Question

Arpan cut a cake in two equal parts and one of these two pieces is again cut into equal parts. Each small part is 20 grams. If there are seven parts of the whole cake, what was the weight of the original cake?
(a) 140 grams
(b) 280 grams
(c) 240 grams
(d) 300 grams

Answer 1

Hint: First, we have to suppose the weight of the original cake is x grams . Then, we will find the weight of the one equal half of the cake. Then, further we will divide the one part of the cake into 6 equal parts to get the total parts as 7.Then, we will equate the weight of the small piece of the cake with the value given in the question which is 40 gram.

Complete step-by-step answer:
In this question, we have to suppose the weight of the original cake is x grams.
Now, the first condition is applied to get the cake being cut into two equal halves and get the two parts of the cake.
So, if we cut the cake into two equal parts then the weight of one part of the cake will become in grams as:
$\dfrac{1}{2}\times x=\dfrac{x}{2}$
Again, we have the condition of cutting one of the part of the cake into equal parts so as to get the total 7 parts of the entire cake.
So, we should cut the one half of the cake in 6 equal pieces so as to get the total of 7 pieces of the entire original cake.
Now, we will cut the one part of the cake into 6 equal parts.
To get the one half be in 6 equal parts we need to divide it by 6 and get the result as the weight of the small part as:
$\dfrac{1}{6}\times \dfrac{x}{2}=\dfrac{x}{12}$

So, it gives the weight of the smallest part of the cake which is $\dfrac{x}{12}$grams.
Now, in the question the value of the small part of the cake is given as 20 grams.
So, equate it with the above calculated expression to get the total weight of the original cake as:
$\begin{align}
  & \dfrac{x}{12}=20 \\
 & \Rightarrow x=240 \\
\end{align}$
So, the weight of the original cake is 240 grams.
Hence, option (c) is correct.

Note: The other approach to this type of question is on diving the parts as if we start dividing it by an assumption that firstly we divide the cake into two equal parts as $\dfrac{x}{2}$ due to the fact the total original cake is x grams. Then, those two equal parts are further divided into four equal parts as $\dfrac{x}{4}$ . Then, divide one of the four equal parts into further 3 parts which gives the smallest part value as:
$\dfrac{1}{3}\times \dfrac{x}{4}=\dfrac{x}{12}$
Now, in the question the value of the small part of the cake is given as 20 grams.
So, equate it with the above calculated expression to get the total weight of the original cake as:
$\begin{align}
  & \dfrac{x}{12}=20 \\
 & \Rightarrow x=240 \\
\end{align}$
So, the weight of the original cake is 240 grams.