Question

Express 180 to 145 as a ratio.

Answer 1

So here we can use the concept of GCF (Greatest Common Factor) which is the largest positive number that divides evenly and thus giving zero remainder. Expressing two numbers as a ratio simply means expressing the two numbers as a fraction. So to express the given numbers as a ratio we have to find their GCF, divide the numerator and denominator with the GCF and then rewrite the fraction.

Complete step by step solution:
Given
$180\;{\text{to}}\;145................................\left( i \right)$
Now we need to express the given numbers 180 to 145 as a ratio. We also know that expressing two numbers as a ratio simply means expressing the two numbers as a fraction.
So on converting the statement (i) into fraction we get:
$180\;{\text{to}}\;145 = \dfrac{{180}}{{145}}......................\left( {ii} \right)$
We can see that (ii) can be simplified, so simplifying the fraction that we got in (ii):
Such that we need to find its GCF.
The GCF for $180\;{\text{and}}\;145\;{\text{is}}\;5$ according to the definition of GCF.
Now dividing the numerator and denominator with their GCF, such that (ii) becomes:

$$\Rightarrow \dfrac{{180}}{{145}} = \dfrac{{\dfrac{{180}}{5}}}{{\dfrac{{145}}{5}}} \\\ \Rightarrow \dfrac{{\dfrac{{180}}{5}}}{{\dfrac{{145}}{5}}} = \dfrac{{36}}{{29}}..........................\left( {iii} \right) \\\$$

On observing (iii) we can say that it cannot be further simplified since the GCF of $36\;{\text{and}}\;29\;{\text{is}}\;1$.
Now as a ratio $\dfrac{{36}}{{29}}$ can also be expressed as $36:29$.
Therefore on expressing $180\;{\text{to}}\;145$ as a ratio we get: $\dfrac{{36}}{{29}}\;{\text{or}}\;36:29$

Note: We can also use the method prime factorization here to simplify the terms. When one uses the prime factorization method the result that one obtains is the same and is equally accurate. If one is using the prime factorization method one should take prime numbers only and avoid composite numbers. In simple terms an example of the prime factorization process is given below:
$12 = 2 \times 2 \times 3$