Question

During a sale, colour pencils were being sold in packs of 24 each and crayons in pack of 32 each. If you want full packs of both and the same number of pencils and crayons, how many of each would you need to buy?

Answer 1

Hint: We start solving the problem by recalling the definition of L.C.M (Least Common Multiple) of two or more numbers as the smallest positive number which is divisible by all the given numbers. We then find the L.C.M of 24 and 32 as we need to buy an equal number of pencils and crayons. We then find the total number by dividing the obtained L.C.M with the number of pencils in each pack to get the number of packs of pencils. We then find the total number by dividing the obtained L.C.M with the number of crayons in each pack to get the number of packs of crayons.

Complete step by step answer:
According to the problem, we are given that color pencils were being sold in packs of 24 each and crayons in packs of 32 each during a sale. We need to find the total number of packs that we need to buy if we want full packs of both and the same number of pencils and crayons.
Since we need the same number of pencils and crayons, we find the L.C.M (Least Common Multiple) of 24 and 32.
We know that L.C.M (Least Common Multiple) of two or more numbers is defined as the smallest positive number which is divisible by all the given numbers.
So, the process of finding the L.C.M of 24 and 32 is as shown below:
 $ \begin{align}
  & 2\left| \!{\underline {\,
  24,32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12,16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6,8 \,}} \right. \\
 & \left| \!{\underline {\,
  3,4 \,}} \right. \\
\end{align} $ .
So, the L.C.M of 24 and 32 is $ 2\times 2\times 2\times 3\times 4=96 $ .
We need to buy 96 pencils and 96 crayons. Now, we need to find the number of packs that were required to buy in order to have 96 pencils and 96 crayons.
We have 24 pencils in each pack. So, the total number of packs of pencils we need to buy is $ \dfrac{96}{24}=4 $.
We have 32 crayons in each pack. So, the total number of packs of crayons we need to buy is $ \dfrac{96}{32}=3 $.
 $ \, therefore, $ We need to buy 4 packs of pencils and 3 packs of crayons in order to have a same number of pencils and crayons.

Note:
 Here, we have assumed that we need to buy the minimum number of packs to have an equal number of pencils and crayons. We should not stop solving the problem by finding the L.C.M of 32 and 24 which is the most common mistake done by students. We can also find the L.C.M by dividing the product of the given numbers with the G.C.D (Greatest Common divisor). Similarly, we can expect problems to find the total number that we need to buy an equal number of pencils and crayons whose number lies between 100 and 200.