, going up $3$ floors means adding $3$ : $-5 + 3 = -2$ . The elevator ends up on floor $-2$ , which represents the second basement level. Looking at the wrong answers: Choice A ( $-5$ ) represents where the elevator was after the second move but before the final upward movement of $3$ floors. This is a common mistake when students forget to complete all the steps. Choice C ( $2$ ) might result from incorrectly treating the downward movement as positive, calculating $0 + 7 + 12 + 3 = 22$ , or from some other sign error that leads to a positive result. Choice D ( $8$ ) could come from adding all the numbers without considering direction: $7 + 12 + 3 - 0 = 22$ , or from other calculation errors. When working with elevator problems, always establish your convention first (up is positive, down is negative), then track your position after each move rather than trying to do all calculations at once. Writing out each step helps you avoid sign errors and ensures you don't miss any movements.
If $k$ is an integer and $|k| < 4$ , what is the sum of all possible values of $k$ ?
Answer: $0$ This question tests your understanding of absolute value and how to find all integers that satisfy an inequality involving absolute value. When you see $|k| < 4$ , you need to find all integers $k$ whose distance from zero on the number line is less than 4. The absolute value $|k|$ represents the distance from $k$ to zero, regardless of whether $k$ is positive or negative. Since $|k| < 4$ , we need $k$ to be closer to zero than 4 units away. This means $k$ can be any integer from $-3$ to $3$ , inclusive. Let's list them: $k = -3, -2, -1, 0, 1, 2, 3$ . To find the sum: $(-3) + (-2) + (-1) + 0 + 1 + 2 + 3 = 0$ . Notice that each positive integer pairs with its negative counterpart, so they cancel out, leaving just zero. Looking at the wrong answers: Choice A ( $-6$ ) might result from incorrectly thinking only negative values satisfy the inequality or making an arithmetic error. Choice C ( $6$ ) could come from only considering positive values or miscalculating the sum. Choice D ( $10$ ) might result from incorrectly including $\pm 4$ in your list (since $|4| = 4$ , not less than 4) and then summing $-4 + (-3) + (-2) + (-1) + 0 + 1 + 2 + 3 + 4 = 0$ , or from other calculation mistakes. Remember: when dealing with absolute value inequalities, always consider both positive and negative values, and pay careful attention to whether the inequality is strict ( $<$ ) or includes equality ( $\leq$ ).
Which of the following correctly shows the order of the integers -6, -2, and 4 from greatest to least?
Answer: 4, -2, -6 When ordering integers from greatest to least, you need to understand how negative numbers work on the number line. Negative numbers are always less than positive numbers, and among negative numbers, those closer to zero are actually greater. Let's place our three integers on a number line: -6, -2, and 4. Moving from left to right (least to greatest), we have -6, then -2, then 4. This means 4 is the greatest number, -2 is in the middle, and -6 is the least. Therefore, ordering from greatest to least gives us: 4, -2, -6. Looking at the answer choices: Choice A (-6, -2, 4) shows the numbers from least to greatest, which is the opposite of what the question asks for. Choice C (-2, 4, -6) incorrectly places -6 as the greatest number, when it's actually the smallest. Choice D (-6, 4, -2) also starts with -6 as if it were the greatest, showing a fundamental misunderstanding of negative numbers. Only choice B (4, -2, -6) correctly orders the numbers from greatest to least. The key insight is that negative numbers follow a counterintuitive pattern: -2 is greater than -6 because -2 is closer to zero. Think of temperature or debt - owing $2 is better than owing $6. Study tip: When comparing negative numbers, remember that the number with the smaller absolute value is actually the greater number. Practice visualizing a number line to avoid the common trap of thinking -6 is greater than -2.
On a number line, the distance between integers $a$ and $b$ is $12$ . If $a = -5$ , which of the following could be the value of $b$ ?
Answer: $-17$ or $7$ When you see a question about distance on a number line, remember that distance is always positive and represents how far apart two points are, regardless of direction. The key insight is that if two points are a certain distance apart, there are typically two possible locations for the second point. Since $a = -5$ and the distance between $a$ and $b$ is 12, you need to find all possible values of $b$ . Distance on a number line is calculated as $|a - b|$ , so you have $|-5 - b| = 12$ . This absolute value equation gives you two cases to consider: - Case 1: $-5 - b = 12$ , which gives $b = -5 - 12 = -17$ - Case 2: $-5 - b = -12$ , which gives $b = -5 + 12 = 7$ You can verify this by checking: the distance from $-5$ to $-17$ is $|-5 - (-17)| = |12| = 12$ , and the distance from $-5$ to $7$ is $|-5 - 7| = |-12| = 12$ . Both work! Choice A) gives only $-17$ , missing the positive solution. Choice B) gives only $7$ , missing the negative solution. Choice C) incorrectly suggests $17$ , which would be 22 units away from $-5$ , not 12. Choice D) correctly includes both possible values. Strategy tip: Distance problems on number lines almost always have two solutions unless one endpoint is already at zero or the distance extends beyond a given boundary. Always solve the absolute value equation completely to find both possibilities.
The integers $a$ , $b$ , and $c$ satisfy $a < b < c$ . If $a + c = 4$ , $b = 1$ , and $c > 5$ , which of the following could be the value of $a$ ?
Answer: $-3$ Since $a + c = 4$ , $c = 4 - a$ . The condition $c > 5$ means $4 - a > 5$ , so $a < -1$ . The ordering $a < 1 < c$ must also hold. Test $a = -3$ : then $c = 4 - (-3) = 7$ , so $-3 < 1 < 7$ and $7 > 5$ . Every condition is met. $a = -1$ gives $c = 5$ , which is not greater than 5. $a = 0$ gives $c = 4$ ; the ordering $0 < 1 < 4$ is fine, but 4 fails $c > 5$ . $a = 2$ breaks $a < b$ , because 2 is not less than 1, and $a = 5$ makes $c = -1$ , which breaks the ordering too. Strategy tip: Rewrite every condition in terms of one variable, then test each choice against all of the conditions, not just the first one that comes to mind.
Temperatures are integers on a number line. Left means smaller (colder), and right means larger (warmer). A report shows: -6, -5, -4, 4, 5. What is the correct order of these integers on the number line?
Answer: -6, -5, -4, 4, 5 This question tests the ability to compare and order integers on a number line, a key skill in understanding number properties at the middle school level. Integers include both positive and negative whole numbers, and ordering them involves arranging them from least to greatest or vice versa. In this question, the number line shown includes integers from -10 to 10, providing a visual representation to guide ordering. The correct answer is choice A because it accurately lists the integers in the order from least to greatest based on the number line. Choice B is incorrect because it reverses the order, a common mistake when not paying attention to the direction of the number line. To help students avoid such errors, encourage them to visualize the number line mentally and practice placing real-world contexts, like temperatures or bank balances, on it. This strategy can solidify their understanding of integer order.
If $a < b < 0 < c < d$ and all variables represent integers, which of the following expressions represents the smallest value?
Answer: $a + c$ When you encounter inequalities with variables on both sides of zero, think systematically about how positive and negative numbers behave when added together. Given that $a < b < 0 < c < d$ , you know that $a$ and $b$ are negative integers, while $c$ and $d$ are positive integers. Since $a < b$ , the value $a$ is more negative (farther from zero) than $b$ . Similarly, since $c < d$ , the value $c$ is smaller than $d$ . To find the smallest sum, you want to combine the most negative number with the smallest positive number. That means pairing $a$ (the most negative) with $c$ (the smallest positive). This gives you $a + c$ , which represents the correct answer (A). Let's see why the other options are larger: Choice (B) pairs $a + d$ , but since $d > c$ , this sum is larger than $a + c$ . Choice (C) gives you $b + c$ , but since $b > a$ (meaning $b$ is less negative), this sum is also larger than $a + c$ . Choice (D) combines $b + d$ , using both the less negative number and the larger positive number, making it the largest of all four expressions. For inequality problems involving positive and negative numbers, remember this key strategy: to minimize a sum, pair the most negative value with the smallest positive value. This principle helps you quickly identify which combination will yield the smallest result without having to test specific numbers.
Which of the following lists shows integers arranged in order from least to greatest?
Answer: $-12, -8, -3, 0, 5$ When ordering integers from least to greatest, you need to understand how negative numbers work on the number line. Negative numbers get smaller as their absolute value increases, while positive numbers get larger as their absolute value increases. To arrange these integers correctly, visualize them on a number line. Starting from the left (smallest): $-12$ is the smallest because it's the farthest from zero in the negative direction. Next comes $-8$ , then $-3$ (closer to zero), then $0$ , and finally $5$ as the largest positive number. This gives us: $-12, -8, -3, 0, 5$ , which matches choice B. Choice A places $-12$ at the end, treating it as if it were the largest number. This is a common error where students focus on the digit 12 and forget that the negative sign makes it smaller than all other numbers in the list. Choice C starts with $0$ and then lists negative numbers in decreasing order (getting more negative), which is backwards. This shows a misunderstanding of how negative numbers relate to zero and each other. Choice D arranges the numbers from greatest to least instead of least to greatest, giving the completely opposite order from what's requested. Remember this key principle: when comparing negative integers, the one with the larger absolute value is actually the smaller number. So $-12 < -8 < -3 < 0 < 5$ . Always place negative numbers to the left of zero and positive numbers to the right when visualizing the number line.
Which of the following integers is greater than $-\frac{7}{2}$ but less than $\frac{5}{2}$ ?
Answer: $0$ Convert the bounds to decimals: $-\frac{7}{2} = -3.5$ and $\frac{5}{2} = 2.5$ . You need an integer $x$ with $-3.5 < x < 2.5$ , so the integers that work are $-3, -2, -1, 0, 1, 2$ . Of the choices, only $0$ is in that list. Both $-4$ and $-5$ are less than $-3.5$ , so they are too small; a common slip is to think $-4$ is greater than $-3.5$ because 4 is greater than 3.5. The integer $3$ is greater than $2.5$ , and $4$ is even farther above the upper bound.
If three consecutive integers have a sum of $-21$ , what is the smallest of these integers?
Answer: $-8$ When you encounter problems about consecutive integers with given sums, set up an algebraic equation using the middle integer as your variable. Consecutive integers differ by 1, so if the middle integer is $n$ , then the three consecutive integers are $n-1$ , $n$ , and $n+1$ . Setting up the equation: $(n-1) + n + (n+1) = -21$ . Simplifying the left side, you get $3n = -21$ , so $n = -7$ . This means the three consecutive integers are $-8$ , $-7$ , and $-6$ . The smallest is $-8$ . Looking at the wrong answers: Choice A gives $-9$ , which would make the three integers $-9$ , $-8$ , and $-7$ . Their sum would be $-24$ , not $-21$ . Choice C gives $-7$ , which is actually the middle integer in our correct sequence, not the smallest. If you mistakenly used $-7$ as the smallest, your three integers would be $-7$ , $-6$ , and $-5$ , giving a sum of $-18$ . Choice D gives $-6$ , which is the largest integer in the correct sequence. Using $-6$ as the smallest would give you $-6$ , $-5$ , and $-4$ , with a sum of $-15$ . Remember that with negative numbers, the number with the largest absolute value is actually the smallest. Also, always verify your answer by checking that your three integers actually sum to the given total. This catches computational errors and ensures you identified the correct integer from your sequence.
An integer $n$ satisfies the condition $-10 < n < -3$ . How many possible values are there for $n$ ?
Answer: $6$ When you encounter inequalities with integers, you need to identify all whole numbers that satisfy the given conditions. The inequality $-10 < n < -3$ means $n$ must be greater than $-10$ and less than $-3$ simultaneously. To find the possible values, list all integers between these boundaries. Since $n > -10$ , the integer $-10$ itself is not included. Since $n < -3$ , the integer $-3$ is also not included. The integers that satisfy both conditions are: $-9, -8, -7, -6, -5, -4$ . Counting these values gives us $6$ possible integers, confirming answer choice B. Let's examine why the other options are incorrect. Choice A ( $5$ ) likely results from miscounting or accidentally excluding one of the valid integers. Choice C ( $7$ ) probably comes from incorrectly including one of the boundary values, either $-10$ or $-3$ , in your count. Choice D ( $8$ ) suggests including both boundary values $-10$ and $-3$ , which violates the strict inequality conditions (the "less than" and "greater than" symbols don't include the endpoints). Remember that strict inequalities use $<$ and $>$ symbols, which exclude the boundary values themselves. If the problem had used $\leq$ or $\geq$ , then you would include the endpoints. Always double-check by listing out the integers systematically when dealing with small ranges—this prevents counting errors and ensures you correctly handle the boundary conditions.
Temperatures are integers (whole numbers that can be negative, positive, or zero). On a number line, numbers to the left are smaller and numbers to the right are larger. A student checks five temperatures during a week: -10 $^\circ$ C, -6 $^\circ$ C, -1 $^\circ$ C, 3 $^\circ$ C, 10 $^\circ$ C. Integers help show changes above and below $0^\circ$ C. What is the correct order of these integers on the number line?
Answer: -10, -6, -1, 3, 10 This question tests the ability to compare and order integers on a number line, a key skill in understanding number properties at the middle school level. Integers include both positive and negative whole numbers, and ordering them involves arranging them from least to greatest or vice versa. In this question, the number line shown includes integers from -10 to 10, providing a visual representation to guide ordering. The correct answer is choice A because it accurately lists the integers in the order from least to greatest based on the number line. Choice B is incorrect because it reverses the order, a common mistake when not paying attention to the direction of the number line. To help students avoid such errors, encourage them to visualize the number line mentally and practice placing real-world contexts, like temperatures or bank balances, on it. This strategy can solidify their understanding of integer order.
If $x$ and $y$ are integers such that $x < -5$ and $y > 3$ , which of the following must be true about $x + y$ ?
Answer: $x + y > x$ Because $x$ and $y$ are integers, $x < -5$ means $x$ is $-6$ or less, and $y > 3$ means $y$ is $4$ or more. Try a few allowed pairs: $x = -6, y = 4$ gives $x + y = -2$ ; $x = -100, y = 4$ gives $-96$ ; $x = -6, y = 100$ gives $94$ . The sum can be very negative or very positive, so nothing about its size is guaranteed. What is guaranteed is that $y$ is positive, and adding a positive number to $x$ always gives a result larger than $x$ . So $x + y > x$ must be true. The statement $x + y > -2$ is the tempting trap: it comes from adding the boundary values $-5 + 3$ , but with integers the smallest allowed pair, $-6 + 4$ , gives exactly $-2$ , which is not greater than $-2$ . The statements $x + y < 0$ and $x + y > 0$ each fail for one of the examples above ( $94$ and $-96$ ). Strategy tip: For "must be true" questions, test extreme allowed values; one counterexample eliminates a choice.
If $m$ and $n$ are integers with $m < n$ , and there are exactly $4$ integers between $m$ and $n$ (not including $m$ and $n$ ), what is $n - m$ ?
Answer: $5$ When you encounter problems about integers "between" two numbers, you need to be careful about what "between" means and how to count the gaps. Let's work with a concrete example. If $m = 2$ and there are exactly 4 integers between $m$ and $n$ , those integers would be 3, 4, 5, and 6. This means $n = 7$ , since we don't include $n$ itself in our count. So $n - m = 7 - 2 = 5$ . You can visualize this on a number line: $2 \rightarrow 3 \rightarrow 4 \rightarrow 5 \rightarrow 6 \rightarrow 7$ . There are 4 integers between 2 and 7, and the difference is 5. Here's why each wrong answer represents a common mistake: A) 4 assumes you're simply counting the integers between $m$ and $n$ , but this ignores that you need to account for the endpoints being excluded. C) 6 likely comes from counting both endpoints plus the 4 integers between them, but the question asks for $n - m$ , not the total count of numbers. D) 7 might result from incorrectly including both $m$ and $n$ in your count and then adding the 4 between them. The correct answer is B) 5 because when there are exactly 4 integers between two integers $m$ and $n$ , the difference $n - m$ must be 5. Strategy tip: For "between" problems, try a simple example first. Pick small numbers like 1, 2, 3 to see the pattern, then apply the same logic to the abstract problem.
On a number line marked with consecutive integers, the integer at position n is -3. What integer is at position n + 7?
Answer: 4 This question tests your understanding of position and movement on a number line. When you see problems involving positions and shifts, think about the relationship between where you start and where you end up. You're told that position $n$ contains the integer $-3$ . To find what's at position $n + 7$ , you need to move 7 positions to the right on the number line. Since each position represents consecutive integers, moving 7 positions right means adding 7 to your starting value: $-3 + 7 = 4$ . Looking at the wrong answers: Choice (A) gives $-10$ , which would result from subtracting 7 from $-3$ instead of adding it—this represents moving left instead of right on the number line. Choice (B) gives $-4$ , which you'd get by adding only 1 to $-3$ , suggesting a misunderstanding of how far to move. Choice (D) gives $10$ , which might come from incorrectly thinking you add 7 to the position number $n$ rather than to the integer value at that position, or from some other computational error. The correct answer is (C) 4. Remember this key insight: when a problem asks about moving positions on a number line with consecutive integers, the change in position equals the change in value. Moving right means adding, moving left means subtracting. Always pay attention to the direction of movement and apply the operation to the given integer value, not the position variable.
Which integer is closest to the middle of $-15$ and $7$ on a number line?
Answer: $-4$ When you need to find the point exactly halfway between two numbers on a number line, you're looking for their average or midpoint. This requires adding the two numbers and dividing by 2. To find the middle of $-15$ and $7$ , calculate: $\frac{-15 + 7}{2} = \frac{-8}{2} = -4$ . Since the question asks which integer is closest to this middle point, and $-4$ is exactly the midpoint, the answer is $-4$ . Let's examine why the other choices are incorrect. Choice (A) $-8$ is the sum of the two numbers before dividing by 2 — this represents a common error where students forget the division step in finding an average. Choice (B) $-5$ might result from incorrectly calculating the sum as $-15 + 7 = -10$ and then dividing by 2, which shows an arithmetic mistake in handling negative numbers. Choice (D) $-3$ is close to the correct midpoint but represents rounding or calculation errors that might occur when working hastily with negative numbers. You can verify this makes sense by checking distances: from $-15$ to $-4$ is 11 units, and from $-4$ to $7$ is also 11 units, confirming $-4$ is exactly in the middle. Remember that finding the midpoint between any two numbers always uses the same formula: add them and divide by 2. This works whether the numbers are positive, negative, or mixed, so practice this reliable method rather than trying to estimate visually.
The temperature at 6 AM was $-8°F$ . By noon, it had risen $15°F$ , and by 6 PM it had dropped $9°F$ from the noon temperature. At midnight, the temperature was $3°F$ warmer than it was at 6 PM. What was the temperature at midnight?
Answer: $1°F$ When you encounter temperature change problems, you need to track each change step by step, being careful with positive and negative values. Start at 6 AM with $-8°F$ . By noon, the temperature rose $15°F$ , so you add: $-8 + 15 = 7°F$ . From noon to 6 PM, it dropped $9°F$ , so you subtract: $7 - 9 = -2°F$ . Finally, from 6 PM to midnight, it warmed $3°F$ : $-2 + 3 = 1°F$ . Looking at the wrong answers: Choice A ( $-5°F$ ) likely comes from incorrectly calculating the 6 PM temperature as $-8°F$ instead of $-2°F$ , then adding $3°F$ to get $-5°F$ . This error occurs when students subtract the noon rise from the starting temperature instead of adding it. Choice B ( $-2°F$ ) is the temperature at 6 PM, not midnight - this happens when you forget the final step of adding the $3°F$ warming from 6 PM to midnight. Choice D ( $4°F$ ) results from miscalculating the 6 PM temperature as $1°F$ instead of $-2°F$ , then adding $3°F$ . The correct answer is C ( $1°F$ ). For temperature problems, always work chronologically through each time period and double-check your arithmetic with negative numbers. Remember that "rising" means adding and "dropping" means subtracting, regardless of whether you're working with positive or negative temperatures. Writing out each step helps prevent calculation errors.

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