Question

Express the ratio \[2.5:6.5:8\] in simplest form:

Answer 1

Hint: We use the concept of ratio of three numbers and write them in fraction form. Cancel all possible factors between numerators and denominators of the fraction.
* Ratio of three numbers \[a:b:c\] can be written in fraction form as \[\dfrac{a}{b} \times \dfrac{b}{c}\]

Complete step by step solution:
We are given ratio of three numbers \[2.5:6.5:8\]
Write the given ratio in form of fraction
\[ \Rightarrow 2.5:6.5:8 = \dfrac{{2.5}}{{6.5}} \times \dfrac{{6.5}}{8}\].............… (1)
Now use the conversion of decimal number to fraction form to write the RHS of equation (1)
Since all the decimal numbers in the fraction have one digit after the decimal point, we remove the decimal and divide that number by 10
\[ \Rightarrow 2.5:6.5:8 = \dfrac{{25 \times 10}}{{10 \times 65}} \times \dfrac{{65}}{{8 \times 10}}\]
Cancel the same factors from numerator and denominator of fraction such that the denominator of the first fraction remains equal to the numerator of the second fraction.
\[ \Rightarrow 2.5:6.5:8 = \dfrac{5}{{13}} \times \dfrac{{13}}{{8 \times 2}}\]
Multiply the remaining terms in the fraction
\[ \Rightarrow 2.5:6.5:8 = \dfrac{5}{{13}} \times \dfrac{{13}}{{16}}\]
Write the fraction in right hand side in ratio form
\[ \Rightarrow 2.5:6.5:8 = 5:13:16\]

\[\therefore \]Simplest form of the ratio \[2.5:6.5:8\] is \[5:13:16\]

Note:
Alternate method:
We are given the ratio of three numbers \[2.5:6.5:8\]
We write the terms in ratio in simple form by converting the decimal number to fraction form
\[ \Rightarrow 2.5:6.5:8 = \dfrac{{25}}{{10}}:\dfrac{{65}}{{10}}:8\]
We write the last term of ratio by multiplying 10 to numerator and denominator
\[ \Rightarrow 2.5:6.5:8 = \dfrac{{25}}{{10}}:\dfrac{{65}}{{10}}:\dfrac{{80}}{{10}}\]
Since the denominator in each fraction in ratio terms on the right hand side is the same, we can cancel the same factor in the denominator.
\[ \Rightarrow 2.5:6.5:8 = 25:65:80\]...................… (1)
Now we calculate greatest common divisor of three numbers 25, 65 and 80
Since \[25 = 5 \times 5;65 = 5 \times 13;80 = 5 \times 2 \times 2 \times 2 \times 2\]
And the greatest common factor between three numbers is 5
\[ \Rightarrow \]GCD of 25, 65 and 80 is 5
We know the simplest form of ratio can be obtained by dividing all the terms in ratio by GCD of the three numbers
\[ \Rightarrow 2.5:6.5:8 = \dfrac{{25}}{5}:\dfrac{{65}}{5}:\dfrac{{80}}{5}\]
Cancel same factors from numerator and denominator of fractions in right hand side
\[ \Rightarrow 2.5:6.5:8 = 5:13:16\]
\[\therefore \]Simplest form of the ratio \[2.5:6.5:8\] is \[5:13:16\]