Positive & Negative Rational Number Operations — Challenge

Negative numbers are directions, not just "less than nothing". Subtraction is adding the opposite, and sign rules come from patterns you can see.

What these problems cover

These questions use 3 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Problem 1

What is (-8)+3?

  1. -11
  2. -5
  3. -24
  4. 5
  5. 11
Worked answer

-5

Adding a positive number to a negative one moves right along the number line, towards zero.

Starting at -8 and moving 3 to the right lands on -5. The answer stays negative because 8 is larger than 3.

Problem 2

What is (-12)÷(-3)-5?

  1. -9
  2. -1
  3. 1
  4. 9
  5. -5
Worked answer

-1

Division comes before subtraction. A negative divided by a negative is positive: (-12)÷(-3) = 4.

Then 4-5 = -1.

Problem 3

What is (-2)3+(-3)2?

  1. -17
  2. -1
  3. 1
  4. 17
  5. -8
Worked answer

1

An odd power of a negative number is negative: (-2)3 = -8.

An even power of a negative number is positive: (-3)2 = 9.

-8+9 = 1.

Problem 4

What is -23-(-34)?

  1. -1712
  2. -112
  3. 112
  4. 1712
  5. 17
Worked answer

112

Subtracting a negative is adding a positive, so this is -23+34.

Over the common denominator 12: -812+912 = 112.

The result is positive because 34 is the larger of the two fractions.

Problem 5

The temperature rises from -7 degrees to 5 degrees. By how many degrees did it rise?

  1. 2
  2. 7
  3. 12
  4. -12
  5. 35
Worked answer

12

The change is the final value minus the starting value: 5-(-7).

Subtracting a negative adds, so the rise is 5+7 = 12 degrees.

On a number line, going from -7 to 0 is 7 degrees and 0 to 5 is 5 more.

Problem 6

What is -5-(-9)?

  1. -14
  2. -4
  3. 4
  4. 14
  5. -45
Worked answer

4

Subtracting -9 is the same as adding 9: -5+9.

Starting at -5 and moving 9 to the right gives 4.

Problem 7

What is (-12)3?

  1. -18
  2. -32
  3. 18
  4. -16
  5. 38
Worked answer

-18

Cubing means multiplying three copies: (-12)(-12)(-12).

Three negative factors give a negative result, and 12 cubed is 18.

So the answer is -18, which is closer to zero than -12 because multiplying by a fraction shrinks the size.

Problem 8

Which of these numbers is the least?

  1. -3
  2. -2.5
  3. -12
  4. 0
  5. 14
Worked answer

-3

On a number line the least number is the one furthest to the left, and all three negatives lie left of zero.

Among negatives, the one with the largest size is the least: -3 is left of -2.5, which is left of -12.

So -3 is the least.

Problem 9

What is (-3)(4)(-2)?

  1. -24
  2. -14
  3. 14
  4. 24
  5. -9
Worked answer

24

Multiply in order: (-3)(4) = -12.

Then (-12)(-2) = 24, positive because two negative factors were used.

In general an even number of negative factors gives a positive product.

Problem 10

What is -22+(-2)2?

  1. -8
  2. -4
  3. 0
  4. 4
  5. 8
Worked answer

0

The two terms are not the same. In -22 the exponent applies only to the 2, so it means -(22) = -4.

In (-2)2 the brackets put the minus sign inside, so it means (-2)(-2) = 4.

-4+4 = 0. The brackets are the whole question.

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