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This lesson will cover methods for multiplying fractions by whole numbers, fractions, and mixed numbers. Along with these methods, estimation strategies will be used to assess whether the products are reasonable.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

Paulina runs for two-sixths of her free time each day.

External credits: pikisuperstar

If Paulina has $5$ hours of free time per day, how many hours does she run in $4$ days? Give the exact answer.

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Discussion

The product of two fractions is equal to the product of the numerators divided by the product of the denominators. The resulting fraction is then simplified to its lowest terms, if possible.

$ba ⋅dc =b⋅da⋅c $

$65 ⋅43 $

The result of this multiplication can be found in three steps.
1

Multiply the Numerators

2

Multiply the Denominators

The product of the denominators is $6⋅4=24.$

3

Simplify if Possible

Note that $3$ is the greatest common factor of $15$ and $24.$ Divide both the numerator and the denominator by $3$ to simplify the fraction.
Therefore, the product of $65 $ and $43 $ simplified to its lowest terms is $85 .$

Example

Paulina drinks one-third of the water in her bottle before PE class. During class, she drinks ten-twelfths of the remaining water.

a Write a numeric expression to represent the amount of water in the bottle that Paulina drinks during class.

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b Which of the following is the best estimate for the value of the expression written in Part A?

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c Find the value of the expression found in Part A. Simplify the answer if possible.

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a What fraction of the bottle is full before the class? What mathematical operation must be performed to find ten twelfths of a number?

b If a fraction is greater than or equal to $21 ,$ round the fraction to $1.$ If a fraction is less than $21 ,$ round it to $0.$

c To multiply fractions, start with the multiplication of the numerators, followed by the multiplication of the denominators. Then, simplify the resulting fractions.

a Start by finding what fraction of the bottle is full before the physical education class. To do so, subtract one-third from $1.$

$Expression1210 ×32 $

A number line can be used to help with this concept. First, divide the number line between $0$ and $1$ into thirds. Fill in $32 ,$ representing the water that remains in the bottle before class. Then, divide this section into $12$ smaller equal parts and color in $10$ of them.
Notice that each third is divided into six smaller equal parts. The other third can also be divided into six parts. This will make it easier to find what fraction of the whole the red part represents.

The red part shows what fraction of the water bottle Paulina drinks during class.

b Estimation is a great tool to get a rough idea of the result of an operation — in this case, the result of a multiplication. Consider the value of each fraction in the expression separately.

$1210 ×32 $

Now compare each fraction with $21 .$ These fractions do Rewrite | Compare with $21 ,$ or $126 $ | |
---|---|---|

$1210 $ | $1210 $ | $1210 >126 $ |

$32 $ | $3⋅42⋅4 =128 $ | $128 >126 $ |

$Product:Estimate: 1210 ↓1 ×× 32 ↓1 $

Since $1$ is the identity element of multiplication, this product is also equal to $1.$ However, that does not say much about the original product, and $1$ is not one of the possible answer options. Now think of rounding only one of the fractions to $1$ one at a time. Estimate for $1210 ×32 $ | |
---|---|

Round $1210 $ to $1$ | $1×32 =32 $ |

Round $32 $ to $1$ | $1210 ×1=1210 $ |

The fraction $1210 $ can be simplified to $65 .$ The answer is either $32 $ or $65 .$ The fractions in the options are in simplest form and $32 $ is among the options. Therefore, the answer is $32 .$

c To multiply the fractions, start with the multiplication of the numerators, followed by the multiplication of the denominators. Then, simplify the resulting fraction.

$Result95 ≈ Estimate96 $

Since the actual result and the estimate are close to each other, the answer $95 $ is reasonable.
In addition to a number line, the product can also be represented by an area model. In this model, the denominators are used to divide a rectangular diagram into smaller parts. Consider modeling the following product.

$1210 ×32 $

The denominators are $12$ and $3.$ The rectangle is then divided into $12$ columns and $3$ rows.
The numerators determine which parts will be shaded. Since $10$ is the numerator of $1210 ,$ $10$ of the $12$ columns will be shaded. Similarly, $2$ of the $3$ rows will be shaded.

In this model, the overlapping region represents the product. For this example, the product is $3620 =95 .$

Pop Quiz

The product of two fractions is equal to the product of the numerators divided by the product of the denominators. Practice finding the product of fractions. Simplify the answer to its lowest terms.

Discussion

When multiplying fractions by whole numbers or mixed numbers, both factors should be in the form of a proper fraction or an improper fraction.

$9×272 $

Rewrite

Rewrite $9$ as $19 $

$19 ×272 $

$2718 $

ReduceFrac

$ba =b/9a/9 $

$27/918/9 $

SimpQuot

Simplify quotient

$32 $

$354 {whole number:3proper fraction:54 $

The mixed fraction must be converted into an improper fraction before it can be multiplied by a fraction. The multiplication process can be better understood with the help of an example.
$354 ×31 $

Write mixed number as a fraction

$519 ×31 $

$1519 $

$1519 $

Write mixed number as a fraction

Rewrite

Rewrite $19$ as $15+4$

$1515+4 $

WriteSumFrac

Write as a sum of fractions

$1515 +154 $

CalcQuot

Calculate quotient

$1+154 $

Rewrite

Rewrite $1+154 $ as $1154 $

$1154 $

Example

Paulina's PE class lasts for $1125 $ hours. The table shows what fraction of the class time is allocated for various activities.

Activity | Fraction |
---|---|

Warm-up | $51 $ |

Instruction | $21 $ |

Playing a game | $101 $ |

Cool-down | $51 $ |

a Find how many minutes are devoted to warming up and cooling down.

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b Find how many minutes are devoted to the activities other than instruction.

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a Start by determining what fraction of the class time is allocated for warm-up and cool-down. To do so, add the fractions corresponding to the activities. Multiply the portion of the lesson spent on those activities by the total class time. Finally, use the fact that $1$ hour is $60$ minutes.

b Determine what fraction of the class time is allocated for the activities other than instruction.

a The fractions for both the warm-up and the cool-down to find the amount of time spent on them. Take a look at the given table.

Part of $1125 $-hour Class | ||
---|---|---|

Activity | Fraction | |

$Warm-up$ | $51 $ | |

Instruction | $21 $ | |

Playing game | $101 $ | |

$Cool-down$ | $51 $ |

$51 +51 =52 $

Next, the time spent on warming up and cooling down will be found by multiplying this fraction number by the lesson time $1125 .$ This will give the amount of time in hours.
$52 ×1125 $

This is a multiplication of a fraction by a mixed number. Before performing the multiplication, the mixed number will be rewritten as an improper fraction.
$52 ×1125 $

MixedToFrac

$acb =ca⋅c+b $

$52 ×121⋅12+5 $

MultByOne

$a⋅1=a$

$52 ×1212+5 $

AddTerms

Add terms

$52 ×1217 $

$60/234/2 =3017 $

The fraction simplifies to $3017 .$ This means that $3017 $ of an hour is spent on warming up and cooling down. Now the number of minutes spent on these activities will be found by multiplying the fraction by $60$ because $1$ hour is $60$ minutes.
$60×3017 $

This is a multiplication of a whole number by an improper fraction. Start by rewriting $60$ as an improper fraction, then multiply the fractions.
$60×3017 $

Rewrite

Rewrite $60$ as $160 $

$160 ×3017 $

MultFrac

Multiply fractions

$1×3060×17 $

Multiply

Multiply

$301020 $

CalcQuot

Calculate quotient

$34$

b Consider the given table again, this time focusing on the instruction time.

Part of $1125 $-hour Class | ||
---|---|---|

Activity | Fraction | |

$Warm-up$ | $51 $ | |

$Instruction$ | $21 $ | |

Playing game | $101 $ | |

$Cool-down$ | $51 $ |

$21 ×1125 $

Rewrite the mixed number as an improper fraction to find the product.
$21 ×1125 $

Write mixed number as a fraction

$21 ×1217 $

$60×2417 $

This is a multiplication of a whole number by a fraction. An easy way to find this product is to move the whole number to the numerator of the fraction as a factor.