Question

Choose the correct answer from the alternatives given
A soap cake is of size \[8cm\times 5cm\times 4cm\] . The number of such soap cakes that can be packed in a box having dimensions \[56cm\times 35cm\times 28cm\] is
A.49
B.196
C.243
D.343

Answer 1

Hint: We have to find the number of soap cakes with dimensions \[8cm\times 5cm\times 4cm\] which can be packed in a box having the dimensions \[56cm\times 35cm\times 28cm\] . As all the smaller soap cake blocks are to be packed in a the bigger box of dimensions \[56cm\times 35cm\times 28cm\] , so the total volume of all small soap cake blocks will be equal to the volume of the box which has dimension \[56cm\times 35cm\times 28cm\] . Assume, the number of smaller soap cake blocks is x. Then solve it further and find the value of x.

Complete step-by-step answer:

Let the number of smaller soap cakes having dimensions equal to 8cm, width equal to 5 cm, and height equal to 4 cm that can be packed in a box having dimensions equal to 56cm, width equal to 35 cm, and height equal to 28 cm.
Dimensions of each soap cake = 8 cm.
Width of each soap cake = 5 cm.
Height of each soap cake = 4 cm.
Volume of one smaller soap cakes = \[8cm\times 5cm\times 4cm=160c{{m}^{3}}\]
The total volume of all soap cakes = \[160x\,c{{m}^{3}}\] …………………….(1)
Dimensions of the box where the soaps are to be packed = 56 cm.
Width of the box where the soaps are to be packed = 35 cm.
Height of the box where the soaps are to be packed = 28 cm.
Volume of the box where the soaps are to be packed = \[56cm\times 35cm\times 28cm=54880\,c{{m}^{3}}\] …………………..(2)
As all the smaller soap cakes are to be packed in box of dimensions \[8cm\times 5cm\times 4cm\] , so total volume of all soap cakes will be equal to the volume of the box where the soap are to be packed and the box is having dimension \[56cm\times 35cm\times 28cm\] .
Volume of all smaller soap cakes = Volume of the box where the soaps are to be packed ………….(3)
From equation (1) and equation (2), we have the volume of the soap cakes and the volume of the box.
Now, from equation (1), equation (2), and equation (3), we have
     \[\begin{align}
  & Volume\text{ }of\text{ }all\text{ }smaller\text{ }soap\text{ }cakes\text{ }=\text{ }Volume\text{ }of\text{ }the\text{ }box\text{ }where\text{ }the\text{ }soaps\text{ }are\text{ }to\text{ }be\text{ }packed \\
 & \Rightarrow 160x\,c{{m}^{3}}=54880c{{m}^{3}} \\
 & \Rightarrow x=\dfrac{54880}{160} \\
 & \Rightarrow x=343 \\
\end{align}\]
Hence, the total number of soap cakes having the dimensions \[8cm\times 5cm\times 4cm\] that can be packed in a box having the dimensions \[56cm\times 35cm\times 28cm\] is 343.

Note: We can also solve this question by using direct formula,
The total number of soap cakes = \[\dfrac{Volume\,of\,the\,box}{Volume\,of\,one\,soap\,cake}\] .
Volume of the box is \[56cm\times 35cm\times 28cm\] .
Volume of soap cake is \[8cm\times 5cm\times 4cm\] .
Now, the total number of soap cakes = \[\dfrac{56cm\times 35cm\times 28cm}{8cm\times 5cm\times 4cm}=7\times 7\times 7=343\] .
Hence, the total number of soap cakes having the dimensions \[8cm\times 5cm\times 4cm\] that can be packed in a box having the dimensions \[56cm\times 35cm\times 28cm\] is 343.