SOLUTION: The total value of quarters and dimes in a coin bank is $6.90. If the quarters were dimes and the dimes were quarters the total value of the coins would be $7.80. a) set up a s

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: The total value of quarters and dimes in a coin bank is $6.90. If the quarters were dimes and the dimes were quarters the total value of the coins would be $7.80. a) set up a s      Log On


   

Question 1132844: The total value of quarters and dimes in a coin bank is $6.90.
If the quarters were dimes and the dimes were quarters the total value of the coins would be $7.80.
a) set up a system of equations to represent the situation.
b) find the number of quarters in the bank.
Found 3 solutions by josmiceli, ikleyn, Alan3354:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +d+ = the number of dimes
Let +q+ = number of quarters
-----------------------------------------
(1) +10d+%2B+25q+=+690+ ( in cents )
(2) +10q+%2B+25d+=+780+ ( in cents )
---------------------------------------------
Multiply both sides of (1) by +5+
Multiply both sides of (2) by +2+
Subtract (2) from (1)
(1) +50d+%2B+125q+=+3450+
(2) +-50d+-+20q+=+-1560+
--------------------------------------
+105q+=+1890+
+q+=+18+
and
(1) +10d+%2B+25%2A18+=+690+
(1) +10d+=+690+-+450+
(1) +10d+=+240+
(1) +d+=+24+
-----------------------
There are 24 dimes and 18 quarters
--------------------------------------------
check:
(2) +10q+%2B+25d+=+780+
(2) +10%2A18+%2B+25%2A24+=+780+
(2) +180+%2B+600+=+780+
(2) +780+=+780+
OK

Answer by ikleyn(52799) About Me  (Show Source):
You can put this solution on YOUR website!
.
The system of equations is

    10x + 25y = 690   cents     (1)   (x = # of dimes; y = # of quarters)

    25x + 10y = 780   cents     (2)


To solve the system, multiply equation (1) by 5 (both sides) and multiply equation (2) by 2. You will get


    50x + 125y = 5*690           (3)

    50x + 20y = 2*780            (4)


Now subtract eq(4) from eq(3). The terms "50x" will cancel each other, so you eliminate the unknown "x" and get
a single equation for the unknown y in this way (it is how the Elimination method works)

          125y - 20y = 5*690 - 2*780

          105y = 1890     ==============>  y = 1890%2F105 = 18.


ANSWER.  The number of quarters in the bank is 18.

Solved.

-------------------

On the way, you learned from my post on how the Elimination method works.


Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The total value of quarters and dimes in a coin bank is $6.90.
If the quarters were dimes and the dimes were quarters the total value of the coins would be $7.80.
------------------------
25q + 10d = 690
---------
Each swap of a dime for a quarter adds 15 cents.
780 - 690 = 90
90/15 = 6 swaps, which is the difference between the # of d's and q's
Since the total increases, d > q
----
d - q = 6
---
10d + 25q = 690 --> 5q + 2d = 138
2d + 5q = 138
2d - 2q = 12
-------------------- Subtract
7q = 126
q = 18 quarters
d = 24 dimes