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# Multiplication of Rational Numbers

## Multiply fractions: multiply straight across

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Multiplication of Real Numbers

Jacob received tips of $4.00 each from three of his paper route customers. How much did he receive in total? ### Multiplying Real Numbers Multiplication of two integers with the same signs produces a positive result and multiplication of two integers with unlike signs results in a negative answer. These rules can be applied to the multiplication of all real numbers. To multiply fractions, you multiply the numerators and then you multiply the denominators. The product of the numerators over the product of the denominators is the answer to the problem. Sometimes the answer can be expressed as an equivalent fraction. The rules for multiplying integers also apply to multiplying decimals. The sum of the number of digits after the decimal points determines the placement of the decimal point in the answer. #### Let's multiply real numbers, given an application: Sam spent$2.00 for a bottle of chocolate milk at the school cafeteria every school day. At the end of the week, how does this affect his net worth?

The result of \begin{align*}(+5) \times (-2)\end{align*} is –10. The product of a positive integer and a negative integer is always negative.

#### Now, let's practice multiplying real numbers:

What is \begin{align*}(-2) \times (-3)\end{align*}?

The result of \begin{align*}(-2) \times (-3)\end{align*} is +6. The product of two negative integers is always positive.

#### Let's practice multiplying fractional numbers:

1. \begin{align*}\left(\frac{2}{3}\right) \times \left(\frac{5}{7}\right)\end{align*}
• Remember, there are three simple steps to follow to multiply fractions:
• Multiply the numerators of the fractions
• Multiply the denominators of the fractions.
• Simplify the fraction if necessary.

\begin{align*}&= \frac{2 \times 5}{3 \times 7}\\ &= \frac{10}{21}\end{align*}

1. \begin{align*}\left(\frac{7}{8}\right) \times \left(3 \frac{3}{4}\right)\end{align*}

Express the mixed number as an improper fraction

\begin{align*}&= \left(\frac{7}{8}\right) \times \left(\frac{15}{4}\right) \rightarrow \frac{(4 \times 3)+3}{4}\\ &= \frac{7 \times 15}{8 \times 4}\\ &= \frac{105}{32}=3 \frac{9}{32} \end{align*}

1. \begin{align*}\left(5 \frac{3}{4}\right) \times \left(2 \frac{3}{5}\right)\end{align*}

Express the mixed numbers as improper fractions.

\begin{align*}&= \left(\frac{23}{4}\right) \times \left(\frac{13}{5}\right) \rightarrow \frac{(4 \times 5)+3}{4} \ \text{and} \ \frac{(5 \times 2)+3}{5}\\ &= \frac{23 \times 13}{4 \times 5}\\ &= \frac{299}{20}=14 \frac{19}{20} \end{align*}

#### Finally, let's multiply decimal real numbers:

\begin{align*}(14.65) \times (2.7)\end{align*}

Multiply the numbers as you would whole numbers. To place the decimal point in the answer, count the number of digits after the decimal points in the problem. There are two digits after the decimal point in 14.65 and one digit after the decimal point in 2.7. This is a total of three digits after the decimal points. From the right of the answer, count three places to the left and insert the decimal point.

\begin{align*}& 14.65\\ & \underline{\times \; 2.7 \;\;}\\ & \ \ 10255\\ & \underline{+29300}\\ & \ \ \underset{\quad \ {\color{red}\longleftarrow}}{39 {\color{red}.} 555}\end{align*}

### Examples

Earlier, you were told that Jacob received tips of 4.00 each from three of his paper route customers. How much did he receive in total? The product of \begin{align*}(+3) \times (+4)\end{align*} is +12. The product of two positive integers is always positive. #### Example 2 Multiply the following fractions: \begin{align*}\left(\frac{5}{9}\right) \times \left(\frac{-4}{7}\right)\end{align*} Multiply the numerators. Multiply the denominators. Simplify the fraction. \begin{align*}& \left(\frac{5}{9}\right) \times \left(\frac{-4}{7}\right)=\frac{5 \times (-4)}{9 \times 7}=-\frac{20}{63}\end{align*} The answer can be written as \begin{align*}\frac{-20}{63}\end{align*} or \begin{align*}-\frac{20}{63}\end{align*}. #### Example 3 Multiply the following fractions: \begin{align*}\left(3\frac{2}{3}\right) \times \left(4 \frac{1}{5}\right)\end{align*} Write the two mixed numbers as improper fractions. Multiply the denominator and the whole number. Add the numerator to this product. Write the answer over the denominator. Follow the steps for multiplying fractions. Simplify the fraction if necessary. \begin{align*}& \left(3 \frac{2}{3}\right) \times \left(4 \frac{1}{5}\right)\\ & \left(\frac{11}{3}\right) \times \left(\frac{21}{5}\right)\\ & \left(\frac{11}{3}\right) \times \left(\frac{21}{5}\right)=\frac{231}{15}=15 \frac{2}{5}\end{align*} #### Example 4 Multiply the real numbers: \begin{align*}(-135.697) \times (-34.32)\end{align*} Multiply the numbers as you would whole numbers. Remember the rule for multiplying integers. When you multiply two integers that have the same sign, the product will always be positive. \begin{align*}& \ \ -135.697\\ & \underline{\times \; -34.32 \;\;\;\;}\\ & \quad \quad \ \ 271394\\ & \quad \quad \ 407091 {\color{blue}0}\\ & \quad \ \ 542788 {\color{blue}00}\\ & \underline{\;\;\;\; 407091 {\color{blue}000} \;}\\ & \quad \underset{\quad \ {\color{red}\longleftarrow}}{4657{\color{red}.}12104}\end{align*} There are three digits after the decimal point in 135.697 and two digits after the decimal point in 34.32. Beginning at the right of the product, count five places to the left and insert the decimal point. ### Review Multiply. 1. \begin{align*}(-7) \times (-2)\end{align*} 2. \begin{align*}(+3) \times (+4)\end{align*} 3. \begin{align*}(-5) \times (+3)\end{align*} 4. \begin{align*}(+2) \times (-4)\end{align*} 5. \begin{align*}(+4) \times (-1)\end{align*} Match each numbered phrase with the correct letter for the corresponding multiplication statement. Then, determine each product. 1. take away six groups of 3 balls 2. net worth after losing seven5 bills
3. take away nine sets of 8 forks
4. take away four sets of four plates
5. receive eight groups of 4 glasses
6. buy seven sets of 12 placemats

1. \begin{align*}(+8) \times (+4)\end{align*}
2. \begin{align*}(+7) \times (-5)\end{align*}
3. \begin{align*}(-4) \times (+4)\end{align*}
4. \begin{align*}(-9) \times (+8)\end{align*}
5. \begin{align*}(+7) \times (+12)\end{align*}
6. \begin{align*}(-6) \times (+3)\end{align*}

Use the rules that you have learned for multiplying real numbers to answer the following problems.

1. \begin{align*}(-13) \times (-9)\end{align*}
2. \begin{align*}(-3.68) \times (82.4)\end{align*}
3. \begin{align*}\left(\frac{4}{9}\right) \times \left(\frac{5}{7}\right)\end{align*}
4. \begin{align*}\left(7 \frac{2}{3} \right) \times \left(6 \frac{1}{2}\right)\end{align*}
5. \begin{align*}(15.734) \times (-8.1)\end{align*}

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### Vocabulary Language: English

Associative Property

The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).

Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example $a+b=b+a \text{ and\,} (a)(b)=(b)(a)$.

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.

multiplicative identity property

The product of any number and one is the number itself.

Numerator

The numerator is the number above the fraction bar in a fraction.