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Find the product, using suitable properties: $625\times \left( -35 \right)+\left( -625 \right)\times 65$

Last updated date: 24th Feb 2024
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IVSAT 2024
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Hint: At first try to analyse the expression then use distributive law which states that $a\times \left( b+c \right)=a\times b+a\times c$ and try to define and give an example and then use to get the results instead of doing the products.

Complete step-by-step solution:
In the question we have to solve the expression $625\times \left( -35 \right)+\left( -625 \right)\times 65$ using suitable properties.
Now to find the value of the expression we will use the distributive property which lets one multiply a sum by multiplying each addend separately and then add the products. It is represented as$a\times \left( b+c \right)=a\times b+a\times c$.
Let’s take an example of the distributive which is as follows,
$3(2+4)=3\times 2+3\times 4$
In LHS, $3(2+4)$ is equal to \[3\times 6\] which becomes \[18\]. So here LHS is \[18\].
In RHS, $3\times 2+3\times 4$ can be written as $6+12$ which equals $18$. So here RHS is $18$.
Hence LHS=RHS so the property satisfies and hence we can use it in expression to solve it.
Now in the expression we are given that,
$625\times \left( -35 \right)+\left( -625 \right)\times 65\ldots \ldots (1)$
which can be expressed as,
$625\times \left( -35 \right)+625\times \left( -65 \right)\ldots \ldots (2)$
As we know $\left( -a \right)\times b=a\times \left( -b \right)$. Hence we can write expressions on (1) and (2).
So now we will use the distributive property which is
$a\times \left( b+c \right)=a\times b+a\times c$
where we will put a = 625, b = -35, c = -65 in expression (2) to get,
  & 625\times \left( -35 \right)+625\times \left( -65 \right) \\
 & =625\times \left( \left( -35 \right)+\left( -65 \right) \right) \\
 & =625\times \left( -100 \right) \\
 & =-62500 \\
So the value of expression by using the distributive law is -62500.
Hence, the answer is -62500.
Note: In these types of problems instead of thinking they directly find the value of the two products differently and add them up. The distributive law or any other laws such as associative where $a\times \left( b+c \right)=a\times b+a\times c$ is made to ease down calculations instead doing the hectic work. Hence the student shall learn all the laws by heart.