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Multiplication of Rational Numbers

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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?

Watch This

Khan Academy Multiplying Positive and Negative Numbers

Khan Academy Multiplying Fractions

Khan Academy Multiplication 8: Multiplying Decimals

Guidance

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.

Example A

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?

Solution: The result of (+5) \times (-2) is –10. The product of a positive integer and a negative integer is always negative.

Example B

What is (-2) \times (-3) ?

Solution: The result of (-2) \times (-3) is +6. The product of two negative integers is always positive.

Example C

i) \left(\frac{2}{3}\right) \times \left(\frac{5}{7}\right)

ii) \left(\frac{7}{8}\right) \times \left(3 \frac{3}{4}\right)

iii) \left(5 \frac{3}{4} \right) \times \left(2 \frac{3}{5}\right)

Solution: Remember, there are three simple steps to follow to multiply fractions:

  1. Multiply the numerators of the fractions
  2. Multiply the denominators of the fractions.
  3. Simplify the fraction if necessary.

i) \left(\frac{2}{3}\right) \times \left(\frac{5}{7}\right)

&= \frac{2 \times 5}{3 \times 7}\\&= \frac{10}{21}

ii) \left(\frac{7}{8}\right) \times \left(3 \frac{3}{4}\right) Express the mixed number as an improper fraction.

&= \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}

iii) \left(5 \frac{3}{4}\right) \times \left(2 \frac{3}{5}\right) Express the mixed numbers as improper fractions.

&= \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}

Example D

(14.65) \times (2.7)

Solution: 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.

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

Concept Problem Revisited

Jacob received tips of $4.00 each from three of his paper route customers. How much did he receive in total?

The result of (+3) \times (+4) is +12. The product of two positive integers is always positive.

Vocabulary

Denominator
The denominator of a fraction is the number on the bottom that indicates the total number of equal parts in the whole or the group. \frac{5}{8} has denominator 8.
Fraction
A fraction is any rational number that is not an integer.
Improper Fraction
An improper fraction is a fraction in which the numerator is larger than the denominator. \frac{8}{3} is an improper fraction .
Integer
All natural numbers, their opposites, and zero are integers . A number 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} .
Numerator
The numerator of a fraction is the number on top that is the number of equal parts being considered in the whole or the group. \frac{5}{8} has numerator 5.

Guided Practice

Multiply the following fractions:

1. \left(\frac{5}{9}\right) \times \left(\frac{-4}{7}\right)

2. \left(3\frac{2}{3}\right) \times \left(4 \frac{1}{5}\right)

3. Determine the answer to (-135.697) \times (-34.32)

Answers:

1. Multiply the numerators. Multiply the denominators. Simplify the fraction.

& \left(\frac{5}{9}\right) \times \left(\frac{-4}{7}\right)=\frac{5 \times (-4)}{9 \times 7}=-\frac{20}{63}

The answer can be written as \frac{-20}{63} or -\frac{20}{63} .

2. 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.

& \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}

3. 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.

& \ \ -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}

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.

Practice

Multiply.

  1. (-7) \times (-2)
  2. (+3) \times (+4)
  3. (-5) \times (+3)
  4. (+2) \times (-4)
  5. (+4) \times (-1)

Match each given phrase with the correct multiplication statement. Then, determine each product.

  1. take away six groups of 3 balls
  2. net worth after losing seven $5 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. (+8) \times (+4)
  2. (+7) \times (-5)
  3. (-4) \times (+4)
  4. (-9) \times (+8)
  5. (+7) \times (+12)
  6. (-6) \times (+3)

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

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

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