Kevin W. answered • 04/29/19
Math, English, and Writing (with subject flexibility)
Our ordered pairs will be in the form (nickels, dimes)
This problem requires some trial and error, but we know some facts:
We know we're going to need an odd number of nickels, since our total ends in a 5. We also know that every 2 nickels equal the value of 1 dime. This will save us some time.
Because of the odd number of nickels, we're going to need an even number of dimes, since even + odd = odd, and 39 is an odd number.
Attempt 1! Let's try 10 dimes. That gives us $1.00, Thus, does 29 nickels equal the $1.35 we're missing?
No, but it's close! 29 x 0.05 = 1.45, slightly over.
Attempt 2! Following our logic, let's try 12 dimes. $1.20. Does 27 nickels equal the 1.15 we're missing?
No, it's 27 x 0.05 = 1.35, which is a bit worse!
Attempt 3! Let's try the other direction: 8 dimes. 0.80. Does 31 nickels equal the 1.55 we're missing?
YES! 31 x 0.05 = 1.55. Our first ordered pair is (31,8)
Looking at our 3 attempts, we find that every 2 dimes we add, we must subtract 2 nickels to satisfy the 39 coins needed, and every 2 dimes we subtract, we must add 2 nickels. But not every combination will work. Following is the different ordered pairs that satisfy the 39 coins, and the ones in bold are the ones that satisfy both the 39 coins and the $2.35 limit.
39,0 (1.95 + 0)
37,2 (1.85 + 0.20)
35,4 (1.75 + 0.40)
33,6 (1.65 + 0.60)
31,8 (1.55 + 0.80)
29,10 (1.45 + 1.00)
27,12 (1.35 + 1.20)
25,14 (1.25 + 1.40)
23,16 (1.15 + 1.60)
21,18 (1.05 + 1.80)
19,21 (0.95 + 2.00)
18,23 (0.85 + 2.20) *This is the end of the road, as having 25 dimes goes over our dollar limit)
As we can see, the only combination that fits is 31 Nickels and 8 dimes. Any other will not fit the dollar limit.
