Lesson More Coin problems

More Coin problems

This lesson follows the previous one, Coin problems, of this module.
The methods of solution remain the same: reducing the problem to the linear equation or to the system of linear equations.

Problem 1

There are 2 times as many quarters as dimes in Eric’s pocket. He has $1.20 in coins totally.
How many coins of each type does Eric have?

Solution 1
Lets us denote as the number of dimes Eric has in his pocket.
Then the number of quarters is equal to , as it is given.
So, Eric has cents in dimes and cents in quarters.
Since Eric has $1.20 totally, that is 120 cents, this leads to the equation
.

Simplify this equation combining like terms at the left side. You get
.
Now divide both sides by 60. You get
.
So, Eric has 2 dimes in his pocket.
The number of quarters Eric has is in 2 times more, that is 4 quarters.

Check.
Now, check the total amount of money Eric has.
It is equal to cents.
So, the solution is correct.

Answer. Eric has 2 dimes and 4 quarters in his pocket.

Solution 2
You can solve the problem by reducing it to the system of two linear equations in two unknowns.

Denote the number of dimes Eric has in his pocket as and the number of quarters as .
Since the number of quarters is twice as number of dimes, you can write the first equation
.

Now, Eric has cents in dimes and in quarters.
Since Eric has $1.20 totally, you can write the second equation
.

So, you have the system of two linear equations in two variables


Solve it using the substitution method.
Substitute to the first equation of the system. You get the equation
.
Simplify it
.

The rest of the solution is the same as in Solution 1.

Answer. Eric has 2 dimes and 4 quarters in his pocket.

Problem 2

Monica has 34 coins in nickels, dimes, and quarters. The coins worth $5.50.
The number of dimes exceeds the number of nickels by 6.
How many nickels, dimes and quarters does Monica have?

Solution
Denote as the number of nickels Monica has.
Then the number of dimes is equal to , as it is given.
Hence, the number of quarters is equal to
.

Now, Monica has cents in nickels, cents in dimes and cents in quarters.
Since Monica has $5.50 totally, that is 550 cents, this leads to the equation
.

Simplify this equation step by step.
(after brackets opening)
(after collecting variable terms at the left side and collecting constant terms at the right side)
                       (after combining like terms)
                                  (after dividing both sides by )

So, Monica has 6 nickels.
Since the number of dimes is in 6 more, it is equal to .
Hence, the number of quarters is equal to .

Check
To check the solution, calculate the total amount Monica has based on these quantities:
5*6 + 10*12 + 25*16 = 30 + 120 + 400 = 550 cents = $5.50.
So, the solution is correct.

Answer. Monica has 6 nickels, 12 dimes and 16 quarters.