SSAT Upper Level

Integers & Negatives

Q: Do I need to memorize the rules for negative numbers for the SSAT?

Yes! Knowing that 'a negative times a negative is a positive' will save you tons of time. You will see these concepts all over the test.

Q: What exactly is an integer? Are fractions integers?

Integers are only whole numbers and their negatives (like `-2`, `0`, and `5`). Fractions like `\frac{1}{2}` and decimals are NOT integers!

Q: Is zero positive or negative?

Zero is the referee of the number line—it is neither positive nor negative! It's just a neutral integer.

Q: How many negative number questions are on the SSAT?

You'll see them everywhere! They don't just appear in basic arithmetic; they pop up in algebra and word problems too, making them super important to master.

Operations with negative numbers and integer properties

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Have you ever owed your friend money for a slice of pizza? Or played a video game where your score dropped below zero? Welcome to the wild world of negative numbers! 🍕🎮 Integers are just whole numbers (no fractions or decimals allowed), but they have a secret alter ego: negatives. Think of a number line like a giant tug-of-war. Zero is the referee standing right in the middle. Positive numbers pull to the right, and negative numbers pull to the left. If you have 5 dollars in your piggy bank and want to buy a cool toy for 8 dollars, you don't just have zero dollars—you actually owe 3 dollars. In math language, we write that as $- 3$ . 🏦 On the SSAT, test-makers love to see how well you can juggle these positive and negative numbers. Adding a negative is just like taking away points in a video game. But subtracting a negative? That’s like a referee taking away a penalty—it actually gives you points back! 🚀 So, whenever you see two minus signs right next to each other, simply smash them together to make a giant plus sign. Master these simple rules, and you'll be an integer ninja ready to crush the math section!

Key Formula

Subtracting a negative is adding a positive:

a - (- b) = a + b

. For multiplying and dividing: Same signs make a positive (

- a \cdot - b = a \cdot b

), different signs make a negative (

- a \cdot b = - ab

Practice Questions

4 practice questions for SSAT Upper Level

Q1 Hard

Evaluate the expression:

(- 2)^{3} - 5 \cdot (- 3)^{2} + ∣ - 10 - (- 4) ∣

A -65

B -59

C -53

D -47

E -41

Show Solution

To evaluate the expression, follow the order of operations (PEMDAS/BODMAS):
1. Parentheses/Exponents:

• $(- 2)^{3} = - 8$ (negative base raised to an odd power remains negative)
• $(- 3)^{2} = 9$ (negative base raised to an even power becomes positive)
2. Absolute Value:

• $∣ - 10 - (- 4) ∣ = ∣ - 10 + 4∣ = ∣ - 6∣ = 6$ (Subtracting a negative is equivalent to adding a positive. The absolute value of -6 is 6.)
3. Multiplication:

• $5 \cdot (- 3)^{2} = 5 \cdot 9 = 45$
4. Rewrite the expression with the calculated values:

• The expression becomes $(- 8) - 45 + 6$
5. Addition/Subtraction (from left to right):

• $- 8 - 45 = - 53$
• $- 53 + 6 = - 47$
The final value of the expression is -47.

Answer: D

Q2 Hard

A submarine begins at a depth of 250 feet below sea level. It then descends an additional 120 feet, ascends 80 feet, and finally descends another 170 feet. What is the submarine's final depth, in feet, relative to sea level?

A 160 feet below sea level

B 220 feet below sea level

C 460 feet below sea level

D 520 feet below sea level

E 620 feet below sea level

Show Solution

Let's represent depths below sea level as negative numbers and ascents/descents as changes to this number.
1. Starting depth: 250 feet below sea level is $- 250$ feet.
2. Descends 120 feet: This means subtracting 120 from the current depth.

• $- 250 - 120 = - 370$ feet.
3. Ascends 80 feet: This means adding 80 to the current depth.

• $- 370 + 80 = - 290$ feet.
4. Descends another 170 feet: This means subtracting 170 from the current depth.

• $- 290 - 170 = - 460$ feet.
The final depth of the submarine is $- 460$ feet, which means 460 feet below sea level.

Answer: C

Q3 Hard

a

and

b

are non-zero integers such that

a \cdot b < 0

and

a - b > 0

, which of the following statements must be true?

a < 0

and

b > 0

a > 0

and

b < 0

a + b = 0

∣ a ∣ < ∣ b ∣

a \cdot b

is an even integer

Show Solution

Let's analyze the given conditions:
1. $a \cdot b < 0$ : This condition tells us that the product of $a$ and $b$ is negative. For a product of two non-zero numbers to be negative, one number must be positive and the other must be negative. So, $a$ and $b$ have opposite signs.
2. $a - b > 0$ : This condition can be rewritten by adding $b$ to both sides as $a > b$ . This means $a$ is greater than $b$ .

Now, let's combine these two pieces of information:
• Case 1: $a$ is positive and $b$ is negative.
• If $a$ is positive (e.g., $a = 5$ ) and $b$ is negative (e.g., $b = - 2$ ), then $a \cdot b = 5 \cdot (- 2) = - 10$ , which is less than 0. This satisfies the first condition.
• Also, $a - b = 5 - (- 2) = 5 + 2 = 7$ , which is greater than 0. This satisfies the second condition.
• This case (a > 0 and b < 0) is consistent with both conditions.
• Case 2: $a$ is negative and $b$ is positive.
• If $a$ is negative (e.g., $a = - 5$ ) and $b$ is positive (e.g., $b = 2$ ), then $a \cdot b = - 5 \cdot 2 = - 10$ , which is less than 0. This satisfies the first condition.
• However, $a - b = - 5 - 2 = - 7$ , which is NOT greater than 0. This violates the second condition.
Since only Case 1 is consistent with both given conditions, it must be true that $a$ is positive and $b$ is negative.
Let's check the choices:
• A) $a < 0$ and $b > 0$ : This contradicts our finding.
• B) $a > 0$ and $b < 0$ : This matches our finding.
• C) $a + b = 0$ : This is not necessarily true (e.g., $a = 5, b = - 2$ gives $a + b = 3$ ).
• D) $∣ a ∣ < ∣ b ∣$ : This is not necessarily true (e.g., $a = 5, b = - 2$ means $∣ a ∣ = 5$ and $∣ b ∣ = 2$ , so $∣ a ∣ > ∣ b ∣$ ).
• E) $a \cdot b$ is an even integer: While $a \cdot b$ is negative, it could be odd if both $a$ and $b$ are odd (e.g., $a = 3, b = - 1$ , then $a \cdot b = - 3$ , which is odd). So, this is not necessarily true.
Therefore, the statement that must be true is $a > 0$ and $b < 0$ .

Answer: B

Q4 Hard

Over a four-day period, the daily high temperatures recorded were

5^{\circ} C

- 3^{\circ} C

- 7^{\circ} C

, and

x^{\circ} C

. If the average high temperature over these four days was

- 1^{\circ} C

, what was the temperature

x

on the fourth day?

0^{\circ} C

1^{\circ} C

- 1^{\circ} C

- 3^{\circ} C

- 5^{\circ} C

Show Solution

To find the average of a set of numbers, you sum the numbers and then divide by the count of the numbers. We are given the average and three of the four temperatures.
1. Set up the average formula:

$A v er a g e = (S u m o f t e m p er a t u r es) / (N u mb er o f t e m p er a t u r es)$
2. Plug in the given values:

$A v er a g e = - 1^{\circ} C$
$T e m p er a t u r es = 5^{\circ} C, - 3^{\circ} C, - 7^{\circ} C, x^{\circ} C$
$N u mb er o f t e m p er a t u r es = 4$
So, $- 1 = (5 + (- 3) + (- 7) + x) /4$
3. Simplify the sum of known temperatures:

$5 - 3 - 7 = 2 - 7 = - 5$
4. Rewrite the equation:

$- 1 = (- 5 + x) /4$
5. Solve for $x$ :

• Multiply both sides by 4:
$- 1 \cdot 4 = - 5 + x$
$- 4 = - 5 + x$
• Add 5 to both sides:
$- 4 + 5 = x$
$x = 1$
Therefore, the temperature on the fourth day was $1^{\circ} C$ .

Answer: B

Tips & Strategies

Draw a quick number line on your scratch paper! It helps you physically count hops left and right so you don't get your positive and negative directions mixed up.
Think of money! If a problem has lots of plus and minus signs, pretend you are earning and spending dollars. It makes the math feel much more real and easier to solve.

Common Mistakes

Watch out for the 'double minus' trap! When you see $5 - (- 3)$ , don't subtract. Taking away a negative penalty is a good thing, so it becomes a plus: $5 + 3 = 8$ .
Don't confuse the rules for adding and multiplying! Adding two negatives makes a bigger negative ( $- 2 + - 2 = - 4$ ), but multiplying two negatives makes a positive ( $- 2 \cdot - 2 = 4$ ).

Frequently Asked Questions

Do I need to memorize the rules for negative numbers for the SSAT?

Yes! Knowing that 'a negative times a negative is a positive' will save you tons of time. You will see these concepts all over the test.

What exactly is an integer? Are fractions integers?

Integers are only whole numbers and their negatives (like $- 2$ , $0$ , and $5$ ). Fractions like $\frac{1}{2}$ and decimals are NOT integers!

Is zero positive or negative?

Zero is the referee of the number line—it is neither positive nor negative! It's just a neutral integer.

How many negative number questions are on the SSAT?

You'll see them everywhere! They don't just appear in basic arithmetic; they pop up in algebra and word problems too, making them super important to master.

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