was given a piggy bank on her seu 39. SAVINGS Janice birthday, and she put it to

Question

was given a piggy bank on her seu 39. SAVINGS Janice birthday, and she put it to use immediately. Each bank, she keeps tra puts one or more coins into the pigey bank, she kehe e of the number of coins she has collected to date accumulated value of her collection. Janice collectso and the nickels, dimes, and quarters. Six months after her seventh birthday, Janice looked at her record and ascertained tha she had collected 480 coins, which were worth $60 a. How many combinations of coins are possible in Janice's collection? b. Janice counted 100 quarters in her savings. How nickels and dimes are in her collection?

Explanation / Answer

Back-up Theory

With one equation with 2 unknown variables, there can be infinite number of solutions. However, if there is a restriction on the variables that they can take only integer values. It may be possible to get a finite set of solutions, in some cases, even unique solution is possible.

Method to evaluate the finite set of solutions

Let the equation be ax + by = c, where x and y can assume only positive integer values.

Substituting the minimum most integer value possible for x, say x1, find the possible and acceptable integer value for y, say y1.

Then, values of y are: y1, y1- a, y1- 2a, y1- 3a, ………. and the corresponding values of x are: x1, x1+ b, x1+ 2b, x1+ 3b, ……….

Now to solve the given problem,

Part (a)

Let x = number of nickel coins and y = number of dimes coins. Then, given total number of coins is 480, number of quarter coins = (480 – x - y).

Total value of these coins = 5x + 10y + 25(480 – x - y) = 6000 or

x + 2y + 5(480 – x - y) = 1200 or

4x + 3y = 1200 ………………………………………………………………………(1)

At x = 0, y = 400 and at y = 0, x = 300.

Thus, the values x can take are: 0, 3, 6, ……., 300. Clearly, there are 101 possible values x can take and hence y also can take 101 possible values. Thus, there are

101 possible combinations of coins in the collection ANSWER

Part (b)

100 quarters value $25 and so x nickel and y dimes coins would value $35 and our equation becomes

5x + 10y = 3500 or

x + 2y = 700

Following the same method as in Part (a), x = 0, 2, 4, ……., 700. Thus, there are

351 possible combinations. ANSWER

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