Question

Find the ratio of 5m and 1500cm.

Answer 1

Hint: We convert the given values into same units either in meter or centimetre. Let them be $A$ and $B$. We find the greatest common divisor(GCD) of $A$ and $B$ and then divide them by the GCD say $n$. The resulting ratio $\dfrac{A}{n}:\dfrac{B}{n}$ will be our result.

Complete step-by-step solution:
A ratio symbolized as $p:q$ is a fraction with both numerator and denominator as positive integers expressed in the standard form of a fraction which is $\dfrac{p}{q}$ where both $p$ and $q$ are positive integers and the highest common factor of $p$ and $q$ is 1. It means $p$ and $q$ are co-prime or relative prime.\[\]
The ratio between two positive integers numbers $A$ and $B$ is written as $a:b$ and is given as
\[A:B=\dfrac{\dfrac{A}{n}}{\dfrac{B}{n}}=\dfrac{a}{b}=a:b\]
Where $n$ is the greatest common divisor of $A$ and $B$.
If two numbers $A$ and $B$ are in a ratio $a:b$ and then for some positive integer $k$ if multiplied, $kA$ and $kB$ will so have the same ratio $a:b$.\[\]
If two numbers $A$ and $B$ are in a ratio $a:b$ and then for some positive integer $k$, if divided $\dfrac{A}{k}$ and $\dfrac{B}{k}$ will have the same ratio $a:b$ where k is a factor of both $A$ and $B$.\[\]
Ratios are used to compare when the values in numerator and denominator are of the same type and in the same units. The ratio is always without units. The given two values are 5 meters (m) and 1500 centimeter (cm). Let $A=5$ metre and B=1500cm. We see that the given values are in different units and we have to convert them into the same units. We know that 1 metre=100cm or $1\text{cm}=\dfrac{1}{100}$m. So we have \[B=1500\text{cm}=1500\times \dfrac{1}{100}=15\]m. Now we have
\[A:B=\dfrac{A}{B}=\dfrac{5\text{m}}{15\text{m}}=\dfrac{5}{15}\]
The greatest common divisor(GCD) of 5 and $15=5\times 3$ is 5. Let us divide the GCD 5 in numerator and denominator to get the ratio. Then we get,
\[A:B=\dfrac{A}{B}=\dfrac{\dfrac{5}{5}}{\dfrac{15}{5}}=\dfrac{1}{3}=1:3\]

Note: We can also find the ratio by converting 5m to 500 centimeters but then the calculation of GCD of 500 and 1500 will be a little difficult. The ratios are called also proportion when it involves more than one ratio for example $A:B::C:D\Rightarrow \dfrac{A}{B}=\dfrac{C}{D}$. The ratios with more than two numbers are called continued proportion, for example, $A:B: C$.