What is the value of : 16*67*45/45(65+67*65)

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Here's another way to solve this problem:

There is a common factor of 45 in the numerator and denominator; those divide out to equal 1.  Then we have (16*67)/(65+67*65).

Although there is a 67 in both the numerator and denominator, this is NOT a common factor, because there is something else added to the product in the denominator.  So we have to leave it as is and do some multiplication.  16*67 = 1072, and 67*65 = 4355, then we add the 65 to get 4420.

Now, our fraction is 1072/4420.  If you know the rules for divisibility by 4 (if the last two digits of a number are a number that is a multiple of 4, then the entire number is a multiple of 4.  [see below the answer for an explanation of why that works.]

Well, 72 = 4*18, so 1072 has a factor of 4. (We also know that's true because one of the factors was 16, or 4*4.)  20 = 4*5, so 4420 also is divisible by 4.  1072 = 4*268 and 4420=4*1105, so the simplified fraction = 268/1105.  This fraction doesn't simplify further because 268=2*2*67 and 1105=5*221.

Now, why does the divisibility rule for 4's work? Obviously, if a two-digit number is divisible by 4, that takes care of the divisibility of that part of the number.  100 is divisible by 4 (e.g., 4 quarters = $1.00); therefore, any multiple of 100--including 1,000 (10^3), 10^4, etc.--will be divisible by 4.

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To find the value of 16*67*45/45(65+67*65), use the proper precedence of operations.

First evaluate what is within the brackets. Remember that multiplication is done before addition

65 + 67*65 = 65 + 4355 = 4420

16*67*45/45(65+67*65)

=> 16*67*45/45*4420

cancel 45 which is common in the numerator and denominator

=> 16*67/4420

=> 4*67/1105

=> 268/1105

The required result is 268/1105

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