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Guias de estudo > Mathematics for the Liberal Arts Corequisite

Modeling and Finding Equivalent Fractions

Learning Outcomes

  • Use fraction tiles or visual aids to create equivalent fractions
  • Find an equivalent fraction given a fraction
Let’s think about Andy and Bobby and their favorite food again. If Andy eats 12{\Large\frac{1}{2}} of a pizza and Bobby eats 24{\Large\frac{2}{4}} of the pizza, have they eaten the same amount of pizza? In other words, does 12=24?{\Large\frac{1}{2}}={\Large\frac{2}{4}}? We can use fraction tiles to find out whether Andy and Bobby have eaten equivalent parts of the pizza.

Equivalent Fractions

Equivalent fractions are fractions that have the same value.
Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of the fraction tiles shown earlier and extend it to include eighths, tenths, and twelfths. Start with a 12{\Large\frac{1}{2}} tile. How many fourths equal one-half? How many of the 14{\Large\frac{1}{4}} tiles exactly cover the 12{\Large\frac{1}{2}} tile?

One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into four pieces, each labeled as one fourth. Since two 14{\Large\frac{1}{4}} tiles cover the 12{\Large\frac{1}{2}} tile, we see that 24{\Large\frac{2}{4}} is the same as 12{\Large\frac{1}{2}}, or 24=12{\Large\frac{2}{4}}={\Large\frac{1}{2}}.

How many of the 16{\Large\frac{1}{6}} tiles cover the 12{\Large\frac{1}{2}} tile?

One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into six pieces, each labeled as one sixth. Since three 16{\Large\frac{1}{6}} tiles cover the 12{\Large\frac{1}{2}} tile, we see that 36{\Large\frac{3}{6}} is the same as 12{\Large\frac{1}{2}}.

So, 36=12{\Large\frac{3}{6}}={\Large\frac{1}{2}}. The fractions are equivalent fractions.  

Example

Use fraction tiles to find equivalent fractions. Show your result with a figure.
  1. How many eighths (18{\Large\frac{1}{8}}) equal one-half (12{\Large\frac{1}{2}})?
  2. How many tenths (110{\Large\frac{1}{10}}) equal one-half (12{\Large\frac{1}{2}})?
  3. How many twelfths (112{\Large\frac{1}{12}}) equal one-half (12{\Large\frac{1}{2}})?
Solution 1. It takes four 18{\Large\frac{1}{8}} tiles to exactly cover the 12{\Large\frac{1}{2}} tile, so 48=12{\Large\frac{4}{8}}={\Large\frac{1}{2}}. One long, undivided rectangle is shown, labeled 1. Below it is an identical rectangle divided vertically into two pieces, each labeled 1 half. Below that is an identical rectangle divided vertically into eight pieces, each labeled 1 eighth. 2. It takes five 110{\Large\frac{1}{10}} tiles to exactly cover the 12{\Large\frac{1}{2}} tile, so 510=12{\Large\frac{5}{10}}={\Large\frac{1}{2}}. One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into ten pieces, each labeled as one tenth. 3. It takes six 112{\Large\frac{1}{12}} tiles to exactly cover the 12{\Large\frac{1}{2}} tile, so 612=12{\Large\frac{6}{12}}={\Large\frac{1}{2}}. One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into twelve pieces, each labeled as one twelfth.
Suppose you had tiles marked 120{\Large\frac{1}{20}}. How many of them would it take to equal 12{\Large\frac{1}{2}}? Are you thinking ten tiles? If you are, you’re right, because 1020=12{\Large\frac{10}{20}}={\Large\frac{1}{2}}. We have shown that 12,24,36,48,510,612{\Large\frac{1}{2},\frac{2}{4},\frac{3}{6},\frac{4}{8},\frac{5}{10},\frac{6}{12}}, and 1020{\Large\frac{10}{20}} are all equivalent fractions.

Try it

[ohm_question height="270"]146001[/ohm_question]

Find Equivalent Fractions

We used fraction tiles to show that there are many fractions equivalent to 12{\Large\frac{1}{2}}. For example, 24,36{\Large\frac{2}{4},\frac{3}{6}}, and 48{\Large\frac{4}{8}} are all equivalent to 12{\Large\frac{1}{2}}. When we lined up the fraction tiles, it took four of the 18{\Large\frac{1}{8}} tiles to make the same length as a 12{\Large\frac{1}{2}} tile. This showed that 48=12{\Large\frac{4}{8}}={\Large\frac{1}{2}}. See the previous example. We can show this with pizzas, too. Image (a) shows a single pizza, cut into two equal pieces with 12{\Large\frac{1}{2}} shaded. Image (b) shows a second pizza of the same size, cut into eight pieces with 48{\Large\frac{4}{8}} shaded. Two pizzas are shown. The pizza on the left is divided into 2 equal pieces. 1 piece is shaded. The pizza on the right is divided into 8 equal pieces. 4 pieces are shaded. This is another way to show that 12{\Large\frac{1}{2}} is equivalent to 48{\Large\frac{4}{8}}. How can we use mathematics to change 12{\Large\frac{1}{2}} into 48{\Large\frac{4}{8}}? How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as:

1⋅42⋅4=48{\Large\frac{1\cdot\color{blue}{4}}{2\cdot\color{blue}{4}}}={\Large\frac{4}{8}}

These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

Equivalent Fractions Property

If a,ba,b, and cc are numbers where b≠0b\ne 0 and c≠0c\ne 0, then ab=a⋅cb⋅c{\Large\frac{a}{b}}={\Large\frac{a\cdot c}{b\cdot c}}
When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.

1⋅32⋅3=36{\Large\frac{1\cdot\color{blue}{3}}{2\cdot\color{blue}{3}}}={\Large\frac{3}{6}} so 12=36{\Large\frac{1}{2}}={\Large\frac{3}{6}}

1⋅22⋅2=24{\Large\frac{1\cdot\color{blue}{2}}{2\cdot\color{blue}{2}}}={\Large\frac{2}{4}} so 12=24{\Large\frac{1}{2}}={\Large\frac{2}{4}}

1⋅102⋅10=1020{\Large\frac{1\cdot\color{blue}{10}}{2\cdot\color{blue}{10}}}={\Large\frac{10}{20}} so 12=1020{\Large\frac{1}{2}}={\Large\frac{10}{20}}

So, we say that 12,24,36{\Large\frac{1}{2},\frac{2}{4},\frac{3}{6}}, and 1020{\Large\frac{10}{20}} are equivalent fractions.

Example

Find three fractions equivalent to 25{\Large\frac{2}{5}}.

Answer: Solution To find a fraction equivalent to 25{\Large\frac{2}{5}}, we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by 2,32,3, and 55. 2⋅25⋅2=410[/latex]     [latex]2⋅35⋅3=615[/latex]     [latex]2⋅55⋅5=1025{\Large\frac{2\cdot\color{blue}{2}}{5\cdot\color{blue}{2}}}={\Large\frac{4}{10}}[/latex]          [latex]{\Large\frac{2\cdot\color{blue}{3}}{5\cdot\color{blue}{3}}}={\Large\frac{6}{15}}[/latex]          [latex]{\Large\frac{2\cdot\color{blue}{5}}{5\cdot\color{blue}{5}}}={\Large\frac{10}{25}} So, 410,615{\Large\frac{4}{10},\frac{6}{15}}, and 1025{\Large\frac{10}{25}} are equivalent to 25{\Large\frac{2}{5}}.

 

Try it

Find three fractions equivalent to 35{\Large\frac{3}{5}}.

Answer: Correct answers include 610,915,and 1220{\Large\frac{6}{10},\frac{9}{15}},\text{and }{\Large\frac{12}{20}}.

  Find three fractions equivalent to 45{\Large\frac{4}{5}}.

Answer: Correct answers include 810,1215,and 1620{\Large\frac{8}{10},\frac{12}{15}},\text{and }{\Large\frac{16}{20}}.

 

Example

Find a fraction with a denominator of 2121 that is equivalent to 27{\Large\frac{2}{7}}.

Answer: Solution To find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in 2121. Since we can multiply 77 by 33 to get 2121, we can find the equivalent fraction by multiplying both the numerator and denominator by 33.

27=2⋅37⋅3=621{\Large\frac{2}{7}}={\Large\frac{2\cdot\color{blue}{3}}{7\cdot\color{blue}{3}}}={\Large\frac{6}{21}}

 

Try it

[ohm_question height="270"]146005[/ohm_question]
In the following video we show more examples of how to find an equivalent fraction given a specific denominator. https://youtu.be/8gJS0kvtGFU

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