Question

Smita made a large chocolate cake. She gave 1/2 of the cake to her daughter, 1/3 to her son, 3 pieces to her son's friend and 2 pieces to her daughter's friend. Find the total number of pieces she divided the cake into and her sons and daughter's shares.

Answer 1

Hint: Assume the total number of pieces of the cake equal to a variable and then divide them among people according to the question.
The sum of the divided pieces will be equal to the total number of pieces of the cake.

Complete step-by-step answer:
Let the total pieces of large cake be x
Dividing the pieces of the cake according to the question:
Daughter $\dfrac{1}{2}$ of cake ($\dfrac{1}{2}$ X total pieces) = $\dfrac{1}{2}x$
Son $\dfrac{1}{3}$ of cake ($\dfrac{1}{3}$ X total pieces) = $\dfrac{1}{3}x$
Son’s friend 3 pieces = 3
Daughter’s friend 2 pieces = 2
Now,
The sum of all the divided pieces will be equal to the total number of pieces:
$\dfrac{1}{2}x$ + $\dfrac{1}{3}x$ + 3 + 2 = x
Taking LCM :
$
  \dfrac{{3x + 2x + 18 + 12}}{6} = x \\
  3x + 2x + 18 + 12 = 6x \\
 $
Calculating for x:
5x – 6x = - 30
- x = - 30
$ \Rightarrow x = 30$ _____________ (1)
Therefore, the total number of pieces of cake are 30
Calculating the number of pieces of the cake came into and her son's and daughters shares:
Son = $\dfrac{1}{3}x$
Substituting the value of x from (1), we get:
$\dfrac{1}{3}x = \dfrac{1}{3} \times 30$
  = 10
Thus, her son’s share is 10 pieces
Daughter = $\dfrac{1}{2}x$
Substituting the value of x from (1), we get:
$\dfrac{1}{2}x = \dfrac{1}{2} \times 30$
 = 15
Thus, her daughter’s share is 15 pieces

Therefore, it can be concluded that when Smita divided a large chocolate cake, she gave her son and daughter 10 and 15 pieces respectively out of the total 30 pieces and the rest were distributed among their friends.

Note: Son’s and daughter’s share were in terms of x because their shares were given with respect to the cake and not with respect to the pieces and we required all the quantities as pieces so that we could sum them up equal to the total number of pieces.
While others shares were already with respect to the pieces.