How to Divide and Multiply by Negative Numbers

Co-authored by Taylor Klein

Last Updated: December 26, 2024 References

Negative numbers have a value of less than zero. These numbers are found when working with the numbers on the left side of the number line. Negative numbers can be added, subtracted, multiplied, and divided just like positive numbers. However, special rules apply when carrying out operations with negative numbers. It is important to pay close attention to the numbers' signs when you divide and multiply by negative numbers.

Part 1

Part 1 of 3:

Dividing with Negative Numbers

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1
Divide a positive number by a negative number. To do this, divide the integers as usual, then place a negative sign in front of the quotient. A positive number divided by a negative number is always negative. This also is the rule when dividing a negative number by a positive number.^{[1]
X
Research source}
- For example:
  $10\div 5=2$
  $10\div -5=-2$
  $-10\div 5=-2$
2
Divide a negative number by a negative number. To do this, divide the integers as usual, and ignore the negative signs. A negative divided by a negative always equals a positive.^{[2]
X
Research source}
- For example:
  $10\div 5=2$
  $-10\div -5=2$
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$Step 3 Divide a positive fraction by a negative number.$

3
Divide a positive fraction by a negative number. To do this, divide the numbers as usual, then add a negative sign to the quotient. A positive number divided by a negative number will always be negative, regardless of whether the number is a whole number or a fraction. The same is true when dividing a negative number by a positive number. Remember that dividing by a number is the same as multiplying by its reciprocal.^{[3]
X
Research source}
- For example:
  ${\frac {5}{8}}\div -4$
  $={\frac {5}{8}}\div {\frac {-4}{1}}$
  $={\frac {5}{8}}\times {\frac {-1}{4}}$
  $={\frac {-5}{32}}$
$Step 4 Divide a negative fraction by a negative number.$

4
Divide a negative fraction by a negative number. To do this, divide the numbers as usual, and ignore the negative signs. A negative number divided by a negative number will always be positive, regardless of whether the number is a whole number or a fraction. Remember that dividing is the same as multiplying by the reciprocal.^{[4]
X
Research source}
- For example:
  ${\frac {-5}{8}}\div -4$
  $={\frac {-5}{8}}\div {\frac {-4}{1}}$
  $={\frac {-5}{8}}\times {\frac {-1}{4}}$
  $={\frac {5}{32}}$

Part 2

Part 2 of 3:

Multiplying with Negative Numbers

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1
Multiply a positive number by a negative number. To do this, multiply the integers as usual, then add a negative sign to the product. A positive number multiplied by a negative number is always negative.^{[5]
X
Research source}
- For example:
  $10\times 5=50$
  $-10\times 5=-50$
  $10\times -5=-50$
2
Multiply a negative number by a negative number. To do this, multiply the integers as usual, and ignore the negative signs. A negative number multiplied by a negative number is always positive.^{[6]
X
Research source}
- For example:
  $10\times 5=50$
  $-10\times -5=50$
$Step 3 Multiply a positive fraction by a negative number.$

3
Multiply a positive fraction by a negative number. To do this, multiply the numbers as usual, then add a negative sign to the product. A positive number times a negative number will always be negative, regardless of whether the number is a whole number or a fraction.^{[7]
X
Research source}
- For example:
  ${\frac {5}{8}}\times -4$
  $={\frac {5}{8}}\times {\frac {-4}{1}}$
  $={\frac {-20}{8}}$
$Step 4 Multiply a negative fraction by a negative number.$

4
Multiply a negative fraction by a negative number. To do this, multiply the numbers as usual, and ignore the negative signs. A negative number times a negative number will always be positive, regardless of whether the number is a whole number or a fraction.^{[8]
X
Research source}
- For example:
  ${\frac {-5}{8}}\times -4$
  $={\frac {-5}{8}}\times {\frac {-4}{1}}$
  $={\frac {20}{8}}$

Part 3

Part 3 of 3:

Solving Sample Problems

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1
Try this problem. $224\div -7$ $224\div -7$
- Remember that a positive number divided by a negative number will equal a negative number.
- Since $224\div 7=32$ , you know that $224\div -7=-32$ .
2
Try this problem. A peregrine falcon can dive (lose height) at a rate of 320 km/hr. Assuming it can sustain this rate indefinitely, how long would it take a peregrine to reach a height of -240 km?
- Remember that a negative number (-240km) divided by a negative number (-320km/hr) will equal a positive number (number of hours).
- Since $240\div 320=.75$ , you know that $-240\div -320=.75$ . So the falcon would take 0.75 hours, or about 45 minutes, to dive 240 km.
3
Try this problem. ${\frac {7}{10}}\div -6$ ${\frac {7}{10}}\div -6$
- Remember that a positive fraction divided by a negative number will equal a negative number.
- Since ${\frac {7}{10}}\div 6={\frac {7}{10}}\times {\frac {1}{6}}={\frac {7}{60}}$ , you know that ${\frac {7}{10}}\div -6={\frac {-7}{60}}$ .
4
Try this problem. ${\frac {-5}{6}}\div -3$ ${\frac {-5}{6}}\div -3$
- Remember that a negative fraction divided by a negative number will equal a positive number.
- Since ${\frac {5}{6}}\div 3={\frac {5}{6}}\times {\frac {1}{3}}={\frac {5}{18}}$ , you know that ${\frac {-5}{6}}\div -3={\frac {5}{18}}$ .
5
Try this problem. Jason spends 5 dollars on donuts every morning. How much money will he lose on donuts after 5 days?
- Remember that a positive number (5 days) multiplied by a negative number (-5 dollars) will equal a negative number (money lost).
- Since $5\times 5=25$ , you know that $5\times -5=-25$ . So Jason loses $25 after 5 days of buying donuts.
6
Try this problem. $-12\times -5$ $-12\times -5$
- Remember that a negative number times a negative number will always equal a positive number.
- Since $12\times 5=60$ , you know that $-12\times -5=60$ .
7
Try this problem. Rebecca has a whole pie in her refrigerator. Over the course of three days, her house guest sneaks into the kitchen and eats ${\frac {1}{6}}$ ${\frac {1}{6}}$ of the pie. How much pie has Rebecca lost?
- Remember that a negative fraction ( ${\frac {-1}{6}}$ of a pie) times a positive number (3 days), will equal a negative number (amount of pie eaten).
- Since ${\frac {1}{6}}\times 3={\frac {3}{6}}={\frac {1}{2}}$ , you know that ${\frac {-1}{6}}\times 3={\frac {-1}{2}}$ . So Rebecca has lost half of her pie.
8
Try this problem. ${\frac {-4}{7}}\times -7$ ${\frac {-4}{7}}\times -7$
- Remember that a negative fraction times a negative number will equal a positive number.
- Since ${\frac {4}{7}}\times 7={\frac {28}{7}}=4$ , you know that ${\frac {-4}{7}}\times -7=4$

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How is 240 possibly divided by 320?

Donagan

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When a smaller number is divided by a larger number, the quotient is less than 1.

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What about a fraction multiplied by a negative number?

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Put a "one" under the negative number, leaving the negative sign ON TOP, and then multiply the fraction as usual.

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How do you multiply a negative number by a negative number?

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Multiplying a negative number by a negative number results in a positive number. For example, (-4)(-5) = 20 (that is, +20).

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Multiplying two negative signs creates a positive sign. Therefore, you can count the number of negative signs when you have a list of numbers to multiply. The answer will be positive if you have an even number of negative signs. The answer will be negative if you have an odd number of negative signs.

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