 # 2.04 Practical problems with fraction operations

Lesson

We use fractions to solve many everyday problems. For example, in recipes, ingredients are often measured in fractions of a cup. If we wanted to know the total volume of the ingredients, we could use fraction addition.

Here are some tips for applying fractions to real world problems:

• When we describe equal parts out of a whole, we can write the situation as a fraction. For example, if a prize was split $6$6 ways, each recipient would get $\frac{1}{6}$16 of the total.
• The denominator is the total number of parts. In some cases, we can find it by adding together all of the parts. For example, if Mick picked $4$4 strawberries and Rachel picked $5$5, then Mick picked $\frac{4}{4+5}=\frac{4}{9}$44+5=49 of the strawberries.
• If we want to find a fraction of a quantity, we can multiply the fraction by the quantity. This works if the quantity is a fraction as well. For example, if we want to find $\frac{2}{3}$23 of $\frac{1}{10}$110 of a minute in seconds, we would find $\frac{2}{3}\times\frac{1}{10}\times60$23×110×60.
• Fractions are also a way to write division. If a piece of timber was divided into $5$5 parts, each part would be $\frac{1}{5}$15 of the original piece.
• Improper fractions and mixed numbers can be used to represent more than one whole. For example, if Francisco ran one lap around a track and then ran another third of the track, he has run $\frac{4}{3}$43 or $1\frac{1}{3}$113 laps.

#### Worked example

##### question 1

Juan wants to bring some of his electronics to a friends house in his backpack.  His iPad weighs $1\frac{3}{5}$135 pounds, his PS4 weighs $7\frac{1}{10}$7110 pounds, and his laptop weighs $3\frac{1}{2}$312 pounds  If the bag itself only weighs $\frac{1}{4}$14 of a pound, how many pounds will his backpack be with the iPad, PS4, and the laptop in it?

Think:  We will need to add up the weights of all of these items to determine the total weight.  So we will need to find the least common denominator for each fraction. We have denominators of $10$10, $5$5, $2$2 and $4$4. If we consider the multiples of all of these, we would find that $20$20 is the least common multiple.

Do:

 $1\frac{3}{5}+7\frac{1}{10}+3\frac{1}{2}+\frac{1}{4}$135​+7110​+312​+14​ $=$= $1\frac{12}{20}+7\frac{2}{20}+3\frac{10}{20}+\frac{5}{20}$11220​+7220​+31020​+520​ The LCD for the four fractions is $20$20. $=$= $1+7+3+\frac{12}{20}+\frac{2}{20}+\frac{10}{20}+\frac{5}{20}$1+7+3+1220​+220​+1020​+520​ Adding the whole numbers and fractions separately $=$= $11+\frac{29}{20}$11+2920​ Sum of the whole numbers and fractions that now have a LCD $=$= $12\frac{9}{20}$12920​ Simplify by changing the improper fraction to a mixed number

So, the backpack with all of Juan's equipment will weigh $12\frac{9}{20}$12920 pounds

Reflect:  Notice that the weight of the bag itself needed to be factored into this.  So, we were adding four mixed numbers.  Also recall that to express a mixed number in simplest form, the fractional component must be a proper fraction.  That required $\frac{29}{20}$2920to be rewritten as $1\frac{9}{20}$1920 and then the whole number $1$1 was added to the whole number $11$11 in order to get the final answer.

#### Practice questions

##### Question 2

Carl has $\frac{3}{7}$37m of ribbon. After he uses some ribbon for a present, he has $\frac{1}{4}$14m left.

How much ribbon did he use on the present?

##### Question 3

At a party, Bill makes a drink by combining $5\frac{1}{3}$513L of water with $1\frac{1}{2}$112L of cordial.

What is the total amount of the drink as a mixed number?

##### Question 4

Jack is making bags for his friends. He has $3\frac{1}{2}$312m of fabric.

If each bag requires $\frac{2}{5}$25m of fabric, how many bags can he make?

### Outcomes

#### 6.NS.1

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g. By using visual fraction models and equations to represent the problem.

#### 6.NS.2

Fluently divide multi-digit numbers using the standard algorithm