1.5 Visualize fractions (Page 2/12)

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Equivalent fractions property

If $a, b, c$ are numbers where $b \neq 0, c \neq 0,$ then

\frac{a}{b} = \frac{a \cdot c}{b \cdot c}

If we had cut the pizza differently, we could get

$An image shows three rows of fractions. In the first row are the fractions “1, times 2, divided by 2, times 2, equals two fourths”. Next to this is the word “so” and the fraction “one half, equals two fourths. The second row reads “1, times 3, divided by 2 times 3, equals three sixths”. Next to this is the word “so” and the fraction “one half equals, three sixths”. The third row reads “1 times 10, divided by 2 times 10, ten twentieths”. Next to this is the word “so” and the fraction “one half equals, ten twentieths”.$

So, we say $\frac{1}{2}, \frac{2}{4}, \frac{3}{6}, and \frac{10}{20}$ are equivalent fractions.

Doing the Manipulative Mathematics activity “Equivalent Fractions” will help you develop a better understanding of what it means when two fractions are equivalent.

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

Solution

To find a fraction equivalent to $\frac{2}{5},$ we multiply the numerator and denominator by the same number. We can choose any number, except for zero. Let’s multiply them by 2, 3, and then 5.
$A row of fractions reads “2 times 2, divided by 5 times 2, equals four tenths”. Next to this is “2, times 3, divided by 5 times 3, equals six fifteenths”. Next to this is “2 times 5, divided by 5 times 5, equals ten twenty-fifths”.$
So, $\frac{4}{10}, \frac{6}{15}, and \frac{10}{25}$ are equivalent to $\frac{2}{5} .$

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

$\frac{6}{10}, \frac{9}{15}, \frac{12}{20};$ answers may vary

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

$\frac{8}{10}, \frac{12}{15}, \frac{16}{20};$ answers may vary

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Simplify fractions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.

For example,

$\frac{2}{3}$ is simplified because there are no common factors of 2 and 3.
$\frac{10}{15}$ is not simplified because $5$ is a common factor of 10 and 15.

Simplified fraction

A fraction is considered simplified if there are no common factors in its numerator and denominator.

The phrase reduce a fraction means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

In [link] , we used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.

Equivalent fractions property

If $a, b, c$ are numbers where $b \neq 0, c \neq 0,$

then \frac{a}{b} = \frac{a \cdot c}{b \cdot c} and \frac{a \cdot c}{b \cdot c} = \frac{a}{b}

Simplify: $- \frac{32}{56} .$

Solution

	$- \frac{32}{56}$
Rewrite the numerator and denominator showing the common factors.
Simplify using the equivalent fractions property.	$- \frac{4}{7}$

Notice that the fraction

- \frac{4}{7}

is simplified because there are no more common factors.

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Simplify: $- \frac{42}{54} .$

$- \frac{7}{9}$

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Simplify: $- \frac{45}{81} .$

$- \frac{5}{9}$

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Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime number s. Then divide out the common factors using the equivalent fractions property.

How to simplify a fraction

Simplify: $- \frac{210}{385} .$

Solution

A table is shown with three columns and three rows. The first row of the left column reads “Step 1. Rewrite the numerator and denominator to show the common factors. If needed, use a factor tree”. Next to this in the middle column, it reads “rewrite 210 and 285 as the product of the primes. Next to this in the right column, it reads “negative 210 divided by 385.” Under this, is the equation “two times three times five times seven.” The five and 7 are blue and red respectively.

$The next row down reads “Step 2. Simplify using the equivalent fractions property by dividing out common factors.” Next to this in the middle column, it reads, “Mark the common factors 5 and 7.” Next to this in the right column, it has the equation 2 times, three times five, times seven over 5 times seven times 11. Both the 5 and the 7 are crossed out as common factors. Under this is the equation “negative two times 3 divided by 11.”$ The next row reads, “Step 3. Multiply the remaining factors, if necessary.” Next to this in the right column is negative six elevenths.

The next row reads, “Step 3. Multiply the remaining factors, if necessary.” Next to this in the right column is negative six elevenths.

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Simplify: $- \frac{69}{120} .$

$- \frac{23}{40}$

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Simplify: $- \frac{120}{192} .$

$- \frac{5}{8}$

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We now summarize the steps you should follow to simplify fractions.

Simplify a fraction.

Rewrite the numerator and denominator to show the common factors.
If needed, factor the numerator and denominator into prime numbers first.
Simplify using the equivalent fractions property by dividing out common factors.
Multiply any remaining factors, if needed.

Simplify: $\frac{5 x}{5 y} .$

Solution

	$\frac{5 x}{5 y}$
Rewrite showing the common factors, then divide out the common factors.
Simplify.	$\frac{x}{y}$

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Simplify: $\frac{7 x}{7 y} .$

$\frac{x}{y}$

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Simplify: $\frac{3 a}{3 b} .$

$\frac{a}{b}$

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Multiply fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.

Questions & Answers

What are the factors that affect demand for a commodity

Florence Reply

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

hi guys good evening to all

Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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