Question

Complete the diagrams and use each of them to find 253x31

Answer 1

- Before we begin solving, let us first know what the result of the original product is

`253*31=7843`

- Now, let us analyze what was done on both diagrams.

Diagram A:

- In this diagram, 253 was divided into 3 numbers and 31 was divided into two numbers.

- We can visualize this better if we wrote them in the product form as shown above:

`\begin{gathered} 253*31 \\ 253=200+50+3 \\ 31=30+1 \\ \\ \therefore253*31=(200+50+3)*(30+1) \\ \text{ Expanding the brackets, we have:} \\ \\ 253*31=200(30)+200(1)+50(30)+50(1)+3(30)+3(1) \\ \\ \text{Now, notice that} \\ 200(30)\text{ is associated with the first box at the top left of the entire diagram,} \\ \text{where 200 and 30 me}et \\ \\ 200(1)\text{ depicts where 200 and 1 me}et\text{ on the diagram} \\ \\ 50(30)\text{ depicts where 50 and 30 me}et\text{ on the diagram} \\ \\ \text{And so on}\ldots \end{gathered}`

- We can therefore see that the empty boxes simply represent the product of these numbers stated above.

- Thus, we can easily populate the boxes if we just perform the products associated with them.

- That is:

- Adding up all the numbers in all 6 squares, we have

`6000+1500+90+200+50+3=7843`

- This is the same result as the original product

Diagram B:

- For this diagram, we take a similar approach

`\begin{gathered} 253*31=253*(30+1) \\ =253(30)+253(1) \end{gathered}`

- Placing these products in their respective boxes, we have:

- Again, Adding up these numbers in the boxes we have:

`7590+253=7843`