How to Do Mixed Integer Computations

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Integers are whole numbers and their opposites: \(\ldots,-3,-2,-1,0,1,2,3,\ldots\). Mixed integer computations combine addition, subtraction, multiplication, division, exponents, and grouping symbols.

For addition, compare signs. Same signs: add the absolute values and keep the common sign. Different signs: subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

For subtraction, add the opposite: \(a-b=a+(-b)\). For multiplication and division, same signs give a positive result, while different signs give a negative result.

Example 1

Evaluate \(-12 + 5 - (-9)\). Change subtracting a negative to addition: \(-12 + 5 + 9\). Then \(-12 + 5 = -7\), and \(-7 + 9 = 2\).

Example 2

Evaluate \((-6)(-4) - 30 \div (-5)\). Multiply and divide before subtracting: \((-6)(-4)=24\) and \(30 \div (-5)=-6\). Then \(24 - (-6)=30\).

Mixed Integer Computations

Think of this lesson as more than a rule to memorize. Mixed Integer Computations is about number-line meaning, signs, and distance from zero. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Integers are easiest when you picture direction on a number line. Positive numbers move right, negative numbers move left, and absolute value measures distance from zero.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Read what is given and what is being asked.
Choose the rule that connects them.
Substitute carefully and simplify in small steps.
Check the final answer against the original question.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Mixed Integer Computations

1) \((-6) + 9 =\)

2) \(8 - 13 =\)

3) \((-4)(7) =\)

4) \((-36) \div 6 =\)

5) \((-11) + (-15) =\)

6) \(14 - (-9) =\)

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

8) \(63 \div (-7) =\)

9) \((-18) + 5 - (-7) =\)

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

11) \(42 \div (-3) - 8 =\)

12) \((-5)^2 - 30 =\)

13) \(4[-3 + (-8)] =\)

14) \((-72) \div [6 + (-14)] =\)

15) \(18 - 3(-7 + 2) =\)

16) \((-4)(-6) - 5(3 - 9) =\)

17) \(96 \div [(-2)(8 - 14)] =\)

18) \((-3)^3 + 4(-5) - (-12) =\)

19) \(7[(-4)^2 - 20] \div (-2) =\)

20) \(48 \div (-6) + (-3)[5 - (-7)] - (-10) =\)

Answers

1) \((-6) + 9\). The signs differ, so subtract magnitudes: \(9 - 6 = 3\). The larger magnitude is positive, so the result is \(\color{red}{3}\).

2) \(8 - 13\). Rewrite as \(8 + (-13)\). Subtract magnitudes: \(13 - 8 = 5\). The larger magnitude is negative, so the result is \(\color{red}{-5}\).

3) \((-4)(7)\). A negative times a positive is negative. Multiply magnitudes: \(4 \times 7 = 28\). Result: \(\color{red}{-28}\).

4) \((-36) \div 6\). A negative divided by a positive is negative. Divide magnitudes: \(36 \div 6 = 6\). Result: \(\color{red}{-6}\).

5) \((-11) + (-15)\). Same signs mean add magnitudes: \(11 + 15 = 26\). Keep the negative sign: \(\color{red}{-26}\).

6) \(14 - (-9)\). Subtracting a negative means add: \(14 + 9 = \color{red}{23}\).

7) \((-8)(-5)\). A negative times a negative is positive. Multiply magnitudes: \(8 \times 5 = \color{red}{40}\).

8) \(63 \div (-7)\). A positive divided by a negative is negative. Divide magnitudes: \(63 \div 7 = 9\). Result: \(\color{red}{-9}\).

9) \((-18) + 5 - (-7)\). Rewrite subtraction of a negative: \(-18 + 5 + 7\). Combine left to right: \(-18 + 5 = -13\), then \(-13 + 7 = \color{red}{-6}\).

10) \((-6)(4) + 10\). Multiply first: \((-6)(4) = -24\). Then add: \(-24 + 10 = \color{red}{-14}\).

11) \(42 \div (-3) - 8\). Divide first: \(42 \div (-3) = -14\). Then subtract: \(-14 - 8 = \color{red}{-22}\).

12) \((-5)^2 - 30\). Square first: \((-5)^2 = 25\). Then subtract: \(25 - 30 = \color{red}{-5}\).

13) \(4[-3 + (-8)]\). Inside brackets: \(-3 + (-8) = -11\). Multiply: \(4 \times (-11) = \color{red}{-44}\).

14) \((-72) \div [6 + (-14)]\). Brackets first: \(6 + (-14) = -8\). Divide: \((-72) \div (-8) = \color{red}{9}\).

15) \(18 - 3(-7 + 2)\). Parentheses: \(-7 + 2 = -5\). Multiply: \(3(-5) = -15\). Subtract: \(18 - (-15) = 18 + 15 = \color{red}{33}\).

16) \((-4)(-6) - 5(3 - 9)\). Multiply: \((-4)(-6)=24\). Parentheses: \(3 - 9 = -6\), so \(5(-6)=-30\). Then \(24 - (-30) = \color{red}{54}\).

17) \(96 \div [(-2)(8 - 14)]\). Parentheses: \(8 - 14 = -6\). Brackets: \((-2)(-6)=12\). Divide: \(96 \div 12 = \color{red}{8}\).

18) \((-3)^3 + 4(-5) - (-12)\). Exponent: \((-3)^3=-27\). Multiply: \(4(-5)=-20\). Subtracting \(-12\) means add \(12\): \(-27 - 20 + 12 = \color{red}{-35}\).

19) \(7[(-4)^2 - 20] \div (-2)\). Exponent: \((-4)^2=16\). Brackets: \(16 - 20 = -4\). Multiply and divide left to right: \(7(-4)=-28\), then \(-28 \div (-2)=\color{red}{14}\).

20) \(48 \div (-6) + (-3)[5 - (-7)] - (-10)\). Brackets: \(5 - (-7)=12\). Divide: \(48 \div (-6)=-8\). Multiply: \((-3)(12)=-36\). Subtracting \(-10\) means add \(10\): \(-8 - 36 + 10 = \color{red}{-34}\).

How to Do Mixed Integer Computations