8 coins none are half-dollar add upto$1.45 you lose one of them if the chances of losing any single coin are equal which denomination of coin is most likely los

Question

8 coins none are half-dollar that add up to $1.45

 

you lose one of them.

 

The chances of losing any single coin are equal which denomination of coin is most likely lost?

Answer 1

Hi Katy!

 

We'll begin working through this question by translating as much of it into algebraic equations as we can.  We'll then apply a little critical thinking, and finally test possible solutions (guess and check).

 

To begin with, we know that the total amount of money that we have is $1.45, and that it is made up of quarters ($0.25), dimes ($0.10), nickles ($0.05), and pennies ($0.01).  

 

That means $1.45=$0.25*q+$0.10*d+$0.05*n+$0.01*p where q is the number of quarters, d dimes, n nickles and p pennies.

 

We also know that we have 8 coins, so 8=q+d+n+p.  We could start our guess and check method to find the answer now, but it would take a long time, so we want to know if we can eliminate some answers without much work.  The answer is yes!

 

Let's look at pennies first.  We know that there has to be either 0 pennies or 5 pennies.  All of the other coins end in a 0 or 5, so if we had a different number of pennies then we couldn't get to $1.45. (For example if we had 2 pennies, the last number in whatever amount of money we have would have to be 2 or 7.)  This doesn't seem very helpful at first, but lets just do some quick guessing with pennies.  Let's say that we have 5 pennies.  That gives us $0.05 so far, and three more coins.  Can we get to $1.45?  No!  Even if all of the other coins were quarters, we would end up with only $0.80.  That means there are 0 pennies!

 

Next, we move on to quarters and nickles.  The number of quarters plus the number of nickles that we have must be 1, 3, 5, or 7.  If we had an even number of quarters plus nickles, our amount would end in a 0.  Dimes also end in a zero, so we could never get to $1.45 (remember we have no pennies).

 

Now we can really start some guessing!  What happens if we have one quarter?  That gives us $0.25 and 7 more coins.  If all of the other coins were dimes, we would get $0.25+$0.70=$0.95.  That means we can't have just one quarter or one dime (because we would never get to $1.45).  What if we have three quarters?  Using the same reasoning as with one quarter, we can only get up to $1.25.  Okay, well how about five quarters?  That gives us $1.25, and 3 coins left.  We either have 0 nickles or 2 nickles (to make sure that the number of quarters plus the number of nickles is 5 or 7) and either 3 or 1 dimes.  Well, if we have 0 nickles, then we have 3 dimes and all of the coins add up to $1.55.  If we have 2 nickles, we must have one dime, and it all adds up to $1.45.

 

That means we have 5 quarters, 1 dime and 2 nickles.

 

We have more quarters than anything else, so we're most likely to lose a quarter ($0.25)!

 

Whenever doing one of these problems, start by making as much information into equations as you can.  Make sure that you have all of the variables you need.  Then, see if you can rule out any answers.  Sometimes you can solve the problem right away by substituting one equation into another, but when there are more variables than equations you have to use guess and check.

 

Please let me know if this explanation helped you.