Question

(i) Are the ratios $$25g:30g$$ and $$40g:48g$$ in proportion.
(ii) Do the ratios $$15cm:2m$$ and $$10s:30min$$ form a proportion.

Answer 1

Hint: For solving these problems first we take the ratios to the same units and simplify the ratio to its simplest form and then apply the definition to check the proportion of two ratios. The definition of proportion is if $$a:b$$ and $$c:d$$ are said to be in proportion if and only if $$a+d=b+c$$. By using this definition we can say whether the given two ratios are in proportion or not.

Complete step-by-step solution
Let us solve the first part
(i) We are given that the two ratios are $$25g:30g$$ and $$40g:48g$$
Let us take the first ratio and simplify to its simplest form as
$$\Rightarrow 25:30=5:6$$
Similarly by taking the second ratio and making it to simplest form we get
$$\Rightarrow 40:48=5:6$$
We know that the definition of proportion is given as if $$a:b$$ and $$c:d$$ are said to be in proportion if and only if $$a+d=b+c$$.
By comparing the definition to above two ratios let us find $$a+d$$ as
$$\begin{array}{rl}& \Rightarrow a+d=5+6\\ & \Rightarrow a+d=11\end{array}$$
Similarly the value of$$b+c$$is
$$\begin{array}{rl}& \Rightarrow b+c=5+6\\ & \Rightarrow b+c=11\end{array}$$
Here, we can see that $$a+d=b+c$$ so we can say that the given two ratios are in proportion.
Therefore, the ratios $$25g:30g$$ and $$40g:48g$$ are in proportion.
Now, let us solve the second part.
(ii) We are given that the two ratios are $$15cm:2m$$ and $$10s:30min$$
Here, the two sides of a single ratio are not in the same units. So let us convert them to the same units and then make it into its simplest form.
Let us take the first ratio and simplify to its simplest form by using $$1m=100cm$$ as
$$\begin{array}{rl}& \Rightarrow 15cm:2m=15cm:2\times 100cm\\ & \Rightarrow 15cm:2m=15:200\\ & \Rightarrow 15cm:2m=3:40\end{array}$$
Similarly by taking the second ratio and making it to simplest form by using $$1min=60s$$ we get
$$\begin{array}{rl}& \Rightarrow 10s:30min=10s:30\times 60s\\ & \Rightarrow 10s:30min=10:1800\\ & \Rightarrow 10s:30min=1:180\end{array}$$
We know that the definition of proportion is given as if $$a:b$$ and $$c:d$$ are said to be in proportion if and only if $$a+d=b+c$$.
By comparing the definition to above two ratios let us find $$a+d$$ as
$$\begin{array}{rl}& \Rightarrow a+d=3+180\\ & \Rightarrow a+d=183\end{array}$$
Similarly the value of $$b+c$$ is
$$\begin{array}{rl}& \Rightarrow b+c=40+1\\ & \Rightarrow b+c=41\end{array}$$
Here, we can see that $$a+d\ne b+c$$ so we can say that the given two ratios are not in proportion.
Therefore, the ratios $$15cm:2m$$ and $$10s:30min$$ are not in proportion.

Note: Students may make mistakes in taking the definition of proportion for any ratio. But the definition is applicable to only ratios, which are in their simplest forms. First, we need to convert the ratio to the simplest form then only we can go for a proportion definition. Also, the ratios will hold only for the quantities of the same units. We can’t apply the ratios to different units. That is directly taking $$15cm:2m$$ as a ratio will be wrong.