Question

Find the ratio of the first quantity to the second
60 paise,1 rupee

Answer 1

Hint: Convert rupees to paisa or paise to rupees. Keep the units the same and reduce the ratio in lowest terms by finding the HCF of the involving terms and dividing each term by HCF. When we divide two numbers by their HCF, we remove all the common factors. In fact, it can be proven that if HCF (a,b) =g then $\text{HCF}\left( \dfrac{a}{g},\dfrac{b}{g} \right)=1$. Hence when we divide the numerator and denominator by their HCF, we convert the given fraction in lowest terms.

Complete step-by-step solution -

When finding the ratio of two quantities, it is important to have the same units. So, we need to convert 1 rupee in paise or 60 paise in rupees. Since rupees is a larger quantity than paise, we will convert rupees into paise.
We know that 1 rupee = 100 paise
Hence, we will find the ratio of 60 paise to 100 paise.
We need to write the ratio 60:100 in lowest terms.
For that, we find the HCF (60,100) using the Long Division Method.
We have $100=60\times 1+40$
Remainder $\ne 0$
So, we continue
$60=40\times 1+20$
Remainder $\ne 0$
So, we continue
$40=20\times 2+0$
Remainder = 0. So, HCF (60,100) = 20.
Now $\dfrac{60}{20}=3$ and $\dfrac{100}{20}=5$
Hence 60:100 :: 3:5
Hence the ratio of 60 paise to 1 rupee = 3:5

Note: Ratio can be thought of as fraction, e.g. 3:5 can be thought of as $\dfrac{3}{5}$.
So first we convert the fraction in lowest terms as then write the corresponding ratio.
When we divide both the numerator and denominator of a fraction by the gcd of the numerator and denominator, we remove all the common factors and hence the fraction is reduced to lowest terms. As mentioned in the hint if HCF (a,b) =g then $\text{HCF}\left( \dfrac{a}{g},\dfrac{b}{g} \right)=1$. This fact arises from a simple observation that if $\text{HCF}\left( \dfrac{a}{g},\dfrac{b}{g} \right)=k$ then $k|\dfrac{a}{g}$ and $k|\dfrac{b}{g}$
Hence we have $kg|a$ and $kg|b$. In other words, kg is a common factor of a and b. Hence $kg\le g$. But since $k\ge 1$ we have $kg\ge g$. Hence $kg=g$, i.e. k =1