Lesson 3: Multiplying and Dividing Rational Numbers

Property

The final position of an object starting at zero can be found by multiplying its velocity by the time it travels.
Velocity is a signed number representing speed and direction, while time is a positive quantity.

Property

The multiplication $a \times b$ can be modeled on a number line.
The expression represents $|a|$ groups of size $b$. Start at 0 and make $|a|$ jumps of size $b$.
If $a$ is positive, the final position is in the same direction as the jumps.
If $a$ is negative, the final position is in the opposite direction of the jumps.

Examples

Property

Just as $3 \times 5$ can be understood as $(5) + (5) + (5) = 15$, so $3 \times (-5)$ can be understood as $(-5) + (-5) + (-5) = -15$.
For a product like $(-3) \times 5$, one way is to recognize that it is the same as $5 \times (-3)$ (e.g. the commutative property).
Another is to understand $-3$ as the 'opposite' of 3, so $(-3) \times 5$ is the opposite of $3 \times 5$, which is $-15$.

Examples

• Calculate $6 \times (-4)$. This is equivalent to adding $-4$ six times: $(-4) + (-4) + (-4) + (-4) + (-4) + (-4) = -24$.
• To find $(-8) \times 5$, you can find the opposite of $8 \times 5$. Since $8 \times 5 = 40$, the opposite is $-40$. Therefore, $(-8) \times 5 = -40$.
• A submarine's depth increases by 40 feet per minute (represented as $-40$). After 4 minutes, its total change in depth is $4 \times (-40) = -160$ feet.

Explanation

When multiplying a positive and a negative integer, the result is always negative. You can think of it as repeated addition of a negative number, or as finding the opposite of what the positive product would be. The signs are different, so the answer is negative.

Property

Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number.
Since division by 5 is the inverse of multiplication by 5, the equation $(-15) \div 5 = x$ tells us that the solution is some number $x$, which when multiplied by 5 gives us $-15$.

Examples

• Calculate $(-48) \div 8$. This problem asks, 'What number multiplied by $8$ equals $-48$?' Since $8 \times (-6) = -48$, the answer is $-6$.
• Calculate $(-63) \div (-9)$. Since a negative divided by a negative results in a positive, the answer is positive. $63 \div 9 = 7$, so $(-63) \div (-9) = 7$.
• A company lost a total of 5,000 dollars over 10 months. To find the average monthly loss, calculate $(-5000) \div 10 = -500$. The average loss was 500 dollars per month.

Explanation

Division asks, 'What number do I multiply the divisor by to get the dividend?' The rules for signs in division are the same as in multiplication. For example, to solve $(-30) \div 6$, you ask $6 \times ? = -30$. The answer is $-5$.