Grade 7

Intermediate

7 min

Lesson 3 of 12

Multiplying and Dividing Rational Numbers

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Master the rules for multiplying and dividing signed rational numbers and understanding the properties of operations.

Imagine you are a deep-sea explorer. If your submarine descends at a rate of 10.5 feet per minute, where will you be in 5 minutes? Understanding how to multiply and divide 'signed' numbers allows you to predict the future of your bank account, your location, and even the temperature!

Learning Objectives

1.Determine the sign of a product or quotient based on the signs of the factors
2.Multiply and divide fractions and decimals with positive and negative signs efficiently
3.Explain why division by zero is undefined in the context of rational numbers

The Secret Code of Signs

When multiplying or dividing rational numbers, the most important step is determining the sign of your answer. Think of it as a simple code: if the signs are the same, the result is always positive. If the signs are different, the result is always negative. This rule applies whether you are working with whole numbers, fractions, or decimals. For example, multiplying two negative numbers like $(-3) \times (-4)$ results in a positive $12$ . Conversely, dividing a negative by a positive, like $-10 \div 2$ , results in a negative $-5$ . This consistency makes it easier to handle complex calculations without getting lost in the symbols.

Calculate the product of $(-5) \times 6$ .

1. Identify the signs: The first factor is negative $(-5)$ and the second is positive $(6)$ . 2. Determine the final sign: Since the signs are different, the answer must be negative. 3. Multiply the absolute values: $5 \times 6 = 30$ . 4. Combine: The final answer is $-30$ .

Quick Check

What is the sign of the result when you divide a negative number by another negative number?

Remember: Same signs result in a positive outcome.

Answer

Positive

Fractions and Decimals in Motion

Rational numbers aren't always clean integers; they often appear as fractions or decimals. To multiply fractions, simply multiply the numerators together and the denominators together:

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

. To divide, we multiply by the reciprocal (flip the second fraction):

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

. For decimals, multiply as you would with whole numbers and then place the decimal point by counting the total decimal places in the factors. Always apply the sign rules after you've calculated the numerical value.

Solve: $-\frac{3}{4} \div \frac{1}{2}$

1. Keep the first fraction: $-\frac{3}{4}$ . 2. Change division to multiplication and flip the second fraction: $\times \frac{2}{1}$ . 3. Multiply the numerators: $-3 \times 2 = -6$ . 4. Multiply the denominators: $4 \times 1 = 4$ . 5. Simplify: $-\frac{6}{4} = -\frac{3}{2}$ or $-1.5$ .

Quick Check

True or False: Multiplying $0.5 \times (-0.2)$ will result in a positive number.

Check if the signs are the same or different.

Answer

False

The 'No-Go' Zone: Division by Zero

In mathematics, division by zero is considered undefined. Why? Think of division as splitting a pie into groups. If you have 10 slices and want to put them into groups of 2, you get 5 groups. But if you try to put 10 slices into groups of size 0, you can never finish the task—no matter how many groups you 'create,' you still have 10 slices left! In the context of rational numbers,

\frac{p}{q}

is only a rational number if

q \neq 0

. Attempting to divide by zero breaks the logic of multiplication, as there is no number

x

such that

0 \times x = 10

Evaluate: $(-2.5) \times (\frac{4}{5}) \div (-2)$

1. Convert the decimal to a fraction for easier calculation: $-2.5 = -\frac{5}{2}$ . 2. Multiply the first two terms: $(-\frac{5}{2}) \times (\frac{4}{5}) = -\frac{20}{10} = -2$ . 3. Now divide by the last term: $-2 \div (-2)$ . 4. Apply sign rules: Since both are negative (same signs), the result is positive. 5. Final result: $1$ .

Key Takeaways

1Same signs (positive/positive or negative/negative) always result in a positive product or quotient.
2Different signs (positive/negative) always result in a negative product or quotient.
3To divide fractions, multiply by the reciprocal of the divisor.
4Division by zero is undefined because it is logically impossible to group items into sets of zero.

Quick Check

Multiple Choice

What is the value of $(-0.4) \times (-0.5)$ ?

Multiple Choice

Solve: $\frac{-2}{3} \div \frac{4}{9}$

True/False

The expression $\frac{5}{0}$ results in 0.

What's Next?

Review Tomorrow

In 24 hours, try to explain to a friend why a negative times a negative equals a positive, and why you can't divide by zero.

Practice Activity

Look at a weather report for a cold region. If the temperature drops by $2.5^{\circ}$ every hour for $4$ hours, use multiplication to find the total change.

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Multiplying and Dividing Rational Numbers

Learning Objectives

The Secret Code of Signs

Fractions and Decimals in Motion

The 'No-Go' Zone: Division by Zero

Key Takeaways

Quick Check

What's Next?

Multiplying and Dividing Rational Numbers

Learning Objectives

The Secret Code of Signs

Fractions and Decimals in Motion

The 'No-Go' Zone: Division by Zero

Key Takeaways

Quick Check

What's Next?