Dozens of Nuts and Bolts Consider the same “reaction” above in terms of dozens of hardware pieces: B + 3N = BN3 One dozen bolts combines with three dozen nuts to make one dozen BN3.
The answer to each should be “dozen.” These problems may be solved exactly as those above. The coefficients are simply scaled up to dozens.
5. How many dozen nuts (N) are needed to combine with 9 dozen bolts (B)?
6. How many dozen BN3 can be made from 56 dozen nuts (N)?
7. How many dozen B are needed to combine with 36 dozen N?
To solve these problems, we need to consider the coefficients and scale them up to dozens.
5. How many dozen nuts (N) are needed to combine with 9 dozen bolts (B)?
In the given equation, we have B + 3N = BN3. By scaling up to dozens, we can rewrite this equation as 1B + 3N (dozens) = 1BN3 (dozens).
So we can say that 9B + 3N = 9BN3 (both are in dozens).
Since we know that we have 9 dozen bolts (B), we need to determine the number of dozen nuts (N). From the equation, we can solve for N by subtracting 9B from both sides:
9BN3 - 9B = 3N
9B(N3 - 1) = 3N
N = 9B(N3 - 1)/(3(N3))
N = 3B(N3 - 1)
Therefore, we need 3 times the number of dozen bolts (B) to get the required number of dozen nuts (N). In this case, we have 9 dozen bolts, so we need 3 * 9 = 27 dozen nuts. Hence, 27 dozen nuts (N) are needed to combine with 9 dozen bolts (B).
6. How many dozen BN3 can be made from 56 dozen nuts (N)?
From the equation B + 3N = BN3, we can rewrite it as 1B + 3N (dozens) = 1BN3 (dozens).
Now, we can determine the number of BN3 by substituting the value of N.
1B + 3(56) = 1BN3
1B + 168 (dozens) = BN3 (dozens)
Since we know that 1B + 168 = BN3, we can say that 1 dozen bolts (B) combines with 168 dozens of nuts (N) to make 1 dozen BN3 (dozens).
So, from 56 dozens of nuts (N), we can make 56/BN3 dozen BN3 (dozens).
7. How many dozen B are needed to combine with 36 dozen N?
We know that B + 3N = BN3 (dozens).
To find how many dozen B are needed to combine with 36 dozen N, we need to solve for B.
B + 3(36) = BN3
B + 108 (dozens) = BN3 (dozens)
From the equation, we can say that 1 dozen bolts (B) combines with 108 dozen nuts (N) to make 1 dozen BN3 (dozens).
Thus, we need 108 dozen bolts (B) to combine with 36 dozen nuts (N).