Multiplying and Dividing Integers: Additional Practice (Set 2)

Seven minutes of a $-3$ meter change each is repeated addition of a negative, which is multiplication.

$7 \times (-3) = -21$

The depth changes by $-21$ meters, that is, $21$ meters downward from the start.

Multiply before adding. The products are $(-9) \times 6 = -54$ and $4 \times (-5) = -20$.

$-54 + (-20) = -74$

Both products are negative, so their distances add and the sum stays negative.

Work left to right. First the quotient: $\frac{-90}{-9} = 10$, since same signs give a positive result.

$10 \times (-2) = -20$

Then multiplying the positive $10$ by $-2$ gives $-20$.

The product $72$ is positive while $-8$ is negative, so the missing factor must be negative to make the signs match. Divide the sizes.

$72 \div (-8) = -9$

Check: $(-9) \times (-8) = 72$, so the missing factor is $-9$.

Multiply the sizes and count the negatives. The sizes give $1 \times 2 \times 3 \times 4 \times 5 = 120$, and there are five negative factors, an odd count.

$(-1) \times (-2) \times (-3) \times (-4) \times (-5) = -120$

An odd number of negatives makes the product negative.

The descent is $5 \times (-15) = -75$ meters, and the rise adds $25$ meters.

$-75 + 25 = -50$

The total change is $-50$ meters, that is, $50$ meters lower than where it started.

Multiply before subtracting. The products are $(-7) \times 3 = -21$ and $(-4) \times (-5) = 20$.

$-21 - 20 = -41$

Subtracting $20$ from $-21$ moves farther left to $-41$.

Multiply the sizes and count the negatives. The sizes give $2 \times 2 \times 2 \times 2 \times 2 = 32$, and there are five negative factors, an odd count.

$(-2)\;\text{multiplied five times} = -32$

An odd number of negatives makes the product negative.

Substitute the values, then multiply before subtracting. The product $a \times b = (-4) \times (-3) = 12$.

$12 - (-3) = 12 + 3 = 15$

Subtracting $b = -3$ adds $3$, giving $15$.

Dividing by zero asks for a number that, times $0$, gives $-9$. No number times zero equals $-9$, so no quotient exists.

$\frac{-9}{0} \;\text{is undefined}$

The others are all defined: $\frac{-9}{3} = -3$, $\frac{-9}{-3} = 3$, and $\frac{0}{-9} = 0$. Only division by zero fails.

Finding the unknown factor is a division: divide the product by the known factor.

$(-36) \div (-9) = 4$

The signs match, so the quotient is positive. Check: $(-9) \times 4 = -36$, so the other integer is $4$.

Finding the unknown factor is a division: divide the product by the known factor.

$(-48) \div 6 = -8$

The signs differ, so the quotient is negative. Check: $6 \times (-8) = -48$, so the other integer is $-8$.