IV. COIN PROBLEMS
When
solving coin problems (and mixture problems of various types later on!), it is
helpful to form a three-column chart. This
chart is simply a convenient way to organize the information you need to write
the equation. While the number of
rows in the chart varies from one problem to another, there will always be three
columns. For coin problems, the
first column is the number of coins (of each type!).
The second column is the value of each type of coin (in dollars or in
cents--cents is recommended!). The
third column is the “values” column. The
third column is obtained by taking the product of the quantities in the first
two columns. The equation will
always be in the third column. The
examples that follow illustrate this method.
EXAMPLE
12. A
certain number of quarters and four times as many pennies are worth $1.45.
How many of each coin are there?
Solution:
Let x = number of quarters
4x = number of
pennies
Remember
to fill in the first column of the chart, then the second column of the chart.
Finally, multiply the quantities in the first two columns to obtain the
third column.
No.
Coins ×
Each (¢)
= Values
|
Q
|
x
|
25
|
25(x)
|
|
P
|
4x
|
1
|
1(4x)
|
|
|
|
|
145¢
|
Write the equation:
(The equation will always be found in the third column.)
25(x) + 1(4x) =
145
Solve the equation: 25x
+ 4x = 145
29x = 145
x
= 5
Answer
the question:
x = 5 Quarters
4x = 20 Pennies
Check:
5 Quarters = $ 1.25
20
Pennies = .20
$ 1.45
EXERCISES:
50.
A certain number of dimes and four times as many pennies are worth $.98.
How many of each are there?
Solution: Let x
= number of __________
_____ = number of
__________
No.
Coins
×
Each (¢)
= Values
51.
A certain number of quarters and three more dimes than quarters are worth
$7.30. How many of each are there?
Solution: Let x
= number of __________
_____ = number of
__________
No. Coins ×
Each (¢)
= Values
52.
A certain number of dimes and three less pennies than dimes are worth
$7.67. How many of each are there?
Solution:
Let x = number of __________
_____ = number of
__________
No. Coins ×
Each (¢)
= Values
53.
A certain number of nickels and some dimes are worth $7.20.
The number of dimes is three less than twice the number of nickels. How many of each are there?
Solution:
Let x = number of __________
_____ = number of
__________
No. Coins ×
Each (¢)
= Values
EXAMPLE
13. A
box contains 30 coins, in nickels and dimes, worth $2.40.
How many of each coin are there?
Solution: Let x
= number of nickels
30 – x = number of
dimes
No. Coins
× Each (¢)
=
Values
Write
the equation:
5(x) + 10(30 - x) = 240
Solve the equation:
5(x) + 10(30 - x) = 240
5x
+ 300 - 10x = 240
-5x
+ 300 = 240
-5x
= -60
x
= 12 Nickels
Answer the question: x
= 12 Nickels
Check: 12(.05)
= $ .60
30-x = 18
Dimes
18(.10) = 1.80
$2.40
[Note:
You may also let x=number of dimes, and 30-x= number of nickels.
See Example 14.]
EXAMPLE
14. A
box contains 30 coins, in nickels and dimes, worth $2.40.
How many of each coin are there? [Note: This time, let x=no. dimes,
30-x=no. nickels.]
Solution: Let
x =
number of dimes and
30 – x = number of nickels
No. Coins
× Each (¢)
=
Values
Write
the equation:
10(x) + 5(30 - x) = 240
Solve the equation:
10(x) + 5(30 - x) = 240
10x
+ 150 - 5x = 240
5x + 150
= 240
5x
= 90
x
= 18 Dimes
Answer the question:
x
= 18 Dimes
Check:
18(.10) = $ 1.80
30-x = 12 Nickels
12(.05) = .60
$2.40
EXERCISES.
54.
A box contains 20 coins in quarters and dimes worth $2.90. How many
of each coin are there?
Solution: Let
x = number of
__________
_____
= number of __________
No. Coins ×
Each (¢)
= Values
55.
A box contains 20 coins in quarters and dimes worth $3.80.
How many of each coin are there?
56. A box contains 35 coins in quarters and
nickels worth $3.15. How many of
each coin are there?
EXAMPLE
15. A
box contains nickels, dimes, and quarters worth a total of $2.10.
There are twice as many dimes as quarters, and the number of nickels is
two less than the number of dimes. How
many of each coin are there?
Solution
No. Coins
Each (¢)
Values
|
Q
|
x
|
25
|
25x
|
|
D
|
2x
|
10
|
10(2x)
|
|
N
|
2x
- 2
|
5
|
5(2x
– 2)
|
|
|
|
|
210¢
|
Equation:
25x +
20
x + 10 x - 10 =
210
&n