Як добувати еквівалентні дроби

Equivalent Fractions

There are many different ways we can write fractions that are equal to each other. You might have seen the following statement: $$\frac12=\frac48$$ This can made more clear by thinking about a pizza with $8$ slices. If you eat $4$ slices of an $8$ slice pizza, you have eaten one half ($\frac12$) of the pizza. Notice that $4$ is $4\cdot 1$, and $8$ is $4\cdot 2$. We have multiplied both the numerator and denominator by $4$. We can write this out as a fraction as well. $$\frac12 \cdot \frac44=\frac48$$ We can make any equivalent fraction by multiplying the numerator and denominator by the same number. This is because a fraction with the same numerator and denominator is equal to $1$. Remember that fractions show the same thing as division. A number divided by itself is $1$, so multiplying a fraction by another fraction with the same numerator and denominator is the same as multiplying by $1$. Let's write this out in a general form $$\frac{n}{d} \cdot \frac{c}{c}= \frac{n}{d}$$ In this equation, $n$ is your numerator, $d$ is your denominator, and $c$ is any number.

Equivalent Fraction Problems

Let's look at a problem that asks you to work with equivalent fractions.

We are looking for fractions where $\frac43$ has been multiplied by the same number in the numerator and denominator. There are $2$ answers that show this. $$\begin{align} \frac43 \cdot \frac22 &= \frac86\\ \frac43 \cdot \frac33 &=\frac{12}{9}\end{align}$$

Some problems will be word problems that give you some background information or context to let you think through the process, rather than just giving you the first steps immediately.

We can write our original fraction as follows: $$\frac{2\text{ Cups of Water}}{1 \text{ Cup of Rice}}$$ We're going to multiply the top and bottom by $5$ to find our equivalent fraction. $$\frac21 \cdot \frac55 = \frac{10}{5}$$ We will have $10$ cups of water for $5$ cups of rice.

(Videos) Equivalent Fractions

by mathman1024

Here's a video by mathman1024 that explains equivalent fractions.

$\large{\text{Summary}}$

Equivalent fractions are fractions with different denominators that have the same meaning.

Let's look at a picture to understand this.

Here we have a representation of the fraction $\frac12$. But watch what happens if I draw a horizontal line through it.

Now, we have the fraction $\frac24$. The fraction itself changed but the amount shaded has not. That is because these two fractions, $\frac12$ and $\frac24$, are equivalent. So, we compare them with an equals sign:

$$\frac12=\frac24$$

Let's look at the fraction $\frac13$ now. If we multiply $\frac13$ by $1$ nothing happens to it. $$\frac13\cdot1=\frac13$$ Now, let's multiply it by the fraction $\frac55$. The fraction $\frac55$ is the same as the number $1$. $$\frac13\cdot{\color{#29A603}{\frac55}}=\frac5{15}$$ Therefore $\frac13$ and $\frac5{15}$ are equivalent fractions. So, $$\frac13=\frac5{15}$$

Say we have the fraction $\frac25$ and we want to make it a fraction out of $20$. What would be that fraction? $$\frac25=\frac{}{20}$$ Well we know that $5$ goes into $20$, $4$ times. So if we multiply the numerator and denominator we will get a fraction that's equivalent and out of $20$. $$\frac25\cdot{\color{#29A603}{\frac44}}=\frac{8}{20}$$

Try these next ones on your own and then watch the video to see the answers!

1. $\frac{6}{7}=\frac{}{42}$
2. $\frac{11}{10}=\frac{}{90}$

Here's another video where he shows you how to go the other direction to find equivalent fractions.

$\large{\text{Summary}}$

Let's look at the fraction $\frac{14}{20}$ now. If we divide $\frac{14}{20}$ by $1$ nothing happens to it. $$\frac{14}{20}\div1=\frac{14}{20}$$ Now, let's divide it by the fraction $\frac22$. The fraction $\frac22$ is the same as the number $1$. $$\frac{14}{20}\div{\color{#29A603}{\frac22}}=\frac7{10}$$ Therefore, $\frac{14}{20}$ and $\frac7{10}$ are equivalent fractions. So, $$\frac{14}{20}=\frac7{10}$$

Now, say we have the fraction $\frac{55}{100}$ and we want it out of $20$. What would the new fraction look like? $$\frac{55}{100}=\frac{}{20}$$

This question very much resembles a proportions problem.

To get from $100$ to $20$ we have to divide by $5$. In terms of equations, we are trying to solve the equation $\frac{100}{x} = 20$, which has solution $x = 5$.

So, if we divide both the numerator and denominator by $5$, we will get an equivalent fraction that's out of $20$. $$\frac{55}{100}\div{\color{#29A603}{\frac55}}=\frac{11}{20}$$

Try this next one on your own and then watch the video to see the answer.

1. $\frac{25}{100}=\frac{}{4}$

Let's try finding an equivalent fraction that's in lowest terms. A fraction is in lowest terms when the numerator and denominator don't have any common factors.

Let's look at the fraction $\frac{12}{30}$. Both the numerator and denominator have a factor of $2$ so we immediately know this is not in lowest terms. Let's find the largest number that goes into both of these numbers. The answer is $6$. If we divide both the numerator and denominator by this largest common factor, we get a fraction that's in lowest terms. $$\frac{12}{30}\div{\color{#29A603}{\frac66}}=\frac{2}{5}$$ As you can see, $2$ and $5$ don't have any common factors. Therefore, it's in lowest terms. What we just did was reduce this fraction. We reduced $\frac{12}{30}$ to $\frac25$.

Just multiply

Fractions are equivalent if their simplest forms are the same. So, to make equivalent fractions, all you have to do is multiply the top number (the numerator) and the bottom number (the denominator) by the same number—and it can be any number, as long as you use the same one for both the top and bottom. This will change the individual numbers themselves, but leave the basic form of the fraction—the simplest form—the same.

To visualize this, let's think (as we always do) of a pizza. If we start out with $2$ slices of pepperoni, the pizza will look like this:

$2$ out of $8$ ($\frac{2}{8}$) slices of this pizza have pepperoni on them, which is equal to $\frac{1}{4}$ of the whole pizza.

If we cut every single slice in half but don't change anything else, the pizza will look like this:

What we have here is a pizza of the same size as before. There are just more slices, that are smaller.

Now $4$ out of $16$ ($\frac{4}{16}$) slices have pepperoni on them—but it's still $\frac{1}{4}$ of the whole pizza. That means that $\frac{4}{16} = \frac{2}{8} = \frac{1}{4}$.