in a bag of coins, 2/5 are pennies, 1/3 are nickels, 1/4 are dimes, and there are 5 quarters. if there are no other coins in the bag, what is the total number of coins?

Answer 1

This is a good Problem Solving question, and to crack it, we need to make sense of those fractions. Namely, where do 2/5, 1/3, and 1/4 cross paths? You can multiply the denominators together to find a common multiple, and in this case, that also happens to be the least common multiple: 60. Would the numbers add up if we assumed that there were 60 coins in the bag?

Pennies: 2/5 becomes 24/60 by a multiplier of 12/12

Nickels: 1/3 becomes 20/60 by a multiplier of 20/20

Dimes: 1/4 becomes 15/60 by a multiplier of 15/15

Since 24/60 + 20/60 + 15/60 = 59/60, that leaves only 1/60, or a single quarter, to fill the remaining slot. This tells us that there must be more coins in the bag. We could do more guesswork and try the next multiple of 60, then the next, and so on, but if we understand that setting the total number of coins to 60 leaves room for just 1 quarter when we need 5, then it makes sense to multiply that number by 5 itself to see what we come up with, remembering to alter our denominators in like fashion:

Pennies: (24 x 5)/(60 x 5) = 120/300

Nickels: (20 x 5)/(60 x 5) = 100/300

Dimes: (15 x 5)/(60 x 5) = 75/300

There are now 295 coins (i.e. 295 out of 300 coins are accounted for), leaving exactly 5 empty slots for our quarters. Since we are told that there are, in fact, 5 quarters, the answer must be 300 total coins. It all adds up.

Good luck with your studies.

Answer 2

Let x = total number of coins

Then (2/5)x + (1/3)x + (1/4)x + 5 = x

Multiply the equation by the LCD, 60 to obtain:

24x + 20x + 15x + 300 = 60x

x = 300