Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios form a proportion.
[a] 25cm:1m and Rs. 40:Rs. 160
[b] 39 litres : 65 litres and 6 bottles: 10 bottles
[c] 2kg : 80kg and 25 g: 625g
[d] 200 ml:2.5 litre and Rs. 4 : Rs. 50

Hint: We know that a: b:: c: d if and only if ad = bc, i.e. product of extreme terms = product of middle terms. Alternatively, you can express the ratios in the lowest terms and equate them if and only if both the ratios have the same lowest term expression.

Complete step-by-step answer:

[a] 25cm:1m and Rs. 40:Rs. 160
Here a = 25 cm, b = 1m = 100cm , c = Rs. 40 and d = Rs. 160
Extreme terms: a = 24cm and d = Rs. 160
Middle terms : b = 100cm and c = Rs. 40
Product of extreme terms = ad $=25\times 160=4000\text{ cm Rs}$
Product of middle terms =bc $=100\times 40=4000\text{cm Rs}$
Since the product of extreme terms = The product of middle terms, we have 25cm:1m:: Rs. 40: Rs. 160.
[b] 39 litres : 65 litres and 6 bottles: 10 bottles
Here a = 39 litres, b = 65 litres, c = 6 bottles and d = 10 bottles.
Extreme terms: a = 39 litres and d = 10 bottles
Middle terms : b = 65 litres and c = 6 bottles
Product of extreme terms = ad $=39\times 10=390\text{ litres bottles}$
Product of middle terms =bc $=65\times 6=390\text{ litres bottles}$
Since the product of extreme terms = The product of middle terms, we have 39 litres : 65 litres:: 6 bottles: 10 bottles
[c] 2kg : 80kg and 25 g: 625g
Here a = 2kg, b = 80 kg, c =25g and d = 625g
Extreme terms: a = 2kg and d = 625 g
Middle terms : b = 80 kg and c = 25 g
Product of extreme terms = ad $=2\times 625=1250\text{ g kg}$
Product of middle terms =bc $=80\times 25=2000\text{ }\text{g kg}$
Since the product of extreme terms $\ne $ The product of middle terms, we have \[2\text{kg : 80kg}\]is not in proportion with \[\text{ }25\text{g : 625g}\]
[d] 200 ml:2.5 litre and Rs. 4 : Rs. 50
Here a = 200 ml , b = 2.5 litres = 2500 ml, c = Rs. 4 and d = Rs. 50
Extreme terms: a= 200ml and d = Rs. 50
Middle terms: b = 2500ml and c = Rs. 4
Product of extreme terms = ad $=200\times 50=10000\text{ ml Rs}$
Product of middle terms = bc $=2500\times 4=10000\text{ ml Rs}$
Hence we have the product of middle terms = The product of extreme terms. Hence 200ml:2.5 litre :: Rs. 4 : Rs. 50.
Note: Alternatively, we can express the ratios in lowest terms and compare.
$\begin{align}
  & 25\text{cm:1m=}\dfrac{25cm}{100cm}=\dfrac{25}{100}=\dfrac{1}{4}=1:4 \\
 & \text{Rs}\text{. 40: Rs 160=}\dfrac{\text{Rs 40}}{\text{Rs 160}}=\dfrac{40}{160}=\dfrac{1}{4}=1:4 \\
\end{align}$
Since 1:4::1:4, we have
25cm:1m::Rs. 40 : Rs.60
Similarly,
$\begin{align}
  & 39\text{ litres : 65 litres}=\dfrac{39\text{ litres}}{65\text{ litres}}=\dfrac{39}{65}=\dfrac{3}{5}=3:5 \\
 & 6\text{ bottles : 10 bottles = }\dfrac{6\,\text{bottles}}{10\text{ bottles}}=\dfrac{6}{10}=\dfrac{3}{5}=3:5 \\
\end{align}$
Since 3:5::3:5, we have 39 litres: 65 litres:: 6 bottles:10 bottles
$\begin{align}
  & 2\text{ kg : 80 kg=}\dfrac{2\text{ kg}}{80\,\text{kg}}=\dfrac{2}{80}=\dfrac{1}{40}=1:40 \\
 & 25\text{ g : 625 g =}\dfrac{25\text{ g}}{625\text{ g}}=\dfrac{25}{625}=\dfrac{1}{25}=1:25 \\
\end{align}$
Since \[1\text{ : 40 }\]is not in proportion with \[1\text{ : 25}\] we have \[2\text{kg : 80kg }\]is not in proportion with \[25\text{g : 625g}\]
$\begin{align}
  & 200\text{ ml : 2}\text{.5 litres = }\dfrac{200\text{ ml}}{2.5\text{ litres}}=\dfrac{200\text{ ml}}{2500\text{ ml}}=\dfrac{2}{25}=2:25 \\
 & \text{Rs 4 :Rs}\text{. 50 =}\dfrac{\text{Rs}\text{. 4}}{\text{Rs}\text{. 50}}=\dfrac{4}{50}=\dfrac{2}{25}=2:25 \\
\end{align}$
Since 2:25 :: 2:25, we have 200ml:2.5litres :: Rs. 4 : Rs. 50