7 junio, 2024

Fractions equivalent to 3/4 (examples and solved exercises)

The fractions equivalent to 3/4 are those in which, dividing the numerator by the denominator, results in the decimal number 0.75.

It is always possible to express a fraction as an equivalent decimal number, making the quotient of the numerator between the denominator. If the result of this operation is equal to 0.75, the fraction is equivalent to 3/4, for example, the fraction 6/8:

Now, the fraction 6/8 was obtained by multiplying by 2 both the numerator and the denominator of ¾. By simultaneously multiplying the numerator and the denominator by the same quantity, the decimal value of a given fraction is not altered, but it allows obtaining fractions that are equivalent to a given one.

Another way to find a fraction equivalent to another would be by dividing the numerator and denominator by the same quantity. However, in the case of ¾, it is not possible to find a number such that it divides both 3 and 4 and the result is an integer. This is because 3 and 4 are prime to each other, so they do not have common divisors.

When the numerator and denominator of a fraction are mutually prime, the fraction is said to be irreducible. Therefore, ¾ is irreducible.

Ways to find a fraction equivalent to another

There are two very easy ways to find a fraction equivalent to another given fraction: the first is by reduction and the second is by amplification.

Reduction and amplification of fractions

Reduction

This procedure consists of finding a number that is a divisor of both the numerator and the denominator. Once found, we proceed to divide both the numerator and denominator by said value and immediately obtain a fraction equivalent to the original. It is verified that this is the case by making the numerator quotient between the denominator and the compare.

When you want to find the irreducible fraction of another, divide the numerator and denominator of said fraction by the greatest common divisor (GCD) of both. The fraction thus obtained is irreducible.

The fraction ¾ is irreducible, as was said before, since 3 and 4 are prime to each other, but the following method allows finding infinitely many fractions equivalent to ¾.

Amplification

To amplify a given fraction, the numerator and denominator must be multiplied by the same amount, regardless of whether it is a positive or negative number. For example, the fraction 6/8 was obtained by amplifying ¾ by the factor 2:

Although the fractions have different numerators and denominators, they are both the same.

Observe the following figure, which contains two identical circles, divided into equal parts, although of different sizes. Observing carefully, the green and purple areas have the same size, but the green area has been subdivided into 3 parts, out of 4 in total that make up the circle on the left. Instead, the circle on the right has been subdivided into 8 equal parts, and the purple area equals 6 of them.

In this way, it can be verified graphically that ¾ is equivalent to 6/8, since both fractions represent the same quantity.

In general, if you multiply the fraction ¾ by the number n, you get as many fractions equivalent to it as you want:

It is important to note that n can never be equal to 0, since division by 0 is undefined. No fraction can have 0 in its denominator.

How to know if a fraction is equivalent to 3/4?

As explained at the beginning, one way to know if a fraction is equal to ¾ is by making the quotient between the numerator and the denominator. If it turns out to be 0.75, the fraction is equivalent to ¾, but there are a couple of other ways to find out that don’t require you to do the division directly:

Method 1

Suppose the fraction a/b, and you want to know if it is equivalent to ¾, that is, if it is true that:

For them to be equivalent, product 4a must be equal to product 3b:

4a= 3b

Method 2

If the fraction a/b is equivalent to ¾, dividing a and b by their greatest common divisor GCD, the result must be ¾.

To clarify the use of these methods, see the following examples.

examples

Example 1

Determine if the fraction 150/200 is equivalent to ¾:

By method 1

In this case a = 150 and b = 200, it must be true that:

4a= 3b

4 × 150 = 600 3 × 200 = 600

It is concluded that 150 / 200 is equivalent to ¾.

By method 2

The greatest common factor of 150 and 300 divides them both exactly. It is found by decomposing both quantities into their prime factors and then multiplying the common factors with their smallest exponent:

150 = 2 × 52 × 3,200 = 23 × 52

The 2 and 5 are common, they are multiplied by choosing the lowest power with which they appear:

GCD (150, 200) = 2 × 52 = 2 × 25 = 50

Now we proceed to divide:

solved exercises

Exercise 1

Write by amplification five fractions equivalent to ¾, multiplying the numerator and denominator each time by the following integers:

a) 3, b) 5, c) (-2), d) 10 and e) 20

Solution to

solution b

solution c

Note that there is no difference when amplifying the fraction by 2 or by –2, since according to the rule of signs, the indicated quotient between two negative quantities is positive.

solution d

solution and

Note that this fraction is the same as that obtained by amplifying the previously obtained fraction by 2:

Exercise 2

Check if the following fractions are equivalent to ¾:

a) 18/24; b) 21/28; c) 24/32; d) 27/38; d) 33/44

Solution to

Using method 1 described above:

4a= 3b

For the fraction 18/24 we have that a = 18 and b = 24, then:

Therefore, 18/24 and 3/4 are equivalent.

solution b

According to method 2, you have to find the greatest common divisor (GCD) of 21 and 28, then divide both by the result, and if you get the fraction 3/4, they are equivalent:

21 = 3×7
28 = 4×7 = 22×7

The common factor is 7, therefore GCD (21,28) = 7, then:

solution c

For this exercise, it is checked if the quotient between 24 and 32 is 0.75 or not:

24 ÷ 32 = 0.75

So 24/32 is equivalent to 3/4.

solution d

In the fraction 27/38 it is observed that 38 is not a multiple of 4, therefore, it is not equivalent to 3/4. In any case, the quotient between 27 and 38 is carried out:

27 ÷ 38 = 0.710526

From which it is concluded that 27/38 is not equivalent to 3/4.

solution and

It is easy to see that the fraction 33/44 is obtained by multiplying the numerator and denominator of 3/4 by 11, like this:

Therefore, they are equivalent.

Deja una respuesta

Tu dirección de correo electrónico no será publicada. Los campos obligatorios están marcados con *