Question

How do you express the ratio \[76:38\] into simplest form?

Answer 1

Hint: In this problem we will express the given ratio \[76:38\] into simplest form. First of all we will find the HCF (Highest Common Factor) of the given number in ratio. Then we will express the given ratio in fraction form then we need to divide the given ratio by HCF . The results obtained are ratios in simplest form.

Complete step-by-step solution:
The given question is based on the simplest form of ratio. A fraction is said to be in simplest form if its numerator and denominator are relatively prime to each other. i.e. They have no common factor other than \[1\].
A fraction can be expressed in the form of ratio. For example: \[\dfrac{a}{b}\] can also be written in the form of a ratio as \[a:b\].
Consider the given ratio \[76:38\].
First we find the HCF of the given numbers in ratio \[76:38\] using a common factor method.
Factors of \[76\] are: \[1\] , \[2\] , \[4\] , \[19\] , \[38\] and \[76\]
Factors of \[38\] are: \[1\] , \[2\] , \[19\] and \[38\]
Common factor of \[76\] and \[38\] are \[1\] , \[2\] , \[19\] and \[38\]
Hence the highest common factor of \[76\] and \[38\] is \[38\].
Hence HCF of \[76\] and \[38\] is \[38\].
Now we express the given ratio in fraction
i.e. \[76:38 = \dfrac{{76}}{{38}}\]
On dividing the numerator and denominator by HCF, we have
\[ \Rightarrow 76:38 = \dfrac{{76 \div 38}}{{38 \div 38}}\]
On solving in further, we have
\[ \Rightarrow 76:38 = \dfrac{2}{1}\].
The given fraction \[\dfrac{2}{1}\] is in simplest form as it has no common factor other than \[1\].
Hence, \[76:38\] in simplest form can be written as \[2:1\].

Note: A fraction can be expressed as ratio and vice versa. For example : \[\dfrac{5}{6} = 5:6\]
A fraction in simplest form has no common factor other than one. Hence it is relatively prime to each other.
We can use prime factorization method, long division method and factor method to find the HCF of any given number.
HCF of any number is always less than or equal to the number.