.
There are different methods to solve the problem.
I will show you 3 (three) basic methods in this post.
1- Algebra solution, 2 equations approach
Let d be the number of dimes and q be the number of quarters.
Then you have this system of two equations
d + q = 77, (1) (counting dimes)
10d + 25q = 1280. (2) (counting cents)
To solve it, multiply eq(1) by 10. You will have
10d + 10q = 770, (1')
10d + 25q = 1280. (2')
Next subtract (1') from (2'). You will get
25q - 10q = 1280 - 770,
15q = 510 ====> q = 
= 34.
The parking meter contains 34 quarters and 77-34 = 43 dimes.
Check. 43*10 + 34*25 = 33*10 + 34*25 = 1280. ! Correct !
Answer. 34 quarters and 43 dimes.
2- Algebra solution, 1 equation approach
Let q be the number of quarters.
Then the number of dimes is (77-q).
The quarters contribute 25q cents toward the total.
The dimes contribut 10*(77-q) toward the total.
The total is 25q + 10*(77-q).
From the other side, it is 1280 cents, according to the condition.
It gives you an equation
25q + 10*(77-q) = 1280.
Simplify and solve for q:
25q + 770 - 10q = 1280
15q = 1280 - 770
15q = 510 ====> q = = 34.
The parking meter contains 34 quarters and 77-34 = 43 dimes.
You got the same answer.
2- Logical analysis (MENTAL solution without using equations)
Imagine for a minute that all coins in the parking meter are DIMES.
Then you would have 10*77 = 770 cents in the parking meter.
But in reality the money amount there is 1280 cents, which is 1280-770 = 510 cents more than 770 cents.
Why we have this difference ? - But of course, because we counted 25-cent quarters as 10-cent coins.
Then it is clear, that the number of quarters is 
= = 30 + 4 = 34.
And you get the same answer.
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Congratulations ! You are now familiar with 3 basic methods for solving typical coin problems.
I suggest that algebraic methods will be your basic methods for such problems,
and the logical analysis method will allow you to solve the problems MENTALLY without using equations.
I will be happy if it will make your horizon wider.
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To see other similar solved coin problems, look in the lessons
- Coin problems
- More Coin problems
- Solving coin problems without using equations
- Typical coin problems from the archive
- Solving coin problems mentally by grouping without using equations
in this site.
To see how the logical method works for other similar problems, look into the lessons
- Problem on two-wheel and three-wheel bicycles
- Problem on animals at a farm
- Problem on pills in containers
- What type of problems are these?
in this site.
