Equivalent Fractions

Here we will learn about equivalent fractions.

$equivalent fractions$ Suppose you have a cake. You decide to divide it into 3 pieces and eat one piece. In this case you eat $\frac{{1}}{{3}}$ (1 out of 3) of the cake.

But you see that the piece is too big, so you divide each piece into two pieces, so there are ${3}\cdot{2}={6}$ pieces now.

You eat one piece, and the cake is so tasty that you decide to eat another piece. It appears that you eat $\frac{{2}}{{6}}$ (2 out of 6) of the cake.

But it is the same as if you eat $\frac{{1}}{{3}}$ of the cake!

This means that $\frac{{1}}{{3}}$ and $\frac{{2}}{{6}}$ have same value - we can write that $\frac{{1}}{{3}}=\frac{{2}}{{6}}$ .

In the similar manner we can guess that $\frac{{1}}{{3}}=\frac{{3}}{{9}}$ (if we divide cake into 9 pieces and eat 3 of them).

Fractions are called equivalent if they have same value.

How will we obtain fraction that is equivalent to the given fraction?

If we multiply (or divide) both numerator and denominator of the given fraction by the same integer, we will get equivalent fraction.

For example, we can multiply both numerator and denominator of $\frac{{1}}{{3}}$ by 2 to obtain equivalent fraction $\frac{{{1}\cdot{\color{red}{{{2}}}}}}{{{3}\cdot{\color{red}{{{2}}}}}}=\frac{{2}}{{6}}$ .

Warning 1. This works only for multiplication and division. This doesn't work for addition or subtraction. For example, $\frac{{1}}{{3}}$ is not equivalent to $\frac{{{1}+{2}}}{{{3}+{2}}}=\frac{{3}}{{5}}$ .

Warning 2. Make sure that when you divide numerator and denominator by the integer, both numerator and denominator remain integer.

Now, let's solve a couple of problems.

Example 1. Find a fraction that is equivalent to the fraction $\frac{{2}}{{5}}$ and has denominator 15.

OK, here we need to decide by what number to multiply numerator and denominator. Given fraction has denominator 5 and required fraction has denominator 15. So, we need to multiply fraction by $\frac{{15}}{{5}}={3}$ (indeed, when we multiply 5 by 3 we obtain 15).

So, $\frac{{2}}{{5}}=\frac{{{2}\cdot{\color{red}{{{3}}}}}}{{{5}\cdot{\color{red}{{{3}}}}}}=\frac{{6}}{{15}}$ .

Thus, answer is $\frac{{6}}{{15}}$ .

Now, let's see how to divide to obtain equivalent fractions.

Example 2. Find a fraction that is equivalent to the fraction $\frac{{10}}{{55}}$ and has numerator 2.

We need to decide by what number to divide numerator and denominator, because given numerator is greater than required. Given fraction has numerator 10 and required fraction has numerator 2. So, we need to divide fraction by $\frac{{10}}{{2}}={5}$ .

So, $\frac{{10}}{{55}}=\frac{{\frac{{10}}{{\color{red}{{{5}}}}}}}{{\frac{{55}}{{\color{red}{{{5}}}}}}}=\frac{{2}}{{11}}$ .

Thus, answer is $\frac{{2}}{{11}}$ .

Now, let's see how to determine whether fractions are equivalent.

Example 3. Determine whether fractions $\frac{{5}}{{7}}$ and $\frac{{20}}{{28}}$ are equivalent.

By what number we need to multiply 5 to get 20? By $\frac{{20}}{{5}}={4}$ .

By what number we need to multiply 7 to get 28? By $\frac{{28}}{{7}}={4}$ .

Since we multiply numerator by 4 and denominator by the the same number 4 then fractions are equivalent.

Next example.

Example 4. Determine whether fractions $\frac{{21}}{{8}}$ and $\frac{{7}}{{2}}$ are equivalent.

By what number we need to multiply 7 to get 21? By $\frac{{21}}{{7}}={3}$ .

By what number we need to multiply 8 to get 2? By $\frac{{8}}{{2}}={4}$ .

Since we multiply numerator by 3 and denominator by another number 4 then fractions are not equivalent.

Next example.

Example 5. Determine whether fractions $\frac{{45}}{{7}}$ and $\frac{{9}}{{5}}$ are equivalent.

By what number we need to multiply 9 to get 45? By $\frac{{45}}{{9}}={5}$ .

By what number we need to multiply 5 to get 7? There is no such integer.

So, fractions are not equivalent.

Now, do a couple of exercises.

Exercise 1. Find a fraction that is equivalent to the $\frac{{12}}{{8}}$ and has numerator 60.

Answer: $\frac{{60}}{{40}}$ .

Next exercise.

Exercise 2. Find a fraction that is equivalent to the $\frac{{8}}{{15}}$ and has numerator 20.

Answer: there is no such fraction.

Next exercise.

Exercise 3. Find a fraction that is equivalent to the $\frac{{5}}{{16}}$ and has denominator 48.

Answer: $\frac{{15}}{{48}}$ .

Next exercise.

Exercise 4. Determine whether fractions $\frac{{12}}{{11}}$ and $\frac{{60}}{{55}}$ are equivalent.

Answer: Yes. Hint: multiply numerator and denominator of the first fraction by 5.

Next exercise.

Exercise 5. Determine whether fractions $\frac{{5}}{{7}}$ and $\frac{{12}}{{28}}$ are equivalent.

Answer: No.